Evaluating the Einstein-Hilbert action

Question

The Einstein-Hilbert action is given by,

$$I = \frac{1}{16\pi G} \int_{M} \mathrm{d}^d x \, \sqrt{-g} \, R \, \, + \, \, \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{d-1}x \, \sqrt{-h} \, K$$

including the Gibbons-Hawking-York boundary term. A well-known derivation of the entropy of the Schwarzschild metric requires the evaluation of the boundary term. However, one must introduce a radial regulator $R$, and subtract off a counter-term which is the Gibbons-Hawking action of emtpy space with the same boundary. The final result is finite as $R\to\infty$.

I am attempting to calculate the full action for a solution for which $R\neq0$, hence I need to also compute the pure Einstein-Hilbert piece. However, I must introduce a regulator, and the final action is not finite as I take it to infinity. My question: is there an analogous procedure to "tame" the infinity of the pure Einstein-Hilbert piece, perhaps similar to the treatment of the Gibbons-Hawking term?

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I actually have to introduce two regulators. For my solution, the Ricci scalar is independent of a particular coordinate, $x_1$, so I get a factor of $x_1$ after integration evaluated at $\pm\infty$, so I introduced regulators, such that $L^{-} < x_1 < L^{+}$. As I take $L^{\pm}\to \pm \infty$, of course it is divergent.

Answer 1

There are, in fact, much better procedures than the background subtraction used by Gibbons and Hawking. For their procedure to work one must be able to embed the relevant regulating surface into an appropriate "background" spacetime. In some cases you can do this, but in general it is not possible for spacetime dimension greater than 3. To complicate matters, the "correct" background may not even be clear. However, one can always construct an intrinsic procedure for a given class of boundary conditions that addresses these divergences. This involves adding suitable surface terms to the action which do not affect the equations of motion, but render the action finite. The necessary surface terms can usually be written as integrals of local functions of the "boundary data" on the surface that you use to regulate the calculation.

(Note that the real problem isn't finiteness of the action; it's whether or not the variation of the action really vanishes "on shell" for arbitrary field variations that preserve the boundary conditions. The divergences you encountered are actually a symptom of this deeper problem. This was first studied for the Hamiltonian formulation of GR by Regge and Teitelboim in their paper "Role of Surface Integrals in the Hamiltonian Formulation of General Relativity", http://dx.doi.org/10.1016/0003-4916(74)90404-7 . Once this issue is addressed the resulting action will always give sensible results.)

The sort of procedure I'm describing was first understood for asymptotically de Sitter spacetimes; see Balasubramanian and Kraus (http://arxiv.org/abs/hep-th/9902121) or Emparan, Johnson, and Myers (http://arxiv.org/abs/hep-th/9903238). In asymptotically flat spacetimes the work of Regge and Teitelboim was generalized to the Lagrangian formulation of GR by Mann and Marolf in http://arxiv.org/abs/hep-th/0511096 (see also http://arxiv.org/abs/arXiv:0804.2079). Since then, the technique has been extended to a wide variety of theories with different asymptotics. There are many recent examples -- a whole literature, really -- including things like non-relativistic Lifshitz spacetimes (http://arxiv.org/abs/arXiv:1107.4451 and http://arxiv.org/abs/1107.5792), and wide classes of theories that can be reduced to two-dimensional models (http://arxiv.org/abs/hep-th/0703230 and http://arxiv.org/abs/1406.7007).