Introduction–Higher-curvature contributions, built from the Riemann tensor and its covariant derivatives, are expected to play a central role in any viable quantum theory of gravity. In particular, in the low-energy limit of string theory, the gravitational Lagrangian contains infinitely many curvature invariants and their derivatives. The corresponding field equations are typically extremely complicated and, in general, do not admit generic Einstein metrics as solutions.

Nevertheless, certain special metrics in General Relativity retain an extraordinary property: they continue to solve the field equations of all higher–derivative gravity theories constructed from curvature and its covariant derivatives, independently of the detailed form of the Lagrangian. Understanding these “quantum-protected” or universal metrics is therefore of considerable conceptual and technical interest.

Beginning with Deser [1] and subsequent works [2]–[3], plane waves have been shown to be completely insensitive to quantum corrections: all higher-order curvature contributions vanish identically in their effective field equations. For a long time, plane waves were believed to be the only universal metric. This picture changed when \(pp\)-waves were shown to share the same property [4]–[6].

Both plane waves and \(pp\)-waves belong to the Kerr–Schild family and enjoy additional structural simplifications. The metrics to be studied here have analogous properties; let us recap some properties of the \(pp\)-waves and Kerr-Schild-Kundt (KSK) metrics. The \(pp\)-waves take their simplest form in the Brinkmann coordinates, \[ds^{2}=2\,du\,dv+dx^{2}+dy^{2}+2V(u,x,y)\,du^{2},\label{ks1}\tag{1}\] for which the Einstein field equations reduce to \(\Box V=0\), where \(\Box\) denotes the Beltrami–Laplace operator of the flat background, that is \(V=0\) in 1 . The plane waves correspond to the choice \(V= S_{ab}(u)x^{a}x^{b}\) with \(\mathrm{trace}\,S=0\); hence, the Einstein field equations are satisfied identically. Therefore, the plane wave metrics are said to be universal. A defining property of \(pp\)-waves is that the null vector \(\lambda_{\mu}=\partial_{\mu}u\) is covariantly constant: \(\nabla_{\mu}\lambda_{\nu}=0\). Such metrics do not automatically satisfy the field equations of a generic gravity theory; hence, they are not universal. However, they are almost universal since the field equations of a generic gravity theory reduce to, \[\sum_{n=1}^{N}a_{n}\,\Box^{n}V=0.\label{cond1}\tag{2}\] where \(a_{n}\) depend on the coupling constants of the chosen higher–curvature theory. In this flat background, the Kerr–Schild metric \[g_{\mu\nu}^{\mathrm{AU}}=\eta_{\mu\nu}+2V(u,x^{a})\,\lambda_{\mu}\lambda_{\nu},\label{un1}\tag{3}\] naturally distinguishes between naturally universal (Minkowski) backgrounds and almost universal (AU) metrics, which obey one additional scalar equation of the form 2 .

Beyond the flat spacetime, de Sitter (dS) and anti-de Sitter (AdS) spacetimes are also naturally universal. This observation led to the generalization of almost universal metrics to KSK metrics [6]–[7]–[11] with constant–curvature backgrounds, which take the form \[g_{\mu\nu}=\bar{g}_{\mu\nu}+2V\,\lambda_{\mu}\lambda_{\nu},\label{un2}\tag{4}\] where \(\bar{g}_{\mu\nu}\) is the (A)dS metric, and the null vector \(\lambda_{\mu}\) satisfies \[\nabla_{\mu}\lambda_{\nu}=\xi_{\mu}\lambda_{\nu}+\xi_{\nu}\lambda_{\mu},\] with \(\xi_{\mu}\), defined by this equation, is orthogonal to \(\lambda_{\mu}\); and \(\lambda^{\mu}\partial_{\mu}V=0\). For the generic class of theories defined by \[I=\int d^{D}x\,\sqrt{-g}\,f\!\left(g,\mathrm{Riem},\nabla \mathrm{Riem},\ldots\right),\label{eq:Generic95gravity95theories}\tag{5}\] where \(f\) is a smooth function. The field equations of (5 ) take the form \[G_{\alpha\beta}+\Lambda_{0}g_{\alpha\beta}+H_{\alpha\beta}=\kappa T_{\alpha\beta},\label{eq:EoM95generic}\tag{6}\] where \(G_{\alpha\beta}\equiv R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R\), and \(H_{\alpha\beta}\) represents all the higher derivative terms. The KSK ansätz 4 reduces the source-free field equations to: (i) algebraic relations among the couplings, the cosmological constant, and the background curvature, and (ii) a single linear partial differential equation for \(V\) of the form \[\sum_{n=1}^{N}a_{n}\,\Box^{n}V=0,\label{cond2}\tag{7}\] hence they are almost universal.

