1 Introduction↩︎

Einstein’s general relativity is usually formulated as a geometric theory of gravitation . It is also well known, however, that the Einstein field equations can be approached from a field-theoretic point of view, starting from a massless spin-two field on a flat Minkowski background. This line of thought, developed in different forms by Gupta, Kraichnan, Feynman, Deser and others , shows that the nonlinear structure of general relativity is strongly constrained by the consistency requirements of a self-interacting spin-two field. See also Ref.[1].

Let \(h_{\mu\nu}\) be a symmetric tensor field on a Minkowski background, universally coupled to matter. The Lagrangian density for the symmetric tensor field \(L\) is determined by Feynman’s consistency condition \[\begin{align} g_{\beta\nu}\partial_\mu \Big( \frac{\delta L}{\delta h_{\mu\nu}} \Big) + [\mu\nu,\beta] \frac{\delta L}{\delta h_{\mu\nu}} =0 ,\label{1.1} \end{align}\tag{1}\] with \(g_{\mu\nu}:=\eta_{\mu\nu}+2\lambda h_{\mu\nu}\) and \([\mu\nu,\beta]:=\lambda \left( \partial_\nu h_{\mu\beta} +\partial_\mu h_{\nu\beta} -\partial_\beta h_{\mu\nu} \right)\). Here, \(\lambda\) is the coupling constant and \(\delta/\delta h_{\mu\nu}\) is the Euler-Lagrange derivative. Feynman derived this condition by using the equation of motion for a point mass and the field equation for \(h_{\mu\nu}\) .

We derive Feynman’s consistency condition by imposing Poincaré invariance, conservation of the total energy-momentum tensor \[\begin{align} \partial_\mu \Big(U^{\mu\nu}+\tau^{\mu\nu}\Big)=0 \end{align}\] and the field equation \(\delta L/\delta h_{\mu\nu}=\lambda \tau^{\mu\nu}\). Here, \(\tau^{\mu\nu}\) is the matter energy-momentum tensor and \(U^{\mu\nu}\) is the Belinfante-Rosenfeld tensor [2], [3] for \(h_{\mu\nu}\). Furthermore, we will show that when the same method is applied to a vector field coupled to a system of point masses, no conditions are imposed on the field’s Lagrangian density.

The paper is organized as follows. In Section 2, we review the electromagnetic energy-momentum tensor in order to motivate the use of the Belinfante-Rosenfeld tensor. In Section 3, we derive the general Belinfante-Rosenfeld construction for a Poincaré-invariant field theory. In Section 4, we apply the method to the vector field and show that total energy-momentum conservation does not determine the vector-field Lagrangian. In Section 5, we turn to a symmetric rank-two field and show that the same conservation principle yields Feynman’s consistency condition and uniquely selects the Einstein Lagrangian density. Appendix 7 recalls Feynman’s derivation of the consistency condition. In Appendix 8, we prove the uniqueness of the solution of Feynman’s consistency condition. In Appendix 9, we compute the Belinfante-Rosenfeld tensor for the Einstein Lagrangian and show that it coincides with Papapetrou’s gravitational energy-momentum pseudotensor.

We emphasize that our derivation does not fall into the “from gravitons to gravity" category [1], given that it does not start from the Fierz-Pauli Lagrangian to couple the field to its own energy-momentum tensor in successive iterations to converge to the gravitational action. Instead, the requirement of total energy-momentum conservation of the system is realized in a single step, imposing this condition and turning it into a functional equation for the field Lagrangian that drastically constrains its form. Consequently, we are not dealing with a bootstrapping process in which we introduce a flat-space Lagrangian into a box to open it later and find that it has transmuted into the general relativity one: in our case, there is no such thing as an initial Lagrangian, and consequently, it cannot be iteratively modified; what we have instead is a single condition on an initially arbitrary Lagrangian.

2 The electromagnetic energy-momentum tensor↩︎

As mentioned above, the analysis developed in this paper is based on the conservation of energy and momentum. For this reason, it is necessary to define precisely and justify the mathematical object to which we will link this conservation. For the interaction field, we take this object to be the Belinfante-Rosenfeld tensor; to motivate and justify its use, let us begin by reviewing the well-known electromagnetic case: We depart from the standard Lagrangian density \[\begin{align} L &\!\!\!=\!\!\!&-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} \;\;(F_{\mu\nu}:= \partial_\mu A_\nu-\partial_\nu A_\mu) \label{2.1}. \end{align}\tag{2}\] We seek an object \(U^{\mu\nu} [A_\kappa]\) satisfying the following conditions:

1. \(\partial_\mu U^{\mu\nu} = 0\) when the interaction is off, i.e. when \(A_\mu = A^{0}_\mu\), where \(A^{0}_\mu\) is the solution of the free field equation. This condition ensures energy-momentum conservation for the free field.

2. \(U^{\mu\nu} [A^0_\kappa]=U^{\nu\mu}[A^0_\kappa]\) so that the angular momentum can be generated, in the free case, from \(U^{\mu\nu}\).

3. Similarly, in order to generate the angular momentum correctly in the interacting case, we assume that \(U^{\mu\nu}[A^I_\kappa] =U^{\nu\mu} [A^I_\kappa]\) where \(A^I_\kappa\) is the dynamical solution to the interaction field equation.

4. Conservation of the total energy-momentum of the system is achieved by imposing \(\partial_\mu ( U^{\mu\nu} + \tau^{\mu\nu} ) = 0\) when the interaction is on, i.e. when \(A_\mu = A^{I}_\mu\). \(\tau^{\mu\nu}\) stands for the matter energy-momentum tensor. Note that this property is not determined solely by the form of the field Lagrangian density \(L\); it also depends on the matter and interaction Lagrangians.

We begin by examining candidate objects to describe the energy and momentum content starting from the canonical energy-momentum tensor: \[\begin{align} \Theta^\mu_{\;\nu} &\!\!\!:=\!\!\! &\frac{\partial L}{\partial(\partial_\mu A_\rho )} \partial_ \nu A_\rho - \delta^\mu_\nu L. \label{2.2} \end{align}\tag{3}\] Direct calculations show that \[\begin{align} \Theta^\mu_{\;\nu} &\!\!\!=\!\!\!&-F^{\mu\rho}\partial_\nu A_\rho + \frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\delta^\mu_\nu. \label{2.3} \end{align}\tag{4}\] \(\Theta ^ \mu_{\;\nu}\) satisfies condition a), i.e. \(\partial_\mu \Theta ^{ \mu \nu} = 0\) but it fails to meet property b) as \(\Theta ^{ \mu \nu} [A^{0}_\kappa] \neq \Theta ^{ \nu \mu} [A^{0}_\kappa]\). We know that the object \(T^{\mu\nu}\) defined by \[\begin{align} T^{\mu\nu}&\!\!\!:=\!\!\! &\Theta^{\mu\nu} -\partial_\sigma f^{\mu\sigma\nu} \;\;(f^{\mu\sigma\nu} = - f^{\sigma\mu\nu})\label{2.4} \end{align}\tag{5}\] is just as valid an energy-momentum tensor candidate as \(\Theta ^{ \mu \nu}\), since \(\partial_\mu \Theta ^ \mu_{\;\nu} = \partial_\mu T ^ \mu_{\;\nu}\) and both satisfy condition a). If we choose \(f^{\mu\sigma\nu}\) as \[\begin{align} f^{\mu\sigma\nu} = -F^{\mu\sigma}A^\nu \label{2.6} , \end{align}\tag{6}\] we obtain \[\begin{align} T^{\mu\nu} &\!\!\!=\!\!\!&F^\mu_{\;\sigma}F^{\sigma\nu} + \frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\eta^{\mu\nu}+\partial_\sigma F^{\mu\sigma}A^\nu \label{2.7}. \end{align}\tag{7}\] By the free-field equation, the last term in (7 ) vanishes when the interaction is off and \(T^{\mu\nu}[A^0_\kappa] =T^{\nu\mu} [A^0_\kappa]\), but when the interaction is on \(\partial_\sigma F^{\mu\sigma}\neq 0\) and \(T^{\mu\nu} [A^I_\kappa] \neq T^{\nu\mu}[A^I_\kappa]\).

