Consider the standard Einstein-Hilbert action in $D \ne 2$ dimensions spacetimes : \begin{equation} S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x, \end{equation} where $\Omega$ is an arbitrary region of the spacetime manifold. Arbitrary variations of the metric components give two terms : \begin{equation} \delta S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} G_{\mu \nu} \; \delta g^{\mu \nu} \, \sqrt{- g} \; d^D x + \frac{1}{2 \kappa} \int_{\Omega} \partial_{\lambda} \big( \sqrt{- g} \; (g^{\mu \nu} \, g^{\lambda \kappa} - g^{\mu \kappa} \, g^{\nu \lambda}) \, \nabla_{\kappa} \; \delta g_{\mu \nu} \big) \, d^D x. \end{equation} The last term can be transformed into a surface integral, by Gauss theorem, and lead to an apparent problem. If we ask that the metric variations vanish on the boundary ; $\delta g_{\mu \nu} = 0$ on $\partial \, \Omega$, the surface integral does not vanish and the variational principle cannot be applied. We can't ask that the partial derivatives $\partial_{\kappa} \; \delta g_{\mu \nu}$ vanishes too on the boundary surface. Usually, this forces us to introduce the Gibbons-Hawking-York surface integral into the gravitationnal action to remove that variational issue. I don't like that. I think that the GHY term is very "unatural" and artificial (feel like a patching work), and would prefer another way. Apparently, there's one other way but it seems to work only when $D = 4$ (see below).
It is well known that the EH lagrangian can be decomposed into two terms : a bulk term and a surface (divergence) term : \begin{equation} R = g^{\mu \nu} (\Gamma_{\lambda \kappa}^{\lambda} \; \Gamma_{\mu \nu}^{\kappa} - \Gamma_{\mu \kappa}^{\lambda} \; \Gamma_{\lambda \nu}^{\kappa}) + \frac{1}{\sqrt{- g}} \; \partial_{\mu} \big( \sqrt{- g} \; (g^{\mu \nu} \, g^{\lambda \kappa} - g^{\mu \lambda} \, g^{\nu \kappa}) \, \partial_{\nu} \; g_{\lambda \kappa} \big). \end{equation} Lets call the first term $\mathscr{L}_{H}$ (quadratic in first partial derivatives of $g_{\mu \nu}$). The second term involves second derivatives of the metric. Using this decomposition, we could verify this remarkable identity : \begin{equation} \frac{1}{2 \kappa} \; R = \mathscr{L}_H - \frac{2}{D - 2} \, \frac{1}{\sqrt{- g}} \; \partial_{\mu} \Big( \sqrt{- g} \; g_{\lambda \kappa} \; \frac{\partial \mathscr{L}_H}{\partial (\partial_{\mu} \, g_{\lambda \kappa})} \Big). \end{equation} Now, this is very similar to a classical system with the following lagrangian (but only if $D = 4$) : \begin{equation} L' = L(q, \dot{q}) - \frac{d}{dt} \Big( q^i \, \frac{\partial L}{\partial \dot{q}^i} \Big). \end{equation} It is easy to prove that this classical lagrangian gives exactly the same equations as $L(q, \dot{q})$ under arbitrary variations $\delta q^i$, if we just impose the canonical momentum $p_i = \partial L / \partial \dot{q}^i$ to be fixed at the boundaries ; $\delta p_i = 0$ at $t_1$ and $t_2$ (while $\delta q^i$ stay arbitrary) : \begin{align} \delta S &= \int_{t_1}^{t_2} \Big( \frac{\partial L}{\partial q^i} \; \delta q^i + \frac{\partial L}{\partial \dot{q}^i} \; \delta \dot{q}^i \Big) \, dt - \delta (q^i \, p_i) \Big|_{t_1}^{t_2} \\[12pt] &= \int_{t_1}^{t_2} \Big[ \frac{\partial L}{\partial q^i} - \frac{d}{dt} \Big( \frac{\partial L}{\partial \dot{q}^i} \Big) \Big] \, \delta q^i \; dt + \int_{t_1}^{t_2} \frac{d}{dt} \Big( \frac{\partial L}{\partial \dot{q}^i} \; \delta q^i \Big) \, dt - \delta (q^i \, p_i) \Big|_{t_1}^{t_2} \\[12pt] &= (\text{usual Euler-Lagrange variation of } S) + \big(p_i \; \delta q^i - \delta (q^i \; p_i)\big)\Big|_{t_1}^{t_2} \\[12pt] &= (\ldots) - q^i \; \delta p_i \Big|_{t_1}^{t_2}. \end{align} The last term cancels if we ask that $\delta p_i(t_1) = \delta p_i(t_2) = 0$, and we get the usual Euler-Lagrange equation for an arbitrary variation $\delta q^i$. For the gravitationnal action above, we can ask the same ; $\delta P^{\mu \lambda \kappa} = 0$ on $\partial \, \Omega$, where \begin{equation} P^{\mu \lambda \kappa} = \sqrt{- g} \; \frac{\partial \mathscr{L}_H}{\partial (\partial_{\mu} \, g_{\lambda \kappa})}, \end{equation} while $\delta g_{\mu \nu}$ is still arbitrary on the boundary. Apparently, this works but ONLY WHEN $D = 4$ (so $\frac{2}{D - 2} = 1$ in the lagrangian identity above) !