Now, we move to the discussion of \(pp\)-wave metrics with non-flat transverse two-space. Recently, a new family of four-dimensional solutions of quadratic gravity was constructed in [12] that are of the form \[ds^{2}=2\,du\,dv+h_{ab}(x^{c})\,dx^{a}dx^{b}+2V(u,x^{a})\,du^{2},\label{eq:pp-wave}\tag{8}\] where \((u,v)\) are null coordinates and \(h_{ab}\) (\(a=2,3\)) is the metric of a two-dimensional constant-curvature surface \(S^{2}\) or \(H^{2}\). In particular, \(h_{ab}\) depends only on \(x^{a}\) and not on \(u\) or \(v\). These metrics do not solve Einstein’s equations with a cosmological constant and are therefore genuinely non-Einsteinian \(pp\)-wave solutions of quadratic gravity. Let us first focus on the corresponding background metric, \[ds^{2}=2\,du\,dv+h_{22} (dx^{2})^2+ 2 h_{23}\,dx^2 dx^3+h_{33}\,(dx^3)^{2},\label{eq:gBar95components}\tag{9}\] to show that it is almost universal and solves the cosmological Einstein-Maxwell equations. Its Riemann tensor has a nontrivial part only in the transverse two-space: \[\bar{R}_{abcd} =\frac{\bar{R}}{2}\left(h_{ac}h_{bd}-h_{ad}h_{bc}\right),\label{eq:Riembar95in95max95sym95const95curv95form}\tag{10}\] and therefore, the nonzero components of the Ricci and Einstein tensors are \[\bar{R}_{ab} =\frac{\bar{R}}{2}\,h_{ab},\quad\bar{G}_{ab} =\frac{\bar{R}}{2}\,\left(h_{ab}-g_{ab}\right),\quad \bar{R}\equiv 4\Lambda,\label{eq:Background95two-tensors}\tag{11}\] where we defined \(\Lambda\) in the last equation in terms of the constant scalar curvature. With a null \(\lambda_{\mu}=\delta_{\mu}^{0}\) satisfying \(\nabla_{\mu}\lambda_{\nu}=0\), the metric tensor can be written in the form \[\bar{g}_{\mu\nu}=n_{\mu}\,\lambda_{\nu}+n_{\nu}\,\lambda_{\mu}+h_{\mu\nu},\label{eq:gbar95in95n95and95lambda}\tag{12}\] where \(n_{\mu}=\delta_{\mu}^{1}\) satisfies \(\nabla_{\mu}\,n_{\nu}=0\). Here, \(h_{\mu\nu}\) has the non-zero components \(h_{ab}\); therefore, it satisfies \(\lambda^{\mu} h_{\mu\nu}=0\) and \(n^{\mu} h_{\mu\nu}=0\). Let us show that \(\bar{g}_{\mu\nu}\) is a solution to the cosmological Einstein-Maxwell theory with the action \[I = \int d^{4}x\,\sqrt{-g}\left[\frac{1}{\kappa}\left(R-2\Lambda_{0}\right)-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right].\label{eq:Einstein-Maxwell95action}\tag{13}\] Assuming the field potential and its field strength tensor are as follows: \[A_{\mu}=\varepsilon(u)\,v\lambda_{\mu}\Longrightarrow F_{\mu\nu}=\varepsilon(u)\left(n_{\mu}\,\lambda_{\nu}-n_{\nu}\,\lambda_{\mu}\right),\label{eq:Vector95field}\tag{14}\] representing an electric field in the null direction as \(F_{01}=E=-\varepsilon(u)\). The corresponding energy-momentum tensor \(T_{\mu\nu}^{(F)} \equiv F_{\mu}\,^{\sigma}\,F_{\nu\sigma}-\frac{1}{4}\,\bar{g}_{\mu\nu}\,F^{\rho\sigma}F_{\rho\sigma}\) becomes \[\frac{T_{\mu\nu}^{(F)}}{\varepsilon^{2}}= \frac{\bar{g}_{\mu\nu}}{2}-n_{\mu}\,\lambda_{\nu}-n_{\nu}\,\lambda_{\mu} = h_{\mu\nu}-\frac{1}{2}\,\bar{g}_{\mu\nu}.\label{en2}\tag{15}\] since \(\lambda^{\mu}\,n_{\mu}=1\) and \(n_{\mu}\,\lambda_{\nu}+n_{\nu}\lambda_{\nu}=\bar{g}_{\mu\nu}-h_{\mu\nu}\). Using this result together with the Ricci tensor and scalar curvature from (11 ) in the cosmological Einstein-Maxwell field equations \(G_{\mu\nu}+\Lambda_{0}g_{\mu\nu}=\kappa T_{\mu\nu}^{(F)}\) yields \[\left(2\Lambda-\kappa\varepsilon^{2}\right)h_{\mu\nu}+\left(-2\Lambda+\Lambda_{0}+\frac{\kappa\varepsilon^{2}}{2}\right)\,\bar{g}_{\mu\nu}=0.\] Therefore, the algebraic equations \[2\Lambda-\kappa\varepsilon^{2} =0,\quad -2\Lambda+\Lambda_{0}+\frac{\kappa\varepsilon^{2}}{2} =0,\label{eq:epsilon-Lambda95095eqn}\tag{16}\] must be satisfied for a solution to exist. Note that only positive scalar curvature spacetimes are possible, and \(\varepsilon\) must be a constant. Solving these algebraic equations yields the cosmological constant in terms of the electric field as \[\Lambda=\Lambda_{0}=\kappa \frac{\varepsilon^{2}}{2}=\kappa \frac{E^{2}}{2}.\] Therefore, the electromagnetic energy density determines the cosmological constant.