The construction of an object satisfying c) is straightforward: all we have to do is get rid of the last term in (7 ) and define \[\begin{align} U^{\mu\nu} &\!\!\!:=\!\!\! &T^{\mu\nu} -\partial_\sigma F^{\mu\sigma}A^\nu \nonumber\\ &\!\!\!=\!\!\!&F^\mu_{\;\sigma}F^{\sigma\nu} + \frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\eta^{\mu\nu}. \label{2.8} \end{align}\tag{8}\]

In summary, in the transition \(\Theta^{\mu\nu} \rightarrow T^{\mu\nu} \rightarrow U^{\mu\nu}\), we have arrived at an object \(U^{\mu\nu}\) that satisfies requirements a), b), and c). It remains to be determined whether d) is satisfied. As previously discussed, the fulfillment of this condition depends not only on the form (2 ) of the field Lagrangian density, but it is also constrained by the forms of the matter and interaction Lagrangians. With the standard choice of these Lagrangians (or with any other dynamically equivalent one), it is easy to see that d) is satisfied , i.e. \[\begin{align} \partial_\mu ( U^{\mu\nu} + \tau^{\mu\nu} ) = 0. \label{2.9} \end{align}\tag{9}\]

3 The Belinfante-Rosenfeld energy-momentum tensor↩︎

3.1 General theory↩︎

In the previous section, we constructed an object \(U^{\mu\nu}\). The transition \(\Theta^{\mu\nu} \rightarrow T^{\mu\nu} \rightarrow U^{\mu\nu}\), in which each step added one further desired property, was carried out in an ad hoc manner; we will now seek to generalize this study for an arbitrary Lagrangian density. In this direction, we consider the action \(S\) for a dynamical field \(\psi^A\), defined by \[\begin{align} S = \int L (\psi^A, \partial_\mu \psi^A)\,\, d^4 x \label{3.1} \end{align}\tag{10}\] which we assume to be invariant under the action of the Poincaré group \[\begin{align} \delta x^\mu &\!\!\!=\!\!\!&{\omega^\mu} _\nu\, x^\nu + a^\mu \tag{11} ,\\ \delta\psi^A &\!\!\!=\!\!\!&\frac{1}{2}\omega_{\mu\nu}(S^{\mu\nu})^A_{\;B}\psi^B \tag{12} \end{align}\] where \(\omega_{\mu \nu} = - \omega_{\nu \mu}\) and \((S^{\mu\nu})^A_{\;B}\) determines the infinitesimal transformation of the field \(\psi^B\). A direct calculation shows that \[\begin{align} \Theta^\mu_{\;\nu} &\!\!\!:=\!\!\! &\frac{\partial L}{\partial(\partial_\mu \psi^A)}\partial_\nu \psi^A-\delta^\mu_\nu L \label{3.4} \end{align}\tag{13}\] generalizes (3 ) and satisfies condition a), i.e. \(\partial_\mu \Theta^\mu_{\;\nu} = 0\). When the interaction is off and unless \(\psi^A\) is a scalar field (and \((S^{\mu\nu})^A_{\;B}= 0\)), we generally have \(\Theta^{\mu \nu} \neq \Theta^{\nu \mu}\). Some algebra demonstrates that invariance of (10 ) under (11 ) and (12 ) implies that the quantity we call the Belinfante energy-momentum tensor, defined by \[\begin{align} T^{\mu\nu} &\!\!\!:=\!\!\! &\Theta^{\mu\nu} -\partial_\sigma f^{\mu\sigma\nu} \label{3.5} \end{align}\tag{14}\] with \[\begin{align} f^{\mu\sigma\nu} :=\frac{1}{2}\Big( \frac{\partial L}{\partial(\partial_\mu \psi^A)} (S^{\sigma\nu})^A_{\;B} +\frac{\partial L}{\partial(\partial_\sigma\psi^A)} (S^{\nu\mu})^A_{\;B} -\frac{\partial L}{\partial(\partial_\nu \psi^A)} (S^{\mu\sigma})^A_{\;B} \Big)\psi^B \end{align}\] satisfies condition a), i.e. \(\partial_\mu T^\mu_{\;\nu} = 0\) for the free field. Here, \(f^{\mu\sigma\nu} = - f^{\sigma\mu\nu}\) holds. For (2 ), \(f^{\mu\sigma\nu}\) becomes (6 ).

We now study the behavior of the antisymmetric part of this object in order to determine whether \(T^{\mu\nu}\) satisfies b) and c). We consider again variations (11 ) and (12 ), while the variation of the field derivatives is given by \[\begin{align} \delta(\partial_\lambda\psi^A) &\!\!\!=\!\!\!&\frac{1}{2}\omega_{\mu\nu}(S^{\mu\nu})^A_{\;B}\partial_\lambda\psi^B+\omega_\lambda^{\;\rho}\partial_\rho \psi^A. \label{3.7} \end{align}\tag{15}\] The Lagrangian density varies in the following manner: \[\begin{align} \delta L &\!\!\!=\!\!\!&\frac{1}{2}\omega_{\mu\nu} \Big[ \frac{\partial L}{\partial\psi^A} (S^{\mu\nu})^A_{\;B}\psi^B + \frac{\partial L}{\partial(\partial_\lambda\psi^A)} (S^{\mu\nu})^A_{\;B}\partial_\lambda\psi^B \nonumber\\ &&\quad+ \frac{\partial L}{\partial(\partial_\lambda\psi^A)} [\delta^\mu_\lambda\partial^\nu \psi^A-\delta^\nu_\lambda\partial^\mu \psi^A]\Big]. \label{3.8} \end{align}\tag{16}\] We define \([L]_A\) as \[\begin{align} [L]_A &\!\!\!:=\!\!\! &\frac{\partial L}{\partial\psi^A}-\partial_\lambda\frac{\partial L}{\partial(\partial_\lambda\psi^A)} \label{3.9} \end{align}\tag{17}\] and rewrite (16 ) as \[\begin{align} \delta L &\!\!\!=\!\!\!&\frac{1}{2}\omega_{\mu\nu} \Big( [L]_A(S^{\mu\nu})^A_{\;B}\psi^B +\partial_\lambda\frac{\partial L}{\partial(\partial_\lambda\psi^A)} (S^{\mu\nu})^A_{\;B}\psi^B + \frac{\partial L}{\partial(\partial_\lambda\psi^A)} (S^{\mu\nu})^A_{\;B}\partial_\lambda\psi^B \nonumber\\ && +\Theta^{\mu\nu}-\Theta^{\nu\mu}\Big) \nonumber\\ &\!\!\!=\!\!\!&\frac{1}{2}\omega_{\mu\nu} \Big( [L]_A(S^{\mu\nu})^A_{\;B}\psi^B +\partial_\lambda\Big\{ \frac{\partial L}{\partial(\partial_\lambda\psi^A)} (S^{\mu\nu})^A_{\;B}\psi^B \Big\} +\Theta^{\mu\nu}-\Theta^{\nu\mu}\Big). \label{3.10} \end{align}\tag{18}\] If the Lagrangian density \(L\) is invariant under Lorentz transformations, then we have \(\delta L =0\): \[\begin{align} [L]_A(S^{\mu\nu})^A_{\;B}\psi^B +\partial_\lambda\Big\{ \frac{\partial L}{\partial(\partial_\lambda\psi^A)} (S^{\mu\nu})^A_{\;B}\psi^B \Big\} +\Theta^{\mu\nu}-\Theta^{\nu\mu} = 0. \label{3.11} \end{align}\tag{19}\] Using (19 ) and (14 ), we conclude that \[\begin{align} T^{\mu\nu} - T^{\nu\mu} &\!\!\!=\!\!\!&\Theta^{\mu\nu}-\Theta^{\nu\mu} -\partial_\sigma\Big\{\frac{\partial L}{\partial(\partial_\sigma\psi^A)} (S^{\nu\mu})^A_{\;B}\psi^B \Big\} \nonumber\\ &\!\!\!=\!\!\!&-[L]_A(S^{\mu\nu})^A_{\;B}\psi^B . \label{3.12} \end{align}\tag{20}\] (20 ) not only shows that in the free field scenario (\([L]_A=0\)) b) is fulfilled but also provides the key to derive the expression for \(U^{\mu\nu}\), the Belinfante-Rosenfeld energy-momentum tensor, that satisfies the properties a), b) as well as c).

Once the object \(U^{\mu\nu}\) has been obtained, the next step is to determine whether condition d) fixes the general form of the Lagrangian density \(L\) that generates \(U^{\mu\nu}\). We will address this point in Section 4 and Section 5. However, before doing so, let us examine some results that will prove useful for our calculations. First, and from (14 ) and \(f^{\mu\sigma\nu} = - f^{\sigma\mu\nu}\), it is straightforward to deduce that \[\begin{align} \partial _\mu T^{\mu\nu} = \partial_\mu \Theta^{\mu\nu}. \label{3.13} \end{align}\tag{21}\] Secondly, we study the 4-divergence of a translationally invariant canonical energy-momentum tensor: The variation of the field Lagrangian density can be written as \[\begin{align} \delta L= [L]_A (\delta\psi^A-\delta x^\mu \partial_\mu \psi^A)+\partial_\mu \Big(\frac{\partial L}{\partial(\partial_\mu \psi^A)} \delta\psi^A-\Theta^\mu_{\;\nu}\delta x^\nu \Big). \label{3.14} \end{align}\tag{22}\] For an infinitesimal global transformation \[\begin{align} \delta x^\mu =\epsilon^\mu , \quad\delta\psi^A =0 , \quad\delta L=0, \end{align}\] (22 ) becomes \[\begin{align} 0 = -\epsilon^\mu [L]_A \partial_\mu \psi^A-\epsilon^\nu \partial_\mu \Theta^\mu_{\;\nu}, \end{align}\] which leads to \[\begin{align} \partial_\mu \Theta^{\mu}_{\;\beta} &\!\!\!=\!\!\!&-[L]_A\partial_\beta\psi^A. \label{3.20} \end{align}\tag{23}\]

3.2 The vector field and the spin-2 field↩︎

We now determine the form of the Belinfante-Rosenfeld energy-momentum tensor for a vector field \(A_\mu\). We start by using (20 ) to write \[\begin{align} T^{\mu\nu} - T^{\nu\mu} &\!\!\!=\!\!\!&-[L]^\rho{(S^{\mu\nu})_\rho}^{\lambda}A_\lambda \end{align}\] where \[\begin{align} [L]^\rho := \frac{\delta L}{\delta A_{\rho}} \end{align}\] and \((S^{\mu\nu})_\rho^{\;\lambda}\) is now given by \[\begin{align} (S^{\mu\nu})_\rho^{\;\lambda} &\!\!\!=\!\!\!& \delta^\mu_{\rho}\eta^{\nu\lambda} - \delta^\nu_{\rho}\eta^{\mu\lambda} . \end{align}\] All of this leads us to define the Belinfante-Rosenfeld energy-momentum tensor of the vector field \(A_\mu\) as \[\begin{align} U^{\mu\nu}\,:=\, T^{\mu\nu} + [L]^\mu A^\nu .\label{4.7} \end{align}\tag{24}\] It is easy to check that (8 ) is a particular case of (24 ).

