Boundary effects in General Relativity with tetrad variables

Abstract

Varying the gravitational Lagrangian produces a boundary contribution that has various physical applications. It determines the right boundary terms to be added to the action once boundary conditions are specified, and defines the symplectic structure of covariant phase space methods. We study general boundary variations using tetrads instead of the metric. This choice streamlines many calculations, especially in the case of null hypersurfaces with arbitrary coordinates, where we show that the spin-1 momentum coincides with the rotational 1-form of isolated horizons. The additional gauge symmetry of internal Lorentz transformations leaves however an imprint: the boundary variation differs from the metric one by an exact 3-form. On the one hand, this difference helps in the variational principle: gluing hypersurfaces to determine the action boundary terms for given boundary conditions is simpler, including the most general case of non-orthogonal corners. On the other hand, it affects the construction of Hamiltonian surface charges with covariant phase space methods, which end up being generically different from the metric ones, in both first and second-order formalisms. This situation is treated in the literature gauge-fixing the tetrad to be adapted to the hypersurface or introducing a fine-tuned internal Lorentz transformation depending non-linearly on the fields. We point out and explore the alternative approach of dressing the bare symplectic potential to recover the value of all metric charges, and not just for isometries. Surface charges can also be constructed using a cohomological prescription: in this case we find that the exact 3-form mismatch plays no role, and tetrad and metric charges are equal. This prescription leads however to different charges whether one uses a first-order or second-order Lagrangian, and only for isometries one recovers the same charges.

Appendices

Appendix A: Conventions

We denote by $\underset{\widetilde{}}{\epsilon }{} _{\mu \nu \rho \sigma }$ the completely antisymmetric spacetime density with $\underset{\widetilde{}}{\epsilon }{} _{0123}=1$, and $\tilde{\epsilon }^{\mu \nu \rho \sigma } \underset{\widetilde{}}{\epsilon }{} _{\mu \nu \rho \sigma }=-4!$. It is related to the volume 4-form by

$$\begin{aligned} \epsilon :=\frac{1}{4!}\epsilon _{\mu \nu \rho \sigma }~dx^\mu \wedge dx^\nu \wedge dx^\rho \wedge dx^\sigma , \qquad \epsilon _{\mu \nu \rho \sigma }:=\sqrt{-g} \, \underset{\widetilde{}}{\epsilon }{} _{\mu \nu \rho \sigma }. \end{aligned}$$

(A.1)

For the internal Levi-Civita $\epsilon _{IJKL}$ the density notation is unnecessary, and we use the same convention, ${\epsilon }_{0123}=1$. Hence the tetrad determinant is

$$\begin{aligned} e = -\frac{1}{4!}\epsilon _{IJKL}~\tilde{\epsilon }^{\mu \nu \rho \sigma } ~e_\mu ^I e_\nu ^J e_\rho ^K e_\sigma ^L. \end{aligned}$$

(A.2)

Accordingly,

$$\begin{aligned} 4e^{[\mu }_Ie^{\nu ]}_J&= - \epsilon _{IJKL}\epsilon ^{\mu \nu \rho \sigma } e_\rho ^Ke_\sigma ^L, \end{aligned}$$

(A.3a)

$$\begin{aligned} 6e^{[\mu }_Ie^{\nu }_Je^{\rho ]}_K&= - \epsilon _{IJKL}\epsilon ^{\mu \nu \rho \sigma } e_\sigma ^L, \end{aligned}$$

(A.3b)

which are used in the main text.

For the Hodge dual operator, $\star : \Lambda ^p\mapsto \Lambda ^{n-p}$ satisfies

$$\begin{aligned} \star ^2 \omega ^{(p)}= -(-1)^{p(n-p)}\omega ^{(p)}, \qquad \omega ^{(p)}\wedge \star \theta ^{(q)} = \omega ^{(p)}\lrcorner \theta _{(q)} \sqrt{-g} d^nx. \end{aligned}$$

(A.4)

In components,

$$\begin{aligned} (\star \omega ^{(p)})^{\mu _1..\mu _{n-p}}&:= \frac{1}{ p!} \omega ^{(p)}_{\alpha _1..\alpha _p} \epsilon ^{\alpha _1..\alpha _p\mu _{1}..\mu _{n-p}}, \end{aligned}$$

(A.5a)

$$\begin{aligned} \omega ^{(p)}_{\alpha _1..\alpha _p}&:=- \frac{1}{(n-p)!}\epsilon _{\alpha _1..\alpha _p\mu _1..\mu _{n-p}} (\star \omega ^{(p)})^{\mu _1..\mu _{n-p}}, \end{aligned}$$