With the topology \(\mathbb{R}^{1,1}\times S^{2}\) , \(\bar{g}_{\mu\nu}\) is a critical point between the Nariai solution \({\rm dS}^{2}\times S^{2}\) and the Bertotti-Robinson solution \({\rm AdS}^{2}\times S^{2}\); further properties of this metric were discussed in [13].

In [14], we showed that 17 solves the generic gravity field equations (6 ); hence, it is almost universal. This property means that for the metric (17 ), any rank-two symmetric tensor built from the metric, its curvature, and covariant derivatives is a linear combination of \(\bar{g}_{\mu\nu}\) and \(h_{\mu\nu}\). Therefore, we have the following theorem:

Then we have the next theorem:

This reduction in the field equations suggests that for the metric (17 ), the field equations of any generic gravity theory reduce to the Einstein-Maxwell field equations with a cosmological constant since the Einstein tensor given in (11 ) with \(\bar{R}=4\Lambda\), that is \(G_{\mu\nu}=2\Lambda h_{\mu\nu}-2\Lambda\bar{g}_{\mu\nu}\), becomes \[G_{\mu\nu}=\left(\kappa\varepsilon^{2}-e_{1}\right)h_{\mu\nu}-\left(\Lambda_{0}+\frac{\kappa\varepsilon^{2}}{2}+e_{0}\right)\bar{g}_{\mu\nu},\] where we used (19 ). Then, writing \(h_{\mu\nu}\) in terms of the energy-momentum tensor of an electromagnetic field \(A_{\mu}=\Upsilon \,v\lambda_{\mu}\) as \[h_{\mu\nu}=\frac{1}{\Upsilon^{2}}T_{\mu\nu}^{(F)}+\frac{1}{2}\bar{g}_{\mu\nu},\] yields \[G_{\mu\nu}=\frac{\kappa\varepsilon^{2}-e_{1}}{\Upsilon^{2}}T_{\mu\nu}^{(F)}-\left(\Lambda_{0}+e_{0}+\frac{e_{1}}{2}\right)\bar{g}_{\mu\nu}.\label{eq:Einstein951}\tag{20}\] This equation reduces to the Einstein-Maxwell field equations with updated parameters, \[\Upsilon^{2} \equiv\varepsilon^{2}-\frac{e_{1}}{\kappa},\quad \bar{\Lambda}_{0} \equiv\Lambda_{0}+e_{0}+\frac{e_{1}}{2},\label{eq:beta-LambdaBar95defn}\tag{21}\] and we have the following corollary.