For the symmetric field \(h_{\alpha\beta}\), the infinitesimal variation of the field is given by \[\begin{align} (S^{\mu\nu})_{\alpha\beta}^{\;\;\;\gamma\delta} &\!\!\!=\!\!\!&\frac{1}{2}\Big[ \delta^\mu_\alpha \bigl(\eta^{\nu\gamma}\delta^\delta_\beta +\eta^{\nu\delta}\delta^\gamma_\beta\bigr) + \delta^\mu_\beta \bigl(\eta^{\nu\gamma}\delta^\delta_\alpha +\eta^{\nu\delta}\delta^\gamma_\alpha\bigr) \nonumber\\ &&\quad- \delta^\nu_\alpha \bigl(\eta^{\mu\gamma}\delta^\delta_\beta +\eta^{\mu\delta}\delta^\gamma_\beta\bigr) - \delta^\nu_\beta \bigl(\eta^{\mu\gamma}\delta^\delta_\alpha +\eta^{\mu\delta}\delta^\gamma_\alpha\bigr) \Big]. \label{5.6} \end{align}\tag{25}\] Substituting in (20 ), we obtain \[\begin{align} T^{\mu\nu} - T^{\nu\mu} &\!\!\!=\!\!\!&-[L]^{\alpha\beta} (S^{\mu\nu})_{\alpha\beta}^{\;\;\;\gamma\delta}h_{\gamma\delta} \nonumber\\ &\!\!\!=\!\!\!&-2[L]^{\mu\alpha}h_\alpha^{\;\nu} + 2[L]^{\nu\alpha}h_\alpha^{\;\mu} . \label{5.8} \end{align}\tag{26}\] Here, \[\begin{align} [L]^{\alpha\beta} := \frac{\partial L}{\partial h_{\alpha\beta}}-\partial_\lambda\frac{\partial L}{\partial(\partial_\lambda h_{\alpha\beta})} . \end{align}\] (26 ) suggests the definition of the Belinfante-Rosenfeld energy-momentum tensor \(U^{\mu\nu}\) for \(h_{\alpha\beta}\): \[\begin{align} U^{\mu\nu} &\!\!\!:=\!\!\! &T^{\mu\nu}+2[L]^{\mu\alpha}h_\alpha^{\;\nu}. \label{5.10} \end{align}\tag{27}\]

3.3 The Hilbert tensor↩︎

The Belinfante-Rosenfeld tensor coincides with the Hilbert tensor. We consider the Poincaré-invariant Lagrangian density \(L=L(\psi^A, \partial_\gamma\psi^A; \eta)\) where \(\eta\) stands for the Minkowski metric. We assume that there exists a Lagrangian density \(\mathfrak{L}=L(\psi^A, \nabla_\gamma\psi^A; f)\), obtained by replacing the background metric \(\eta\) with a general metric \(f\), which reduces to the original Lagrangian density when \(f \to \eta\). Here, \(\nabla_\lambda\) is the covariant derivative associated with the metric \(f\). Let \(\mathfrak{L}\) be called the covariantization of \(L\). The Hilbert energy-momentum tensor associated with \(L\) is defined by \[\begin{align} -2\frac{\delta\mathfrak{L}}{\delta f_{\mu\nu}}\Big \vert_{f \to \eta}. \end{align}\] The Belinfante-Rosenfeld tensors (24 ) and (27 ) coincide with the Hilbert tensors [4].

The preceding construction gives a definite meaning to the field contribution entering the total energy-momentum tensor. We can now ask the converse question: once this Belinfante-type tensor has been specified, does total energy-momentum conservation constrain the Lagrangian density from which it is built?

The two applications below show that the answer is field-dependent. For a vector field coupled in the standard way to a point particle, the conservation condition is satisfied identically and imposes no restriction on the vector-field Lagrangian. For a symmetric rank-two field universally coupled to the matter energy-momentum tensor, however, the same requirement leads to Feynman’s consistency condition and, within the class of Lagrangians considered here, uniquely determines the Einstein Lagrangian density up to a total divergence. This contrast is central to the argument of the paper.

4 Does conservation of the total energy-momentum tensor determine the form of the Lagrangian density of the vector field \(A_\mu\)?↩︎

Our aim is to study whether the condition d) \[\begin{align} \partial_\mu ( U^{\mu\nu}\,(L(A)) + \tau^{\mu\nu} ) = 0 \label{B4.1} \end{align}\tag{28}\] suffices to fix the form of \(L(A)\), where \(L(A)\in \mathbf{L}(A)\) and \(\mathbf{L}(A)\) is the set of all Lagrangian densities that satisfy the requirements of locality, Poincaré-invariance and quadratic in first derivatives of \(A_\mu\). We have already mentioned that the fulfillment of the conservation condition d) may not only depend on the field Lagrangian density \(L\) but on the matter and interaction Lagrangians as well. Therefore, in the problem of determining which values of \(L\) are compatible with (28 ), we have to consider a total action, for example \[\begin{align} S = -m \int {\sqrt{\eta_{\mu\nu}dz^\mu dz^\nu}} -e \int A_\mu(z)\, dz^\mu+ \int L(A)\, d^4x,\label{4.2} \end{align}\tag{29}\] and then perform the relevant calculations, involving the determination of \(U^{\mu\nu}\) and the motion and field equations, to determine whether the condition (9 ) is satisfied or not.

It is not hard to see that, using the 4-dimensional vector current \(j^\mu\), the second term in the action can be written as a space-time integral instead of a line integral, \[\begin{align} S = - m\,\int {\sqrt{\eta_{\mu\nu}dz^\mu dz^\nu}} -\int A_\mu\, j^\mu \, d^4 x+ \int L(A)\, d^4x.\label{4.3} \end{align}\tag{30}\]

We are now in a position to study condition (28 ) in detail; to do so, we will begin deriving the value of \(\partial_\mu \tau^{\mu\nu}\) for the energy-momentum tensor of a point particle of mass \(m\) and trajectory \(z^\mu\). The particle’s energy-momentum tensor \(\tau^{\mu\nu}\) is defined as \[\begin{align} \tau^{\mu\nu}(x) := \, m\int \delta^4(x-z)\frac{dz^\mu}{d\tau}\frac{dz^\nu}{d\tau}d\tau . \label{def_tau} \end{align}\tag{31}\] It is not hard to realize (see ) that the condition of mass conservation plus the equations of motion (which are independent of the field Lagrangian density \(L\) appearing in (30 )) allows us to write \[\begin{align} \partial_\mu \tau^{\mu\nu}\,=\, {F^\nu}_\mu\, j^\mu \end{align}\] and (28 ) can be rewritten as \[\begin{align} \partial_\mu U^{\mu\nu}\,+ {F^\nu}_\mu\, j^\mu= 0.\label{4.11} \end{align}\tag{32}\]

