Boundary Terms in Tetrad and Connection Gravity

Boundary terms in first-order gravity are selected by the fields held fixed, not by the bulk equations alone. A bare Palatini or Holst action naturally produces a connection variation at the boundary. Fixing the coframe instead requires a different completion, and fixing only the induced metric leaves Lorentz-gauge variations that must also be controlled. Classical equivalence with the metric action therefore includes boundary conditions, gauge freedom, corners, and reference terms.

Required background. Frames, Coframes, and Anholonomy supplies boundary-adapted frames and Lorentz gauge, The Gibbons-Hawking-York Boundary Term supplies the metric Dirichlet comparison, and Presymplectic Potentials and Currents supplies differentiability, symplectic flux, and charge criteria.

Helpful background. The Einstein Equation supplies the shared bulk dynamics, Hypersurface Normals and Tangential Projectors supplies normal and corner geometry, and Formulations of General Relativity places these surface choices alongside metric Hamiltonians.

The boundary variation of a first-order action

Use the chapter conventions

$$\Sigma_{IJ}=\star(e_I\wedge e_J), \qquad \Pi_{IJ}^{(\gamma)} =\Sigma_{IJ}-\frac1\gamma e_I\wedge e_J,$$

and consider

$$S_{\rm bulk} =\frac1{2\kappa}\int_{\mathcal M} \Pi_{IJ}^{(\gamma)}\wedge F^{IJ}, \qquad \kappa=8\pi G.$$

The cosmological term has no connection derivative and does not alter the following surface analysis. Since $\delta F^{IJ}=D\delta\omega^{IJ}$,

$$\delta S_{\rm bulk} =\text{bulk equations} +\frac1{2\kappa}\int_{\partial\mathcal M} \Pi_{IJ}^{(\gamma)}\wedge\delta\omega^{IJ}.$$

The bare action is differentiable if the pullback of the connection is fixed, $\delta\omega^{IJ}|_{\partial\mathcal M}=0$, and the action is finite. That is not the gravitational Dirichlet problem usually desired. To fix the pullback of the coframe instead, add, for the displayed bulk sign,

$$S_{\partial}^{(e)} =-\frac1{2\kappa}\int_{\partial\mathcal M} \Pi_{IJ}^{(\gamma)}\wedge\omega^{IJ}.$$

Its variation cancels the $\Pi\wedge\delta\omega$ term and leaves

$$\left.\delta(S_{\rm bulk}+S_{\partial}^{(e)})\right|_{\partial\mathcal M} =-\frac1{2\kappa}\int_{\partial\mathcal M} \delta\Pi_{IJ}^{(\gamma)}\wedge\omega^{IJ},$$

which vanishes when the coframe is fixed. This elementary cancellation is the first-order analogue of removing normal derivatives of $\delta g_{\mu\nu}$ from the Einstein–Hilbert variation. The Palatini and Holst surface variations are derived in Corichi, Rubalcava-García, and Vukašinac 2016, §§IV–V, pp. 21–42, PDF.

Metric data are weaker than coframe data

Fixing $e^I$ fixes the metric, but fixing the metric permits

$$\delta_\lambda e^I=\lambda^I{}_Je^J, \qquad \lambda_{IJ}=-\lambda_{JI}.$$

The simple $\Pi\wedge\omega$ boundary functional is not invariant under an arbitrary Lorentz transformation that remains nontrivial at the boundary, because $\omega$ transforms inhomogeneously. One must therefore specify which Lorentz transformations are proper gauge and then do at least one of the following: fix an adapted tetrad gauge, restrict boundary gauge transformations, introduce an internal normal, use a reference connection so that a difference of connections appears, or add boundary degrees of freedom that restore gauge invariance.

On the torsion-free branch and in a boundary-adapted gauge, the Palatini surface term reduces to the Gibbons–Hawking–York term with the orientation and normal sign appropriate to the chosen timelike or spacelike boundary. This agreement requires the same induced metric data. It is not an identity between uncompleted actions for arbitrary frames.