(A.5b)

$$\begin{aligned} (\star \omega ^{(p)})_{\mu _1..\mu _{n-p}}&:= \frac{1}{p!} \omega ^{(p)}{}^{\alpha _1..\alpha _p} \epsilon _{\alpha _1..\alpha _p\mu _{1}..\mu _{n-p}}, \end{aligned}$$

(A.5c)

$$\begin{aligned} \omega ^{\alpha _1..\alpha _{p}}&:= - \frac{1}{(n-p)!} \epsilon ^{\alpha _1..\alpha _p\mu _{1}..\mu _{n-p}} (\star \omega ^{(p)})_{\mu _1..\mu _{n-p}}. \end{aligned}$$

(A.5d)

Appendix B: Dressing 2-form

To make some manipulations with the dressing 2-form of [1] more manifest, we report here some useful explicit formulas. First, we have

$$\begin{aligned} \star (e_I\wedge \delta e^I)&= -\frac{1}{2}\epsilon _{IJKL}~ e^I\wedge e^J\, (e^{K\alpha }\delta e_{\alpha }^L) = \epsilon _{\mu \nu }^{\;\;\;\;\rho \sigma }e_{I\rho }\delta e_{\sigma }^I \, dx^\mu \wedge dx^\nu , \end{aligned}$$

(B.1a)

$$\begin{aligned} e_I\wedge \delta e^I&= -\eta _{IJKL} ~e^I\wedge e^J\, (e^{K\alpha }\delta e_{\alpha }^L) = e_{I[\mu }\delta e_{\nu ]}^I \, dx^\mu \wedge dx^\nu . \end{aligned}$$

(B.1b)

Second, the explicit expression of the Hodge dual of the exact 3-form $d\alpha (\delta )$ is, in the general case including the Barbero-Immirzi term,

$$\begin{aligned} (\star d\alpha )^\mu&= \frac{1}{3!} (d\alpha )_{\nu \alpha \beta } \epsilon ^{\nu \alpha \beta \mu } = \frac{1}{2} \partial _\nu \alpha _{\alpha \beta } \epsilon ^{\nu \alpha \beta \mu } = \frac{1}{2}\partial _\nu \left( \epsilon _{\alpha \beta }^{\;\;\;\;\rho \sigma }e_{I\rho }\delta e_{\sigma }^I +\frac{2}{\gamma }e_{I[\alpha }\delta e_{\beta ]}^I\right) \epsilon ^{\nu \alpha \beta \mu } \nonumber \\&= \frac{1}{2}\partial _\nu \left( e e_{I\rho }\delta e_{\sigma }^I - \frac{1}{2\gamma } e_{I\gamma }\delta e_{\delta }^I \tilde{\epsilon }^{\;\;\;\;\gamma \delta }_{\rho \sigma }\right) \tilde{\epsilon }_{\alpha \beta }^{\;\;\;\;\rho \sigma }\epsilon ^{\nu \alpha \beta \mu }\nonumber \\&= -2\partial _\nu \left( e e^{I[\nu } \delta e^{\mu ]I} + \frac{1}{2\gamma } \tilde{\epsilon }^{\mu \nu \gamma \delta } e_{I\gamma }\delta e_{\delta }^I \right) , \end{aligned}$$

(B.2)

where in the final step we used

$$\begin{aligned} e^\nu _I g^{\mu \sigma }\delta e_{\sigma }^I = - e^\mu _I \delta e^{I\nu }. \end{aligned}$$

(B.3)

Finally when torsion is present, we have the following additional relations

$$\begin{aligned}&\partial _\sigma (2e e^{[\sigma }_I \delta e^{\mu ]I}) = \nabla _{\sigma }\left( 2e^{[\sigma }_I \delta e^{\mu ]I}\right) - T^\mu {}_{\nu \rho }e^\nu _I\delta e^{I\rho }, \end{aligned}$$

(B.4a)

$$\begin{aligned}&\partial _\sigma (\tilde{\epsilon }^{\mu \nu \rho \sigma }e_{I\nu }\delta e^I_\rho ) = \epsilon ^{\mu \nu \rho \sigma } \Big [\nabla _\sigma (e_{I\nu }\delta e^I_\rho ) - \frac{1}{2} T^\lambda {}_{\sigma \nu }(\delta e_{I\lambda }e^I_\rho - e_{I\lambda }\delta e_\rho ^I) \Big ], \end{aligned}$$

(B.4b)

in agreement with (8.3).