Let us now work out the appearance of the first term in (32 ). Using (24 ), we obtain \[\begin{align} \partial_\mu U^{\mu\nu} &\!\!\!=\!\!\!&\partial_\mu \Theta^{\mu\nu} + \partial_\mu [L]^\mu A^\nu + [L]^\mu \,\partial_\mu A^\nu. \end{align}\] We used (21 ). (23 ) becomes \[\begin{align} \partial_\mu \Theta^{\mu}_{\;\nu} &\!\!\!=\!\!\!&-[L]^\rho \,\partial_\nu A_\rho. \label{4.8} \end{align}\tag{33}\] Thus, we obtain \[\begin{align} \partial_\mu U^{\mu\nu} &\!\!\!=\!\!\!&-[L]^\mu \eta^{\sigma\nu}\partial_\sigma A_\mu + \partial_\mu [L]^\mu A^\nu + [L]^\mu \,\partial_\mu A^\nu \nonumber\\ &\!\!\!=\!\!\!&[L]^\mu \,{F_\mu}^\nu+ \partial_\mu [L]^\mu A^\nu .\label{4.14} \end{align}\tag{34}\] Substituting (34 ) into (32 ), we obtain \[\begin{align} [L]^\mu \,{F_\mu}^\nu+ \partial_\mu [L]^\mu A^\nu + {F^\nu}_\mu\, j^\mu = 0. \label{4.15} \end{align}\tag{35}\] But, regardless of the value of \(L\) in (30 ), the field equations determine that \[\begin{align} [L]^\mu = j^\mu . \end{align}\] The first and last terms of (35 ) cancel out automatically, while the second term vanishes as a consequence of charge conservation. Therefore, condition (28 ) is satisfied identically for any Lagrangian density \(L(A)\) and therefore provides no information about its form. Hence, the answer to the question posed in the title of this section is “no, the total energy-momentum tensor conservation does not determine the form of the Lagrangian density of \(A_\mu\)", or, schematically,

\[\begin{align} \boxed{ \partial_\mu ( U^{\mu\nu}(L(A)) + \tau^{\mu\nu} ) = 0 \qquad \forall L(A)\in {\mathbf{L}}(A). }\label{ALP} \end{align}\tag{36}\]

5 Does conservation of the total energy-momentum tensor determine the form of the Lagrangian density for the symmetric field \(h_{\alpha\beta}\)?↩︎

In the following analysis, we restrict attention to local Poincaré-invariant Lagrangian densities that are analytic in \(h_{\mu\nu}\), are quadratic in first derivatives of \(h_{\mu\nu}\), contain no non-derivative potential terms, and admit an expansion; in a manner analogous to how we proceeded in section 4, we will call the set of Lagrangian densities that satisfy these requirements \(\mathbf{L} (h)\).

With the Belinfante-Rosenfeld energy-momentum tensor \(U^{\mu\nu}\) in hand, together with the results derived at the end of the previous section, we now address the central question of this paper: does imposing condition d) determine the form of the Lagrangian density \(L\) that generates \(U^{\mu\nu}[h_{\alpha\beta}]\)? We assume that \(h_{\alpha\beta}\) is a symmetric field that is taken to describe the gravitational interaction. To address this question, we begin by determining the general form of the action to be considered. We consider the total action \[\begin{align} S &\!\!\!=\!\!\!&\int L_{\rm{matter}}\, d^4x + \int L(h)\, d^4x. \label{5.2} \end{align}\tag{37}\] Here, \(L_{\rm{matter}}\) is the matter Lagrangian density including the interaction with \(h_{\mu\nu}\) and \(L(h) \in \mathbf{L} (h)\).

The field equation of (37 ) is given by \[\begin{align} -\lambda\tau^{\mu\nu}+\frac{\delta L}{\delta h_{\mu\nu}}=0\label{5.3} \end{align}\tag{38}\] where \[\begin{align} \tau^{\mu\nu} := -\frac{1}{\lambda}\frac{\delta L_{\rm{matter}}}{\delta h_{\mu\nu}}. \end{align}\] We study whether the conservation condition \[\begin{align} \partial_\mu ( U^{\mu\nu}\,(L(h)) + \tau^{\mu\nu} ) = 0, \label{5.4} \end{align}\tag{39}\] where \(L(h)\in \mathbf{L}(h)\), imposes any restriction on the form of the Lagrangian density \(L(h)\) itself. Taking into account (38 ), we can write our restriction (39 ) in the following manner \[\begin{align} \partial_\mu \Bigl( U^{\mu\nu} (L(h))\Bigr )= \frac{-1}{\lambda} \partial_\mu \Biggl(\frac{\delta L(h)}{\delta h_{\mu\nu}}\Biggr) .\label{5.5} \end{align}\tag{40}\] We now address the central question: does the conservation condition determine the form of the Lagrangian density \(L(h)\) in (40 )? Before answering this question, let us see what the Belinfante-Rosenfeld energy-momentum tensor looks like for our case.

Using (27 ), we have \[\begin{align} \eta_{\beta\nu}\partial_\mu U^{\mu\nu} &\!\!\!=\!\!\!&\partial_\mu \Theta^{\mu}_{\;\beta}+2\partial_\mu [L]^{\mu\alpha}h_{\alpha\beta}+2[L]^{\mu\alpha}\partial_\mu h_{\alpha\beta}. \end{align}\] We used (21 ). Next, we write (23 ) as \[\begin{align} \partial_\mu \Theta^{\mu}_{\;\beta} &\!\!\!=\!\!\!&- [L]^{\alpha\mu}\partial_\beta h_{\alpha\mu}. \label{5.13} \end{align}\tag{41}\] Then, we have \[\begin{align} \eta_{\beta\nu}\partial_\mu U^{\mu\nu} &\!\!\!=\!\!\!&2 h_{\beta\nu}\partial_\mu [L]^{\mu\nu} +\frac{1}{\lambda}\bigl[\mu\nu ,\beta \bigr][L]^{\mu\nu} \label{5.14} \end{align}\tag{42}\] where \[\begin{align} [\mu\rho ,\nu ]:= \lambda\,\,\bigl[\partial_\rho h_{\mu\nu} +\partial_\mu h_{\rho\nu}-\partial_\nu h_{\mu\rho}\bigr].\label{5.15} \end{align}\tag{43}\]

(42 ) gives the general expression for the four-divergence of the Belinfante-Rosenfeld energy-momentum tensor; we proceed to make use of it in the conservation condition (40 ) written as \[\begin{align} \partial_\mu U^{\mu\nu} &\!\!\!=\!\!\!&{\frac{-1}{\lambda}}\,\partial_\mu [L]^{\mu\nu}.\label{5.17} \end{align}\tag{44}\] Substituting (44 ) into (42 ), we obtain \[\begin{align} g_{\beta\nu}\, \partial_\mu [L]^{\mu\nu} +\bigl[\mu\nu ,\beta \bigr][L]^{\mu\nu}&\!\!\!=\!\!\!&0\label{5.21} \end{align}\tag{45}\] where \[\begin{align} g_{{\beta\nu}}\,:=\,\,\eta_{\beta\nu} +2\lambda h_{\beta\nu} .\label{5.20} \end{align}\tag{46}\] (45 ) is the consistency condition (1 ) obtained by Feynman in Ref.. At this point, we have recovered Feynman’s consistency condition from the conservation of the total energy-momentum tensor. It remains to determine which field Lagrangian \(L(h)\in \mathbf{L} (h)\) satisfies this condition. Note that Feynman’s approach arrives at the same condition (45 ) by requiring consistency between the equations of motion and the field equations, see Appendix 7.

It should be noted that the fact that the consistency condition (45 ) can be recovered from different assumptions is not particularly surprising once one realizes that this condition is closely related to the contracted second Bianchi identity, since (45 ) may be rewritten as \[\begin{align} \nabla_\mu \left( \frac{[L]^{\mu\nu}}{{\sqrt{-g}}} \right) =\,0 \label{Bianchi} \end{align}\tag{47}\] where \(\nabla_\mu\) is the covariant derivative naturally associated to (46 ) and \(g:= \det(g_{\mu\nu})\). The equivalence of conditions (39 ) and (47 ) highlights the deep connection between field theory (through the conservation of the Belinfante-Rosenfeld tensor) and the geometric framework underlying the covariant condition (47 ).

The Einstein Lagrangian density \[\begin{align} L_G := -\gamma\sqrt{-g}\mathcal{G} \label{defF} \end{align}\tag{48}\] identically satisfies (45 ). Here, \[\begin{align} \mathcal{G} := g^{\mu\nu}\Big[ \Gamma^\rho_{\;\gamma\nu}\Gamma^\gamma_{\;\mu\rho}-\Gamma^\rho_{\;\gamma\rho}\Gamma^\gamma_{\;\mu\nu} \Big], \label{defG} \end{align}\tag{49}\] where \(\Gamma^\rho_{\;\mu\nu}:= g^{\rho\sigma}[\mu\nu, \sigma]\) and \(g^{\mu\nu}\) is the inverse matrix of \(g_{\mu\nu}\). Note that \[\begin{align} \sqrt{-g}R &\!\!\!=\!\!\!&\sqrt{-g}\mathcal{G}+\partial_\mu [\sqrt{-g}(g^{\rho\nu}\Gamma^\mu_{\;\rho\nu}-g^{\rho\mu}\Gamma^\nu_{\;\rho\nu})]\label{HE} \end{align}\tag{50}\] where \(R\) is the scalar curvature. We can show that the Einstein Lagrangian density is the unique solution in \(\mathbf{L} (h)\) of (45 ) (Appendix 8). Namely, \(L \stackrel{\mathrm{w}}{=}L_G\). Here, \(A \stackrel{\mathrm{w}}{=}B\) means that there exists \(C^\mu\) such that \(A=B+\partial_\mu C^\mu + constant\).

