1 Introduction↩︎

1.1 Background and Motivation↩︎

The discovery of deep structural connections between gauge theory and gravity has reshaped our understanding of field theory and spacetime. One of the most striking of these connections is the double copy, a correspondence in which gravitational field theories emerge as “squares” of gauge theories. This idea was originally motivated by the Kawai–Lewellen–Tye (KLT) relations in string theory [@Kawai:1986arb], and was further refined by the Bern–Carrasco–Johansson (BCJ) color-kinematics duality [@Bern:2010yg; @Bern:2019nnu; @Bern:2019prr]. In recent years, the double copy has been extended to classical field configurations, inspiring a wide range of research programs aimed at exploring its algebraic foundations and physical implications.

While the earliest incarnations of the double copy were discovered in the context of string scattering amplitudes [@Dunbar:1994bn; @Kawai:1986arb], successful applications have since been found in mathematics [@Alkac:2021bav; @Coll:2000rm; @Easson:2023dbk], particle physics [@Adamo:2020qru; @Bern:2019nnu; @Dunbar:1994bn; @Kawai:1986arb; @Monteiro:2015bna], black hole physics [@Ayon-Beato:2015nvz; @Gonzo:2021drq], supersymmetry and supergravity [@Anastasiou:2014qba; @Anastasiou:2016csv; @Anastasiou:2017nsz; @Cardoso:2016ngt; @Cardoso:2016amd], and quantum gravity [@Bern:2010ue]. Several frameworks now extend it into the classical regime. These include the self-dual [@Adamo:2020qru; @Anastasiou:2018rdx; @Campiglia:2021srh], convolutional [@Godazgar:2022gfw; @Luna:2020adi], and Kerr–Schild (KS) double copy, first developed by Monteiro, O’Connell, and White [@Monteiro:2014cda; @Monteiro:2015bna]. Each provides a distinct map between gauge fields and gravitational solutions, offering complementary insights into the structure of the correspondence.

The convolutional double copy is perhaps the most algebraically transparent. It constructs linearized gravitational fields by convolving pairs of Yang–Mills fields (including ghosts), preserving both linearity and BRST invariance. Within this framework, BRST symmetry plays a central role: the BRST operator \(\mathcal{Q}\) consistently encodes gauge redundancies, and its cohomology identifies the physical states. Remarkably, the convolutional double copy preserves this cohomological structure, with gauge theory ghosts mapping cleanly to gravitational diffeomorphism ghosts. In this way, the convolutional formalism provides a systematic, symmetry-preserving correspondence between Yang–Mills theory and gravity [@Anastasiou:2018rdx; @Godazgar:2022gfw; @Luna:2020adi]. However, its scope is limited: because the construction is intrinsically linear, it reproduces solutions such as the Schwarzschild metric only in their linearized form. The inability to generate fully non-linear geometries motivates the search for alternative approaches.

The Kerr–Schild double copy, by contrast, is capable of producing exact classical solutions. In this formalism, the gravitational metric is written as \(g_{\mu \nu} = \eta_{\mu \nu} + \varphi(x) k_\mu k_\nu\), where \(\eta_{\mu\nu}\) is a flat background, \(\varphi(x)\) is a scalar profile, and \(k_\mu\) is a null vector with respect to both \(g_{\mu\nu}\) and \(\eta_{\mu\nu}\). The associated (Abelian) gauge field is taken to be \(A_\mu = \Phi(x) k_\mu\), so that the Kerr–Schild tensor \(k_\mu k_\nu\) is replaced by its “single copy” \(k_\mu\).

This correspondence provides a direct map between classical solutions in the two theories. A canonical example is the Schwarzschild–Coulomb correspondence [@Monteiro:2014cda; @Monteiro:2015bna]. Expressed in Kerr–Schild form, the Schwarzschild metric arises from the single copy of the Abelian Coulomb potential. The structural dictionary identifies the gravitational mass \(M\) with the gauge theory electric charge \(Q\), the gravitational coupling \(\kappa\) with the Yang–Mills coupling \(g\), and the Kerr–Schild metric with its vectorial counterpart. This makes the KS formalism particularly powerful: it realizes exact black hole geometries as double copies of simple point-charge configurations.

What remains unclear, however, is whether the Kerr–Schild construction also preserves the underlying residual symmetries. In Yang–Mills theory, gauge transformations that preserve the Kerr–Schild ansatz form infinite-dimensional algebras. In the convolutional double copy, BRST invariance ensures that these residual symmetries lift consistently to diffeomorphisms in gravity, preserving the algebraic structure. For the Kerr–Schild double copy, by contrast, no analogous demonstration exists: while exact spacetimes such as Schwarzschild are faithfully reproduced, the status of the associated residual symmetry algebras remains unresolved.

Addressing this problem is the central aim of this work. On the gauge theory side, we systematically derive the full set of residual transformations preserving the Kerr–Schild ansatz, compute their algebras, and establish their coordinate-independence. On the gravity side, beginning with the Schwarzschild solution in Kerr–Schild form, we derive the corresponding system of PDEs for residual diffeomorphisms, solve them explicitly in the Killing sector, and analyze their algebraic structure. This direct comparison reveals a striking structural mismatch: while gauge theory admits infinite-dimensional residual algebras, the gravitational residual diffeomorphisms reduce to the finite-dimensional global isometries of Schwarzschild.

In parallel, we take a first step toward a BRST formulation of the Kerr–Schild double copy, constructing a consistent ghost sector and nilpotent charge, and demonstrating that, in this sector, the residual symmetry algebra has only a trivial realization in cohomology. Together, these results provide the first systematic derivation and algebraic analysis of residual symmetries in the Kerr–Schild double copy, highlighting both the extent and the limitations of the correspondence at the level of symmetries.

1.2 Outline↩︎

This paper builds on the foundational works of Kerr, Schild, and Debney [@Debney:1969zz; @Kerr:2007dk; @Kerr:1965vyg] and Monteiro et al. [@Monteiro:2014cda; @Monteiro:2015bna], as well as classifications/analyses by Ayón-Beato [@Ayon-Beato:2015nvz], Coll et al. [@Coll:2000rm], Gonzo and Shi [@Gonzo:2021drq], and Ridgway and Wise [@Ridgway:2015fdl], but the results presented here are, to the best of our knowledge, original: we provide the first explicit PDE derivation of residual diffeomorphisms preserving the Kerr–Schild form of the double copy in Schwarzschild geometry and establish their precise algebraic structure. We also lay some fundamental groundwork in the construction of the BRST formalism for the Kerr-Schild double copy program, with the Schwarzschild solution serving as a natural starting point.

The structure of this paper is as follows:

• Section 2: we derive the residual symmetries for Abelian and non-Abelian gauge fields in the Kerr–Schild ansatz and analyze their underlying Lie algebras. In the Abelian case, we show that the residual transformations correspond to infinite-dimensional families of gauge parameters, whose characteristic curves reflect the causal structure of flat spacetime. Extending to the non-Abelian case, we find that the residual algebra retains its infinite-dimensional character but acquires additional structure from the non-linear terms of Yang–Mills theory. We perform the analysis in both Cartesian and spherical coordinates, and demonstrate explicitly that the resulting symmetries are coordinate-independent, as expected from general covariance. This coordinate invariance serves as an important contrast to the gravitational case studied in Section 3, where the Kerr–Schild ansatz introduces a strong coordinate dependence.

• Section 3: we turn to the gravitational side and derive the residual diffeomorphisms that preserve the Kerr–Schild form of the Schwarzschild metric. The resulting system of PDEs naturally decomposes into angular, radial–temporal, and mixed sectors. The angular subsystem formally admits the full conformal Killing algebra of the round two-sphere. For tractability we restrict to the Killing sector. Within this sector, the mixed equations eliminate all nontrivial radial–temporal dependence, leaving only constant coefficients. The surviving solutions lift consistently to the global isometries of Schwarzschild: time translations and spatial rotations, together with their linear combinations. Thus, the residual algebra in this setting is finite-dimensional, juxtaposing the infinite-dimensional algebras of residual gauge symmetries studied in Section 2. This algebraic mismatch highlights a structural limitation of the Kerr–Schild double copy: while it provides a correspondence between field configurations, it does not preserve the richer algebra of residual transformations. Finally, we show, as a first step, that there is no nontrivial BRST formulation for the Killing subclass of solutions developed in this section.

• Section 4: we conclude by summarizing the main results and their implications. We emphasize that, unlike in the convolutional double copy where residual symmetries are preserved, the Kerr–Schild formalism in Schwarzschild reduces — within the Killing sector — to the global isometries: time translations and spatial rotations. As a result, the residual algebra is finite-dimensional, in sharp contrast to the infinite-dimensional gauge algebras. We close by discussing the implications of this rigidity and outlining possible directions for future work, including a full treatment of the conformal Killing vector sector, extensions to other exact solutions such as Kerr, and alternative double copy formalisms that may preserve richer symmetry structures.

1.3 Conventions↩︎

We adopt the mostly-plus convention \(\begin{pmatrix} -, +, +, + \end{pmatrix}\) in this paper, and, unless explicitly noted, the background is taken to be flat Minkowski space \(\eta_{\mu\nu}\). In Cartesian coordinates \((t,x,y,z)\), the background metric is

\[\eta_{\mu \nu} = \text{diag} \begin{pmatrix} -1, 1, 1, 1 \end{pmatrix}.\]
In spherical coordinates \((t,r,\vartheta,\varphi)\), the background metric is

\[\eta_{\mu \nu} = \text{diag} \begin{pmatrix} -1, 1, r^2, r^2 \sin^2 \vartheta \end{pmatrix}.\]
Because our analysis centers on Schwarzschild geometry, spherical coordinates are the default choice, and it should be assumed that we are working in spherical coordinates unless stated otherwise. Additionally, for each parametrization of the Kerr–Schild vector \(k^\mu\), we take all components of \(k^\mu\) to be positive. The corresponding (co)vector \(k_\mu\) then carries a relative minus sign in the time component. In Cartesian coordinates, \(k^\mu\) is given in Monteiro, O’Connell, and White [@Monteiro:2014cda; @Monteiro:2015bna] as

\[\label{GG} \begin{matrix} k^\mu = \begin{pmatrix} 1, \frac{x^i}{r} \end{pmatrix} & , & k_\mu = \begin{pmatrix} -1, \frac{x^i}{r} \end{pmatrix} \end{matrix}\tag{1}\]
for \(i = 1, 2, 3\) and \(x^i x_i := r^2 = x^2 + y^2 + z^2\). In spherical coordinates, we will explicitly show in Section 2 that

\[\label{HAPPY} \boxed{\begin{matrix} k^\mu = \begin{pmatrix} 1,~1,~0,~0 \end{pmatrix} & , & k_\mu = \begin{pmatrix} -1,~1,~0,~0 \end{pmatrix} \end{matrix}.}\tag{2}\]
Naturally, we adopt the spherical form of \(k^\mu\) throughout, unless stated otherwise.

2 Yang-Mills Symmetries in Schwarzschild Spacetime↩︎

In this section, we examine the residual symmetries of gauge theories that admit a Kerr-Schild-type ansatz, starting with the Abelian case. We identify the class of gauge transformations that preserve the functional form of the gauge potential \(A_\mu\) and show how these symmetries can be characterized using the method of characteristics in spherical coordinates. These transformations form a symmetry algebra, which we compute explicitly, along with the algebra induced on the scalar field \(\Phi\). After establishing the Abelian case, we extend the analysis to non-Abelian gauge theory, where self-interactions modify the symmetry structure and complicate algebraic closure. This section therefore establishes how gauge theoretic residual symmetries behave under the Kerr–Schild constraints, setting the stage for their gravitational counterparts in the double copy framework.

2.1 The Abelian Case↩︎

We begin by analyzing residual gauge symmetries in the Abelian theory, where the structure is simplest. Our goal is to identify the class of gauge transformations that preserve the Kerr-Schild ansatz for the gauge field, which we will introduce shortly. This reduces the problem to finding functions \(\lambda(x)\) such that \(\delta_\lambda A_\mu = \partial_\mu \lambda(x)\) respects the chosen field structure. Solving this in the Abelian case establishes a clear baseline and introduces the methodology we will generalize to the non-Abelian setting.

2.1.1 Residual Symmetries in Spherical Coordinates↩︎

Consider the Kerr–Schild ansatz for the gauge field \(A_\mu\) in Schwarzschild coordinates:

\[\label{A} A_\mu := \Phi(x) k_\mu,\tag{3}\]
where \(k_\mu = (-1,~1,~0,~0)\) is the null vector in spherical coordinates, obtained from its Cartesian form as shown in Appendix A. Under a local gauge transformation with smooth parameter \(\lambda(x)\), the field transforms as

\[\label{B} A_\mu \rightarrow A_\mu' = A_\mu + \partial_\mu \lambda(x).\tag{4}\]
We require that \(A_\mu'\) preserve the Kerr–Schild form 3 , i.e., there exists a scalar field \(\Phi'(x)\) such that

\[\label{C} A_\mu' := \Phi'(x) k_\mu.\tag{5}\]
For an infinitesimal perturbation \(\delta_\lambda\), the transformed field is

\[A_\mu' = A_\mu + \delta_\lambda A_\mu = [\Phi(x) + \delta_\lambda \Phi(x)] k_\mu.\]
Comparing with 4 gives

\[\label{E} \partial_\mu \lambda(x) = \delta_\lambda \Phi(x) k_\mu.\tag{6}\]
Thus, only gauge transformations for which \(\partial_\mu \lambda(x)\) is proportional to \(k_\mu\) preserve the Kerr–Schild form. To solve for \(\lambda(x)\), we exploit the null condition \(k^\mu k_\mu = 0\). Projecting 6 along \(k^\mu\) yields a homogeneous PDE:

\[k^\mu \partial_\mu \lambda(x) = 0.\]
With \(k^t = k^r = 1\) and \(k^\vartheta = k^\varphi = 0\) in spherical coordinates, this reduces to

\[[\partial_t + \partial_r] \lambda(x) = 0.\]
This can be solved via the method of characteristics. Define curves \(s \rightarrow (t(s), r(s))\) such that along these curves \(d\lambda / ds = 0\). Choosing tangent vectors aligned with the PDE coefficients,

\[\frac{dt}{ds} = 1 ~~~~~,~~~~~ \frac{dr}{ds} = 1,\]
we find

\[\frac{dt}{dr} = 1 \implies t - r = \text{constant}.\]
Along these outgoing null curves, \(\lambda\) is constant. Hence, the general solution for the residual gauge parameter is

\[\lambda(t,r) = f(t-r),\]
where \(f\) is an arbitrary smooth function. The residual gauge freedom is therefore “frozen” along outgoing null rays, propagating only in the retarded time \(u = t - r\).

We can illustrate this explicitly by plotting the characteristic curves (\(t-r=\) constant). These curves are tangent to the null vector \(k^\mu\) and therefore coincide with the outgoing radial null geodesics. . Physically, they represent outgoing light rays or radiation in flat spacetime, consistent with the null structure of the Kerr–Schild background. Figure 1 shows these curves for several values of the constant, appearing as straight lines at 45° in the \((t,r)\) plane. These lines lie exactly on the light cone and confirm that the general solution \(\lambda(t, r) = f(t-r)\) propagates rigidly along outgoing null directions.

For completeness, we also show the incoming null geodesics, corresponding to curves \(t + r =\) constant, forming the other branch of the light cone. Only the outgoing branch is physically relevant here, as \(k^\mu\) would instead select the incoming null geodesics. Note that the choice of outgoing versus incoming is a matter of convention: reversing the sign of a component of \(k^\mu\) would instead select the incoming null geodesics.

Plugging \(\lambda(t,r) = f(t-r)\) into \(\eqref{B}\), the nonzero transformed field components become

\[A_t' = \Phi(x) k_t + \partial_t f(t-r) = [\Phi(x) - f_{,u}(u)] k_t ~~~~~,~~~~~A_r' = \Phi(x) k_r + \partial_r f(t-r) = [\Phi(x) - f_{,u}(u)] k_r,\]
where \(f_{,u} = df/du\) and \(u = t-r\). With \(A_\vartheta' = A_\varphi' = 0\), the transformed field can be written compactly as

\[A_\mu' = [\Phi(x) - f_{,u}(u)] k_\mu\]
so the transformed scalar field is

\[\boxed{\Phi'(x) = \Phi(x) - f_{,u}(u) ~~~~~,~~~~~ \delta_\lambda \Phi(x) = - f_{,u}(u).}\]
This characterizes the Abelian residual gauge symmetry that preserves the Kerr–Schild form, showing explicitly how the gauge parameter \(\lambda(x)\) is constrained along outgoing null rays. This result provides a baseline for the analysis we will extend to the non-Abelian theory in Section 2.2. For completeness, the coordinate-independence of the governing PDE \(k^\mu \partial_\mu \lambda(x) = 0\), ensuring that the solution \(\lambda(t,r) = f(t-r)\) is valid in any coordinate system, as one would expect from a gauge theory.

2.1.2 Algebra Generated by Residual Symmetries↩︎

Having determined that an infinitesimal gauge transformation \(\delta_\lambda A_\mu = \partial_\mu \lambda(x)\) preserves the Kerr–Schild form 3 only when

\[\label{I} \partial_\mu \lambda(x) = \delta_\lambda \Phi(x) k_\mu,\tag{7}\]
we now examine the algebraic structure of these residual symmetry transformations. The constraint \(k^\mu \partial_\mu \lambda(x) = 0\) ensures that \(\lambda(x)\) is constant along integral curves of \(k^\mu\), which correspond to outgoing radial null geodesics. Defining the null coordinate \(u = t - r\), the general solution is

In the previous sections, we obtained the constraint \(k^\mu \partial_\mu \lambda(x) = 0\). This is a first-order linear PDE that expresses the fact that \(\lambda(x)\) must be constant along the integral curves of the null vector field \(k^\mu\). These integral curves are the characteristic curves of the PDE. Since the KS ansatz, by our prescription of \(k^\mu\), refers to outgoing radial null geodesics, we defined the null coordinate \(u = t-r\), and the general solution was

\[\label{J} \lambda(x) = f(u) ~~~~~, ~~~~~ f(u) \in C^\infty(\mathbb{R}).\tag{8}\]
The infinitesimal transformation acts as

\[\delta_\lambda A_\mu = \partial_\mu \lambda(x) = f_{,u}(u) k_\mu\]
so that the transformed field is

\[A_\mu \rightarrow A_\mu' = A_\mu + \delta_\lambda A_\mu = [\Phi(x) - f_{,u}(u)] k_\mu.\]
demonstrating that the Kerr–Schild structure is preserved.

We now identify the Lie algebra of these residual transformations. Let \(\mathfrak{g}_{\text{res}}\) be the set of residual gauge transformations \(\delta_\lambda\) with \(\lambda(x) = f(u)\). Each such \(\lambda(x)\) is in one-to-one correspondence with a smooth function \(f(u)\), as given by 8 . Define the map

\[\Psi : \mathfrak{g}_{\text{res}} \rightarrow C^\infty(\mathbb{R}) ~~~~~,~~~~~ \delta_\lambda \mapsto f(u).\]
This map is:

• Linear: for any \(\delta_{\lambda_1}, \delta_{\lambda_2} \in \mathfrak{g}_{\text{res}}\) and \(\alpha, \beta \in \mathbb{R}\),

  \[\Psi(\alpha \delta_{\lambda_{1}} + \beta \delta_{\lambda_{2}}) = \alpha f_1(u) + \beta f_2 (u) = \alpha \Psi(\delta_{\lambda_1}) + \beta \Psi(\delta_{\lambda_2}).\]

• Injective: if \(\Psi(\delta_\lambda) = 0\), then \(f(u) = 0 \implies \lambda(x) = 0 \implies \delta_\lambda = 0\).