Let us show that the background metric (17 ) is of type D. The Weyl tensor \(\bar{C}_{\mu\alpha\nu\beta}\) is found to be \[\begin{align} \bar{C}_{\mu\alpha\nu\beta} & = \frac{\bar{R}}{2}\left(h_{\mu\nu}h_{\alpha\beta}-h_{\mu\beta}h_{\alpha\nu}\right)\nonumber +\frac{\bar{R}}{6}\left(\bar{g}_{\mu\nu}\bar{g}_{\alpha\beta}-\bar{g}_{\mu\beta}\bar{g}_{\alpha\nu}\right) \\ &-\frac{\bar{R}}{4}\left(\bar{g}_{\mu\nu}h_{\alpha\beta}+\bar{g}_{\alpha\beta}h_{\mu\nu}-\bar{g}_{\mu\beta}h_{\alpha\nu}-\bar{g}_{\alpha\nu}h_{\mu\beta}\right). \quad \label{eq:Cbar} \end{align}\tag{22}\] Hence, the background spacetime is not conformally flat since \(\bar{C}_{\mu\alpha\nu\beta}\neq 0\) for \(\bar{R}\neq 0\). To show that \(\bar{C}_{\mu\alpha\nu\beta}\) is of type D, it must satisfy [15] \[\bar{C}_{\mu\alpha\nu[\beta}\lambda_{\sigma]}\lambda^{\alpha}\lambda^{\nu}=0,\quad\bar{C}_{\mu\alpha\nu[\beta}n_{\sigma]}n^{\alpha}n^{\nu}=0.\] Calculating \(\bar{C}_{\mu\alpha\nu\beta}\lambda^{\alpha}\lambda^{\nu}\) and \(\bar{C}_{\mu\alpha\nu\beta}n^{\alpha}n^{\nu}\), one has \[\bar{C}_{\mu\alpha\nu\beta}\lambda^{\alpha}\lambda^{\nu}=\frac{\bar{R}}{6}\,\lambda_{\mu}\lambda_{\beta}, \quad \bar{C}_{\mu\alpha\nu\beta}n^{\alpha}n^{\nu}=\frac{\bar{R}}{6}\,n_{\mu}n_{\beta},\] therefore, \(\bar{C}_{\mu\alpha\nu[\beta}\lambda_{\sigma]}\lambda^{\alpha}\lambda^{\nu}=0\) and \(\bar{C}_{\mu\alpha\nu[\beta}n_{\sigma]}n^{\alpha}n^{\nu}=0\). Therefore, we have the following theorem.

Here, we extend the discussion of the fact that \(\bar{g}_{\mu\nu}\) reduces the generic gravity field equations to the cosmological Einstein-Maxwell field equations, as stated in Theorem 3, to the Kerr-Schild metric \[g_{\mu\nu}=n_{\mu}\,\lambda_{\nu}+n_{\nu}\,\lambda_{\mu}+h_{\mu\nu}+2V(u,x^a)\lambda_{\mu}\,\lambda_{\nu},\label{met1}\tag{24}\] for which the generic gravity field equations reduce to those of cosmological Einstein-Maxwell with null dust. For this, we first extend Theorem 1 by showing that the Kerr-Schild metric (24 ) satisfies the cosmological Einstein-Maxwell field equations with null dust with the energy density \(\Phi\) as \[G_{\mu\nu}+\Lambda_{0}g_{\mu\nu}=\kappa T_{\mu\nu}^{(F)}+\kappa T_{\mu\nu}^{(\Phi)}.\label{eq:EM-null95dust95EoM}\tag{25}\] We show this result starting with the Einstein tensor of (24 ), which has the form \[G_{\mu\nu}=\frac{\bar{R}}{2}\,\left(h_{\mu\nu}-g_{\mu\nu}\right)\,-\,\lambda_{\mu}\,\lambda_{\nu}\,\Box V, \label{ein1}\tag{26}\] where we used (11 ). As in the background case, for the vector field \(A_{\mu}=\varepsilon\,v\lambda_{\mu}\), \(F_{\mu}\,^{\sigma}\) takes the form \(F_{\mu}\,^{\sigma}=\varepsilon\left(n_{\mu}\,\lambda^{\sigma}-\delta_{u}^{\sigma}\,\lambda_{\mu}+2V\lambda_{\mu}\lambda^{\sigma}\right)\); therefore, \[T_{\mu\nu}^{(F)}=\varepsilon^{2}\,\left(-\frac{1}{2}\,g_{\mu\nu}+h_{\mu\nu}\right).\label{en1}\tag{27}\] In addition, let the null dust with energy density \(\Phi\) have the energy-momentum tensor \(T_{\mu\nu}^{(\Phi)} = \Phi \lambda_{\mu} \lambda_{\nu}\). Now, putting (26 ), (27 ), \(T_{\mu\nu}^{(\Phi)} = \Phi \lambda_{\mu} \lambda_{\nu}\), and \(R=\bar{R}=4 \Lambda\) in the field equation (25 ) yields \[\begin{align} 0 = & \left(2\Lambda-\kappa\varepsilon^{2}\right)h_{\mu\nu}+\left(-2\Lambda+\Lambda_{0}+\frac{\kappa\varepsilon^{2}}{2}\right) g_{\mu\nu} \nonumber \\ &-\left(\Box V+\kappa \Phi\right)\lambda_{\mu}\,\lambda_{\nu}, \end{align}\] hence the set of equations, \[2\Lambda-\kappa\varepsilon^{2} =0,\quad -2\Lambda+\Lambda_{0}+\frac{\kappa\varepsilon^{2}}{2} =0,\quad \Box V+\kappa \Phi =0, \label{eq:h-g-Null95coefs}\tag{28}\] must be satisfied. The first equation requires \(\Lambda =\kappa \varepsilon^{2}/2\) which means that only positive scalar curvature is possible. As a result, we have the following theorem.