In summary, within the class of local Poincaré-invariant Lagrangian densities \(\mathbf{L}(h)\) specified above (analytic in \(h_{\mu\nu}\), quadratic in first derivatives, containing no non-derivative potential terms, and admitting an expansion), we have demonstrated that the following equivalence holds \[\begin{align} \boxed{ \partial_\mu ( U^{\mu\nu}(L(h)) + \tau^{\mu\nu} ) = 0 \iff L(h)\stackrel{\mathrm{w}}{=}-\gamma\sqrt{-g} \mathcal{G} . \label{5.34} } \end{align}\tag{51}\] Here, \(U^{\mu\nu}\) is the Belinfante-Rosenfeld energy-momentum tensor of \(L(h)\) and its expression is given by (27 ), where the Belinfante energy-momentum tensor \(T^{\mu\nu}\) is obtained via (14 ). For the Einstein Lagrangian density (48 ), the Belinfante-Rosenfeld energy-momentum tensor satisfies the relation (Appendix 9) \[\begin{align} U^{\mu\nu}/\gamma&\!\!\!=\!\!\!&-2\sqrt{-g}G^{\mu\nu}+P^{\mu\nu}, \\ P^{\mu\nu} &\!\!\!:=\!\!\! &\partial_\alpha\partial_\beta N^{\mu\nu\alpha\beta} ,\\ N^{\mu\nu\alpha\beta} &\!\!\!:=\!\!\! &\sqrt{-g}(g^{\mu\nu}\eta^{\alpha\beta}-g^{\mu\alpha}\eta^{\nu\beta}+g^{\alpha\beta}\eta^{\mu\nu}-g^{\nu\beta}\eta^{\mu\alpha}), \label{def_N} \end{align}\tag{52}\] where \(G^{\mu\nu}\) is the Einstein tensor. Thus, \(U^{\mu\nu}\) coincides with Papapetrou’s gravitational energy-momentum pseudotensor [5], [6] while \(N^{\mu\nu\alpha\beta}\) is its associated superpotential.

6 Conclusions↩︎

We investigated whether the conservation of total energy and momentum constrains, or even determines, the Lagrangian density of the interaction field. This question was formulated only after specifying the Belinfante-Rosenfeld tensor associated with a Poincaré-invariant field Lagrangian, which represents the explicit field contribution to the total energy and momentum tensor. In the case of a vector field, we have shown that the conservation principle generally does not determine the field Lagrangian. For a symmetric spin-2 field \(h_{\mu\nu}\), the conditions for total energy and momentum conservation become Feynman’s consistency conditions for the field Lagrangian. In contrast to the original derivation of Feynman’s conditions, this approach does not derive the conditions by combining specific matter equations of motion with the field equations. Furthermore, it does not depend on any given point-particle matter action. The matter energy-momentum tensor is introduced via a variational definition in terms of \(h_{\mu\nu}\), and the consistency condition follows from the conservation of a clear Belinfante-Rosenfeld type energy-momentum tensor and the field equations. Among the class of local Lorentz-invariant Lagrangian densities considered here (namely, those corresponding to symmetric fields that are analytic, quadratic in first-order derivatives, and contain no non-derivative potential terms), this condition uniquely determines the Einstein Lagrangian density, up to a total divergence.

For the Einstein Lagrangian, the corresponding Belinfante-Rosenfeld tensor coincides with Papapetrou’s gravitational energy-momentum pseudotensor. This relationship is obtained only after the Lagrangian has been fixed by the conservation condition. Therefore, rather than taking the pseudotensor as the starting point of the discussion, this approach bridges the gap between the conservation law approach and the traditional pseudotensor description of gravitational energy and momentum.

Several issues remain unresolved. It would be useful to compare the Belinfante-Rosenfeld tensor obtained here with other gravitational energy-momentum complexes, such as the Landau-Lifshitz pseudotensor, to determine whether this conservation law formulation allows for a fully geometric reformulation. As an alternative approach, one could investigate more systematically under what kinds of matter couplings and field representations energy-momentum conservation acts as a genuine constraint on the Lagrangian density.

Acknowledgment↩︎

This work received no funding.

7 Derivation of the condition (1 ) following Feynman’s approach↩︎

We provide a brief derivation of how to arrive at condition (1 ) following Ref.. We start from the total action of the system \[\begin{align} S(z,h)=\, -\frac{m}{2}\int \eta_{\mu\nu} \frac{dz^\mu}{d\tau}\frac{dz^\nu}{d\tau}d\tau -\lambda\int h_{\mu\nu}(x)\, \tau^{\mu\nu}(x) \;d^4x +\int L\, d^4x . \label{Z5.2} \end{align}\tag{53}\] Here, \(\tau^{\mu\nu}\) is given by (31 ). Its motion equation is \[\begin{align} \bigl(\eta_{\mu\nu}+2\lambda \,h_{\mu\nu}\bigr)\frac{d^2z^\mu}{d\tau^2} +2\lambda \partial_\rho h_{\mu\nu}\frac{dz^\rho}{d\tau}\frac{dz^\mu}{d\tau} -\lambda \partial_\nu h_{\mu\rho}\frac{dz^\mu}{d\tau}\frac{dz^\rho}{d\tau}=0\label{Z2}, \end{align}\tag{54}\] while the field equation gives \[\begin{align} -\lambda \tau^{\mu\nu}+ \frac{\delta L}{\delta h_{\mu\nu}}=0\label{Z4}. \end{align}\tag{55}\] It is not hard to check that (54 ) leads to \[g_{\mu\lambda}\partial_\rho \tau^{\rho\mu}=-[\mu\rho ,\lambda ]\tau^{\mu\rho}\;\label{Z3},\tag{56}\] where, once again, the following notation has been used \[\begin{align} &g_{\mu\nu}:= \eta_{\mu\nu}+2\lambda \,h_{\mu\nu}\,, \nonumber \\ &[\mu\rho ,\nu ]:= \lambda\,\,\bigl[\partial_\rho h_{\mu\nu} +\partial_\mu h_{\rho\nu}-\partial_\nu h_{\mu\rho}\bigr]\;. \nonumber \end{align}\] Using (56 ) and the field equation (55 ), we infer the consistency condition (1 ): \[\begin{align} g_{\mu\lambda}\,\partial_{\rho}\Biggl(\frac{\delta L}{\delta h_{\rho\mu}}\Biggr) =-\bigl[\mu\rho ,\lambda \bigr] \frac{\delta L}{\delta h_{\rho\mu}} \nonumber . \end{align}\]

8 Proof of uniqueness of Einstein’s Lagrangian density↩︎

8.1 Determining \(L(h)\)↩︎

We determine the form of \(L(h)\) that satisfies equation (45 ). We expand \(L(h)\) as [7] \[L(h)=\sum_{j=2}^\infty L^{(j)}(h)\label{defLexp}\tag{57}\] where \(L^{(j)}\) is quadratic in first derivatives and of order \(j\) in the \(h_{\alpha\beta}\), schematically \[L^{(j)}(h)=\sum_{n=1}^{q_j} a^{(j)}_n\,\Bigl( h^{j-2}\partial h\partial h\Bigr)_n \label{defLexp2}.\tag{58}\] For the lower orders, we have \(q_2 = 5\), \(q_3 = 16\), \(q_4 =43\) and so on [7]. Determining the coefficients \(a^{(j)}_n\) in (58 ) for all \(j\) and \(n\) is equivalent to solving (45 ). (45 ) leads to \[\begin{align} \partial_\mu [L^{(2)}]^{\mu\nu} =0 \end{align}\] where \[\begin{align} [L^{(j)}]^{\mu\nu} := \frac{\delta L^{(j)}}{\delta h_{\mu\nu}}. \end{align}\] We can determine \(L^{(2)}\) by assuming its most general form as a linear combination of \[\begin{align} &&\Bigl(\partial h\partial h\Bigr)_1 = \partial_\alpha h_{\mu\nu} \partial^\alpha h^{\mu\nu}, \; \Bigl(\partial h\partial h\Bigr)_2 =\partial_\alpha h_\mu^{\; \nu}\partial_\nu h^{\mu\alpha} ,\; \Bigl(\partial h\partial h\Bigr)_3=(\partial h)^{\mu}\partial_\mu h ,\nonumber\\ &&\Bigl(\partial h\partial h\Bigr)_4=\partial^\mu h \partial_\mu h ,\; \Bigl(\partial h\partial h\Bigr)_5=(\partial h)^{\mu}(\partial h)_\mu \end{align}\] where \(h:= h^\mu_{\;\mu}\) and \((\partial h)^{\nu} := \partial_\mu h^{\mu\nu}\). Here, \((\partial h)^{\mu}(\partial h)_\mu \stackrel{\mathrm{w}}{=}\partial_\alpha h_\mu^{\; \nu}\partial_\nu h^{\mu\alpha}\). We obtain a linear system of equations \(\{a^{(2)}_n\}_{n=1}^5\) as \[\begin{align} a^{(2)}_1=\gamma\lambda^2, \;a^{(2)}_2+a^{(2)}_5=-2\gamma\lambda^2, \;a^{(2)}_3=2\gamma\lambda^2, \;a^{(2)}_4=-\gamma\lambda^2. \label{intro_a_i} \end{align}\tag{59}\] If \(a^{(2)}_5=0\), \(L^{(2)}\) becomes the Fierz-Pauli Lagrangian density \(L_{FP}\). (45 ) becomes \[\begin{align} \eta_{\beta\nu}\partial_\mu [L^{(j)}]^{\mu\nu} = -2\lambda h_{\beta\nu} \partial_\mu [L^{(j-1)}]^{\mu\nu} -[\mu\nu,\beta][L^{(j-1)}]^{\mu\nu} \;\;(j\ge 3). \end{align}\] Thus, once the value of \(L^{(2)}\) is obtained, we can continue iteratively: using these values, it is possible to go to higher orders to determine \(L^{(3)}\), then \(L^{(4)}\) and so on [7].