• Surjective: for any \(f \in C^\infty(\mathbb{R})\) \(\lambda(x) = f(u)\) defines a residual gauge transformation \(\delta_\lambda \in \mathfrak{g}_{\text{res}}\) with \(\Psi(\delta_\lambda) = f(u)\).

This establishes \(\Psi\) as a vector space isomorphism, i.e., \(\mathfrak{g}_{\text{res}} \cong C^\infty(\mathbb{R})\). Since the gauge theory is Abelian, the commutator of two transformations vanishes:

\[[\delta_{\lambda_1}, \delta_{\lambda_2}] A_\mu = 0 ~~~~~,~~~~~[f_1, f_2] = f_1 f_2 - f_2 f_1 = 0.\]

It follows that \(\Psi\) also preserves the Lie algebra structure. We conclude that the Lie algebra of residual gauge symmetries preserving the Abelian Kerr–Schild ansatz is an infinite-dimensional Abelian Lie algebra, isomorphic to the additive Lie algebra \(C^\infty(\mathbb{R})\), the space of smooth real-valued functions on null coordinate \(u=t-r\).

2.1.3 Algebra Induced over the Field \(\Phi(x)\)↩︎

Before moving to the non-Abelian case, it is useful to examine how the residual symmetries act on the scalar field \(\Phi(x)\). The key point is that the Abelian residual gauge transformations correspond to functions \(\lambda(x) = f(u)\) along outgoing null rays, with \(u = t-r\). Physically, this means that the freedom in \(\lambda\) is “frozen” along the direction of light-like propagation: any shift of \(\Phi\) occurs only along the outgoing null congruence defined by \(k^\mu\).

The infinitesimal action of a residual gauge transformation on \(\Phi(x)\) is

\[\delta_f \Phi(u) = - f'(u),\]
where the minus sign arises from the orientation of \(k^\mu\). This describes an additive shift along outgoing null rays, so that the space of scalar field profiles naturally carries a representation of the residual gauge algebra. In other words, each smooth function \(f(u)\) generates a linear operator on the space of \(\Phi\) configurations, shifting the field locally along the null coordinate.

To understand the algebraic structure induced on \(\Phi\), consider two transformations \(\delta_f\) and \(\delta_g\):

\[[\delta_f, \delta_g] \Phi(u) = \delta_f(\delta_g \Phi) - \delta_g(\delta_f \Phi) = 0.\]
The vanishing commutator reflects that these shifts act independently along null rays: applying one transformation does not interfere with the other. Therefore, the induced algebra on the scalar field is Abelian, just like the underlying gauge algebra.

Not all gauge functions produce a nontrivial effect on \(\Phi\). Constant functions \(f(u) = c\) generate \(\delta_f \Phi = 0\), leaving the field unchanged. Removing these trivial transformations gives the physically meaningful algebra:

\[\boxed{\mathfrak{g}_{\text{res}} \cong C^\infty(\mathbb{R}) / \mathbb{R},}\]
which captures precisely the residual gauge freedom that manifests in \(\Phi\) along outgoing null rays. In summary, the Abelian residual gauge transformations act as local shifts of the scalar field along null rays, and these shifts form an infinite-dimensional Abelian algebra modulo constants. This provides a clear, physically intuitive benchmark for understanding the more complicated non-Abelian and gravitational cases that follow.

2.2 The Non-Abelian Case↩︎

In the non-Abelian theory, the situation becomes more intricate due to the presence of self-interactions among the fields, which deform the residual symmetry structure compared to the Abelian case. Unlike the Abelian setting, the gauge transformations no longer commute, and the scalar field \(\Phi^a(x)\) transforms in the adjoint representation, introducing nontrivial structure constants into the algebra. Our goal in this section is to generalize the previous Abelian analysis, identifying the class of residual transformations that preserve the Kerr–Schild form and studying the algebra they generate.

2.2.1 Residual Symmetries in Spherical Coordinates↩︎

Consider next the Kerr–Schild formulation of the non-Abelian gauge field \(A_\mu^a\):

\[\label{NA-KS} A_\mu^a := \Phi^a(x) k_\mu,\tag{9}\]
where \(\Phi^a(x)\) is a Lie-algebra-valued scalar field and \(a = 1,...,N^2 -1\) indexes the components in the adjoint representation of \(SU(N)\). Unlike in the Abelian case, non-Abelian gauge transformations introduce self-interactions among the fields via the structure constants \(f^{abc}\):

\[\label{NA-gauge} A_\mu^a \rightarrow A_{\mu}^{'a} = A_{\mu}^a + \partial_\mu \Lambda^a (x) + g f^{abc} A_\mu^b \Lambda^c(x).\tag{10}\]
with \(g\) the Yang-Mills coupling of the field and \(\Lambda^a\) smooth gauge parameters. The primary goal is to determine the class of \(\Lambda^a\) that preserves the Kerr–Schild form 9 , i.e., for which there exists \(\Phi^a(x)\) such that

\[\label{LX} A_{\mu}^{'a} = \Phi^{'a}(x) k_\mu.\tag{11}\]
As before, we write the infinitesimal transformation as

\[\label{Q} A_{\mu}^a \rightarrow A_{\mu}^{'a} = A_{\mu}^a + \delta_\Lambda A_{\mu}^a = [\Phi^a(x) + \delta_\Lambda \Phi^a(x)] k_\mu.\tag{12}\]
Comparing with 10 , the Kerr–Schild preservation condition requires

\[\delta_\Lambda \Phi^a(x) k_\mu = \partial_\mu \Lambda^a(x) + g f^{abc} \Phi^b(x) k_\mu \Lambda^c(x).\]
Projecting along \(k^\mu\) and invoking the null condition \(k^\mu k_\mu = 0\) eliminates the self-interaction term:

\[\label{O} k^\mu \partial_\mu \Lambda^a(x) = 0.\tag{13}\]
Thus, the PDE governing \(\Lambda^a(x)\) is formally identical to the Abelian case, with an independent equation for each adjoint index \(a\). In spherical coordinates, the solution is

\[\Lambda^a(t,r) = f^a(t-r),\]
where \(f^a(u)\) are arbitrary smooth functions of the retarded time \(u = t -r\). Equivalently, one can factor the adjoint index as

\[\Lambda^a(t,r) := \alpha^a \Lambda(t,r),\]
with constants \(\alpha^a\) and a single smooth function \(\Lambda(t,r) = f(t-r)\). This highlights that the non-Abelian residual symmetry space is spanned by \(N^2 - 1\) independent functional directions, one for each adjoint component.

To verify that this indeed preserves the Kerr–Schild structure, we compute the transformed field components:

\[A_{\mu}^{'a} = \Phi^{a}(x) k_\mu + \partial_\mu f^a(t-r) + g f^{abc} \Phi^b k_\mu f^c(t-r).\]
We proceed to work out each non-vanishing component to show that this solution preserves the structure of 11 . Keeping \(a\) arbitrary, the time component \(A_t^{'a}\) reads:

\[\begin{align} A_{t}^{'a} &= \Phi^{a}(x) k_t + f_{,u}^a(t-r) + g f^{abc} \Phi^b(x) k_t f^c(t-r) \\ &= [\Phi^{a}(x) - f_{,u}^a(t-r) + g f^{abc} \Phi^b f^c(t-r)] k_t. \end{align}\]
where we have used \(k_t = -1\) and defined \(f^a_{,u} := \frac{\partial f^a}{\partial u}\) to avoid confusion with the primed notation on the transformed field, \(A_\mu^{'a}\). Similarly, for the radial component

\[A_{r}^{'a} = [\Phi^{a}(x) - f^{a}_{,u}(t-r) + g f^{abc} \Phi^b(x) f^c(t-r)] k_r.\]
All other components vanish, so the transformed field retains the Kerr–Schild form

\[A_\mu^{'a} = [\Phi^{a}(x) - f^{a}_{,u}(t-r) + g f^{abc} \Phi^b f^c(t-r)] k_\mu = \Phi^{'a} k_\mu\]
with

\[\Phi^{'a}(x) = \Phi^{a}(x) - f^{a}_{,u}(t-r) + g f^{abc} \Phi^b f^c(t-r).\]
We conclude that the non-Abelian residual gauge symmetry is

\[\boxed{\delta_\Lambda \Phi^a(x) = - f^{a}_{,u}(t-r) + g f^{abc} \Phi^b(x) f^c(t-r).}\]
This generalizes the Abelian result by including a term proportional to the structure constants \(f^{abc}\), encoding the self-interactions among gauge components. Despite these interactions, the residual symmetry remains infinite-dimensional, parameterized by \(N^2 - 1\) independent functions of the retarded time. Physically, the correction term reflects how the gauge field components influence each other along outgoing null rays, while the Kerr–Schild structure ensures the linear alignment along \(k_\mu\) is preserved. This formalism sets the stage for analyzing the induced algebra on \(\Phi^a\), as well as for comparison with the Abelian baseline.

2.2.2 Algebra Generated by Residual Symmetries↩︎

Having established the form of the residual gauge transformations in the non-Abelian case,

\[\delta_\Lambda \Phi^a(x) = - f_{,u}^a(u) + g f^{abc} \Phi^b(x) f^c(u), \qquad f^a(u) \in C^\infty(\mathbb{R}),\]
we now examine the algebra they generate. In the non-Abelian theory, residual gauge transformations acquire a richer structure due to the presence of the structure constants \(f^{abc}\).
Let \(\mathfrak{g}_{\text{res}}\) denote the set of infinitesimal transformations \(\delta_\Lambda\) with \(\Lambda^a(x) = f^a(u)\). Define a map

\[\Psi : \mathfrak{g}_{\text{res}} \rightarrow \mathfrak{g} \otimes C^\infty(\mathbb{R}) ~~~~~,~~~~~ \delta_\Lambda \mapsto f^a(u) T^a,\]
where \(\{T^a\}\) are the generators of the Lie algebra \(\mathfrak{g}\).

This map is:

• Linear: for \(\delta_{\Lambda_1}, \delta_{\Lambda_2} \in \mathfrak{g}_{\text{res}}\) and \(\alpha,\beta \in \mathbb{R}\),

  \[\Psi(\alpha \delta_{\Lambda_1} + \beta \delta_{\Lambda_2}) = \alpha f_1^a(u) T^a + \beta f_2^a(u) T^a = \alpha \Psi(\delta_{\Lambda_1}) + \beta \Psi(\delta_{\Lambda_2}).\]

• Injective: if \(\Psi(\delta_\Lambda) = 0\), then \(f^a(u) = 0\) for all \(u\), implying \(\Lambda^a(x) = 0\), so \(\delta_\Lambda = 0\).

• Surjective: for any \(f^a(u) \in C^\infty(\mathbb{R})\), there exists a \(\delta_\Lambda \in \mathfrak{g}_{\text{res}}\) such that \(\Lambda^a(x) = f^a(u)\).

Hence, \(\Psi\) is a vector space isomorphism, i.e., \(\mathfrak{g}_{\text{res}} \cong \mathfrak{g} \otimes C^\infty(\mathbb{R})\). Unlike in the Abelian case, the Lie bracket (non-vanishing) is inherited from \(\mathfrak{g}\) pointwise along \(u\):

\[[f^a T^a, g^b T^b] (u) := f^a g^b [T^a, T^b](u) = f^{abc} f^a(u) g^b(u) T^c,\]
where \([T^a, T^b] := f^{abc} T^c\).

This bracket is:

• Bilinear: follows from linearity of scalar multiplication.

• Antisymmetric: \(f^{abc} = -f^{acb}\);

• Satisfies the Jacobi identity: inherited from \(\mathfrak{g}\).

Hence, the residual gauge algebra forms a current algebra:

\[\boxed{\mathfrak{g}_{\text{res}} \cong \mathfrak{g} \otimes C^\infty(\mathbb{R}).}\]
For example, when \(\mathfrak{g} = \mathfrak{su}(N)\), the algebra is \(\mathfrak{su}(N) \otimes C^\infty(\mathbb{R})\), which indeed describes smooth Lie-algebra-valued functions along outgoing null rays.

2.2.3 Algebra Induced over the Field \(\Phi^a(x)\)↩︎

We now examine the algebra induced on the scalar field \(\Phi^a(x)\) by the residual gauge symmetries preserving the Kerr-Schild ansatz. Although these gauge transformations act on the gauge field \(A_\mu^a\) via a first-order differential operator, they induce a nontrivial transformation on \(\Phi^a(x)\), which encodes information about the gauge-invariant content of the theory.

Define the infinitesimal action of a residual gauge transformation:

\[\delta_f : \Phi^a(u) \mapsto - f_{,u}^a(u) + g f^{abc} \Phi^b(u) f^c(u).\]
Let \(\mathcal{F}\) denote the space of admissible scalar fields \(\Phi^a(u)\). The residual gauge transformations then define a linear map \(\delta_f : \mathcal{F} \rightarrow \mathcal{F}\), yielding a representation of the gauge algebra \(\mathfrak{g}\) on \(\mathcal{F}\).

Since the \(\delta_f\) are linear maps on \(\mathcal{F}\), we may study the Lie algebra they generate via the commutator. Let \(\delta_1\) and \(\delta_2\) be two such transformations, defined by parameter functions \(f^a(u),~h^a(u) \in C^\infty(\mathbb{R})\). Computing the commutator:

\[[\delta_f, \delta_h] := \delta_f \circ \delta_h - \delta_h \circ \delta_f\]
allows us to determine how these residual gauge transformations close under composition and reveals the field-dependent structure of the induced algebra. Acting on \(\Phi^a(u)\), the commutator of two residual gauge transformations is defined by

\[[\delta_f, \delta_h] \Phi^a(u) := \delta_f \begin{pmatrix} \delta_h \Phi^a(u) \end{pmatrix} - \delta_h \begin{pmatrix} \delta_f \Phi^a(u) \end{pmatrix}.\]
This expression measures the non-commutativity of the transformations and will reveal the nontrivial structure of the induced algebra on \(\Phi^a(u)\), in contrast with the Abelian case where the commutator vanishes.

A straightforward but careful computation yields:

\[\begin{align} \label{OM} [\delta_f, \delta_h] \Phi^a(u) &= - g f^{abc} \begin{pmatrix} f_{,u}^b(u) h^c(u) - f^c(u) h_{,u}^b(u) \end{pmatrix} \\ &+ g^2 f^{abc} f^{bde} \Phi^d(u) \begin{pmatrix} f^e(u) h^c(u) - h^e(u) f^c(u) \end{pmatrix}, \end{align}\tag{14}\]
where the first term arises from the derivative of gauge parameters along the null direction, and the second term is field-dependent, reflecting non-Abelian self-interactions. The vanishing of \((\delta_f(h_{,u}^a(u)))\) follows because \(\delta_f\) acts only on the fields \(\Phi^a(u)\), not on the derivatives of the gauge functions.

By relabeling the dummy indices \(b \leftrightarrow c\) in the first term of 14 and using the antisymmetry of the structure constants, \(f^{abc} = - f^{acb}\), the term can be rewritten in the form of a Leibniz rule:

\[f_{,u}^b(u) h^c(u) + f^b(u) h_{,u}^c(u) = \partial_u (f^b(u) h^c(u)).\]
To simplify the second term in 14 , recall that the structure constants satisfy the Jacobi identity:

\[\label{ROX} f^{abe} f^{bcd} + f^{dab} f^{cbe} + f^{abc} f^{bde} = 0,\tag{15}\]
which follows from the Jacobi identity of the generators \(T^a\) of \(\mathfrak{g}\).

Consider the expression

\[g^2 f^{abc} f^{bde} \Phi^d(u) \begin{pmatrix} f^e(u) h^c(u) - h^e(u) f^c(u) \end{pmatrix}.\]
Swapping the dummy indices \(c \leftrightarrow e\) in the second piece and using the antisymmetry of \(f^{abc}\) gives

\[g^2 \begin{pmatrix} f^{abc} f^{bde} + f^{abe} f^{bcd}\end{pmatrix} \Phi^d(u) f^e(u) h^c(u).\]
By the Jacobi identity, \(f^{abe} f^{bcd} + f^{abc} f^{bde} = - f^{abd} f^{bec}\), yielding

\[-g^2 f^{abd} f^{bec} \Phi^d(u) f^e(u) h^c(u).\]
Plugging this back into the second term of the Lie bracket, we have,

\[[\delta_f, \delta_h] \Phi^a(u) = - g f^{abc} \partial_u (f^b(u) h^c(u)) - g^2 f^{abd} f^{bec} \Phi^d(u) f^e(u) h^c(u).\]
Swapping \(c \leftrightarrow e\) and then \(c \leftrightarrow d\) in the second term gives:

\[[\delta_f, \delta_h] \Phi^a(u) = - g f^{abc} \partial_u (f^b(u) h^c(u)) + g^2 f^{abc} f^{bde} \Phi^c(u) f^d(u) h^e(u).\]
Define the pointwise Lie bracket on the gauge parameters:

\[[f,h]^a(u) := g f^{abc} f^b(u) h^c(u).\]
and the induced transformation

\[\delta_{[f,h]} := -\partial_u \begin{pmatrix} [f,h]^a \end{pmatrix} + g f^{abc} [f,h]^b \Phi^c\]
It follows immediately that

\[[\delta_f, \delta_h] \Phi^a(u) = \delta_{[f,h]} \Phi^a \implies [\delta_f, \delta_h] = \delta_{[f,h]},\]
showing closure under the commutator. Consequently, the residual transformations on \(\Phi^a(u)\) form an infinite-dimensional Lie algebra, isomorphic to the current algebra

\[\boxed{\mathfrak{g}_{\text{res}} \cong \mathfrak{g} \otimes C^\infty(\mathbb{R}).}\]
Although \(\mathfrak{g}_{\text{res}}\) acts linearly on the gauge parameters \(f^a(u)\), its induced action on \(\Phi^a(u)\) is nonlinear in the gauge parameters and the coupling \(g\). This nonlinearity originates from the self-interacting nature of non-Abelian gauge theory.

To summarize, for the Abelian case, residual transformations are parametrized by arbitrary smooth functions of \(u\), forming the infinite-dimensional Abelian algebra \(C^\infty(\mathbb{R})\) with trivial commutator. The induced algebra on \(\Phi(x)\) reduces to the quotient \(C^\infty(\mathbb{R}) / \mathbb{R}\), reflecting the physical irrelevance of constant shifts. In the non-Abelian case, the structure constants modify the residual transformations but preserve the essential null dependence. The induced algebra on \(\Phi^a(u)\) is a nonlinear current algebra \(\mathfrak{g} \otimes C^\infty(\mathbb{R})\), highlighting the persistence of infinite-dimensional symmetry even in the presence of self-interactions. This demonstrates the remarkable flexibility of the Kerr-Schild ansatz in gauge theory. In contrast, as we will see in the gravitational case, the residual diffeomorphisms preserving the Kerr-Schild ansatz reduce to a finite-dimensional algebra.