Now we follow the same procedure as we did for the metric \(\bar{g}_{\mu\nu}\). We first generalize Theorem 2 for the metric (8 ), whose almost universal property is shown in [14].

Using Theorem 6, Theorem 3 can be extended for the metric 8 .

This reduction in the field equations suggests that for the metric (29 ), the field equations of any generic gravity theory are equivalent to the cosmological Einstein-Maxwell field equations with null dust. To see this, first, using \(\bar{R}=4\Lambda\) in the Einstein tensor (26 ), and then using (32 ) and (33 ) yields \[\begin{align} G_{\mu\nu}= & \left(\kappa\varepsilon^{2}-e_{1}\right)h_{\mu\nu}-\left(\Lambda_{0}+\frac{\kappa\varepsilon^{2}}{2}+e_{0}\right)g_{\mu\nu}\nonumber \\ & +\left(\kappa\Phi-\sum_{n=1}^{N}c_{n}\square^{n}V\right)\lambda_{\mu}\lambda_{\nu}. \end{align}\] Then, writing \(h_{\mu\nu}\) in terms of the energy-momentum tensor of an electromagnetic field \(A_{\mu}=\Upsilon\,v\lambda_{\mu}\) as \[h_{\mu\nu}=\frac{1}{\Upsilon^{2}}T_{\mu\nu}^{(F)}+\frac{1}{2} g_{\mu\nu},\] yields \[\begin{align} G_{\mu\nu}= & \frac{\kappa\varepsilon^{2}-e_{1}}{\Upsilon^{2}}T_{\mu\nu}^{(F)}-\left(\Lambda_{0}+e_{0}+\frac{e_{1}}{2}\right)g_{\mu\nu}\nonumber \\ & +\left(\kappa\Phi-\sum_{n=1}^{N}c_{n}\square^{n}V\right)\lambda_{\mu}\lambda_{\nu}.\label{eq:Einstein95195f40Riem41} \end{align}\tag{34}\] Finally, with the definitions, \[\Upsilon^{2} \equiv\varepsilon^{2}-\frac{e_{1}}{\kappa},\quad \bar{\Lambda}_{0} \equiv\Lambda_{0}+e_{0}+\frac{e_{1}}{2},\quad \bar{\Phi} \equiv\Phi-\frac{1}{\kappa}\sum_{n=1}^{N}c_{n}\square^{n}V,\label{eq:beta95defn95LambdaBar95defn95PhiBar95defn}\tag{35}\] (34 ) reduces to \[G_{\mu\nu}+\bar{\Lambda}_{{\rm 0}}g_{\mu\nu}=\kappa T_{\mu\nu}^{(F)}+T_{\mu\nu}^{(\bar{\Phi})},\] which are the cosmological Einstein-Maxwell field equations with null dust in terms of the updated parameters.

We now show that the constant-curvature \(pp\)-wave spacetimes 24 are of Petrov type II. First, note that the Weyl tensor \(C_{\mu\alpha\nu\beta}\) can be written as \[C_{\mu\alpha\nu\beta}=\bar{C}_{\mu\alpha\nu\beta}+4\lambda_{[\mu}\Omega_{\alpha][\beta}\lambda_{\nu]},\label{eq:Weyl95tensor}\tag{36}\] where \(\Omega_{\alpha\beta}\) is defined as \[\Omega_{\alpha\beta}\equiv-\left[\nabla_{\beta}\nabla_{\alpha}-\frac{1}{2}h_{\alpha\beta}\left(\square-\frac{\bar{R}}{3}\right)\right]V.\label{eq:Omega}\tag{37}\] Since \(\lambda^{\mu}h_{\mu\nu}=0\), \(\lambda^{\mu}\partial_{\mu}V=0\) and \(\nabla_{\mu}\lambda_{\nu}=0\), we have \(\lambda^{\mu}\Omega_{\mu\nu}=0\). Then, it follows that \(C_{\mu\alpha\nu\beta}\lambda^{\beta}=\bar{C}_{\mu\alpha\nu\beta}\lambda^{\beta}\) which yields \[C_{\mu\alpha\nu[\beta}\lambda_{\sigma]}\lambda^{\alpha}\lambda^{\nu}=\bar{C}_{\mu\alpha\nu[\beta}\lambda_{\sigma]}\lambda^{\alpha}\lambda^{\nu}=0.\label{eq:Type95II95cond}\tag{38}\] On the other hand, the \(n_{\mu}=\delta_{\mu}^{v}\) one-form, which is null with respect to the background spacetime, now has the norm \(n_{\mu}n^{\mu}=-2V\). Therefore, we need to define the other null vector with respect to the constant-curvature \(pp\)-wave spacetime as \(N_{\mu}\equiv n_{\mu}+V\lambda_{\mu}\) for which \(C_{\mu\alpha\nu[\beta}N_{\sigma]}N^{\alpha}N^{\nu}\ne0\) since \(V\) has the coordinate dependence \(V=V\left(u,x^{a}\right)\). Hence, the constant-curvature \(pp\)-wave spacetime is type II, as (38 ) is satisfied [15]–[17].