We expand \(L_G\) as \(L_G^{(2)}+L_G^{(3)}+\cdots\) where \(L_G^{(j)}\) is of \(j\)-th order in \(h_{\mu\nu}\). \(L_G^{(2)} = L_{FP}\) holds. One solution for \(L^{(j)}\) is \(L_G^{(j)}\). The general solution of \(L^{(j)}\) is given by \[\begin{align} L^{(j)} = L_G^{(j)}+\Delta^{(j)} \;\;(j\ge 3) \end{align}\] where \(\Delta^{(j)}\) is the solution of \[\begin{align} \partial_\mu [\Delta^{(j)}]^{\mu\nu}=0. \end{align}\] In §8.2, we show \[\begin{align} [\Delta^{(j)}]^{\mu\nu} = 0 \;\;(j\ge 3). \label{Dl=0} \end{align}\tag{60}\] Because \(\delta\Delta/\delta\psi^A=0\) leads to \(\Delta\stackrel{\mathrm{w}}{=}0\) (see §8.3), we obtain \[\begin{align} L^{(j)} \stackrel{\mathrm{w}}{=}L_G^{(j)} \;\;(j\ge 2). \end{align}\] Thus, \(L_G\) is the unique solution for (45 ).

8.2 Proof of (60 )↩︎

We introduce \[\begin{align} \chi_{(j)}^{\mu\nu}:=\chi_{(j)}^{\mu\nu}[h]:=[\Delta^{(j)}]^{\mu\nu}. \end{align}\] For an arbitrary variation, we have \[\begin{align} \delta\Delta^{(j)}=\chi_{(j)}^{\mu\nu}\delta h_{\mu\nu}+\partial_\lambda\Big( \frac{\partial\Delta^{(j)}}{\partial(\partial_\lambda h_{\mu\nu})} \delta h_{\mu\nu} \Big). \end{align}\] Substituting \(\delta h_{\mu\nu} =\delta_\xi h_{\mu\nu}:=\partial_\mu \xi_\nu + \partial_\nu \xi_\mu\), we have \[\begin{align} \delta_\xi \Delta^{(j)}&\!\!\!=\!\!\!&2\chi_{(j)}^{\mu\nu}\partial_\mu \xi_\nu+\partial_\lambda\Big( 2\frac{\partial\Delta^{(j)}}{\partial(\partial_\lambda h_{\mu\nu})} \partial_\mu \xi_\nu \Big) \nonumber\\ &\!\!\!=\!\!\!&-2\partial_\mu \chi_{(j)}^{\mu\nu}\xi_\nu+\partial_\lambda\Big(2\chi_{(j)}^{\lambda\nu}\xi_\nu + 2\frac{\partial\Delta^{(j)}}{\partial(\partial_\lambda h_{\mu\nu})} \partial_\mu \xi_\nu \Big). \end{align}\] Using \(\partial_\mu \chi_{(j)}^{\mu\nu}=0\), we have \[\begin{align} \delta_\xi \Delta^{(j)}&\!\!\!=\!\!\!&\partial_\lambda\mathcal{B}^\lambda,\; \mathcal{B}^\lambda:= 2\chi_{(j)}^{\lambda\nu}\xi_\nu + 2\frac{\partial\Delta^{(j)}}{\partial(\partial_\lambda h_{\mu\nu})} \partial_\mu \xi_\nu. \end{align}\] Here, \(\partial_\mu \chi_{(j)}^{\mu\nu}=0\) is understood as an off-shell identity, valid for arbitrary \(h_{\mu\nu}\). Since \(\delta_\xi h_{\mu\nu}\) is a transformation independent of the field \(h_{\mu\nu}\), the variation \(\delta_\xi\) commutes with the Euler-Lagrange derivative. Therefore, \(\chi_{(j)}^{\mu\nu}\) is gauge invariant: \[\begin{align} \delta_\xi \chi_{(j)}^{\mu\nu} = \delta_\xi \frac{\delta\Delta^{(j)}}{\delta h_{\mu\nu}}= \frac{\delta\delta_\xi \Delta^{(j)}}{\delta h_{\mu\nu}} =0. \label{key_dl_chi} \end{align}\tag{61}\] By repeating small transformations, a finite transformation can be achieved. Therefore, for any finite \(\xi_\lambda\), \[\begin{align} \chi_{(j)}^{\mu\nu}[h']= \chi_{(j)}^{\mu\nu}[h] \label{chi_inv} \end{align}\tag{62}\] holds ³ for \(h'_{\mu\nu}= h_{\mu\nu}+\partial_\mu \xi_\nu + \partial_\nu \xi_\mu\). Fix an arbitrary point \(x_0\). In a neighborhood of \(x_0\), we choose \[\begin{align} \xi_\lambda&\!\!\!=\!\!\!&-\frac{1}{2}h_{\lambda\rho}(x_0)(x-x_0)^\rho-\frac{1}{2}\Gamma_{\lambda\mu\nu}(x_0)(x-x_0)^\mu(x-x_0)^\nu \nonumber\\ &&\quad+O((x-x_0)^3) \end{align}\] where \[\begin{align} \Gamma_{\lambda\mu\nu} := \frac{1}{2}(\partial_\mu h_{\lambda\nu}+\partial_\nu h_{\lambda\mu}-\partial_\lambda h_{\mu\nu}). \end{align}\] Then, we have \[\begin{align} \partial_\rho \xi_\lambda(x_0)=-\frac{1}{2}h_{\lambda\rho}(x_0) , \quad\partial_\mu\partial_\nu \xi_\lambda(x_0)=-\Gamma_{\lambda\mu\nu}(x_0). \end{align}\] This leads to \[\begin{align} h'_{\mu\nu}(x_0)=0 , \quad\partial_\lambda h'_{\mu\nu}(x_0)=0 \label{h=0=ph} \end{align}\tag{63}\] because of \(\delta_\xi \Gamma_{\lambda\mu\nu}=\partial_\mu \partial_\nu \xi_\lambda\), \(\Gamma'_{\lambda\mu\nu}(x_0)=0\), and \(\partial_\lambda h_{\mu\nu}=\Gamma_{\mu\lambda\nu}+\Gamma_{\nu\lambda\mu}\). Since \[\begin{align} \Delta^{(j)} \sim h^{j-2}(\partial h)(\partial h), \end{align}\] its Euler-Lagrange derivative has the schematic form \[\begin{align} \chi_{(j)}^{\mu\nu} \sim h^{j-2}\partial\partial h +h^{j-3}(\partial h)(\partial h) \;\;(j\ge 3). \end{align}\] Using (63 ) and (62 ), we have \[\begin{align} \chi_{(j)}^{\mu\nu}[h](x_0) = \chi_{(j)}^{\mu\nu}[h'](x_0)=0 \;\;(j\ge 3). \end{align}\] Because \(x_0\) was arbitrary, it follows that \[\begin{align} \chi_{(j)}^{\mu\nu}=0 \;\;(j\ge 3). \end{align}\]

8.3 Proof that \(\delta\Delta/\delta\psi^A=0 \Longrightarrow \Delta\stackrel{\mathrm{w}}{=}0\)↩︎

For a Lagrangian density \(L=L(\psi^A, \partial_\mu \psi^A)\), we assume \[\begin{align} L_A(s) := \Big(\frac{\partial L}{\partial\psi^A}\Big)_s-\partial_\mu\Big( \frac{\partial L}{\partial(\partial_\mu \psi^A)}\Big)_s =0 \label{A} \end{align}\tag{64}\] for \(0\le s \le 1\). Here, \[\begin{align} (X)_s := X \vert_{\psi^A \to s\psi^A, \partial_\mu \psi^A \to s \partial_\mu \psi^A}. \end{align}\] Differentiating \(L_s := L(s\psi^A, s\partial_\mu \psi^A)\) with respect to \(s\), we obtain \[\begin{align} \frac{dL_s}{ds} &\!\!\!=\!\!\!&\psi^A \Big(\frac{\partial L}{\partial\psi^A}\Big)_s + \partial_\mu \psi^A \Big(\frac{\partial L}{\partial(\partial_\mu \psi^A)}\Big)_s \nonumber\\ &\!\!\!=\!\!\!&\psi^A L_A(s) + \partial_\mu \Big[ \psi^A \Big(\frac{\partial L}{\partial(\partial_\mu \psi^A)}\Big)_s \Big]. \end{align}\] Using (64 ), we obtain \[\begin{align} L &\!\!\!=\!\!\!&L(0, 0)+\partial_\mu K^\mu ,\label{L we 0}\\ K^\mu &\!\!\!:=\!\!\! &\psi^A \int_0^1 ds\; \Big(\frac{\partial L}{\partial(\partial_\mu \psi^A)}\Big)_s . \end{align}\tag{65}\] Because \(L(0, 0)\) is a constant, (@{eq:L we 0} ) means \(L \stackrel{\mathrm{w}}{=}0\). The proof of this subsection is based on Ref.[8].