3 Gravitational Symmetries in Schwarzschild Spacetime↩︎

In this section we study the most general class of residual diffeomorphisms \(\xi^\mu\) that preserve the Kerr-Schild form of the Schwarzschild metric. Our aim is twofold: first, to characterize the infinitesimal coordinate transformations generated by \(\xi^\mu\) that leave the Kerr-Schild structure intact, and second, to analyze the algebra formed by these transformations. In spherical coordinates, the system of equations governing \(\xi^\mu\) naturally decomposes into angular, radial-temporal, and mixed subsystems. At the level of the angular equations, two distinct types of solutions appear: the standard Killing vectors of the unit two-sphere, and additional conformal Killing vectors, the latter arising through gradient-type contributions.

Both sectors are consistent with the Kerr-Schild ansatz. However, working with the full conformal solution at once leads to cumbersome expressions that obscure the underlying structure. For clarity, in the present section we restrict to the Killing sector, where the analysis simplifies considerably and the resulting equations close neatly. The conformal Killing sector will be treated separately in a follow-up work, allowing us to isolate its distinctive algebraic features.

Within the Killing sector, the mixed subsystem enforces constancy of the coefficients, thereby eliminating any nontrivial radial-temporal dependence. The surviving solutions extend consistently to the full Schwarzschild metric, where they coincide with the global isometries: time translations and spatial rotations. The main result of this section is therefore that, in the Killing sector, the residual diffeomorphisms preserving the Kerr-Schild Schwarzschild metric form a finite-dimensional algebra isomorphic to the Schwarzschild isometry algebra. This provides a sharp contrast to the infinite-dimensional residual gauge algebra obtained in Section 2.

The structure of the section is as follows:

• Section 3.1 introduces the notion of transformations that preserve the Kerr-Schild structure of the metric. We review the framework of infinitesimal coordinate transformations and define the preservation condition in terms of the Lie derivative of the metric.

• Section 3.2 presents the derivation of the residual diffeomorphisms for the Kerr-Schild ansatz in the Schwarzschild geometry. The system of PDEs is highly constrained but naturally decomposes into angular, radial-temporal, and mixed subsystems. The angular subsystem admits both Killing and conformal Killing vectors of the round two-sphere. In this section we restrict attention to the Killing sector, where the angular equations close in a tractable way. The subsequent mixed equations then enforce that the coefficients are constant, eliminating any nontrivial radial-temporal dependence. Altogether, under this restriction the full solution set reduces to the global isometries of Schwarzschild: time translations and spatial rotations. Sections 3.2.1-3.2.4 provide the details of this analysis.

• Section 3.3 computes the Lie algebra generated by these residual diffeomorphisms, confirming that it is finite-dimensional and coincides with the Schwarzschild isometry algebra.

• Section 3.4 contrasts this result with the infinite-dimensional residual gauge algebra obtained in Section 2, thereby exposing an algebraic mismatch between gravity and gauge theory within the Kerr-Schild framework.

3.1 Diffeomorphisms and the Lie Derivative of the Kerr–Schild Metric↩︎

We begin by recalling the Kerr-Schild (KS) ansatz, which expresses the spacetime metric \(\eta_{\mu\nu}\) as a deformation of a background metric \(\eta_{\mu \nu}\) by a scalar field \(\varphi\) and a null vector field \(k^\mu\):

\[\label{XX} g_{\mu \nu} := \eta_{\mu \nu} + \varphi k_\mu k_\nu.\tag{16}\]
The vector \(k^\mu\) is required to be null with respect to the background, \(\eta_{\mu \nu} k^\mu k^\nu = 0\), which immediately implies it is also null with respect to the full metric, \(g_{\mu \nu} k^\mu k^\nu = 0\). In addition, \(k^\mu\) is geodesic with respect to the full Levi-Civita connection \(\nabla^{(g)}\), i.e., \(k^\mu \nabla^{(g)}_\mu k^\nu = 0\). This guarantees that \(k^\mu\) is affinely parameterized and is also geodesic with respect to the background connection \(\nabla^{(\eta)}\), \(k^\mu \nabla^{(\eta)}_\mu k^\nu = 0\).

It is important to stress that 16 defines an exact spacetime metric, not merely a perturbative expansion around \(\eta_{\mu\nu}\). In particular, the Schwarzschild solution can be written in KS form by choosing

\[\begin{matrix} \varphi := \frac{2GM}{r} & , & k^\mu := \begin{pmatrix} 1, 1, 0, 0 \end{pmatrix}. \end{matrix}\]
in spherical coordinates \((t,r,\vartheta,\varphi)\). This coordinate system is natural for a spherically symmetric solution, and we adopt it throughout this analysis.

We now define precisely what it means for a diffeomorphism to preserve the KS structure. Consider an infinitesimal transformation generated by a vector field \(\xi^\mu\),

\[x^\mu \rightarrow x^{\mu'} = x^\mu + \xi^\mu .\]
under which the metric transforms as

\[g_{\mu \nu} \rightarrow g'_{\mu \nu} = g_{\mu \nu} + \delta_\xi g_{\mu\nu} = g_{\mu \nu} + (\mathcal{L}_\xi g)_{\mu\nu}.\]
Inserting the KS form 16 , we demand that the transformed metric retains the same structure:

\[g_{\mu \nu}' \stackrel{!}{=} \eta_{\mu \nu} + (\varphi + \delta_\xi \varphi) k_\mu k_\nu.\]
so that the effect of the transformation is absorbed into a redefinition of the scalar field, \(\varphi \mapsto \varphi + \delta_\xi \varphi\). Equivalently, the Lie derivative of the metric must satisfy

\[\label{YY} \boxed{(\mathcal{L}_\xi g)_{\mu\nu} \stackrel{!}{=} \alpha(x) k_\mu k_\nu}\tag{17}\]
for some smooth function \(\alpha(x)\). Equation 17 is the residual diffeomorphism condition: it characterizes the infinitesimal transformations that preserve the KS ansatz. Such \(\xi^\mu\) will be referred to as residual diffeomorphisms, in direct analogy with the residual gauge symmetries identified in Section 2.

In components, the Lie derivative of the metric reads

\[\label{QUACK} (\mathcal{L}_\xi g)_{\mu \nu} := \xi^\rho \partial_\rho g_{\mu \nu} + 2 \partial_{(\mu} \xi^\rho g_{\nu) \rho},\tag{18}\]
where \(2 \partial_{(\mu} \xi^\rho g_{\nu) \rho } = \partial_\mu \xi^\rho g_{\rho \nu} + \partial_\nu \xi^\rho g_{\mu \rho}\). Substituting 16 , we obtain a natural decomposition:

\[\label{ZZ} (\mathcal{L}_\xi g)_{\mu \nu} := (\mathcal{L}_\xi \eta)_{\mu \nu} + \mathcal{L}_\xi (\varphi k_\mu k_\nu),\tag{19}\]
with

\[\label{AAA} \begin{matrix} (\mathcal{L}_\xi \eta)_{\mu \nu} := \underbrace{\xi^\rho \partial_\rho \eta_{\mu \nu}}_{(1)} + \underbrace{2 \partial_{(\mu} \xi^\rho \eta_{\nu)\rho}}_{(2)} & , & \mathcal{L}_\xi (\varphi k_\mu k_\nu) := \underbrace{(\mathcal{L}_\xi \varphi) k_\mu k_\nu}_{(3)} + \underbrace{2 \varphi k_{(\mu} \mathcal{L}_\xi k_{\nu)}}_{(4)}. \end{matrix}\tag{20}\]
Each of these terms has a clear geometric interpretation:

• (1) variation of the background metric under \(\xi^\mu\). This vanishes in Cartesian coordinates but not in spherical coordinates, where \(\eta_{\mu\nu}\) has explicit \((r,\vartheta)\) dependence.

• (2) the symmetrized derivative of \(\xi^\mu\) contracted with \(\eta_{\mu\nu}\). This vanishes precisely when \(\xi^\mu\) is a Killing vector of the background, since the expression is just the flat space Killing equation.

• (3) the change of the scalar field \(\varphi\) along the flow of \(\xi^\mu\).

• (4) the change in the null vector \(k_\mu\). In spherical coordinates with \(k^\mu=(1,1,0,0)\), this simplifies but remains nonzero.

To preserve the KS structure of the Schwarzschild metric, the infinitesimal diffeomorphism generated by \(\xi^\mu\) must satisfy 17 , ensuring that the Lie derivative of the metric is proportional to \(k_\mu k_\nu\). The decomposition 19 clarifies the separate roles of the background geometry, the scalar field, and the null congruence in this condition.

In the next section we will solve the resulting system of PDEs. The angular subsystem admits both Killing and conformal Killing vectors of the round two-sphere. For clarity, we will treat the Killing and conformal sectors separately: restricting to the Killing sector yields a tractable system in which the mixed equations enforce constant coefficients, while the conformal sector introduces gradient-type solutions that will be analyzed in parallel.

3.2 Deriving the General Class of Residual Diffeomorphisms↩︎

To identify the diffeomorphisms that preserve the Kerr–Schild form of the Schwarzschild metric, we begin by deriving the system of partial differential equations imposed by condition 17 . This system is highly constrained and can be naturally organized into three subsets: equations involving only angular derivatives, equations involving only radial and temporal derivatives, and mixed equations coupling \(t\) and \(r\) to \(\vartheta\) and \(\varphi\). The strategy of this section is to analyze each subsystem in turn — first isolating the angular solutions, then the radial–temporal ones, and finally imposing the mixed equations as consistency conditions. The angular subsystem formally admits both Killing and conformal Killing vectors of the round 2–sphere. In practice, we restrict attention to the Killing sector, which yields a closed and tractable system. Within this sector the mixed equations enforce constant coefficients, allowing the angular Killing vectors to be consistently “lifted” to global isometries of the full Schwarzschild metric.

Before we begin, consider terms (3) and (4) in 20 . Recall that the Lie derivative of the scalar field \(\varphi\) is just the directional derivative (or equivalently, the action of the vector \(\xi = \xi^\mu \partial_\mu\) on the field \(\varphi\)):

\[\label{FFFF} \mathcal{L}_\xi \varphi := \xi(\varphi) = \xi^\mu \partial_\mu \varphi.\tag{21}\]
Moreover, the Lie derivative of the null (co)vector \(k_\mu\) along \(\xi^\mu\) is:

\[(\mathcal{L}_\xi k)_\mu := \xi^\rho \partial_\rho k_\mu + k_\rho \partial_\mu \xi^\rho.\]
Geometrically, the first term encodes the change of \(k^\mu\) as it is dragged along the flow generated by \(\xi^\mu\). The second term captures the effect of the flow on the underlying coordinate grid, reflecting how stretching or shearing of the coordinates alters the components of the (co)vector \(k_\mu\). Together, these terms ensure that the Lie derivative describes both the intrinsic transport of the field and the deformation of the basis in which its components are expressed.

Equation 21 illustrates the utility in adopting spherical coordinates: since the null vector \(k^\mu\) is constant, it satisfies \(\partial_\mu k_\nu = 0\) for all indices \(\mu, \nu\). As a result, the Lie derivative reduces to

\[\label{QQQQ} (\mathcal{L}_\xi k)_\mu := k_\rho \partial_\mu \xi^\rho.\tag{22}\]
Expanding 19 using 21 , 22 and terms (1) and (2) in 20 , the residual diffeomorphism condition becomes:

\[(\mathcal{L}_\xi g)_{\mu \nu} := (\mathcal{L}_\xi \eta)_{\mu \nu} + (\xi^\rho \partial_\rho \varphi) k_\mu k_\nu + \varphi (k_\rho \partial_\mu \xi^\rho) k_\nu + \varphi k_\mu (k_\rho \partial_\nu \xi^\rho) \stackrel{!}{=} \alpha(x) k_\mu k_\nu.\]
However, notice that \((\xi^\rho \partial_\rho \varphi)k_\mu k_\nu\) is already of the Kerr–Schild form. Therefore, we can subtract it to the right-hand side and define the quantity \(\zeta(x) := \alpha(x) - \xi^\rho \partial_\rho \varphi\), so that:

\[\label{SSSS} \mathcal{H}_{\mu \nu} := (\mathcal{L}_\xi g)_{\mu \nu} - (\xi^\rho \partial_\rho \varphi)k_\mu k_\nu = (\mathcal{L}_\xi \eta)_{\mu \nu} + \varphi (k_\rho \partial_\mu \xi^\rho) k_\nu + \varphi k_\mu (k_\rho \partial_\nu \xi^\rho) \stackrel{!}{=} \zeta(x) k_\mu k_\nu.\tag{23}\]
It is precisely this system of equations that we now solve systematically and analytically for the most general class of \(\xi^\mu\) that preserve the KS ansatz.

3.2.1 The Angular Subsystem: Symmetries of the Two-Sphere↩︎

The focus of this section will be on the angular components, i.e., those equations in which both indices lie in the tangent space of the round two-sphere.

Consider 23 in spherical coordinates. That is, we wish to determine \(\mathcal{H}_{\vartheta \vartheta}\), \(\mathcal{H}_{\varphi \varphi}\), and \(\mathcal{H}_{\vartheta \varphi}\). First, recall that the background metric parametrized in spherical coordinates is:

\[\eta_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2\vartheta \end{pmatrix}.\]
Thus, working first the \(\vartheta \vartheta\) equation, we find:

\[\mathcal{H}_{\vartheta \vartheta} := \xi^\rho \partial_\rho \eta_{\vartheta \vartheta} + 2 \partial_\vartheta \xi^\rho \eta_{\rho \vartheta} + 2 \varphi (\partial_\vartheta \xi^r - \partial_\vartheta \xi^t) k_\vartheta \stackrel{!}{=} \zeta(x) (k_\vartheta)^2.\]
Because \(\eta_{\vartheta \vartheta} = r^2\), the first term is nonzero only when \(\rho = r\). Additionally, because \(\eta_{\mu \nu}\) is diagonal, the second term is only nontrivial when \(\rho = \vartheta\). Consequently, \(k_\vartheta = 0\) in spherical coordinates forces the remaining terms to vanish, and we are left with:

\[\label{XAXA} \mathcal{H}_{\vartheta \vartheta} := \xi^r \partial_r \eta_{\vartheta \vartheta} + 2 \partial_\vartheta \xi^\vartheta \eta_{\vartheta \vartheta} \stackrel{!}{=} 0.\tag{24}\]
Substituting into this \(\eta_{\vartheta \vartheta} = r^2\), we find:

\[\label{SASA} \boxed{\xi^r \stackrel{!}{=} - r \partial_\vartheta \xi^\vartheta.}\tag{25}\]
Working next the \(\varphi \varphi\) equations yields a similar result:

\[\mathcal{H}_{\varphi \varphi} := \xi^\rho \partial_\rho \eta_{\varphi \varphi} + 2 \partial_\varphi \xi^\rho \eta_{\rho \varphi} + 2 \varphi (\partial_\varphi \xi^r - \partial_\varphi \xi^t) k_\varphi \stackrel{!}{=} \zeta(x) (k_\varphi)^2.\]
In this case, \(\eta_{\varphi \varphi} = r^2 \sin^2 \vartheta\), so the first term is nonzero when \(\rho = r, \vartheta\). We sum over these. Additionally, the second term is only non-vanishing when \(\rho = \varphi\). As before, \(k_\varphi = 0\) in spherical coordinates forces the remaining terms to vanish, and we are left with:

\[\label{XIXI} \mathcal{H}_{\varphi \varphi} := \xi^r \partial_r \eta_{\varphi \varphi} + \xi^\vartheta \partial_\vartheta \eta_{\varphi \varphi} + 2 \partial_\varphi \xi^\varphi \eta_{\varphi \varphi} \stackrel{!}{=} 0.\tag{26}\]
Substituting in \(\eta_{\varphi \varphi} = r^2 \sin^2 \vartheta\), this becomes:

\[2 r \xi^r \sin^2 \vartheta + 2 r^2 \xi^\vartheta \sin\vartheta \cos\vartheta + 2 r^2 \sin^2 \vartheta \partial_\varphi \xi^\varphi \stackrel{!}{=} 0.\]
Enforcing 25 here allows us to drop a common factor of \(2 r^2 \sin^2\vartheta\) and yields a purely angular equation:

\[\label{ZAZA} \boxed{- \partial_\vartheta \xi^\vartheta + \xi^\vartheta \cot\vartheta + \partial_\varphi \xi^\varphi \stackrel{!}{=} 0.}\tag{27}\]
Finally, we work out the \(\vartheta \varphi\) equation:

\[\mathcal{H}_{\vartheta \varphi} := \partial_\vartheta \xi^\rho \eta_{\rho \varphi} + \partial_\varphi \xi^\rho \eta_{\rho \vartheta} + \varphi (k_\rho \partial_\vartheta \xi^\rho) k_\varphi + \varphi k_\vartheta (k_\rho \partial_\varphi \xi^\rho) \stackrel{!}{=} \zeta(x) k_\vartheta k_\varphi.\]
The first term survives only when \(\rho = \varphi\), while the second is nonzero if and only if \(\rho = \vartheta\). The remaining terms, which couple to \(k_\vartheta\) and \(k_\varphi\), vanish in spherical coordinates. Hence:

\[\label{XYXY} \mathcal{H}_{\vartheta \varphi} := \partial_\vartheta \xi^\varphi \eta_{\varphi \varphi} + \partial_\varphi \xi^\vartheta \eta_{\vartheta \vartheta} \stackrel{!}{=} 0.\tag{28}\]
Using \(\eta_{\vartheta \vartheta} = r^2\) and \(\eta_{\varphi \varphi} = r^2 \sin^2 \vartheta\) as before, we find:

\[\label{YAYA} \boxed{\sin^2 \vartheta \partial_\vartheta \xi^\varphi + \partial_\varphi \xi^\vartheta \stackrel{!}{=} 0.}\tag{29}\]
For each angular PDE, we have found that because \(k_\mu\) has support only in the \((t,r)\)-plane, the right-hand side of 23 vanishes whenever \(\vartheta\) or \(\varphi\) appear as an index. Thus, it is useful to package results 24 , 26 , and 28 into a covariant form on the round two-sphere. We can write the angular block of the metric as

\[\begin{matrix} g_{AB} := r^2 \gamma_{AB} & , & \gamma_{AB} dx^A dx^B := d\vartheta^2 + \sin^2 \vartheta d\varphi^2. \end{matrix}\]
and denote by \(\nabla_A\) the Levi–Civita connection of \(\gamma_{AB}\). For \(A, B, C \in \{\vartheta, \varphi\}\), we have:

\[\label{OHHI} (\mathcal{L}_\xi g)_{AB} := \xi^\rho \partial_\rho (r^2 \gamma_{AB}) + (\partial_A \xi^C) r^2 \gamma_{BC} + (\partial_B \xi^C) r^2 \gamma_{AC},\tag{30}\] by equation 18 . Here, the index \(C\) is summed over angular directions \(\{\vartheta, \varphi\}\). This reflects the fact that the Lie derivative couples \(\xi^A\) to the geometry of the two-sphere through its connection coefficients: in each term \(\partial_A \xi^C\), the free index \(C\) must be contracted with the angular metric \(\gamma_{BC}\). Summation over \(C\) is therefore required to ensure covariance on \(S^2\) and to reproduce the correct transformation law for tensors under diffeomorphisms of the sphere.