One can proceed without the null dust, in which case the generic gravity field equations we have to solve \[0=\sum_{n=1}^{N}c_{n}\Box^{n}V=c_N \prod_{k=1}^{N-1}(\Box-m_{k}^{2})\,\Box V,\label{kok1}\tag{39}\] where we have factorized the equation in the second equality. Here, \(m_{k}^{2}\) are real or complex constants determined in terms of \(c_n\). The general solution of (39 ) can then be written as \[V=\sum_{k=1}^{N-1}V_{k}+V_{N},\quad (\Box-m_{k}^{2})\,V_{k} =0,\quad \Box V_{N} =0,\] where \(m_{k}\)’s are distinct and nonzero for \(k=1,2,\cdots, N-1\). Note that if some \(m_k\)’s coincide or are zero, then new solutions arise as the second and third equations change. Thus, the single higher order equation 39 is equivalent to a set of \(N\) Klein–Gordon–type equations for the components \(V_{k}\) [6]–[8], [14], [18].

0.0.0.1 Example 1–Quadratic gravity

As an illustration, consider the four-dimensional action of quadratic gravity with Maxwell electrodynamics, \[\mathcal{L}=\frac{1}{\kappa}\left(R-2\Lambda_{0}+\alpha R^{2}+\beta R_{\mu\nu}R^{\mu\nu}\right) -\frac{1}{4}F^{\mu\nu}F_{\mu\nu},\] with constants \(\alpha\), \(\beta\), and the bare cosmological constant \(\Lambda_0\). The quadratic curvature term \(H_{\mu\nu}\) in the field equations was given in [19] or in [20], [21] as \[\begin{align} &H_{\mu\nu}= 2\alpha R\left(R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R\right)+\left(2\alpha+\beta\right)\left(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu}\right)R\nonumber \\ & +\beta\square\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)+2\beta\left(R_{\mu\sigma\nu\rho}-\frac{1}{4}g_{\mu\nu}R_{\sigma\rho}\right)R^{\sigma\rho}. \end{align}\] Let us study the solution of this theory, given the metric and electromagnetic field ansätze as (17 or (12 ) and (14 ), respectively. Using (12 ) in \(H_{\mu\nu}\) yields \[H_{\mu\nu}=8\left(2\alpha+\beta\right)\Lambda^{2}h_{\mu\nu}-4\left(2\alpha+\beta\right)\Lambda^{2}g_{\mu\nu},\label{eq:H95quad95grav}\tag{40}\] therefore, comparing with Theorem 2, \(e_{0} =-4\left(2\alpha+\beta\right)\Lambda^{2}\) and \(e_1=-2 e_0\). As stated in Corollary 1, with the ansätze (12 ) and (14 ), the field equations of any generic gravity theory reduce to the Einstein-Maxwell field equation with a cosmological constant as \[G_{\mu\nu}+\bar{\Lambda}_{0}g_{\mu\nu}=\kappa T_{\mu\nu}^{(F)}\left(\Upsilon\right),\] where the updated parameters \(\Upsilon\) and \(\bar{\Lambda}_{0}\) defined in (21 ) must satisfy the algebraic equations (16 ). Using \(e_{0}\) and \(e_{1}\), we found above in (16 ) yields \(\Lambda =\Lambda_{0}\) and \[8\left(2\alpha+\beta\right)\Lambda^{2}+2\Lambda =\kappa\varepsilon^{2}.\] Here, note that in contrast to Einstein-Maxwell theory, the quadratic gravity augmented with Maxwell electrodynamics can have both positive and negative \(\Lambda\), which allows for both topology \(\mathbb{R}^{1,1}\times S^{2}\) and topology \(\mathbb{R}^{1,1}\times H^{2}\) for the background metric. In the absence of the electromagnetic field, that is, setting the parameter \(\varepsilon\) to zero, the sourceless quadratic gravity admits a constant curvature solution for \(1/\Lambda=-4\left(2\alpha+\beta\right)\) and \(\Lambda=0\), which contradicts the metric ansätz. On the other hand, with the presence of the electromagnetic field, there are two solutions with cosmological constant \[\Lambda=\frac{-1\pm\sqrt{1+8\kappa\varepsilon^{2}\left(2\alpha+\beta\right)}}{8\left(2\alpha+\beta\right)}\label{eq:Lambda95quad95grav-Maxwell},\quad 2\alpha+\beta\neq 0.\tag{41}\] Therefore, the metric (17 ) is a solution of quadratic gravity with Maxwell electrodynamics, where the electromagnetic four potential is (14 ), and the cosmological constant (which is equal to the bare cosmological constant) is determined by the theory parameters and the electromagnetic field, as given in (41 ).