When smooth boundary pieces meet, their adapted normals differ by a Lorentz boost or rotation. A joint term is then required, just as in the metric description. Discarding it changes the action variation and the corner symplectic structure even though it leaves the interior Einstein equation unchanged.

Holst and Nieh–Yan completions

The $\gamma^{-1}$ part of $S_{\partial}^{(e)}$ is

$$+\frac1{2\kappa\gamma} \int_{\partial\mathcal M} e_I\wedge e_J\wedge\omega^{IJ}.$$

Like its Palatini partner, this expression is convenient after a boundary gauge has been selected but is not manifestly Lorentz invariant. The Nieh–Yan identity

$$d(e_I\wedge T^I) =T_I\wedge T^I-e_I\wedge e_J\wedge F^{IJ}$$

provides a gauge-covariant alternative organization. Which completion is appropriate depends on whether torsion is allowed, which fields are fixed, and what asymptotic or internal-boundary conditions are imposed.

For the standard torsion-free asymptotically flat sector analyzed in Corichi and Wilson-Ewing 2010, §§II–IV, pp. 3–13, PDF, the Holst contribution to the symplectic current becomes an exact spatial term and does not change the asymptotic Hamiltonian energy, momentum, or angular momentum. This is a result for those falloffs and parity conditions, not a universal rule for every finite boundary, corner, or torsionful matter coupling.

At an internal horizon, the permitted Lorentz transformations and symplectic flux must be assessed with the horizon boundary conditions. Surface terms can then reorganize horizon variables even when the bulk vacuum equations remain $\gamma$ independent. Their subsequent quantization lies outside this chapter.

Comparing formulations responsibly

Two gravitational actions represent the same boundary-value problem only after matching:

• the bulk invertibility and torsion sector;
• the induced fields and Lorentz gauge data held fixed;
• the orientation of every boundary component and joint;
• reference subtractions or counterterms;
• allowed asymptotic symmetries and symplectic flux.

Adding an exact form can leave the bulk equations unchanged while shifting the presymplectic potential and a representative of the Noether charge. A Hamiltonian charge is unchanged only when the corresponding boundary conditions make the shift integrable and physically trivial.

Exercise: fixed-coframe differentiability

Vary $S_{\rm bulk}+S_{\partial}^{(e)}$ and show directly that its boundary variation vanishes when $\delta e^I=0$, without imposing $\delta\omega^{IJ}=0$.

Solution

The bulk connection variation contributes

$$\frac1{2\kappa}\int_{\partial\mathcal M} \Pi_{IJ}^{(\gamma)}\wedge\delta\omega^{IJ}.$$

The boundary functional varies as

$$\delta S_{\partial}^{(e)} =-\frac1{2\kappa}\int_{\partial\mathcal M} \left( \delta\Pi_{IJ}^{(\gamma)}\wedge\omega^{IJ} +\Pi_{IJ}^{(\gamma)}\wedge\delta\omega^{IJ} \right).$$

The two terms containing $\delta\omega$ cancel. Since $\Pi^{(\gamma)}$ is algebraic and quadratic in the coframe, $\delta e^I=0$ implies $\delta\Pi_{IJ}^{(\gamma)}=0$. The remaining surface variation therefore vanishes. If only the metric is fixed, a Lorentz variation of the coframe can remain and the additional gauge analysis described above is required.

References

• Corichi, Alejandro, and Edward Wilson-Ewing. “Surface Terms, Asymptotics and Thermodynamics of the Holst Action.” Classical and Quantum Gravity 27, no. 20 (2010): 205015. DOI. Open PDF.
• Corichi, Alejandro, Irais Rubalcava-García, and Tatjana Vukašinac. “Actions, Topological Terms and Boundaries in First Order Gravity: A Review.” International Journal of Modern Physics D 25, no. 4 (2016): 1630011. DOI. Open PDF.