The first term in 30 survives when \(\rho = r,~C\). Using

\[\partial_r (r^2 \gamma_{AB}) = 2 r \gamma_{AB} ~~~~~,~~~~~\partial_C (r^2 \gamma_{AB}) = r^2 \partial_C \gamma_{AB},\]
this becomes:

\[\label{UGH} \begin{align}(\mathcal{L}_\xi g)_{AB} &:= 2r \xi^r \gamma_{AB} + r^2 \begin{bmatrix} \xi^C \partial_C \gamma_{AB} + (\partial_A \xi^C) \gamma_{BC} + (\partial_B \xi^C) \gamma_{AC} \end{bmatrix} \\ &= 2r \xi^r \gamma_{AB} + r^2 ( \nabla_A \xi_B + \nabla_B \xi_A), \end{align}\tag{31}\]
where

\[\mathcal{L}_\xi \gamma_{AB} := \nabla_A \xi_B + \nabla_B \xi_A = \xi^C \partial_C \gamma_{AB} + (\partial_A \xi^C) \gamma_{BC} + (\partial_B \xi^C) \gamma_{AC}.\]
This identity follows directly from the general definition of the Lie derivative of the two-sphere metric (see, e.g., Wald [@Wald:1984rg] or Carroll [@Carroll:2019]).

The residual condition in the angular block is \((\mathcal{L}_\xi g)_{AB} = 0\). Solving for \(\nabla_A \xi_B + \nabla_B \xi_A\) and dividing by \(r^2\) yields the compact, covariant form

\[\label{OHBOI} \nabla_A \xi_B + \nabla_B \xi_A = - \frac{2 \xi^r}{r} \gamma_{AB},\tag{32}\]
which is precisely the conformal Killing equation on \((S^2, \gamma)\) with conformal factor \(-2 \xi^r / r\). At this stage, \(\xi^r\) may depend on \((t, r, \vartheta, \varphi)\), so \(\xi^A\) is, in general, a conformal Killing field on each sphere \(S^2\) at fixed \((t,r)\). We note that in the case where \(\xi^r = 0\), equation 32 reduces to the Killing equation on the round two-sphere:

\[\nabla_A \xi_B + \nabla_B \xi_A = 0,\]
so the angular sector of any residual diffeomorphism is a linear combination of the three rotational Killing vectors of the round two-sphere. Taking into account a possible \((t,r)\)-dependence, however, the most general solution of 32 on \(S^2\) is a conformal Killing vector, which decomposes uniquely into a rotational (Killing) part plus a “gradient” part (called proper conformal Killing vectors, or CKVs) [@Besse:1987em; @Obata:1970; @Schottenloher:2008cft]:

\[\label{OHMY} \boxed{\xi^A(t, r, \vartheta,\varphi) = \sum_{i=1}^3 a_i(t,r) \xi_{(i)}^A(\vartheta,\varphi) + \sum_{i=1}^3 b_i(t,r) K_{(i)}^A(\vartheta,\varphi).}\tag{33}\]
The conformal Killing algebra is \(\mathfrak{so}(3,1)\), while the Killing algebra of the two-sphere is three-dimensional, corresponding to the Lie algebra \(\mathfrak{so}(3)\). In standard spherical coordinates \((\vartheta, \varphi)\), a convenient basis for the Killing vectors is given by the generators of rotations about the Cartesian axes:

\[\begin{align} \label{CACA} \xi_{(x)}^{A}(\vartheta,\varphi) &= \begin{pmatrix} \sin\varphi,- \cot\vartheta \cos\varphi \end{pmatrix} \\ \xi_{(y)}^{A}(\vartheta,\varphi) &= \begin{pmatrix} \cos\varphi,-\cot\vartheta \sin\varphi \end{pmatrix} \\ \xi_{(z)}^A(\vartheta,\varphi) &= \begin{pmatrix} 0,1 \end{pmatrix}. \end{align}\tag{34}\]
We derived these results in Appendix A (see equations 81 , 82 , and 83 ). Additionally, proper CKVs can be written as:

\[\begin{matrix} K_{(i)}^A = \gamma^{AB} \partial_B n_i & , & (n_x, n_y, n_z) = (\sin\vartheta \cos\varphi, \sin\vartheta \sin\varphi, \cos\vartheta). \end{matrix}\]
With \(\gamma^{\vartheta \vartheta} = 1\) and \(\gamma^{\varphi \varphi} = \sin^{-2}\vartheta\), the proper CKVs take the form:

\[\begin{matrix} K_{(x)}^\vartheta = \cos\vartheta \cos\varphi & , & K_{(y)}^\vartheta = \cos\vartheta \sin\varphi & , & K_{(z)}^\vartheta = -\sin\vartheta \\ K_{(x)}^\varphi = -\frac{\sin\varphi}{\sin\vartheta} & , & K_{(y)}^\varphi = \frac{\cos\varphi}{\sin\vartheta} & , & K_{(z)}^\varphi = 0. \end{matrix}\]
In components, the general solution is, therefore,

\[\label{MAX} \boxed{\begin{align} \xi^\vartheta(t,r, \vartheta, \varphi) &= -a_1 (t,r) \sin\varphi + a_2(t,r) \cos\varphi + b_1(t,r) \cos\vartheta \cos\varphi \\ &+ b_2(t,r) \cos\vartheta \sin\varphi - b_3(t,r) \sin\vartheta \\ \xi^\varphi(t,r, \vartheta, \varphi) &= -a_1 (t,r) \cot\vartheta \cos\varphi - a_2(t,r) \cot\vartheta \sin\varphi + a_3(t,r) \\ &- b_1(t,r) \frac{\sin\varphi}{\sin\vartheta} + b_2(t,r) \frac{\cos\varphi}{\sin\vartheta}. \end{align}}\tag{35}\]
This is the most general class of angular solutions admissible in the Schwarzschild geometry that also preserves the Kerr–Schild ansatz. It is trivial to show that these solutions satisfy 27 and 29 . However, because of the coupled nature of the full system of PDEs, it is highly nontrivial to determine from these angular solutions the allowed forms of \(\xi^t(t,r,\vartheta,\varphi)\) and \(\xi^r(t,r,\vartheta,\varphi)\). To keep things more tractable, we will consider only the rotational (Killing) class of solutions. That is, we set \(b_i(t,r) = 0\).

\[\label{MARS} \boxed{\begin{align} \xi^\vartheta(t,r, \vartheta, \varphi) &= -a_1 (t,r) \sin\varphi + a_2(t,r) \cos\varphi \\ \xi^\varphi(t,r, \vartheta, \varphi) &= -a_1 (t,r) \cot\vartheta \cos\varphi - a_2(t,r) \cot\vartheta \sin\varphi + a_3(t,r). \end{align}}\tag{36}\]
These solutions, along with \(\eqref{SASA}\), impose a stringent constraint on \(\xi^r\), and make the whole system of PDEs more manageable, analytically-speaking. For the remainder of this section, we focus exclusively on this class of solutions, which correspond to the Killing sector of 35 . This makes the analysis substantially easier to perform while providing crucial insight into the full symmetries of the Kerr–Schild ansatz in the Schwarzschild geometry.

3.2.2 The Radial-Time Subsystem: Constraining \(\xi^r\) and \(\xi^t\)↩︎

The goal of this section is twofold: first, we derive the allowable forms of \(\xi^r\) and \(\xi^t\) for the Killing class of solutions, given by 36 , and show that the former is trivial. We then write down the radial-time subsystem of PDEs from the Lie derivative and show that \(\xi^t\) must be independent of \((t,r)\). We will further constrain this result in the next section using the mixed-angle equations.

As a sanity check, note that by equation 25 ,

\[\xi^r \stackrel{!}{=} - r \partial_\vartheta \xi^\vartheta.\] Because \(\xi^\vartheta(t,r,\vartheta,\varphi)\) is completely independent of \(\vartheta\) in 36 , \(\partial_\vartheta \xi^\vartheta = 0\) implies

\[\boxed{\xi^r \stackrel{!}{=} 0.}\]
Next, we work out the \(tt\), \(rr\), and \(tr\) equations and show that \(\xi^r = 0\) necessarily constrains \(\xi^t\) to be constant. This effectively “lifts” our angular Killing solutions of \(S^2\) to the full Schwarzschild geometry. Consider first:

\[\mathcal{H}_{t t} := \xi^\rho \partial_\rho \eta_{tt} + 2 \partial_t \xi^\rho \eta_{\rho t} + 2 \varphi (k_\rho \partial_t \xi^\rho) k_t \stackrel{!}{=} \zeta(x) (k_t)^2.\]
With \(\eta_{tt} = -1\), the first term vanishes for all \(\rho\). Meanwhile, the second term is only nonzero when \(\rho = t\). Summing over \(\rho = t,r\) in the term in parenthesis, we have,

\[\mathcal{H}_{t t} := 2 \partial_t \xi^t \eta_{t t} + 2 \varphi (k_t \partial_t \xi^t + k_r \partial_t \xi^r) k_t \stackrel{!}{=} \zeta(x) (k_t)^2.\]
Since \(k_t = -1\) and \(k_r = 1\) in spherical coordinates, this simplifies to

\[- 2 \partial_t \xi^t + 2 \varphi (\partial_t \xi^t - \partial_t \xi^r) = - 2(1-\varphi) \partial_t \xi^t - 2 \varphi \partial_t \xi^r\stackrel{!}{=} \zeta(x).\]
Enforcing \(\xi^r = 0\), we have:

\[\label{PAPA} \boxed{\partial_t \xi^t \stackrel{!}{=} - \frac{1}{2(1-\varphi)} \zeta(x).}\tag{37}\]
Next, consider the \(rr\) equation:

\[\mathcal{H}_{r r} := \xi^\rho \partial_\rho \eta_{r r} + 2 \partial_r \xi^\rho \eta_{\rho r} + 2 \varphi (k_\rho \partial_r \xi^\rho) k_r \stackrel{!}{=} \zeta(x) (k_r)^2.\]
Since \(\eta_{rr} = 1\), the first term again vanishes for all values of \(\rho\). The second term is only nontrivial when \(\rho = r\). As before, we sum over \(\rho = t, r\) in the third term and use the fact that \(k_r = 1\) to obtain:

\[2 \partial_r \xi^r + 2 \varphi (\partial_r \xi^r - \partial_r \xi^t) = 2 (1 + \varphi) \partial_r \xi^r - 2 \varphi \partial_r \xi^t \stackrel{!}{=} \zeta(x).\]
Again setting \(\xi^r = 0\) and solving for \(\partial_r \xi^t\), we obtain:

\[\label{UUUU} \boxed{\partial_r \xi^t \stackrel{!}{=} - \frac{1}{2 \varphi} \zeta(x).}\tag{38}\]
Finally, the \(tr\) equation reads:

\[\mathcal{H}_{t r} := \xi^\rho \partial_\rho \eta_{t r} + \partial_t \xi^\rho \eta_{\rho r} + \partial_r \xi^\rho \eta_{\rho t} + \varphi ( k_\rho \partial_t \xi^\rho) k_r + \varphi k_t (k_\rho \partial_r \xi^\rho) \stackrel{!}{=} \zeta(x) k_t k_r.\]
The background metric is diagonal, so \(\eta_{tr} = 0\). Furthermore, the second and third terms are nonzero only when \(\rho = r\) and \(\rho = t\), respectively. Summing over \(\rho = t, r\) in the fourth and fifth terms and taking again \(\eta_{tt} = -1\), \(\eta_{rr} = 1\), \(k_t = -1\), and \(k_r = 1\), we have:

\[\mathcal{H}_{t r} := \partial_t \xi^r - \partial_r \xi^t + \varphi(\partial_t \xi^r - \partial_t \xi^t - \partial_r \xi^r + \partial_r \xi^t) \stackrel{!}{=} - \zeta(x)\] Eliminating \(\xi^r\) and combining like terms, this yields

\[\label{QAQA} \boxed{- (1-\varphi)\partial_r \xi^t - \varphi \partial_t \xi^t \stackrel{!}{=} - \zeta(x).}\tag{39}\]
Plugging 37 and 38 into this, we find that

\[\begin{bmatrix} \frac{(1-\varphi)}{2 \varphi} + \frac{\varphi }{2(1-\varphi)} + 1 \end{bmatrix} \zeta(x) \stackrel{!}{=} 0.\]
Since the term in brackets is nonzero for \(\varphi \neq 1\) (that is, when \(r = 2GM\)) and finite \(r\). it follows that \(\zeta(x) \equiv 0\) on any open region away from the horizon/asymptotic boundary; smoothness then implies \(\zeta(x) \equiv 0\) there. With that, it must be the case that \(\xi^t\) is independent of both \(t\) and \(r\), by 37 and 38 , respectively:

\[\begin{matrix} \partial_t \xi^t = 0 \implies \xi^t = \xi^t(r,\vartheta,\varphi) & , & \partial_r \xi^t = 0 \implies \xi^t = \xi^t(\vartheta,\varphi). \end{matrix}\]
Thus, \(\xi^t\) is purely a function of \((\vartheta,\varphi)\). In the next section, we will use the mixed-angle PDEs to show that \(\xi^t\) is, inevitably, constant, signifying a time translation symmetry of the Kerr–Schild ansatz in the Schwarzschild geometry. This shouldn’t come as a surprise because time translation is a global isometry of the Schwarzschild solution (the solution is static, after all).

3.2.3 The Mixed-Angle Subsystem: Constraining \(\xi^t\) and \(a_i(t,r)\)↩︎

We continue our analysis by considering the mixed-angle PDEs. Recalling that \(\eta_{t \vartheta} = 0\), we begin with the \(t \vartheta\) equation:

\[\mathcal{H}_{t \vartheta} := \partial_t \xi^\rho \eta_{\rho \vartheta} + \partial_\vartheta \xi^\rho \eta_{\rho t} + \varphi (k_\rho \partial_t \xi^\rho) k_\vartheta + \varphi k_t (k_\rho \partial_\vartheta \xi^\rho) = \zeta(x) k_t k_\vartheta.\]
The first term is nonzero when \(\rho = \vartheta\), while the second term is nonzero for \(\rho = t\). The third term and right-hand side vanish as \(k_\vartheta = 0\) in spherical coordinates. Finally, only \(\rho = t\) survives in the fourth term since, so we have:

\[\begin{align} \mathcal{H}_{t \vartheta} &:= \partial_t \xi^\vartheta \eta_{\vartheta \vartheta} + \partial_\vartheta \xi^t \eta_{t t} + \varphi k_t (k_t \partial_\vartheta \xi^t) \\ &= \partial_t \xi^\vartheta \eta_{\vartheta \vartheta} + \partial_\vartheta \xi^t \eta_{t t} + \varphi \partial_\vartheta \xi^t \\ &\stackrel{!}{=}0. \end{align}\]
Setting \(\eta_{\vartheta \vartheta} = r^2\) and \(\eta_{tt} = -1\) and combining like terms, this becomes:

\[\label{EAEA} \boxed{r^2 \partial_t \xi^\vartheta - (1 - \varphi) \partial_\vartheta \xi^t \stackrel{!}{=} 0 \implies \partial_t \xi^\vartheta = \frac{(1-\varphi)}{r^2} \partial_\vartheta \xi^t.}\tag{40}\]
Calculating next the \(t \varphi\) component, we have:

\[\mathcal{H}_{t \varphi} := \partial_t \xi^\rho \eta_{\rho \varphi} + \partial_\varphi \xi^\rho \eta_{\rho t} + \varphi (k_\rho \partial_t \xi^\rho) k_\varphi + \varphi k_t (k_\rho \partial_\varphi \xi^\rho) \stackrel{!}{=} \zeta(x) k_t k_\varphi,\]
since \(\eta_{t \varphi} = 0\) as well. Set \(\rho = \varphi\) in the first term and \(\rho = t\) in the second. The third term and right-hand side vanish as \(k_\varphi = 0\) in spherical coordinates. Once again setting \(\rho = t\) in the fourth term, we have:

\[\begin{align} \mathcal{H}_{t \varphi} &:= \partial_t \xi^\varphi \eta_{\varphi \varphi} + \partial_\varphi \xi^t \eta_{t t} + \varphi k_t (k_t \partial_\varphi \xi^t) \\ &= \partial_t \xi^\varphi \eta_{\varphi \varphi} + \partial_\varphi \xi^t \eta_{t t} + \varphi \partial_\varphi \xi^t \\ &\stackrel{!}{=}0. \end{align}\]
Setting \(\eta_{\varphi \varphi} = r^2 \sin^2\vartheta\) and \(\eta_{tt}=-1\) and combining like terms yields:

\[\label{FAFA} \boxed{r^2 \sin^2\vartheta \partial_t \xi^\varphi - (1 - \varphi) \partial_\varphi \xi^t \stackrel{!}{=} 0 \implies \partial_t \xi^\varphi = \frac{(1-\varphi)}{r^2 \sin^2\vartheta} \partial_\varphi \xi^t.}\tag{41}\]
Consider next the \(r \vartheta\) equation:

\[\mathcal{H}_{r \vartheta} := \partial_r \xi^\rho \eta_{\rho \vartheta} + \partial_\vartheta \xi^\rho \eta_{\rho r} + \varphi (k_\rho \partial_r \xi^\rho) k_\vartheta + \varphi k_r (k_\rho \partial_\vartheta \xi^\rho) \stackrel{!}{=} \zeta(x) k_r k_\vartheta,\]
since \(\eta_{r \vartheta} = 0\). Now, the first term is zero unless \(\rho = \vartheta\). The second term vanishes for all \(\rho\). The third term and right-hand side vanish as before. Setting \(\rho = t\) in the fourth term again, we have:

\[\begin{align} \mathcal{H}_{r \vartheta} &:= \partial_r \xi^\vartheta \eta_{\vartheta \vartheta} + \varphi k_r (k_t \partial_\vartheta \xi^t) \\ &= \partial_r \xi^\vartheta \eta_{\vartheta \vartheta} - \varphi \partial_\vartheta \xi^t \\ &\stackrel{!}{=}0. \end{align}\]
With \(\eta_{rr}\) and \(\eta_{\varphi \varphi}\) given as before, this yields:

\[\label{HOHO} \boxed{r^2 \partial_r \xi^\vartheta - \varphi \partial_\vartheta \xi^t \stackrel{!}{=} 0 \implies \partial_r \xi^\vartheta = \frac{\varphi}{r^2} \partial_\vartheta \xi^t.}\tag{42}\]
Finally, it is trivial to see that the \(r \varphi\) equation is identical to this, but with \(\varphi\) instead of \(\vartheta\), so \(\eta_{\varphi \varphi} = r^2 \sin^2 \vartheta\) replaces \(\eta_{\vartheta \vartheta} = r^2\):

\[\label{HIHI} \boxed{r^2 \sin^2 \vartheta \partial_r \xi^\varphi - \varphi \partial_\varphi \xi^t \stackrel{!}{=} 0 \implies \partial_r \xi^\varphi = \frac{\varphi}{r^2 \sin^2 \vartheta} \partial_\varphi \xi^t.}\tag{43}\]
Thus, we have obtained the mixed-angle subsystem of PDEs. The trick now is to differentiate the \(\vartheta \varphi\) equation, and then use results 40 , 41 , 42 , and 43 to eliminate dependence of \(\xi^t\) on \((\vartheta, \varphi)\) and \(\xi^\vartheta, \xi^\varphi\) on \((t, r)\). Recall the \(\vartheta \varphi\) equation, given by 29 :