Now, let us study the solutions of quadratic gravity for the metric ansätz (24 ). With this metric, \(H_{\mu\nu}\) becomes \[\begin{align} H_{\mu\nu}= & 8\left(2\alpha+\beta\right)\Lambda^{2}h_{\mu\nu}-4\left(2\alpha+\beta\right)\Lambda^{2}g_{\mu\nu}\nonumber \\ & -\lambda_{\mu}\lambda_{\mu}\left(4\left(2\alpha+\beta\right)\Lambda\bar{\square}V-\beta\bar{\square}^{2}V\right), \end{align}\] which is in the form that we have proved in Theorem 6. Therefore, \(e_{0}\) and \(e_{1}\) are the same as the background values we gave after 40 and \[c_{1} =-4\left(2\alpha+\beta\right)\Lambda,\quad c_{2} =\beta.\] The solutions for the sourceless quadratic gravity were studied in [12]. On the other hand, once the electromagnetic and null dust sources are introduced via the electromagnetic vector field (14 ) and the null dust energy momentum tensor \(T_{\mu\nu}^{(\Phi)}=\Phi\lambda_{\mu}\lambda_{\nu}\), respectively, the field equations for quadratic gravity with these sources can be found from \[G_{\mu\nu}+\bar{\Lambda}_{0}g_{\mu\nu}=\kappa T_{\mu\nu}^{(F)}\left(\Upsilon\right)+T_{\mu\nu}^{(\bar{\Phi})},\] since, as stated in Corollary 2, the field equations of any generic gravity theory reduce to the cosmological Einstein-Maxwell field equations involving a null dust source with the metric and the electromagnetic vector field ansätze (24 ) and (14 ). Here, the updated parameters \(\Upsilon\) and \(\bar{\Lambda}_{0}\) are given as (32 ), which are the same as the above background case. On the other hand, the updated energy density of null dust is defined in (33 ). These updated parameters must satisfy the set of equations (28 ). The first two algebraic equations are the same as in the background case; therefore, they yield the same solutions for the cosmological constant in terms of either the bare cosmological constant or the theory parameters and the vector potential parameter. On the other hand, the third equation becomes \[\kappa\Phi=\left(\beta\bar{\square}-\left[1+4\left(2\alpha+\beta\right)\Lambda\right]\right)\bar{\square}V.\label{eq:Phi95quad95grav}\tag{42}\] This equation determines the energy density of the null dust. However, it allows the absence of null dust if \(\bar{\square}V=0\) or \(V=V_{1}+V_{2}\) where \(\bar{\square}V_{1}=0\) and \[\left(\beta\bar{\square}-\left[1+4\left(2\alpha+\beta\right)\Lambda\right]\right)V_{2}=0.\] Or, if \(1+4(2\alpha+\beta)\, \Lambda=0\) then \(\bar{\square}^2\,V=0\) if \(\beta\neq0\).

Therefore, the constant curvature \(pp\)-wave metric is a solution of quadratic gravity with Maxwell electrodynamics, for which the electromagnetic four potential is (14 ), while the null dust has the energy density given by (42 ) for a given metric function \(V\). The cosmological constant determining the constant curvature is equal to the bare cosmological constant, and both are determined by the theory parameters and the electromagnetic field as given in (41 ).