9 The Belinfante-Rosenfeld tensor for the Einstein Lagrangian density↩︎

The covariantization of the Einstein Lagrangian density \(\sqrt{-g}\mathcal{G}\) is given by \(G\), where \[\begin{align} G &\!\!\!:=\!\!\! &\sqrt{-g}\mathcal{G}^\ast , \quad \mathcal{G}^\ast :=g^{\mu\nu}\Big[ B^\rho_{\;\gamma\nu}B^\gamma_{\;\mu\rho}-B^\rho_{\;\gamma\rho}B^\gamma_{\;\mu\nu} \Big] , \\ B^\rho_{\;\mu\nu}&\!\!\!:=\!\!\! &g^{\rho\lambda}B_{\lambda\mu\nu} ,\\ B_{\lambda\mu\nu} &\!\!\!:=\!\!\! &\frac{1}{2}\big[-\nabla_\lambda g_{\mu\nu}+\nabla_\mu g_{\lambda\nu}+ \nabla_\nu g_{\lambda\mu} \big] ,\\ \nabla_\lambda g_{\mu\nu} &\!\!\!=\!\!\!&\partial_\lambda g_{\mu\nu}-C^{\alpha}_{\;\mu\lambda}g_{\alpha\nu}-C^{\alpha}_{\;\nu\lambda}g_{\mu\alpha} ,\\ C^\rho_{\;\mu\nu}&\!\!\!:=\!\!\! &f^{\rho\lambda}C_{\lambda\mu\nu} ,\\ C_{\lambda\mu\nu} &\!\!\!:=\!\!\! &\frac{1}{2}\big[-\partial_\lambda f_{\mu\nu}+\partial_\mu f_{\lambda\nu}+ \partial_\nu f_{\lambda\mu} \big] . \end{align}\] Here, \(g^{\mu\nu}\) is the inverse matrix of \(g_{\mu\nu}=f_{\mu\nu}+2\lambda h_{\mu\nu}\) and \(f^{\mu\nu}\) is the inverse matrix of \(f_{\mu\nu}\). \(\nabla_\lambda\) is the covariant derivative associated with the metric \(f\). Note that \(B^\rho_{\;\mu\nu}=\Gamma^\rho_{\;\mu\nu}-C^\rho_{\;\mu\nu}\) where \[\begin{align} \Gamma^\rho_{\;\mu\nu}&\!\!\!=\!\!\!&g^{\rho\lambda}\frac{1}{2}\big[-\partial_\lambda g_{\mu\nu}+\partial_\mu g_{\lambda\nu}+ \partial_\nu g_{\lambda\mu} \big]. \end{align}\] We define \(\Gamma_\gamma:=\Gamma^{\alpha}_{\;\gamma\alpha}\) and \(B_\gamma:=B^{\alpha}_{\;\gamma\alpha}\). We set \(\lambda=1/2\) only for notational simplicity in this appendix; this amounts to a rescaling of \(h_{\mu\nu}\), and the final relation for \(H^{\mu\nu}\) is expressed in terms of \(g_{\mu\nu}\).