\[\sin^2 \vartheta \partial_\vartheta \xi^\varphi + \partial_\varphi \xi^\vartheta \stackrel{!}{=} 0.\]
Differentiating this with respect to \(t\), we find:

\[\partial_t \partial_\vartheta \xi^\varphi \sin^2 \vartheta + \partial_t \partial_\varphi \xi^\vartheta = \partial_\vartheta (\partial_t \xi^\varphi) \sin^2 \vartheta + \partial_\varphi (\partial_t \xi^\vartheta) = 0.\]
Utilizing 40 and 41 and pulling out a common factor of \((1-\varphi)/r^2\), we obtain:

\[\frac{1-\varphi}{r^2} \begin{bmatrix} \sin^2 \vartheta \partial_\vartheta \begin{pmatrix} \frac{1}{\sin^2\vartheta} \partial_\varphi \xi^t \end{pmatrix} + \partial_\varphi \begin{pmatrix} \partial_\vartheta \xi^t\end{pmatrix} \end{bmatrix} = 0.\] Dropping the factor of \((1-\varphi)/r^2\), applying the product rule to differentiate \(\sin^{-2}\vartheta \partial_\varphi \xi^t\) with respect to \(\vartheta\), and simplifying terms gives after a bit of algebra gives:

\[- \cot\vartheta \partial_\varphi \xi^t + \partial_\vartheta \partial_\varphi \xi^t = 0.\]
This can be solved via the following substitution: let \(g(\vartheta) := \partial_\varphi \xi^t\) so that this reads \(g'(\vartheta) - \cot\vartheta g(\vartheta) = 0\). Using the fact that \(\xi^t\) is independent of \((t,r)\), this simple ODE has the solution:

\[g(\vartheta,\varphi) = C(\varphi) \sin\vartheta\]
for smooth function \(C(\varphi)\). Plugging this into 41 gives:

\[\label{YESSS} \partial_t \xi^\varphi = \frac{(1 - \varphi)}{r^2 \sin^2\vartheta} C(\varphi) \sin\vartheta = \frac{(1 - \varphi)}{r^2 \sin\vartheta} C(\varphi).\tag{44}\]
Recall that \(\xi^\vartheta\) is independent of \(\vartheta\). Differentiating 40 with respect to \(\vartheta\), we find that \(\partial_\vartheta \xi^t\) is also independent of \(\vartheta\):

\[\label{AARDVARK} \partial_\vartheta (\partial_t \xi^\vartheta) = \partial_t \underbrace{(\partial_\vartheta \xi^\vartheta)}_{=0} = \frac{(1-\varphi)}{r^2} \partial_\vartheta (\partial_\vartheta \xi^t) = 0 \implies \partial_\vartheta \xi^t := A(\varphi),\tag{45}\]
where \(A(\varphi)\) is smooth and independent of \(\vartheta\). Consider next the \(\vartheta\) derivative of \(g(\vartheta,\varphi)\):

\[\label{YES} \partial_\vartheta g(\vartheta,\varphi) := \partial_\vartheta \partial_\varphi \xi^t = \partial_\varphi \partial_\vartheta \xi^t := \partial_\varphi A(\varphi)\tag{46}\]
is independent of \(\vartheta\). However, by definition,

\[\label{NO} \partial_\vartheta g(\vartheta,\varphi) := C(\varphi) \cos\vartheta.\tag{47}\]
This expression is independent of \(\vartheta\) if and only if \(C(\varphi) \equiv 0\). Otherwise, we obtain a contradiction between 46 and 47 . Thus we must have \(C(\varphi) = 0\), which immediately implies \(\partial_t \xi^\varphi = 0\) by 44 . Thus, \(\xi^\varphi\) has no \(t\)-dependence. Substituting back into 41 forces \(\partial_\varphi \xi^t = 0\), so \(\xi^t\) is independent of \(\varphi\). Exploiting this independence in 43 then forces \(\partial_r \xi^\varphi = 0\), showing that \(\xi^\varphi\) is independent of both \(t\) and \(r\).

Consequently, the functions \(a_i(t,r)\) appearing in the general solution 36 cannot depend on \((t,r)\) and must be constants. Since \(\xi^\vartheta\) involves the same coefficients \(a_1\) and \(a_2\) as \(\xi^\varphi\), it follows that \(\xi^\vartheta\) is likewise independent of \((t,r)\).

Finally, inserting this result into 40 shows that \(\partial_\vartheta \xi^t = 0\). Together with \(\partial_\varphi \xi^t = 0\), we conclude that \(\xi^t\) has no angular dependence. Therefore, \(\xi^t\) is independent of all coordinates \((t,r,\vartheta,\varphi)\) and must be constant.

Our angular solutions 36 simplify to

\[\boxed{\begin{align} \xi^\vartheta &\equiv \xi^\vartheta(\vartheta, \varphi) = -a_1 \sin\varphi + a_2 \cos\varphi \\ \xi^\varphi &\equiv \xi^\varphi(\vartheta, \varphi) = - a_1 \cot\vartheta \cos\varphi - a_2 \cot\vartheta \sin\varphi + a_3. \end{align}}\] With these angular solutions and our results for \(\xi^t\) and \(\xi^r\), our most general set of solutions is:

\[\begin{align} \xi^t &= c_1 \\ \xi^r &= 0 \\ \xi^\vartheta &= -a_1 \sin\varphi + a_2 \cos\varphi \\ \xi^\varphi &= - a_1 \cot\vartheta \cos\varphi - a_2 \cot\vartheta \sin\varphi + a_3. \end{align}\]
Without loss of generality, let \(c_1 = a_1 = a_2 = a_3 = 1\). Then,

\[\label{SOLSFIN} \boxed{\begin{align} \xi^t &= 1 \\ \xi^r &= 0 \\ \xi^\vartheta &= -\sin\varphi + \cos\varphi \\ \xi^\varphi &= - \cot\vartheta \cos\varphi - \cot\vartheta \sin\varphi + 1. \end{align}}\tag{48}\]
This system of solutions contains nothing more than the time translation (e.g., \(\xi = \partial_t\)) and linear combinations of the rotational Killing vectors of \(S^2\). The former is trivial to see. That \(\xi^\vartheta\) and \(\xi^\varphi\) are spatial rotations can be more readily observed by noting that the rotational vector field \(\xi = \xi^\vartheta \partial_\vartheta + \xi^\varphi \partial_\varphi\) is

\[\label{AWJEEZ} \xi = -\sin\varphi \partial_\vartheta - \cot\vartheta \cos\varphi \partial_\varphi + \cos\varphi \partial_\vartheta -\cot\vartheta \sin\varphi \partial_\varphi + \partial_\varphi,\tag{49}\]
which we identify as the sum over all rotation vectors \(\xi_{(i)}\), for \(i = 1,~2,~3\), about the \(x\)-, \(y\)-, and \(z\)-axes. That is,

\[\xi = \xi_{(x)} + \xi_{(y)} + \xi_{(z)}.\]
It follows that when we consider only the rotational (Killing) class of the angular subsystem of equations - 27 and 29 - the most general (non-conformal) diffeomorphisms \(\xi^\mu\) preserving the KS ansatz are precisely time translation and linear combinations of the rotational Killing vectors of \(S^2\). Within this sector, no further independent solutions arise: the rotational freedom is exhausted by the \(\mathfrak{so}(3)\) algebra of \(S^2\), with constant coefficients fixed by the mixed-angle equations. Thus, the space of residual diffeomorphisms we retain is finite-dimensional, closing under the expected Poincaré subalgebra compatible with spherical symmetry. We will elaborate on this in Section 3.4.

3.3 Explicit Analysis of Residual Diffeomorphisms↩︎

In this section, we will briefly show that the residual diffeomorphisms derived previously do result in a vanishing Lie derivative,

\[\label{YIKES} (\mathcal{L}_\xi g)_{\mu \nu} := \xi^\rho \partial_\rho g_{\mu \nu} + 2 \partial_{(\mu} \xi^\rho g_{\nu) \rho} = 0.\tag{50}\]
This guarantees that our solutions are global isometries of the Schwarzschild geometry. In spherical coordinates, the background metric can be written

\[\eta_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2\vartheta \end{pmatrix}.\] Consequently, the full KS metric is

\[g_{\mu \nu} = \begin{pmatrix} - (1 + \varphi) & \varphi & 0 & 0 \\ - \varphi & (1 + \varphi) & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \vartheta \end{pmatrix},\]
where \(\varphi = \varphi(r)\). We will now systematically show that time translations and spatial rotations are global isometries of the Kerr–Schild metric for the Schwarzschild geometry. To save time and space, we will go through all nontrivial components of the Lie derivative for each of 48 , first for time translations \(\xi = \partial_t\), and then for our rotational Killing solutions.

We will separate solutions 48 into two distinct vector fields, namely:

\[\label{LMFAO} \xi^\mu_{(t)} = (1,~0,~0,~0)\tag{51}\]
for time translations and

\[\label{LMAO} \xi^\mu_{(i)} = (0,~0,- \sin\varphi + \cos\varphi,-\cot\vartheta \cos\varphi - \cot\vartheta \sin\varphi + 1)\tag{52}\]
for spatial rotations, each now embedded in 4-dimensional spacetime. Consider first the time translation. With this form of \(\xi^\mu\), the analysis is made considerably easy by noting that only equations with a \(t\) index and no angular indices are nontrivial - due to the fact that \(\xi^r{(t)} = \xi^\vartheta_{(t)} = \xi^\varphi_{(t)} = 0\). This means we only have to work with two equations: \(tt\) and \(tr\). Working first the \(tt\) equation and considering only potentially non-vanishing terms, we have,

\[(\mathcal{L}_{\xi_{(t)}} g)_{tt} = \xi_{(t)}^t \partial_t g_{tt} + 2 \partial_t \xi_{(t)}^t g_{tt} = 0\]
since \(g_{tt}\) and \(\xi^t_{(t)}\) are both independent of \(t\). Next, the \(tr\) equation reads

\[(\mathcal{L}_{\xi_{(t)}} g)_{tr} = \xi_{(t)}^t \partial_t g_{tr} + \partial_t \xi_{(t)}^t g_{tr} + \partial_r \xi_{(t)}^t g_{tt} = 0\]
because \(g_{tr}\) is independent of \(t\) as well. Hence, all components of \((\mathcal{L}_\xi g)_{\mu \nu}\) are zero. With all else vanishing trivially, we have shown that time translation is a global isometry of the Kerr–Schild ansatz in the Schwarzschild geometry.

Next, we will show that the same holds for \(\xi^\mu_{(i)}\). Note that since \(\xi^\mu_{(i)}\) depends on neither \(t\) nor \(r\), all components of the Lagrangian containing a \(t\) or \(r\) vanish. This means we must check the angular subsystem only. Starting with the \(\vartheta \vartheta\) equation, we have:

\[(\mathcal{L}_{\xi_{(i)}} g)_{\vartheta \vartheta} := \xi^\rho \partial_\rho g_{\vartheta \vartheta} + 2 \partial_\vartheta \xi^\rho g_{\vartheta \rho}.\]
By inspection, the first term is only nonzero when \(\rho = r\) since \(g_{\vartheta \vartheta}\) depends on \(r\). However, \(\xi^r_{(i)} = 0\), so this term vanishes. Moreover, the second term vanishes because \(\xi^\vartheta_{(i)}\) is independent of \(\vartheta\), so indeed

\[(\mathcal{L}_{\xi_{(i)}} g)_{\vartheta \vartheta} := \xi_{(i)}^\rho \partial_\rho g_{\vartheta \vartheta} + 2 \partial_\vartheta \xi_{(i)}^\rho g_{\vartheta \rho} = 0.\] Working the \(\varphi \varphi\) equation, we have:

\[(\mathcal{L}_{\xi_{(i)}} g)_{\varphi \varphi} := \xi_{(i)}^\rho \partial_\rho g_{\varphi \varphi} + 2 \partial_\varphi \xi_{(i)}^\rho g_{\varphi \rho}.\]
Since \(g_{\varphi \varphi}\) depends on both \((r, \vartheta)\), the first term is non-vanishing for \(\rho = \vartheta\). Additionally, the second term is non-vanishing when \(\rho = \varphi\) since \(\xi^\varphi_{(i)}\) does depend on \(\varphi\). Using 52 and \(g_{\varphi \varphi} = r^2 \sin^2\vartheta\), this equation becomes:

\[\begin{align} (\mathcal{L}_{\xi_{(i)}} g)_{\varphi \varphi} &:= \xi_{(i)}^\vartheta \partial_\vartheta g_{\varphi \varphi} + 2 \partial_\varphi \xi_{(i)}^\varphi g_{\varphi \varphi} \\ &= r^2 (- \sin\varphi + \cos\varphi) \partial_\vartheta \sin^2 \vartheta \\ &+ 2 r^2 \sin^2 \vartheta \partial_\varphi (-\cot\vartheta \cos\varphi - \cot\vartheta \sin\varphi + 1) \\ &= 2 r^2 \sin\vartheta \cos\vartheta (- \sin\varphi + \cos\varphi) \\ &+ 2 r^2 \sin^2 \vartheta (\cot\vartheta \sin\varphi - \cot\vartheta \cos\varphi) \\ &= 2 r^2 \sin\vartheta \cos\vartheta (- \sin\varphi + \cos\varphi) \\ &+ 2 r^2 \sin \vartheta \cos\vartheta (\sin\varphi - \cos\varphi) \\ &= 0. \end{align}\]
Finally, the \(\vartheta \varphi\) equation is:

\[(\mathcal{L}_{\xi_{(i)}} g)_{\vartheta \varphi} := \xi_{(i)}^\rho \partial_\rho g_{\vartheta \varphi} + \partial_{\vartheta} \xi_{(i)}^\rho g_{\varphi \rho} + \partial_{\varphi} \xi_{(i)}^\rho g_{\vartheta \rho}.\]
But \(g_{\vartheta \varphi} = 0\), so the first term vanishes. The second term is only nonzero when \(\rho = \varphi\), while the third term is nonzero when \(\rho = \vartheta\). This gives:

\[\begin{align} (\mathcal{L}_{\xi_{(i)}} g)_{\vartheta \varphi} &:= \partial_{\vartheta} \xi_{(i)}^\varphi g_{\varphi \varphi} + \partial_{\varphi} \xi_{(i)}^\vartheta g_{\vartheta \vartheta} \\ &= r^2 \partial_\vartheta (-\cot\vartheta \cos\varphi - \cot\vartheta \sin\varphi + 1) \sin^2 \vartheta + r^2 \partial_\varphi (- \sin\varphi + \cos\varphi) \\ &= r^2 (\csc^2 \vartheta \cos\varphi + \csc^2 \vartheta \sin\varphi) \sin^2 \vartheta - r^2 (\cos\varphi + \sin\varphi) \\ &= r^2 (\cos\varphi + \sin\varphi) - r^2 (\cos\varphi + \sin\varphi) \\ &= 0. \end{align}\]
Hence, we have shown that \(\xi_{(i)}^\mu\) is a symmetry of the full metric, and therefore, a global isometry of the Schwarzschild geometry.

Moreover, the Kerr–Schild structure is preserved consistently under these symmetry transformations when expressed in different coordinate systems, such as Cartesian. In Appendix A we demonstrate this explicitly in Cartesian coordinates, confirming that the solutions found in spherical coordinates transform consistently. Because any regular curvilinear coordinate system can be related to Cartesian or spherical frames by the usual tensor transformation law, this argument generalizes: time translations and spatial rotations form a coordinate-invariant class of solutions. Within the Killing sector, these are therefore the only residual diffeomorphisms that survive in the Schwarzschild Kerr–Schild framework.

The consequence is immediate. Since only the global isometries of Schwarzschild remain in this context, there is no nontrivial residual diffeomorphism sector available to match with the infinite-dimensional residual gauge symmetries identified in Section 2. The two structures are therefore fundamentally incompatible, even though both sets of transformations are consistent with their respective Kerr–Schild ansätze. This highlights a key limitation of the Kerr–Schild double copy: while it relates exact field configurations across gauge and gravity, it fails to preserve symmetry structures at the residual level.

3.4 The Residual Diffeomorphism Algebra↩︎

For the restricted class of solutions \(\xi^\mu\), we now turn to the underlying Lie algebra generated by these diffeomorphisms. The aim here is to emphasize the structural mismatch with the results of Section 2, which confirms that no symmetry-preserving map exists between the gauge theory residuals and those of the Schwarzschild Kerr–Schild framework. We write the generators in a Cartesian basis for clarity, but by the results of Appendix A, these are equivalent to the spherical expressions derived in earlier sections.

Let \(\xi^\mu\) be a vector field such that 50 holds. We define the corresponding Lie algebra as

\[\mathfrak{g} := \{ \xi^\mu \in \Gamma(T\mathcal{M})~|~(\mathcal{L}_\xi g)_{\mu \nu} \propto k_\mu k_\nu \},\]
where \(\Gamma(T\mathcal{M})\) denotes the set of all smooth vector fields on the manifold \(M\). In the Killing sector considered above, the algebra \(\mathfrak{g}\) closes under the usual Lie bracket of vector fields and is generated by the time-translation symmetry together with the three rotational Killing vectors of the round two-sphere:

• Rotations: 3 generators, \(R_i\) for \(i = 1,~2,~3\), forming the algebra \(\mathfrak{so}(3)\);

• Time translations: 1 generator, \(\partial_t\), corresponding to \(\mathbb{R}\).

The Lie bracket is given as usual by

\[[\xi_1, \xi_2]^\mu := \xi^\nu_1 \partial_\nu \xi^\mu_2 - \xi^\nu_2 \partial_\nu \xi^\mu_1\]
and with respect to this bracket, the algebra is closed, finite-dimensional, and matches the isometry algebra of Schwarzschild spacetime. To see this explicitly, let \(\mathfrak{g} \subset \Gamma(T \mathcal{M})\) denote the space of vector fields so that \((\mathcal{L}_\xi g)_{\mu \nu}= 0\); equivalently, \(\xi^\mu\) is a Killing vector of the Schwarzschild metric in Kerr–Schild form.

As established above, the only such vector fields are the global isometries of Schwarzschild: time translation and three spatial rotations. Thus, the space \(\mathfrak{g}\) is 4-dimensional, spanned by:

\[\mathfrak{g} = \text{span}\{ \partial_t, R_i \},\]
where \(R_i\) are the generators of spatial rotations, with \(i = 1, 2, 3\).