0.0.0.2 Example 2–Cubic gravity generated by string theory

As a second example, we study the four-dimensional cubic curvature gravity theory with the Lagrangian density, \[\begin{gather} \kappa \mathcal{L}=R-2\Lambda_{0}-\frac{\kappa}{4}F^{\mu\nu}F_{\mu\nu}\\ +\frac{\left(\alpha^{\prime}\right)^{2}}{24}\left( R_{\alpha\beta}^{\mu\nu}R_{\mu\nu}^{\gamma\lambda}R_{\gamma\lambda}^{\alpha\beta}-2R_{\nu\beta}^{\mu\alpha}R_{\mu\lambda}^{\nu\gamma}R_{\alpha\gamma}^{\beta\lambda}\right), \end{gather}\] augmented with Maxwell electrodynamics. The \(H_{\mu\nu}\) term in the field equations coming from the cubic curvature part is \[\begin{align} H_{\mu\nu}= & \frac{\left(\alpha^{\prime}\right)^{2}}{24}\Biggl[6g_{(\mu|\rho}\nabla^{\lambda}\nabla_{\sigma}\left(-2R_{|\nu)\beta}^{\rho\alpha}R_{\alpha\lambda}^{\beta\sigma}+R_{|\nu)\lambda}^{\alpha\beta}R_{\alpha\beta}^{\rho\sigma}\right)\nonumber \\ & +3R_{\rho\sigma(\mu|}^{\phantom{\rho\sigma(\mu|}\lambda}\left(-2R_{|\nu)\beta}^{\rho\alpha}R_{\alpha\lambda}^{\beta\sigma}+R_{|\nu)\lambda}^{\alpha\beta}R_{\alpha\beta}^{\rho\sigma}\right)\nonumber \\ & -\frac{1}{2}g_{\mu\nu}\left(-2R_{\sigma\beta}^{\rho\alpha}R_{\rho\lambda}^{\sigma\gamma}R_{\alpha\gamma}^{\beta\lambda}+R_{\alpha\beta}^{\rho\sigma}R_{\rho\sigma}^{\gamma\lambda}R_{\gamma\lambda}^{\alpha\beta}\right)\Biggr].\label{eq:H95tensor95for95string95cubic95terms} \end{align}\tag{43}\] This cubic gravity was obtained by using the three- and four-point scattering amplitudes of bosonic strings in [22]. In addition, we introduced the bare cosmological constant \(\Lambda_{0}\).

Let us study the solution of this theory given the metric and electromagnetic field ansätze as (24 ) and (14 ), respectively. Using (24 ) in \(H_{\mu\nu}\) yields \[\begin{align} H_{\mu\nu}= & 6\left(\alpha^{\prime}\right)^{2}\Lambda^{3}h_{\mu\nu}-2\left(\alpha^{\prime}\right)^{2}\Lambda^{3}g_{\mu\nu}\nonumber\\ & -2\left(\alpha^{\prime}\right)^{2}\Lambda^{2}\lambda_{\mu}\lambda_{\nu}\bar{\square}V, \end{align}\] therefore, comparing with Theorem 6 yields \[\begin{align} e_{0} =-2\left(\alpha^{\prime}\right)^{2}\Lambda^{3},\,\, e_{1} = -3 e_0,\,\, c_{1} =-2\left(\alpha^{\prime}\right)^{2}\Lambda^{2}. \end{align}\] As stated in Corollary 2, with the ansätze (24 ) and (14 ), the field equations of any generic gravity theory reduce to the cosmological Einstein-Maxwell field equation involving a null dust source, with the updated parameters \(\Upsilon\), \(\bar{\Lambda}_{0}\) and \(\bar{\Phi}\) are given as (32 ), and (33 ). These updated parameters must satisfy the set of equations (28 ). The first two algebraic equations become: \[\varepsilon^{2} =\frac{2\Lambda}{\kappa}\left(1+2\left(\alpha^{\prime}\right)^{2}\Lambda^{2}\right),\quad \Lambda_{0} =\Lambda\left(1-\left(\alpha^{\prime}\right)^{2}\Lambda^{2}\right),\] determining \(\varepsilon\) and \(\Lambda_{0}\) in terms of the cosmological constant. Here, note that \(\Lambda\) needs to be positive since the left-hand side and the term in parentheses are positive in the first equation. In addition, in the absence of the electromagnetic source, there is no solution for the first equation except \(\Lambda=0\) again, since the term in parentheses is positive, and this \(\Lambda=0\) solution contradicts the constant curvature \(pp\)-wave ansätz. On the other hand, the third equation becomes \[\Phi=-\frac{1}{\kappa}\left(1+2\left(\alpha^{\prime}\right)^{2}\Lambda^{2}\right)\bar{\Box}V,\] determining the energy density of the null dust. However, it allows the absence of null dust if \(\bar{\square}V=0\).

0.0.0.3 Conclusions

Universal and almost universal metrics studied so far in the literature are typically Einsteinian, i.e., they solve field equations derived from the Einstein–Hilbert action. In this Letter, we have identified a new class of almost universal metrics based on pp-waves with constant-curvature transverse spaces, whose backgrounds are themselves non-Einsteinian. For generic higher–curvature theories of gravity, these metrics reduce the field equations to two algebraic relations among the cosmological constant, the curvature of the constant–curvature surface \(S^2\) or \(H^2\), and the coupling constants of the theory, together with a single linear differential equation for the Kerr–Schild profile function \(V\). We have explicitly demonstrated this for quadratic and cubic gravity and outlined how the construction generalizes to more complicated higher–derivative theories.

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