The variation of \(G\) is given by \[\begin{align} \delta G &\!\!\!=\!\!\!&\frac{1}{2}g^{\mu\nu}G(\delta f_{\mu\nu}+\delta h_{\mu\nu}) \nonumber\\ &&\quad- \sqrt{-g}(g^{\mu\alpha}g^{\beta\nu}g^{\rho\kappa}g^{\gamma\delta}+g^{\mu\nu}g^{\rho\alpha}g^{\beta\kappa}g^{\gamma\delta} +g^{\mu\nu}g^{\rho\kappa}g^{\gamma\alpha}g^{\beta\delta}) \nonumber\\ && \times (B_{\kappa\gamma\nu}B_{\delta\mu\rho}-B_{\kappa\gamma\rho}B_{\delta\mu\nu}) (\delta f_{\alpha\beta}+\delta h_{\alpha\beta}) \nonumber\\ &&\quad-\sqrt{-g} (g^{\alpha\nu}g^{\lambda\beta}B^\mu_{\;\alpha\beta} +g^{\alpha\lambda} g^{\mu\beta} B^\nu_{\;\alpha\beta} -g^{\alpha\beta} g^{\lambda\nu}B^\mu_{\;\alpha\beta} -g^{\gamma\lambda} g^{\mu\nu}B_\gamma) h_{\lambda\tau} \nonumber\\ && \times \Big(- C^\epsilon_{\;\mu\nu} f^{\tau\sigma} \delta f_{\sigma\epsilon} +f^{\tau\omega} \frac{1}{2}\big[-\delta\partial_\omega f_{\mu\nu}+\delta\partial_\mu f_{\omega\nu}+\delta\partial_\nu f_{\omega\mu} \big]\Big) \nonumber\\ &&\quad+\sqrt{-g} (g^{\alpha\nu}g^{\lambda\beta}B^\mu_{\;\alpha\beta} +g^{\alpha\lambda} g^{\mu\beta} B^\nu_{\;\alpha\beta} -g^{\alpha\beta} g^{\lambda\nu}B^\mu_{\;\alpha\beta} -g^{\gamma\lambda} g^{\mu\nu}B_\gamma) \nonumber\\ && \times \Big(\frac{1}{2}\big[-\delta\partial_\lambda h_{\mu\nu}+\delta\partial_\mu h_{\lambda\nu}+\delta\partial_\nu h_{\lambda\mu}\big] -C^\tau_{\;\nu\mu}\delta h_{\lambda\tau} \Big). \end{align}\] Define \(K^{\mu\nu}\), \(J^{\omega, \mu\nu}\), \(M^{\mu\nu}\), and \(L^{\lambda, \mu\nu}\) by \[\begin{align} \frac{\partial G}{\partial f_{\mu\nu}} = \sqrt{-g}K^{\mu\nu} , \quad \frac{\partial G}{\partial(\partial_\omega f_{\mu\nu})} = \sqrt{-g}J^{\omega, \mu\nu} ,\\ \frac{\partial G}{\partial h_{\mu\nu}} = \sqrt{-g}M^{\mu\nu}, \quad \frac{\partial G}{\partial(\partial_\lambda h_{\mu\nu})} =\sqrt{-g}L^{\lambda, \mu\nu} . \end{align}\] It follows that \[\begin{align} K^{\mu\nu} &\!\!\!=\!\!\!&\frac{1}{2}\mathcal{G}^\ast g^{\mu\nu} -g^{\alpha\mu}g^{\nu\beta}B^\rho_{\;\gamma\beta}B^\gamma_{\;\alpha\rho} + g^{\alpha\mu}g^{\nu\beta}B_\gamma B^\gamma_{\;\alpha\beta} -g^{\alpha\beta}g^{\rho\mu}B^\nu_{\;\gamma\beta}B^\gamma_{\;\alpha\rho} \nonumber\\ &&\quad-g^{\alpha\beta}g^{\rho\nu}B^\mu_{\;\gamma\beta}B^\gamma_{\;\alpha\rho} +\frac{1}{2}g^{\alpha\beta}g^{\rho\mu}B^\nu_{\;\gamma\rho}B^\gamma_{\;\alpha\beta} +\frac{1}{2}g^{\alpha\beta}g^{\rho\nu}B^\mu_{\;\gamma\rho}B^\gamma_{\;\alpha\beta} \nonumber\\ &&\quad+\frac{1}{2}g^{\alpha\beta}g^{\gamma\mu}B_\gamma B^\nu_{\;\alpha\beta} +\frac{1}{2}g^{\alpha\beta}g^{\gamma\nu}B_\gamma B^\mu_{\;\alpha\beta} +l^{(\mu\nu)} , \\ 2J^{\omega, \mu\nu} &\!\!\!=\!\!\!&g^{\alpha\nu}g^{\lambda\beta}B^\mu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\omega} +g^{\alpha\lambda} g^{\mu\beta} B^\nu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\omega} -\frac{1}{2}g^{\alpha\beta} g^{\lambda\nu}B^\mu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\omega} \nonumber\\ &&\quad-\frac{1}{2}g^{\alpha\beta} g^{\lambda\mu}B^\nu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\omega} -g^{\gamma\lambda} g^{\mu\nu}B_\gamma h_{\lambda\tau}f^{\tau\omega} -g^{\alpha\nu}g^{\lambda\beta}B^\omega_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\mu} \nonumber\\ &&\quad-g^{\alpha\lambda} g^{\omega\beta} B^\nu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\mu} +\frac{1}{2}g^{\alpha\beta} g^{\lambda\nu}B^\omega_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\mu} +g^{\gamma\lambda} g^{\omega\nu}B_\gamma h_{\lambda\tau}f^{\tau\mu} \nonumber\\ &&\quad+\frac{1}{2}g^{\alpha\beta} g^{\lambda\omega}B^\nu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\mu} -g^{\alpha\mu}g^{\lambda\beta}B^\omega_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\nu} -g^{\alpha\lambda} g^{\omega\beta} B^\mu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\nu} \nonumber\\ &&\quad+\frac{1}{2}g^{\alpha\beta} g^{\lambda\mu}B^\omega_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\nu} +g^{\gamma\lambda} g^{\omega\mu}B_\gamma h_{\lambda\tau}f^{\tau\nu} +\frac{1}{2}g^{\alpha\beta} g^{\lambda\omega}B^\mu_{\;\alpha\beta}h_{\lambda\tau}f^{\tau\nu}, \end{align}\] \[\begin{align} M^{\mu\nu} &\!\!\!=\!\!\!&\frac{1}{2}\mathcal{G}^\ast g^{\mu\nu} - g^{\alpha\mu}g^{\nu\beta}B^\rho_{\;\gamma\beta}B^\gamma_{\;\alpha\rho} + g^{\alpha\mu}g^{\nu\beta}B_\gamma B^\gamma_{\;\alpha\beta} - g^{\alpha\beta}g^{\rho\mu}\Gamma^\nu_{\;\gamma\beta}B^\gamma_{\;\alpha\rho} \nonumber\\ &&\quad-g^{\alpha\beta}g^{\rho\nu}\Gamma^\mu_{\;\gamma\beta}B^\gamma_{\;\alpha\rho} +\frac{1}{2}g^{\alpha\beta}g^{\rho\mu}\Gamma^\nu_{\;\gamma\rho}B^\gamma_{\;\alpha\beta} +\frac{1}{2}g^{\alpha\beta}g^{\rho\nu}\Gamma^\mu_{\;\gamma\rho}B^\gamma_{\;\alpha\beta} \nonumber\\ &&\quad+\frac{1}{2}g^{\alpha\beta}g^{\gamma\mu}B_\gamma\Gamma^\nu_{\;\alpha\beta} +\frac{1}{2}g^{\alpha\beta}g^{\gamma\nu}B_\gamma\Gamma^\mu_{\;\alpha\beta} ,\\ L^{\lambda,\mu\nu} &\!\!\!=\!\!\!&\frac{1}{2}g^{\gamma\lambda} g^{\mu\nu}B_\gamma+g^{\alpha\nu}g^{\mu\beta}B^\lambda_{\;\alpha\beta} \nonumber\\ &&\quad-\frac{1}{2}g^{\alpha\beta} g^{\mu\nu}B^\lambda_{\;\alpha\beta}-\frac{1}{2}g^{\gamma\mu} g^{\lambda\nu}B_\gamma -\frac{1}{2}g^{\gamma\nu} g^{\lambda\mu}B_\gamma. \end{align}\] Here, \(l^{(\mu\nu)}:=(l^{\mu\nu}+l^{\nu\mu})/2\) and \[\begin{align} l^{\sigma\epsilon} &\!\!\!:=\!\!\! &(g^{\alpha\nu}g^{\lambda\beta}B^\mu_{\;\alpha\beta} +g^{\alpha\lambda} g^{\mu\beta} B^\nu_{\;\alpha\beta} -g^{\alpha\beta} g^{\lambda\nu}B^\mu_{\;\alpha\beta} -g^{\gamma\lambda} g^{\mu\nu}B_\gamma) h_{\lambda\tau} C^\epsilon_{\;\mu\nu} f^{\tau\sigma} . \end{align}\] We introduce \[\begin{align} \sqrt{-g}G^{\mu\nu} &\!\!\!:=\!\!\! &-\Big(\frac{\partial G}{\partial h_{\mu\nu}}-\partial_\sigma\frac{\partial G}{\partial(\partial_\sigma h_{\mu\nu})} \Big)\Big \vert_{f \to \eta} \nonumber\\ &\!\!\!=\!\!\!&\partial_\sigma\Big(\sqrt{-g}L^{\sigma,\mu\nu} \Big \vert_{f \to \eta} \Big)-\sqrt{-g}M^{\mu\nu} \Big \vert_{f \to \eta}. \label{def_G_tensor} \end{align}\tag{66}\] \(G^{\mu\nu}\) coincides with the Einstein tensor. Using \[\begin{align} K^{\mu\nu}\Big \vert_{f \to \eta} &\!\!\!=\!\!\!&M^{\mu\nu} \Big \vert_{f \to \eta} \end{align}\] and (66 ), we have \[\begin{align} -2 \sqrt{-g}K^{\mu\nu}\Big \vert_{f \to \eta} &\!\!\!=\!\!\!&2\sqrt{-g}G^{\mu\nu} +\partial_\omega j_2^{\omega,\mu\nu} ,\\ j_2^{\omega,\mu\nu} &\!\!\!:=\!\!\! &-2\sqrt{-g}L^{\omega, \mu\nu} \Big \vert_{f \to \eta} . \end{align}\] Then, the Hilbert tensor associated with \(\sqrt{-g}\mathcal{G}\) is given by \[\begin{align} H^{\mu\nu} &\!\!\!:=\!\!\! &-2\frac{\partial G}{\partial f_{\mu\nu}}\Big \vert_{f \to \eta}+2\partial_\omega\frac{\partial G}{\partial(\partial_\omega f_{\mu\nu})}\Big \vert_{f \to \eta} \nonumber\\ &\!\!\!=\!\!\!&2\sqrt{-g}G^{\mu\nu} +\partial_\omega j^{\omega,\mu\nu} ,\\ j^{\omega,\mu\nu} &\!\!\!:=\!\!\! &j_1^{\omega,\mu\nu} + j_2^{\omega,\mu\nu} ,\\ j_1^{\omega,\mu\nu} &\!\!\!:=\!\!\! &2\sqrt{-g}J^{\omega, \mu\nu} \Big \vert_{f \to \eta} . \end{align}\] Consequently, the Belinfante-Rosenfeld tensor corresponding to (48 ) is given by \(U^{\mu\nu}=-\gamma H^{\mu\nu}\). A straightforward calculation yields \[\begin{align} j^{\omega,\mu\nu} &\!\!\!=\!\!\!&\sqrt{-g}(g^{\alpha\nu}\eta^{\omega\beta}\Gamma^\mu_{\;\alpha\beta} +\eta^{\alpha\omega} g^{\mu\beta} \Gamma^\nu_{\;\alpha\beta} -\eta^{\gamma\omega} g^{\mu\nu}\Gamma_\gamma\nonumber\\ &&\quad-g^{\alpha\nu}\eta^{\mu\beta}\Gamma^\omega_{\;\alpha\beta} -\eta^{\alpha\mu} g^{\omega\beta} \Gamma^\nu_{\;\alpha\beta} + g^{\alpha\beta} \eta^{\mu\nu}\Gamma^\omega_{\;\alpha\beta} \nonumber\\ &&\quad+\eta^{\gamma\mu} g^{\omega\nu}\Gamma_\gamma -g^{\alpha\mu}\eta^{\nu\beta}\Gamma^\omega_{\;\alpha\beta} -\eta^{\alpha\nu} g^{\omega\beta} \Gamma^\mu_{\;\alpha\beta} +\eta^{\gamma\nu} g^{\omega\mu}\Gamma_\gamma). \end{align}\] Using (52 ), we have \[\begin{align} j^{\omega,\mu\nu}+\partial_\beta N^{\mu\nu\omega\beta} &\!\!\!=\!\!\!&\sqrt{-g}(-\eta^{\mu\beta}g^{\alpha\nu}\Gamma^\omega_{\;\alpha\beta} -\eta^{\alpha\mu} g^{\omega\beta} \Gamma^\nu_{\;\alpha\beta} +\eta^{\gamma\mu} g^{\omega\nu}\Gamma_\gamma +\eta^{\omega\mu}g^{\beta\gamma}\Gamma^\nu_{\;\gamma\beta} )\nonumber\\ &\!\!\!=\!\!\!&\eta^{\mu\rho}\partial_\rho(\sqrt{-g}g^{\omega\nu})-\eta^{\mu\omega}\partial_\rho(\sqrt{-g}g^{\rho\nu}). \end{align}\] Taking \(\partial_\omega\) of the above equation, the two terms on the right-hand side cancel after relabeling dummy indices and using the commutativity of partial derivatives: \[\begin{align} \partial_\omega j^{\omega,\mu\nu} &\!\!\!=\!\!\!&-\partial_\omega\partial_\beta N^{\mu\nu\omega\beta}. \end{align}\] Finally, we obtain \[\begin{align} H^{\mu\nu} &\!\!\!=\!\!\!&2\sqrt{-g}G^{\mu\nu} -\partial_\alpha\partial_\beta N^{\mu\nu\alpha\beta}=2\sqrt{-g}G^{\mu\nu} -P^{\mu\nu}. \end{align}\]

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process↩︎

During the preparation of this work the author used ChatGPT by OpenAI in order to improve the clarity and readability of the manuscript and to assist in developing the proof presented in §8.2 and §8.3. After using this tool, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article.

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