Next, we define the Lie bracket structure. Let \(f \in C^\infty(\mathcal{M})\) be a test function. The Lie brackets are:

\[\begin{matrix} \label{CCC} [\partial_t, R_i](f) = \partial_t (R_i (f)) - R_1 (\partial_t (f)) = R_i (\partial_t (f)) - R_1 (\partial_t (f)) = 0, \end{matrix}\tag{53}\]
since \(R_i\) are independent of time, and

\[[R_i, R_j] (f) = \varepsilon_{ino} \varepsilon_{jol} x^n \partial_l (f) - \varepsilon_{ilo} \varepsilon_{jkl} x^k \partial_o (f).\]
Permuting \(o \leftrightarrow l\) in each term reverses the sign. Applying the identity:

\[\label{BBB} \varepsilon_{jkl} \varepsilon_{mnl} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}\tag{54}\]
yields

\[[R_i, R_j] (f) = (\delta_{ij} \delta_{ko} - \delta_{ik} \delta_{jo}) x^k \partial_o (f) - (\delta_{ij} \delta_{no} - \delta_{il} \delta_{jn}) x^n \partial_l (f).\]
Relabeling \(k \leftrightarrow n\) and \(o \leftrightarrow l\) in the first term in the first set of parentheses causes a cancellation and we are left with:

\[[R_i, R_j] (f) = \delta_{il} \delta_{jk} x^k \partial_l (f) - \delta_{ik} \delta_{jo} x^k \partial_o (f).\]
Relabeling \(k \leftrightarrow n\) in the first term and \(o \leftrightarrow l\) in the second term, we have:

\[[R_i, R_j] (f) = \delta_{il} \delta_{jk} x^k \partial_l (f) - \delta_{ik} \delta_{jl} x^k \partial_l (f) = (\delta_{il} \delta_{jk} - \delta_{ik} \delta_{jl}) x^k \partial_l (f).\]
By identity 54 , this becomes

\[[R_i, R_j] (f) = \varepsilon_{ijm} \varepsilon_{lkm} x^k \partial_l (f) = \varepsilon_{ijk} \varepsilon_{lmk} x^m \partial_l (f)\]
upon relabeling \(k \leftrightarrow m\). But \(R_k = \varepsilon_{k} x^l \partial_m\). Relabeling \(l \leftrightarrow m\) and permuting \(k \leftrightarrow l \leftrightarrow m\) (so we pick up two minus signs by the definition of the Levi-Civita symbol) gives:

\[\label{DDD} [R_i, R_j] (f) = \varepsilon_{ijk} \varepsilon_{klm} x^l \partial_m (f) = \varepsilon_{ijk} R_k (f) \implies [R_i, R_j] = \varepsilon_{ijk} R_k.\tag{55}\]
We have a classic result of Lie group theory. This is a geometric identity — a direct consequence of the fact that the \(R_i\) form a basis of the Lie algebra \(\mathfrak{so}(3)\), the algebra of infinitesimal rotations. This Lie bracket is not dependent on coordinates — it follows from the group structure of \(SO(3)\). That is, the structure constants \(\varepsilon_{ijk}\) encode the algebra of rotations in Euclidean 3-space, and are intrinsic to the geometry.

Together, Lie brackets 53 and 55 define a Lie algebra structure on \(\mathfrak{g}\) with the subalgebra \(\mathfrak{so}(3) = \text{span} \{R_i\}\) and \(\partial_t\) commuting with all \(R_i\), forming an Abelian ideal. Hence,

\[\mathfrak{g} \cong \mathfrak{so}(3) \oplus \mathbb{R}.\]
To show this explicitly, define the linear map \(\Psi: \mathfrak{g} \rightarrow \mathfrak{so}(3) \oplus \mathbb{R}\) by

\[\begin{matrix} \label{EEE} \Psi(\partial_t) = (0, 1) & , & \Psi(R_i) = (J_i, 0) \end{matrix}\tag{56}\]
where \(J_i \in \mathfrak{so}(3)\) are the standard basis elements and \(\mathbb{R}\) is the 1-dimensional Abelian Lie algebra. This is clearly a linear map, by construction. So we must verify that \(\Psi\) respects the Lie brackets. This is straightforward:

• \(\Psi([\partial_t, R_i]) = \Psi(0) = (0,0) = [\Psi(\partial_t), \Psi(R_i)]\);

• \(\Psi([R_i, R_j]) = \Psi(\varepsilon_{ijk} R_k) = (\varepsilon_{ijk} J_k, 0) = [J_i, J_j] = [\Psi(R_i), \Psi(R_j)]\).

So \(\Psi\) is a Lie algebra homomorphism. To check that it is an isomorphism, note that \(\Psi\) is necessarily injective because the kernel is zero. It is also surjective because its image spans all of \(\mathfrak{so}(3) \oplus \mathbb{R}\).

To show the former, define the map \(\Phi: \Gamma(T \mathcal{M}) \rightarrow \text{Der}(\mathcal{F})\) that sends a vector field \(\xi^\mu \in \Gamma(T \mathcal{M})\) to the differential operator \(\mathcal{L}_\xi\), acting on some field space \(\mathcal{F}\) (like metrics, gauge fields, etc.). The kernel \(\ker \Phi \subseteq \Gamma (T \mathcal{M})\) is the set of vector fields such that \(\mathcal{H}\) vanishes for arbitrary tensor field \(T\).

Let’s suppose \(\mathcal{H} = 0\) for all tensor fields \(T\) on the manifold \(\mathcal{M}\). Then we can show that \(\xi^\mu = 0\) identically.

Proof of injectivity.
Let \(\xi \in \Gamma(T \mathcal{M})\) be a smooth vector field on a smooth manifold \(\mathcal{M}\). Suppose that \(\mathcal{H} = 0\) for all tensor fields \(T\) on \(\mathcal{M}\). Additionally, let \(f \in C^\infty(\mathcal{M})\) be an arbitrary smooth function. The Lie derivative of a scalar is:

\[\mathcal{L}_\xi f = \xi^\mu \partial_\mu f.\]
This is just the directional derivative of \(f\) along \(\xi\). So, if \(\mathcal{L}_\xi f = 0\) for all smooth functions \(f\), then:

\[\begin{matrix} \xi^\mu \partial_\mu f = 0 & & \forall~f \in C^\infty(\mathcal{M}). \end{matrix}\]
Now, at any point \(p \in \mathcal{M}\), let’s fix coordinates \((x^1, ..., x^n)\). We can choose functions \(f = x^i\) in the neighborhood of \(p\). Then:

\[\xi^\mu \partial_\mu x^i = \xi^\mu {\delta^i}_\mu = \xi^i = 0.\]
Thus, all components of \(\xi^\mu\) vanish at \(p\). Since was arbitrary, this implies that \(\xi^\mu = 0\) everywhere on \(\mathcal{M}\). The only vector field with vanishing Lie derivative on all functions is the zero vector field, so the kernel is zero. Hence, the Lie algebra of vector fields acts faithfully on the tensor algebra, with no nontrivial vector field acting trivially. Since \(\mathcal{L}_\xi = 0 \implies \xi^\mu = 0\), \(\Psi\) is injective. \(\square\)

In order to show that \(\Psi\) is surjective, we remark that each image in \(\eqref{EEE}\) is an element of \(\mathfrak{so}(3) \oplus \mathbb{R}\) by the definition of \(\Psi\). With that, consider the following proof.

Proof of surjectivity.
Let \((X, a) \in \mathfrak{so}(3) \oplus \mathbb{R}\). Then there exist real numbers \(\alpha^i\) and \(\beta\) such that

\[(X, a) = \begin{pmatrix} \alpha^i e_i, \beta \end{pmatrix}.\]
where \(\{e_i\}\) form a basis for \(\mathfrak{so}(3)\). Consider the element \(\xi = \alpha^i R_i + \beta \partial_t \in \mathfrak{g}\). Then,

\[\Psi(\xi) = \Psi \begin{pmatrix} \alpha^i R_i + \beta \partial_t \end{pmatrix} = \Psi(\alpha^i R_i) + \beta \Psi(\partial_t) = (X, 0) + (0, a) = (X, a).\] Thus, \((X, a) \in \text{im}(\Psi)\), so the image of \(\Psi\) spans the whole \(\mathfrak{so}(3) \oplus \mathbb{R}\) and \(\Psi\) is surjective. \(\square\)

This establishes that \(\Psi\) is a vector space isomorphism, i.e.,

\[\label{FFF} \boxed{\mathfrak{g} \cong \mathfrak{so}(3) \oplus \mathbb{R}.}\tag{57}\]
The Poincaré group is the isometry group of Minkowski spacetime \((\mathbb{R}^{1,3}, \eta)\) with corresponding Lie algebra

\[\mathfrak{p} \cong \mathfrak{so}(1,3) \rtimes \mathbb{R}^{1,3},\]
where \(\mathfrak{so}(1,3)\) is the Lorentz subalgebra (rotations + boosts) and \(\mathbb{R}^{1,3}\) is the Abelian subalgebra of spacetime translations. We have previously shown that the residual symmetries generate the algebra 57 , where \(\mathfrak{so}(3)\) is the algebra of rotations in 3-space. Clearly, \(\mathfrak{so}(3) \subset \mathfrak{so}(1,3)\).

On top of this, the 1-dimensional Abelian Lie algebra \(\mathbb{R} \subset \mathbb{R}^{1,3}\), so it trivial to see that \(\mathfrak{g} \subset \mathfrak{p}\). Hence, the residual diffeomorphisms form a Lie subalgebra of the Poincaré group. This contrasts sharply with the infinite-dimensional residual symmetry algebras found in both Abelian and non-Abelian gauge theories, revealing an algebraic obstruction to symmetry matching in the classical double copy. We discuss this in the next section.

3.5 Algebraic Obstruction to Symmetry Matching↩︎

In the Kerr–Schild formulation of the Schwarzschild geometry, we have shown that the only residual diffeomorphisms preserving the KS structure are the global isometries of the full metric: time translations and spatial rotations. These generate the Lie algebra

\[\mathfrak{g}_{\text{gravity}} \cong \mathfrak{so}(3) \oplus \mathbb{R},\]
which is finite-dimensional and corresponds to the isometry algebra of the Schwarzschild spacetime.

On the other hand, in gauge theory, particularly in the Abelian and non-Abelian settings, the class of residual gauge transformations is vastly richer. There exists an infinite-dimensional space of gauge parameters \(\lambda(x)\) that solve the equation \(k^\mu \partial_\mu \lambda(x) = 0\). This yields a gauge symmetry algebra isomorphic to:

• \(C^\infty (\mathbb{R})\), the smooth, real functions on spacetime (Abelian case), and

• \(\mathfrak{g} \otimes C^\infty (\mathbb{R})\) (non-Abelian case), i.e., a current algebra over \(\mathbb{R}\).

This infinite-dimensional structure is manifestly not isomorphic to \(\mathfrak{so}(3) \oplus \mathbb{R}\). Thus, no Lie algebra homomorphism exists between the residual gauge symmetry algebra and the residual gravitational symmetry algebra in the Kerr–Schild double copy.

This represents an algebraic obstruction to any would-be map between the gauge and gravity residual symmetries. This obstruction reveals a fundamental asymmetry in the symmetry structure of the double copy: while the field-level correspondence between gauge theory and gravity holds via the KS ansatz, the symmetry algebras do not align. This mismatch suggests that:

• The double copy does not extend to a mapping of residual gauge symmetries to residual diffeomorphisms in curved Kerr–Schild backgrounds like Schwarzschild.

• The mismatch is not just coordinate-dependent — it persists even under transformations to other curvilinear coordinate systems.

• Any attempt to extend the double copy to residual symmetries must confront this algebraic incompatibility.

In summary, the residual gauge transformations analyzed in Section 2 form infinite-dimensional algebras, both in the Abelian and non-Abelian cases. By contrast, in the Kerr–Schild Schwarzschild background the residual diffeomorphisms collapse, within the Killing sector, to a finite-dimensional algebra isomorphic to the global isometries, \(\mathfrak{so}(3) \oplus \mathbb{R}\). This sharp difference — infinite-dimensional on the gauge side versus finite-dimensional on the gravity side — constitutes a formal obstruction to identifying residual gauge symmetries with residual diffeomorphisms via the Kerr–Schild double copy. This algebraic mismatch underscores a fundamental limitation: although the double copy provides an elegant map between exact solutions, it fails to preserve the richer algebraic structures of their residual symmetries. Understanding whether this obstruction persists in more general spacetimes (such as Kerr) or in broader formulations of the double copy, remains an important question for future work.

3.6 BRST Formulation for the Killing Class of Symmetries↩︎

Finally, in this section we take the first steps toward a BRST formulation of the Kerr–Schild double copy for the Schwarzschild geometry. In particular, we show that because the only residual diffeomorphisms admitted in the Killing class are global isometries, the Kerr–Schild ansatz admits no nontrivial BRST realization in this sector.
To show this, we introduce Grassmann-odd ghosts \(c^a\) for each generator \(K_a\) of the residual algebra

\[\begin{matrix} \mathfrak{g}_{\text{res}} = \text{span}\{K_a\} = \text{span}\{K_0, K_i\} \cong \mathfrak{so}(3) \oplus \mathbb{R} & , & [K_a, K_b] = {f_{ab}}^c K_c \end{matrix}\]
with the standard structure constants \({f_{ab}}^c\) of \(\mathfrak{so}(3)\), which are antisymmetric in \(a, b\). Here, \(K_0\) generates time translations and \(K_i\) generate standard rotations on \(S^2\).

For any field \(\Psi\) transforming under diffeomorphisms by the Lie derivative, \(\delta_\varepsilon \Psi = \varepsilon^a \mathcal{L}_{K_a} \Psi\) for constant parameters \(\varepsilon^a\), define the BRST operator

\[\label{NEWEQ} \boxed{\begin{matrix} \mathcal{Q} \Psi := c^a \mathcal{L}_{K_a} \Psi & , & \mathcal{Q} c^a := - \frac{1}{2} {f_{bc}}^a c^b c^c, \end{matrix}}\tag{58}\]
which is the standard Chevalley–Eilenberg BRST differential for finite-dimensional Lie algebras [@Figueroa:2006brst]. Nilpotency requires \(\mathcal{Q}^2 = 0\), so we must show that

\[\begin{matrix} \mathcal{Q}^2 \Psi = 0 & , & \mathcal{Q}^2 c^a = 0. \end{matrix}\]

3.6.1 Nilpotency on Fields↩︎

Using the graded Leibniz rule

\[\mathcal{Q}(XY) = (\mathcal{Q}X) Y + (-1)^{|X|} X (\mathcal{Q} Y)\] for fields \(X, Y\), and that \(c^a\) are Grassmann-odd (so \(|c^a|=1\) in the graded Leibniz rule),

\[\mathcal{Q}^2 \Psi = (\mathcal{Q} c^a) \mathcal{L}_{K_a} \Psi - c^a \mathcal{Q} (\mathcal{L}_{K_a} \Psi) = - \frac{1}{2} {f_{bc}}^a c^b c^c \mathcal{L}_{K_a} \Psi - c^a c^b \mathcal{L}_{K_b} \mathcal{L}_{K_a} \Psi.\]
Because \(c^a c^b\) is antisymmetric, we can isolate the commutator part via the identity

\[c^a c^b X_b X_a = \frac{1}{2} c^a c^b (X_b X_a - X_a X_b) = \frac{1}{2} c^a c^b [X_b, X_a].\]
Consequently,

\[c^a c^b \mathcal{L}_{K_b} \mathcal{L}_{K_a} \Psi = \frac{1}{2} c^a c^b [\mathcal{L}_{K_b}, \mathcal{L}_{K_a}] \Psi = \frac{1}{2} {f_{ba}}^c c^a c^b \mathcal{L}_{K_c} \Psi = - \frac{1}{2} {f_{ab}}^c c^a c^b \mathcal{L}_{K_c} \Psi\]
due to the antisymmetry of \({f_{ab}}^c\). Thus,

\[\mathcal{Q}^2 \Psi = \frac{1}{2} {f_{ab}}^c c^a c^b \mathcal{L}_{K_c} \Psi - \frac{1}{2} {f_{bc}}^a c^b c^c \mathcal{L}_{K_a} \Psi.\]
Upon relabeling \(a \leftrightarrow c\) in the second term, we find:

\[\mathcal{Q}^2 \Psi = \frac{1}{2} {f_{ab}}^c c^a c^b \mathcal{L}_{K_c} \Psi - \frac{1}{2} {f_{ba}}^{c} c^b c^a \mathcal{L}_{K_c} \Psi.\]
Swapping \(b \leftrightarrow c\) in the structure constants \({f_{ba}}^{c}\) gives:

\[\label{GHOST95ANTI} \mathcal{Q}^2 \Psi = \frac{1}{2} {f_{ab}}^c c^a c^b \mathcal{L}_{K_c} \Psi + \frac{1}{2} {f_{ab}}^{c} c^b c^a \mathcal{L}_{K_c} \Psi = \frac{1}{2} {f_{ab}}^c (c^a c^b + c^b c^a).\tag{59}\]
The ghosts anticommute, so \(c^a c^b + c^b c^a = 0\), and 59 subsequently vanishes. Hence, \(\mathcal{Q}^2 \Psi = 0\).

3.6.2 Nilpotency on Ghosts↩︎

We now show that \(\mathcal{Q}^2 c^a = 0\), confirming that \(\mathcal{Q}\) is indeed nilpotent as required. Evaluating

\[\mathcal{Q}^2 c^a = - \frac{1}{2} {f_{bc}}^a \mathcal{Q}(c^b) c^c + \frac{1}{2} {f_{bc}}^a c^b \mathcal{Q}(c^c) = \frac{1}{4} {f_{bc}}^a {f_{de}}^b c^d c^e c^c - \frac{1}{4} {f_{bc}}^a {f_{de}}^c c^b c^d c^e.\]
Anticommuting \(c^c, c^e\), then \(c^c, c^d\), and relabeling \(b \leftrightarrow c\) in the first term yields:

\[\mathcal{Q}^2 c^a = - \frac{1}{4} {f_{cb}}^a {f_{de}}^c c^b c^d c^e - \frac{1}{4} {f_{bc}}^a {f_{de}}^c c^b c^d c^e.\]
Finally, permuting \(b \leftrightarrow c\) in \({f_{cb}}^a\) introduces a minus sign to the first term and the terms cancel. Hence, \(\mathcal{Q}^2 c^a = 0\). Therefore, \(\mathcal{Q}\) is nilpotent. \(\square\)

3.6.3 Action on the Kerr–Schild Metric↩︎

For Schwarzschild in Kerr–Schild form, \(g_{\mu \nu}\) takes the form of 16 and the residual generators \(K_a \in \{K_0, K_i \}\) are Killing for \(g\). Therefore,

\[\boxed{ \mathcal{Q} g_{\mu \nu} = c^a \mathcal{L}_{K_a} g_{\mu \nu} = 0.}\]
Moreover, \(\varphi\) is time-independent and spherically symmetric, and \(k^\mu\) is likewise invariant under time translations and spatial rotations. Accordingly,

\[\boxed{ \begin{matrix} \mathcal{Q} \varphi = 0 & , & \mathcal{Q} k^\mu = 0. \end{matrix}}\]
Thus, in the Killing class, the BRST charge acts trivially on all Kerr–Schild fields; there is no nontrivial BRST cohomology associated to residual symmetries beyond the global isometries. This explicit computation shows that, in the Kerr–Schild Schwarzschild sector restricted to Killing generators, the BRST complex encodes no additional gauge-like redundancy. This outcome is physically natural. In the Kerr–Schild ansatz for Schwarzschild, all available gauge freedom is already exhausted by \(\mathfrak{so}(3) \oplus \mathbb{R}\), leaving no further nontrivial structure for the BRST charge to encode.

These results close our analysis of the Kerr–Schild Schwarzschild Killing sector. In the concluding section, we will situate these findings within the wider landscape of double copy research, highlighting both their limitations and avenues for future development.

4 Conclusion and Discussion↩︎

In this paper, we examined the interplay between gauge theory and gravity in the Kerr–Schild double copy, focusing on the fate of residual symmetries in the Schwarzschild solution. Our analysis clarified the scope and limitations of the construction in the Killing sector.