Now my questions are the following :
Is this procedure well defined and rigorous ? Is it really possible to completely throw away the GHY counter-term with this procedure ?
Does it make sense to fix the spacetime region $\Omega$ and its boundary $\partial \, \Omega$, while the metric is still varied on it ($\delta g_{\mu \nu} \ne 0$ on $\partial \, \Omega$) ? (I suspect some issues here.)
If everything is fine, does that mean that we really can't derive the Einstein equation from the EH action (+ matter terms and without the GHY counter-term), when $D \ne 4$ ?
I find all this very surprising ! I firmly believed that the usual Einstein equation $G_{\mu \nu} + \Lambda \, g_{\mu \nu} = -\, \kappa \, T_{\mu \nu}$ was valid for any D dimensions. I'm not so sure anymore.
More details :
Take note that the "canonical momentum" $P^{\mu \lambda \kappa}$ defined above can be seen as a complicated function of $g_{\mu \nu}$ and $\partial_{\lambda} \, g_{\mu \nu}$ (like $p_i$ is a function of $q^i$ and $\dot{q}^i$). The elements $P^{\mu \lambda \kappa}$ do not form a tensor. Explicitely, they have the following shape : \begin{equation} P^{\mu \lambda \kappa} = \frac{1}{4} \; \sqrt{- g} \; \mathcal{M}^{\mu \lambda \kappa \nu \rho \sigma} \; \partial_{\nu} \, g_{\rho \sigma}, \end{equation} where $\mathcal{M}^{\mu \lambda \kappa \nu \rho \sigma}$ is a complicated tensor defined from the metric inverse : \begin{equation} \mathcal{M}^{\mu \lambda \kappa \nu \rho \sigma} = \frac{1}{4 \kappa} \big( g^{\mu \nu} \, g^{\lambda \rho} \, g^{\kappa \sigma} + \ldots \big), \end{equation} with the symetry properties $\mathcal{M}^{\mu \kappa \lambda \nu \rho \sigma} = \mathcal{M}^{\mu \lambda \kappa \nu \sigma \rho} = \mathcal{M}^{\nu \rho \sigma \mu \lambda \kappa} = \mathcal{M}^{\mu \lambda \kappa \nu \rho \sigma}$. We can write the quadratic bulk lagrangian like this : \begin{equation} \mathscr{L}_H = \frac{1}{4} \; \mathcal{M}^{\mu \lambda \kappa \nu \rho \sigma} (\partial_{\mu} \, g_{\lambda \kappa})(\partial_{\nu} \, g_{\rho \sigma}), \end{equation} which looks a bit like the standard lagrangian for a massless scalar field : \begin{equation} \mathscr{L}_{\text{scalar field}} = \frac{1}{2} \; g^{\mu \nu} (\partial_{\mu} \, \phi)(\partial_{\nu} \, \phi). \end{equation} The lagrangian of the electromagnetic field also have a similar structure. We can prove with some labor the following relation to the pesky surface term, which appears under a general variation of the metric, but only if $D = 4$ (remember that $g_{\mu \nu} \, g^{\mu \nu} \equiv D$) : \begin{equation} \frac{1}{2 \kappa} \; \sqrt{- g} \; (g^{\mu \lambda} \, g^{\nu \kappa} - g^{\mu \nu} \, g^{\lambda \kappa}) \, \nabla_{\nu} \, \delta g_{\lambda \kappa} \equiv g_{\lambda \kappa} \; \delta P^{\mu \lambda \kappa}, \end{equation} which can be canceled if we ask $\delta P^{\mu \lambda \kappa} = 0$ on $\partial \, \Omega$, instead of $\delta g_{\mu \nu} = 0$. This is really remarkable, and almost miraculous !
The only author I know who showed some parts of the previous exposition (without discussing the case $D \ne 4$) is T. Padmanabhan. See for example these papers :
http://arxiv.org/abs/1303.1535
http://arxiv.org/abs/gr-qc/0209088
If the answer to all the questions above is really "yes", then I feel that this procedure should be teached in all courses on classical general relativity ! The GHY counter-term could be trashed. But then what about that $D = 4$ restriction ? This is puzzling.
Any opinion on this ?
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