Extracting the expression for the field equations of a diffeomorphism invariant theory of gravity from surface term
Jun-Jin Peng
TL;DR
The paper develops a surface-term perspective for diffeomorphism-invariant gravity, showing that the symmetric tensor $X^{μν}$ arising from the surface term equals the functional derivative $\delta L/\delta g^{μν}$, and that the field equations are $E^{μν}=X^{μν}-\tfrac{1}{2}L g^{μν}$. By replacing the variation operator with the Lie derivative $\mathcal{L}_\zeta$, the surface term decomposes into $2\zeta_ν X^{μν}-∇_ν Q^{μν}$, linking the equations of motion with the Noether charge two-form $Q^{μν}$. The work shows equivalence between two routes to obtain the Euler–Lagrange expression from the surface term and derives the Noether current $J^μ=∇_ν Q^{μν}$, demonstrating a unified framework tying diffeomorphism invariance, conserved currents, and field equations. It proves the generalized Bianchi identity $\nabla_μ E^{μν}=0$ in this surface-term context and provides a generic procedure for extracting field equations from the surface term, extendable to Lagrangians involving arbitrary covariant derivatives of the Riemann tensor. Overall, the results connect surface terms, Noether charges, and the equations of motion in a coherent, generalizable framework beyond Padmanabhan's original formulation.
Abstract
As a contribution towards the understanding for the field equations of diffeomorphism invariant theories of pure gravity, we demonstrate in great detail that the expression for the field equations of such theories can be derived within the perspective of the surface term coming from the variation of the Lagrangian. Specifically, starting with the surface term, we extract a symmetric rank-two tensor together with an anti-symmetric one out of this term with the variation operator replaced with the Lie derivative along an arbitrary vector field. By utilizing an equality stemming from the Lie derivative of the Lagrangian density along an arbitrary vector field, it is proved that the resulting symmetric rank-two tensor is identified with the functional derivative of the Lagrangian density with respect to the metric. Such a result further brings forth the expression for the field equations constructed from the symmetric rank-two tensor, which naturally rules out the derivative of the Lagrangian density with respect to the metric and coincides with the one for the Euler-Lagrange equations of motion. Furthermore, it is illustrated that the construction of the expression for the field equations from the symmetric rank-two tensor must be feasible as long as the variation operator in the variation equation of the Lagrangian is allowed to be substituted by the Lie derivative along an arbitrary vector field. On the other hand, as a byproduct, the anti-symmetric rank-two tensor turns out to be the Noether charge two-form. Our results offer a straightforward support on the proposal in our previous work that the surface term gives a unified description for field equations and Noether charges in the context of theories of gravity admitting diffeomorphism invariance symmetry.