On the gauge theory side, we demonstrated that the residual transformations preserving the Kerr–Schild potential form infinite-dimensional Lie algebras. Conversely, on the gravity side, we showed that the residual diffeomorphisms of the Kerr–Schild Schwarzschild metric, when restricted to the Killing class, collapse entirely to the finite-dimensional global isometries, \(\mathfrak{so}(3) \oplus \mathbb{R}\).

This structural mismatch — infinite-dimensional residual algebras in gauge theory admitting no analogue in the gravitational Killing sector — demonstrates that the Kerr–Schild double copy does not preserve residual symmetry algebras. Our BRST analysis of the Killing sector reinforced this conclusion by showing that while a consistent BRST charge exists, its action on the Kerr–Schild metric is trivial, meaning the residual symmetry algebra has no nontrivial realization in BRST cohomology. This formal result provides the quantum-level justification for the algebraic collapse.

In summary, this work establishes a fundamental structural limitation of the Kerr–Schild double copy: it is powerful in generating exact spacetimes but is algebraically rigid in its failure to map the rich residual symmetry structure of the gauge theory.

To complete the full picture of the Schwarzschild residual symmetries, the analysis must be extended to include the full conformal Killing sector of solutions. This systematic treatment of the proper conformal Killing vectors is the subject of the second paper in this series. Future work will also focus on extending this analysis to Kerr spacetime and comparing with alternative, symmetry-preserving double copy formalisms.

Acknowledgments↩︎

I would like to thank Dr. Silvia Nagy for her guidance and insightful feedback throughout this project. I am also grateful to my previous thesis advisor, Dr. Xavier Calmet, who was gracious enough to endorse this paper. Moreover, I am appreciative of my colleagues and collaborators, including Dr. Martin Bauer and Dr. Edwin Hach, for valuable discussions and support.

Appendix A↩︎

Killing Vectors in Cartesian and Spherical Coordinates↩︎

Sections 2 and 3 were formulated in spherical coordinates, focusing on the Killing sector of residual diffeomorphisms and setting aside the conformal Killing sector for tractability. This appendix verifies that the residual vectors \(\xi^\mu\) identified in this sector are coordinate-independent, representing genuine geometric solutions of the Kerr–Schild Schwarzschild ansatz.

In Section 2, we stated the spherical form of the null vector \(k_\mu\); In this Appendix, we derive it formally. Similarly, in Section 3, the PDEs governing residual diffeomorphisms led to \((\mathcal{L}_\xi g)_{\mu \nu}\). Although coordinate-dependent in form, this condition is geometric: the angular components of \(\xi^\mu\)satisfy the Killing equations of the round two-sphere. By covariance of the Lie derivative, solutions in one coordinate system transform into solutions in any other, confirming that no additional Cartesian solutions exist.

For completeness, we show that the standard Cartesian Killing vectors preserve the Kerr-Schild structure and recover their spherical forms via the usual vector transformation law. These spherical Killing vectors form the explicit basis used in Section 3, where the angular subsystem reduces precisely to them.

Verifying the Spherical Form of \(k^\mu\) in Section 2.1↩︎

In this section, we show that the results obtained previously are retained when expressed in spherical coordinates, and we provide a heuristic argument for why this remains true in any coordinate system. The same is in fact true on the gravity side: although the explicit form of the equations changes drastically when the Kerr–Schild ansatz is expressed in spherical rather than Cartesian coordinates, the set of solutions is coordinate independent. What distinguishes the two cases, and what will become central in Section 3, is that in gauge theory the correspondence between residual symmetries is manifestly covariant, whereas in gravity, the coordinate dependence of the Kerr–Schild decomposition completely obscures this fact.

The aim of this section is to obtain the spherical-coordinate form of the null \(k^\mu\) under the map \((t, x, y, z) \mapsto (t, r, \vartheta, \varphi)\), and then to demonstrate that the gauge parameter \(\lambda(x)\) retains its form. This will serve as evidence of the general covariance of the residual symmetry. To begin, recall the vector transformation rule:

\[k^{\bar{\mu}}_S = \frac{\partial x^{\bar{\mu}}}{\partial x^{\nu}} k^\nu_C,\]
where we have introduced subscripts \(C\) and \(S\) to distinguish Cartesian and spherical forms, respectively. Summing over dummy index \(\nu\) yields:

\[k_S^{\bar{\mu}} = \frac{\partial x^{\bar{\mu}}}{\partial t} k_C^t + \frac{\partial x^{\bar{\mu}}}{\partial x} k_C^x + \frac{\partial x^{\bar{\mu}}}{\partial y} k_C^y + \frac{\partial x^{\bar{\mu}}}{\partial z} k_C^z.\]
Solving first for the time component, \(k_S^t\), we have,

\[\label{AAAA} k_S^{t} = \underbrace{\frac{\partial t}{\partial t} k_C^t}_{=1} + \underbrace{\frac{\partial t}{\partial x}}_{=0} k_C^x + \underbrace{\frac{\partial t}{\partial y}}_{=0} k_C^y + \underbrace{\frac{\partial t}{\partial z}}_{=0} k_C^z = 1,\tag{60}\] since \(k_C^t = 1\). Similarly, the radial component is:

\[\label{OOOO} k_S^{r} = \underbrace{\frac{\partial r}{\partial t}}_{=0} k_C^t + \frac{\partial r}{\partial x} k_C^x + \frac{\partial r}{\partial y} k_C^y + \frac{\partial r}{\partial z} k_C^z.\tag{61}\]
By the definition of \(r\), we have:

\[\frac{dr}{d x^i} = \frac{d}{dx^i} \begin{pmatrix} \sqrt{x^j x_j} \end{pmatrix} = \frac{x_j \delta_{ij}}{\sqrt{x^k x_k}} = \frac{x_i}{r}.\]
Substituting this and the spatial components \(k_C^i = x^i / r\) into 61 , we find that the radial component of \(k_S^r\) is unity:

\[\label{BBBB} k_S^{r} = \frac{x^2}{r^2} + \frac{y^2}{r^2} + \frac{z^2}{r^2} = \frac{x^2 + y^2 + z^2}{r^2} = 1.\tag{62}\]
To determine the angular components \(k_S^\vartheta\) and \(k_S^\varphi\), recall that under the map \((t, x, y, z) \mapsto (t, r, \vartheta, \varphi)\), we have,

\[\label{HHHH} \begin{matrix} t = t & , & x = r \sin\vartheta \cos\varphi & , & y = r \sin\vartheta \sin\varphi & , & z = r \cos\vartheta. \end{matrix}\tag{63}\]
Solving for \(\vartheta\) using \(z = r \cos\vartheta\) gives \(\vartheta = \arccos (z/r)\). We will use this to work out the derivatives of \(\vartheta\) with respect to the Cartesian coordinates \(x^i\). Clearly, \(k_S^\vartheta\) is

\[\label{GGGG} k_S^{\vartheta} = \underbrace{\frac{\partial \vartheta}{\partial t}}_{=0} k_C^t + \frac{\partial \vartheta}{\partial x} k_C^x + \frac{\partial \vartheta}{\partial y} k_C^y + \frac{\partial \vartheta}{\partial z} k_C^z = \frac{\partial \vartheta}{\partial x} k_C^x + \frac{\partial \vartheta}{\partial y} k_C^y + \frac{\partial \vartheta}{\partial z} k_C^z.\tag{64}\]
Note: the first term drops off because \(\vartheta\) is independent of time. By the chain rule,

\[\frac{\partial \vartheta}{\partial x^i} = \frac{\partial \vartheta}{\partial r} \frac{dr}{dx^i},\]
where

\[\label{MOM} \frac{\partial \vartheta}{\partial r} = \frac{z}{r^2 \sqrt{1 - \frac{z^2}{r^2}}} = \frac{\cos\vartheta}{r \sqrt{1 - \cos^2 \vartheta}} = \frac{\cos\vartheta}{r \sin\vartheta},\tag{65}\]
since \(\sin^2 \vartheta = 1 - \cos^2\vartheta\). Furthermore, we have:

\[\label{DAD} \begin{matrix} \frac{\partial r}{\partial x} = \frac{x}{r} = \sin\vartheta \cos\varphi & , & \frac{\partial r}{\partial y} = \frac{y}{r} = \sin\vartheta \sin\varphi & , & \frac{\partial r}{\partial z} = \frac{z}{r} = \cos\vartheta. \end{matrix}\tag{66}\]
Substituting these results into 64 , we find that the \(\vartheta\)-component \(k_S^\vartheta\) vanishes:

\[\begin{align} \label{CCCC} k_S^{\vartheta} &= \frac{\partial \vartheta}{\partial x} k_C^x + \frac{\partial \vartheta}{\partial y} k_C^y + \frac{\partial \vartheta}{\partial z} k_C^z \\ &= \cos\vartheta \sin\vartheta \cos^2\varphi + \cos\vartheta \sin \vartheta \sin^2\varphi - \cos\vartheta \sin\vartheta \\ &= \cos\vartheta \sin \vartheta (\sin^2\varphi + \cos^2\varphi) - \cos\vartheta \sin\vartheta \\ &= \cos\vartheta \sin\vartheta - \cos\vartheta \sin\vartheta \\ &=0. \end{align}\tag{67}\] Next, to determine \(k_S^\varphi\), consider the ratio \(y/x = \tan\varphi\). Then, \(\varphi = \arctan(y/x)\), which has derivatives:

\[\label{SIS} \begin{matrix} \frac{\partial \varphi}{\partial x} = - \frac{y}{x^2 + y^2} = - \frac{ \sin\varphi}{r\sin\vartheta} & , & \frac{\partial \varphi}{\partial y} = \frac{x}{x^2 + y^2} = \frac{\cos\varphi}{r \sin\vartheta} & , & \frac{\partial \varphi}{\partial z} = 0. \end{matrix}\tag{68}\]
It is straightforward to show that \(k_S^\varphi\) vanishes as well:

\[\begin{align} \label{DDDD} k_S^{\varphi} &= \underbrace{\frac{\partial \vartheta}{\partial t}}_{=0} k_C^t + \frac{\partial \varphi}{\partial x} k_C^x + \frac{\partial \varphi}{\partial y} k_C^y + \underbrace{\frac{\partial \varphi}{\partial z}}_{=0} k_C^z \\ &= \frac{\sin\vartheta \cos\varphi \sin\varphi}{r \sin\vartheta} - \frac{\sin\vartheta \cos\varphi \sin\varphi}{r \sin\vartheta} \\ &= 0. \end{align}\tag{69}\]
Combining results 60 , 62 , 67 , and 69 , we obtain the spherical form for \(k^\mu\):

\[\boxed{k_S^{\bar{\mu}} = (1, 1, 0, 0).}\]

Verifying the Results of Section 3.2 in Cartesian Coordinates↩︎

We begin by verifying that the familiar Cartesian Killing vectors of flat space satisfy the condition imposed by the Kerr–Schild ansatz, namely that the Lie derivative of the metric along \(\xi\) preserves the KS structure. Although these vectors are standard results, it is not obvious a priori that they should continue to solve the PDE system derived from \((\mathcal{L}_\xi g)_{\mu \nu} = 0\), since the explicit form of the equations is coordinate-dependent. Demonstrating that the Cartesian Killing fields indeed satisfy the condition therefore provides a direct check of the universality of the residual diffeomorphisms.

Time Translations↩︎

In Cartesian Kerr–Schild coordinates for Schwarzschild spacetime (with background Minkowski \(\eta_{\mu \nu}\) in \((t, x, y, z))\), the time translation Killing vector is just the generator of shifts in the coordinate \(t\):

\[\xi = \partial_t,\]
with components \(\xi^\mu = (1,~0,~0,~0)\). Since \(\xi\) is a Killing vector in Cartesian coordinates, it suffices to show that it satisfies the Cartesian Killing equations for the background metric:

\[\begin{matrix} (\mathcal{L}_\xi \eta)_{t t} = 2 \partial_t \xi_t = 0 & , & (\mathcal{L}_\xi \eta)_{t i} = \partial_t \xi_i + \partial_i \xi_t = 0 & , & (\mathcal{L}_\xi \eta)_{i j} = \partial_i \xi_j + \partial_j \xi_i = 0 \end{matrix}\]
for \(i, j = 1,~2,~3\). This is trivially true for all values of \(i, j\) since \(\xi^t = 1\) and \(\xi^i = 0\), by definition of time translation. Thus, we have:

\[(\mathcal{L}_\xi \eta)_{\mu \nu} = 0.\]
We wish for \(\xi^\mu\) to preserve the KS structure of the metric, i.e.,

\[\label{XOXO} (\mathcal{L}_\xi g)_{\mu \nu} := (\mathcal{L}_\xi \eta)_{\mu \nu} + \mathcal{L}_\xi (\varphi k_\mu k_\nu) = \alpha(x) k_\mu k_\nu.\tag{70}\] We have just shown that the first term vanishes, so we must show that

\[\mathcal{L}_\xi (\varphi k_\mu k_\nu) \stackrel{!}{=} \alpha(x) k_\mu k_\nu,\]
where \(\alpha(x)\) is a smooth function. Expanding this using 21 and 22 , we have:

\[\mathcal{L}_\xi (\varphi k_\mu k_\nu) := (\xi^\rho \partial_\rho \varphi) k_\mu k_\nu + \varphi (\xi^\rho \partial_\rho k_\mu + k_\rho \partial_\mu \xi^\rho) k_\nu + \varphi k_\mu (\xi^\rho \partial_\rho k_\nu + k_\rho \partial_\nu \xi^\rho).\]
Setting this equal to \(\alpha(x) k_\mu k_\nu\), subtracting \((\xi^\rho \partial_\rho \varphi) k_\mu k_\nu\) to the right-hand side, and defining \(\zeta(x) := \alpha(x) - \xi^\rho \partial_\rho \varphi\) gives:

\[\label{ZOZO} \mathcal{L}_\xi (\varphi k_\mu k_\nu) := \varphi (\xi^\rho \partial_\rho k_\mu + k_\rho \partial_\mu \xi^\rho) k_\nu + \varphi k_\mu (\xi^\rho \partial_\rho k_\nu + k_\rho \partial_\nu \xi^\rho) = \zeta(x) k_\mu k_\nu.\tag{71}\]
Since the only nonzero component of \(\xi^\mu\) is \(\xi^t\), this simplifies the Lie derivative to the following:

\[\label{LOLOL} \mathcal{L}_\xi (\varphi k_\mu k_\nu) := \varphi (\xi^t \partial_t k_\mu + k_t \partial_\mu \xi^t) k_\nu + \varphi k_\mu (\xi^t \partial_t k_\nu + k_t \partial_\nu \xi^t) = \zeta(x) k_\mu k_\nu.\tag{72}\]
But \(\xi^t = 1\), so its derivatives vanish. Likewise, recall that in Cartesian coordinates,

\[k^\mu = \begin{pmatrix} 1, \frac{x^i}{r} \end{pmatrix},\]
again for \(i = 1,~2,~3\). The components of \(k^\mu\) are clearly independent of \(t\), so their time derivatives vanish. As such, every term in 72 vanishes, forcing \(\zeta(x) = 0\), leaving:

\[\mathcal{L}_\xi (\varphi k_\mu k_\nu) = 0.\]
Consequently, it follows that the Lie derivative of the whole metric vanishes:

\[(\mathcal{L}_\xi g)_{\mu \nu} = 0.\]
Thus, we retain our conclusion from Section 3 that time translation is a solution that preserves the Kerr–Schild ansatz, as well as a global isometry of the Schwarzschild solution.

Spatial Rotations↩︎

The rotational Killing vectors can be written most efficiently in Cartesian coordinates as

\[\begin{matrix} \xi^i = \omega^{ij} x^j & , & \omega^{ij} = - \omega^{ji}, \end{matrix}\]
where \(\omega^{ij}\) is a constant antisymmetric matrix. This form is equivalent to the familiar generators of rotations (e.g. \(x \partial_y - y \partial_x\)), but is more compact and manifestly encodes all three independent rotational symmetries at once. The antisymmetry of \(\omega^{ij}\) ensures that the vector field generates infinitesimal rotations, preserving both the flat background metric and the radial distance \(r\). This compact notation significantly simplifies calculations in the Kerr–Schild framework, as it allows all angular Killing fields to be treated uniformly without resorting to case-by-case component expressions.

Consider again the Lie derivative 70 . The Lie derivative of the background \(\eta_{\mu \nu}\) is only nontrivial when we take all spatial indices:

\[(\mathcal{L}_\xi \eta)_{i j} := \partial_i \xi^k \eta_{k j} + \partial_j \xi^k \eta_{k i} = \partial_i \xi^k \delta_{k j} + \partial_j \xi^k \delta_{k i},\]
where \(\delta_{ij}\) is the Kronecker delta. However, notice that, by the compact definition of \(\xi^i\):

\[\partial_i \xi^j = \partial_i (\omega^{jk} x^k) = \omega^{jk} \partial_i x^k = \omega^{jk} {\delta^k}_i = {\omega^{j}}_i.\]
Substituting this and contracting over \(k\), we find that the Lie derivative of the background vanishes due to the antisymmetry of \(\omega^{ij}\):

\[(\mathcal{L}_\xi \eta)_{i j} := {\omega^k}_i \delta_{k j} + {\omega^k}_j \delta_{k i} := \omega_{ji} + \omega_{ij} = \omega_{ij} - \omega_{ij} = 0.\]
Next, we show that 71 holds true. But, by the definition of the Lie derivative \((\mathcal{L}_\xi k)_\mu\),

\[(\mathcal{L}_\xi k)_\mu := \omega^{ij} x^j \partial_i k_\mu + k_i \partial_\mu \omega^{ij} x^j = k_i \omega^{ij} {\delta^j}_\mu \propto k_\mu.\]
This result makes sense because \(k^\mu\) depends on \(x^i\), which transforms as a vector under rotations, so the Lie derivative simply rotates \(k^\mu\) within the null congruence: \(k_\mu\) stays a null vector, just rotated. Thus, \(\mathcal{L}_\xi (\varphi k_\mu k_\nu) \propto k_\mu k_\nu\) and the Kerr–Schild structure is preserved. Because the full expansion of the Lie derivative contains the symmetric part \(2 \varphi k_{(\mu} \mathcal{L}_\xi k_{\nu)}\), the sum vanishes due to the antisymmetry of \(\omega^{ij}\). It is clear that time translations and spatial rotations remain solutions preserving the KS structure in Cartesian coordinates. This confirms that the admissible residual diffeomorphism are not an artifact of a particular coordinate system, but genuinely coordinate-invariant symmetries of the Schwarzschild Kerr–Schild form.

Indeed, this makes sense because the Lie derivative is a geometric object. Consequently, any vector field \(\xi^\mu\) that can be related geometrically to any other vector field \(\xi^{\bar{\mu}}\) by the vector transformation rule

\[\xi^{\bar{\mu}} = \frac{\partial x^{\bar{\mu}}}{\partial x^\mu} \xi^\mu\]
is guaranteed to yield the same result. Since in both spherical and Cartesian coordinates the only admissible solutions for the rotational class of general solutions considered in Section 3 are precisely the Killing vectors of the Schwarzschild geometry, it follows that in any other coordinate system the residual diffeomorphisms must also reduce to these same Killing symmetries.

Why Boosts and Spatial Translations Fail↩︎

Before we continue, we must point out precisely why Lorentz boosts and spatial translations fail to preserve the KS ansatz. In particular, boosts do not preserve \(\varphi\), since the scalar field is not Lorentz invariant. Under boosts, \(\varphi\) acquires a time dependence, so \(\partial_t \varphi \neq 0\). Under spatial translations, \(\xi^i = x^i\), the scalar field transforms nontrivially:

\[\xi^i \partial_i \varphi(r) = x^i \frac{d\varphi}{dr} \frac{\partial r}{\partial x^i} = \frac{d\varphi}{dr} \frac{x^i x_i}{r} = r \frac{d\varphi}{dr} \neq 0.\]
Under such spatial translations, \(k_i\) transforms as \(\mathcal{L}_\xi k_i\), and we have:

\[(\mathcal{L}_\xi k)_i = \xi^j \partial_j k_i + k_j \partial_i \xi^j = x^j \partial_j \begin{pmatrix} \frac{x_i}{r} \end{pmatrix} + \frac{x_j}{r} \partial_i x^j = x^j \frac{\delta_{ij}}{r} - \frac{x_i}{r} + \frac{x_j}{r} {\delta^j}_i = \frac{x_i}{r} = k_i.\]
Thus, \(k_i\) transforms as itself and this makes sense because, again, \(k_i\) depends directly on \(x_i\), which transforms like itself under \(x^i \rightarrow x^j \partial_j x^i\). Nevertheless, neither boosts nor spatial translations preserve the full structure of \(\varphi k_\mu k_\nu\), and hence break the KS form. Hence, the only residual diffeomorphisms that preserve both the background \(\eta_{\mu \nu}\) and the KS correction term are time translations and spatial rotations. These are precisely the Killing vectors of the full Schwarzschild geometry. The KS metric is entirely unchanged under this class of symmetries. It seems, then, that in Cartesian coordinates, the Schwarzschild geometry admits no nontrivial residual diffeomorphisms beyond its global isometries - in the case where the proper CKVs of Section 3 are neglected.

Derivation of Rotational Killing Vectors in Spherical Coordinates↩︎

In Section 3, we stated the form of the spherical Killing vectors and used them to show explicitly that they leave the KS ansatz invariant under a general coordinate transformation. Using the results of the previous section, we now derive the spherical Killing vectors explicitly by transforming \(\xi^\mu\) to spherical coordinates via the vector transformation rule. Let \(\xi_S^{\bar{\mu}}\) denote the vector field in spherical coordinates. In order to determine its components, consider the vector transformation law:

\[\label{F4} \xi_S^{\bar{\mu}} = \frac{\partial x^{\bar{\mu}}}{\partial x^\nu} \xi_C^\nu,\tag{73}\]
where \(\xi_C^\mu\) are the Cartesian vectors, as previously defined. We will show that we retain the same results as in the Cartesian case, starting with time translation, i.e., \(\xi_C^\mu = (1,~0,~0,~0)\). Setting \(\mu = t\) and expanding the transformation law gives:

\[\xi_S^t = \underbrace{\frac{\partial t}{\partial t}}_{=1} \xi_C^t + \underbrace{\frac{\partial t}{\partial x^i} \xi^i}_{=0} = \xi_C^t = 1.\]
Since \(\xi_C^i = 0\) for \(i = 1,~2,~3\), it is trivial to show that the remaining terms vanish, so \(\xi_S^\mu = \xi_C^\mu\), and time translations are a symmetry of the Schwarzschild solution in the Kerr–Schild ansatz by the arguments of Section 3.2.1.
Showing that spatial rotations remain a symmetry of the KS ansatz in spherical coordinates is more cumbersome. In order to proceed, recall the Cartesian form of the rotation \(\xi^i = \omega^{ij} x^j\). We can parameterize the \(\omega^{ij}\) by a rotation vector \(\Omega = (\Omega_x, \Omega_y, \Omega_z)\) so that:

\[\omega^{ij} = \varepsilon^{ijk} \Omega^k.\]
Then,

\[\xi^i = \varepsilon^{ijk} \Omega^k x^j := (\Omega \times x)^i,\] by the definition of the cross product. This is the \(i\)th component of the rotation vector \(\xi^i\) in its standard form. In order to make this calculation more precise, we must choose an axis of rotation by picking an appropriate basis for the rotation axis:

• A rotation about the \(x\)-axis corresponds to the basis rotation vector \(\Omega_{(x)}^i = (1,~0,~0)\), and rotation 3-vector components \(\xi_{(x)}^i = (0,-z,~y)\) so that:

  \[\label{IIII} \xi_{(x)} = \xi_{(x)}^i \partial_i = y \partial_z - z \partial_y.\tag{74}\]

• A rotation about the \(y\)-axis corresponds to the basis rotation vector \(\Omega_{(y)}^i = (0,~1,~0)\), and rotation 3-vector components \(\xi_{(y)}^i = (z,~0,-x)\) so that:

  \[\label{JJJJ} \xi_{(y)} = \xi_{(y)}^i \partial_i = z \partial_x - x \partial_z.\tag{75}\]

• A rotation about the \(z\)-axis corresponds to the basis rotation 3-vector \(\Omega_{(z)}^i = (0,~0,~1)\), and rotation vector components \(\xi_{(z)}^i = (-y,~x,-0)\) so that:

  \[\label{KKKK} \xi_{(z)} = \xi_{(z)}^i \partial_i = x \partial_y - y \partial_x.\tag{76}\]

These are the standard rotation 3-vectors from elementary calculus in Cartesian coordinates. For each of these, the corresponding 4-vector (\(\xi_{(x)}^\mu\), for example) contains a zero in the time slot (since axial rotations are independent of time).

Next, we reparameterize these in spherical coordinates by employing the spherical form of \((x,y,z)\), given by 63 :

\[\begin{matrix} t = t & , & x = r \sin\vartheta \cos\varphi & , & y = r \sin\vartheta \sin\varphi & , & z = r \cos\vartheta. \end{matrix}\]
We can substitute these results directly into 74 through 76 , but first, we must determine the spherical forms of the partial derivatives. By the chain rule,

\[\label{BRO} \partial_{i} = \frac{\partial r}{\partial x^i} \partial_r + \frac{\partial \vartheta}{\partial x^i} \partial_\vartheta + \frac{\partial \varphi}{\partial x^i} \partial_\varphi.\tag{77}\]
It follows that \(\frac{\partial r}{\partial x^i} = \frac{x_i}{r}\). Thus, we must determine \(\frac{\partial \vartheta}{\partial x^i}\) and \(\frac{\partial \varphi}{\partial x^i}\). Via another application of the chain rule, we can write \(\frac{\partial \vartheta}{\partial x^i}\) as:

\[\frac{\partial \vartheta}{\partial x^i} = \frac{\partial \vartheta}{\partial r} \frac{\partial r}{\partial x^i} = \frac{x_i}{r} \frac{\cos\vartheta}{r \sin\vartheta}\]
by 65 . Then,

\[\begin{matrix} \frac{\partial \vartheta}{\partial x} = \frac{\cos\vartheta \cos\varphi}{r} & , & \frac{\partial \vartheta}{\partial y} = \frac{\cos\vartheta \sin\varphi}{r} & , & \frac{\partial \vartheta}{\partial z} = - \frac{\sin\vartheta}{r}. \end{matrix}\]
Likewise, recall from 68 that:

\[\begin{matrix} \frac{\partial \varphi}{\partial x} = - \frac{\sin\varphi}{r\sin\vartheta} & , & \frac{\partial \varphi}{\partial y} = \frac{\cos\varphi}{r \sin\vartheta} & , & \frac{\partial \varphi}{\partial z} = 0. \end{matrix}\]
Plugging these results into 77 , we find that the derivatives in spherical coordinates can be expressed as:

\[\begin{align} \partial_x &= \frac{\partial r}{\partial x} \partial_r + \frac{\partial \vartheta}{\partial x} \partial_\vartheta + \frac{\partial \varphi}{\partial x} \partial_\varphi = \sin\vartheta \cos\varphi \partial_r + \frac{\cos\vartheta \cos\varphi}{r} \partial_\vartheta - \frac{r \sin\varphi}{\sin\vartheta} \partial_\varphi, \\ \partial_y &= \frac{\partial r}{\partial y} \partial_r + \frac{\partial \vartheta}{\partial y} \partial_\vartheta + \frac{\partial \varphi}{\partial y} \partial_\varphi = \sin\vartheta \sin\varphi \partial_r + \frac{\cos\vartheta \sin\varphi}{r} \partial_\vartheta + \frac{r \cos\varphi}{\sin\vartheta} \partial_\varphi, \\ \partial_z &= \frac{\partial r}{\partial z} \partial_r + \frac{\partial \vartheta}{\partial z} \partial_\vartheta + \frac{\partial \varphi}{\partial z} \partial_\varphi = \cos\vartheta \partial_r - \frac{\sin\vartheta}{r} \partial_\vartheta. \end{align}\]
Consider a rotation about the \(x\)-axis. Then,

\[\label{F1} \begin{align} \xi_{(x)} &= r \cos\vartheta \sin\vartheta \sin\varphi \partial_r - \sin^2\vartheta \sin\varphi \partial_\vartheta - r \cos\vartheta \sin\vartheta \sin\varphi \partial_r \\ &- \cos^2\vartheta \sin\varphi \partial_\vartheta - r^2 \cot\vartheta \cos\varphi \partial_\varphi \\ &= - (\sin^2\vartheta + \cos^2\vartheta) \sin\varphi \partial_\vartheta - r^2 \cot\vartheta \cos\varphi \partial_\varphi \\ &= - \sin\varphi \partial_\vartheta - r^2 \cot\vartheta \cos\varphi \partial_\varphi. \end{align}\tag{78}\]
A rotation about the \(y\)-direction is, similarly,

\[\label{F2} \begin{align} \xi_{(y)} &= r \cos\vartheta \sin\vartheta \cos\varphi \partial_r + \cos^2 \vartheta \cos\varphi \partial_\vartheta - r^2 \cot\vartheta \sin\varphi \partial_\varphi \\ &- r \cos\vartheta \sin\vartheta \cos\varphi \partial_r + \sin^2\vartheta \cos\varphi \partial_\vartheta \\ &= (\sin^2\vartheta + \cos^2 \vartheta) \cos\varphi \partial_\vartheta - r^2 \cot\vartheta \sin\varphi \partial_\varphi \\ &= \cos\varphi \partial_\vartheta - r^2 \cot\vartheta \sin\varphi \partial_\varphi. \end{align}\tag{79}\]
Finally, consider a rotation about the \(z\)-axis. Then,

\[\label{F3} \begin{align} \xi_{(z)} &= r \sin^2\vartheta \cos\varphi \sin\varphi \partial_r + \cos\vartheta \sin\vartheta \cos\varphi \sin\varphi \partial_\vartheta + r^2 \cos^2\varphi \partial_\varphi \\ &- r \sin^2\vartheta \cos\varphi \sin\varphi \partial_r - \cos\vartheta \sin\vartheta \cos\varphi \sin\varphi \partial_\vartheta + r^2 \sin^2 \varphi \partial_\varphi \\ &= r^2 (\sin^2 \varphi + \cos^2\varphi)\partial_\varphi \\ &= r^2 \partial_\varphi. \end{align}\tag{80}\]
After all of this work, we are finally able to plug 78 through 80 into vector transformation law 73 and determine the components of the vector field \(\xi_{S,(i)}^{\bar{\mu}}\), where the subscript \((i)\) denotes the axis of rotation. We must work out \(\xi\) about all three axes of rotation and then input all three into the Lie derivative 56 to show that each rotation vector remains an isometry of the KS Schwarzschild metric, as it should.

Consider a rotation about the \(x\)-axis, where,

\[\xi_{(x)}^\mu = (0,-z,~y) = (0,-r\cos\vartheta,~r\sin \vartheta \sin\varphi).\]
The time component vanishes as expected:

\[\xi_{S,(x)}^{t} = \frac{\partial t}{\partial t} \underbrace{\xi_{(x)}^t}_{=0} + \underbrace{\frac{\partial t}{\partial x^i}}_{=0} \xi_{(x)}^i = 0.\]
The \(r\)-component is:

\[\xi_{S,(x)}^{r} = \underbrace{\frac{\partial r}{\partial t} \xi_{(x)}^t}_{=0} + \frac{\partial r}{\partial x} \underbrace{\xi_{(x)}^x}_{=0} + \frac{\partial r}{\partial y} \xi_{(x)}^y + \frac{\partial r}{\partial z} \xi_{(x)}^z = -\frac{yz}{r} + \frac{yz}{r} = 0.\]
Working out the \(\vartheta\)-component gives:

\[\begin{align} \xi_{S,(x)}^{\vartheta} &= \underbrace{\frac{\partial \vartheta}{\partial t} \xi_{(x)}^t}_{=0} + \frac{\partial \vartheta}{\partial x} \underbrace{\xi_{(x)}^x}_{=0} + \frac{\partial \vartheta}{\partial y} \xi_{(x)}^y + \frac{\partial \vartheta}{\partial z} \xi_{(x)}^z \\ &= - (\sin^2\vartheta + \cos^2\vartheta) \sin\varphi \\ &= -\sin\varphi. \end{align}\]
Finally, working out the \(\varphi\)-component, we have:

\[\begin{align} \xi_{S,(x)}^{\varphi} &= \underbrace{\frac{\partial \varphi}{\partial t} \xi_{(x)}^t}_{=0} + \frac{\partial \varphi}{\partial x} \underbrace{\xi_{(x)}^x}_{=0} + \frac{\partial \varphi}{\partial y} \xi_{(x)}^y +\underbrace{\frac{\partial \varphi}{\partial z}}_{=0} \xi_{(x)}^z \\ &= \frac{\cos\varphi}{r \sin\vartheta} (-r \cos\vartheta) \\ &= - \cot\vartheta \cos\varphi. \end{align}\]
Thus, for a rotation about the \(x\)-axis:

\[\label{BAA} \boxed{\xi_{S,(x)}^{\bar{\mu}} = \begin{pmatrix} 0,-\sin\varphi,- \cot\vartheta \cos\varphi \end{pmatrix}.}\tag{81}\]
Next, consider a rotation about the \(y\)-axis, where,

\[\xi_{(y)}^\mu = (z,~0,-x) = (~r\cos\vartheta,~0,-r \sin\vartheta \cos\varphi).\]
It is trivial to show that the time component vanishes:

\[\xi_{S,(y)}^{t} = \frac{\partial t}{\partial t} \underbrace{\xi_{(y)}^t}_{=0} + \underbrace{\frac{\partial t}{\partial x^i}}_{=0} \xi_{(y)}^i = 0.\]
Working out the \(r\)-component yields:

\[\xi_{S,(y)}^{r} = \underbrace{\frac{\partial r}{\partial t} \xi_{(y)}^t}_{=0} + \frac{\partial r}{\partial x} \xi_{(y)}^x + \frac{\partial r}{\partial y} \underbrace{\xi_{(y)}^y}_{=0} + \frac{\partial r}{\partial z} \xi_{(y)}^z = \frac{xz}{r} - \frac{xz}{r} = 0.\]
Moreover, the \(\vartheta\)-component reads:

\[\begin{align} \xi_{S,(y)}^{\vartheta} &= \underbrace{\frac{\partial \vartheta}{\partial t} \xi_{(y)}^t}_{=0} + \frac{\partial \vartheta}{\partial x} \xi_{(y)}^x + \frac{\partial \vartheta}{\partial y} \underbrace{\xi_{(y)}^y}_{=0} + \frac{\partial \vartheta}{\partial z} \xi_{(y)}^z \\ &= (\sin^2\vartheta + \cos^2\vartheta) \cos\varphi \\ &= \cos\varphi. \end{align}\]
Finally, the \(\varphi\)-component is:

\[\xi_{S,(y)}^{\varphi} = \underbrace{\frac{\partial \varphi}{\partial t} \xi_{(y)}^t}_{=0} + \frac{\partial \varphi}{\partial x} \xi_{(y)}^x + \frac{\partial \varphi}{\partial y} \underbrace{\xi_{(y)}^y}_{=0} + \underbrace{\frac{\partial \varphi}{\partial z}}_{=0} \xi_{(y)}^z = - \cot\vartheta \sin\varphi.\]
Hence, for a rotation about the \(y\)-axis, we have:

\[\label{BAB} \boxed{\xi_{S,(y)}^{\bar{\mu}} = \begin{pmatrix}~0,~\cos\varphi,-\cot\vartheta \sin\varphi \end{pmatrix}.}\tag{82}\]
To conclude, consider a rotation about the \(z\)-axis, so that:

\[\xi_{(z)}^\mu = (-y,~x,~0) = (-r\sin\vartheta \sin\varphi,~r \sin\vartheta \cos\varphi,~0).\]
It is once again straightforward to show that the time component vanishes:

\[\xi_{S,(z)}^{t} = \frac{\partial t}{\partial t} \underbrace{\xi_{(z)}^t}_{=0} + \underbrace{\frac{\partial t}{\partial x^i}}_{=0} \xi_{(z)}^i = 0.\]
Similarly for the radial component is:

\[\xi_{S,(z)}^{r} = \underbrace{\frac{\partial r}{\partial t} \xi_{(z)}^t}_{=0} + \frac{\partial r}{\partial x} \xi_{(z)}^x + \frac{\partial r}{\partial y} \xi_{(z)}^y + \frac{\partial r}{\partial z} \underbrace{\xi_{(z)}^z}_{=0} = -\frac{xy}{r} + \frac{xy}{r} = 0.\]
Working the \(\vartheta\)-component gives:

\[\begin{align} \xi_{S,(z)}^{\vartheta} &= \underbrace{\frac{\partial \vartheta}{\partial t} \xi_{(z)}^t}_{=0} + \frac{\partial \vartheta}{\partial x} \xi_{(z)}^x + \frac{\partial \vartheta}{\partial y} \xi_{(z)}^y + \frac{\partial \vartheta}{\partial z} \underbrace{\xi_{(z)}^z}_{=0} \\ &= - \cos\vartheta \sin\vartheta \cos\varphi \sin\varphi + \cos\vartheta \sin\vartheta \cos\varphi \sin\varphi \\ &= 0. \end{align}\]
Calculating the \(\varphi\)-component, we have:

\[\begin{align} \xi_{S,(z)}^{\varphi} &= \underbrace{\frac{\partial \varphi}{\partial t} \xi_{(z)}^t}_{=0} + \frac{\partial \varphi}{\partial x} \xi_{(z)}^x + \frac{\partial \varphi}{\partial y} \xi_{(z)}^y + \frac{\partial \varphi}{\partial z} \underbrace{\xi_{(x)}^z}_{=0} \\ &= - \frac{\sin\varphi}{r \sin\vartheta} (-r \sin \vartheta \sin \varphi) + \frac{\cos\varphi}{r \sin\vartheta} (r \sin\vartheta \cos\varphi) \\ &= \sin^2\varphi + \cos^2\varphi \\ &= 1. \end{align}\]
Hence, for a rotation about the \(z\)-axis, we obtain the simplest form of \(\xi_S^{\bar{\mu}}\):

\[\label{BAC} \boxed{\xi_{S,(z)}^{\bar{\mu}} = \begin{pmatrix} 0,~0,~1\end{pmatrix}.}\tag{83}\]
With the explicit spherical forms of the angular Killing vectors now in hand, we recovered the complete set of residual diffeomorphisms considered in Section 3: time translations and spatial rotations.

References↩︎