REVIEW
Field equations from a surface term
General Relativity and Quantum Cosmology · Submitted 24 Mar 2006 · 18 references
Pith's verdict: A careful, explicit negative result: matter action is invariant under the boundary-preserving diffeomorphisms, so the surface-term derivation cannot yield the Einstein equations with matter; the pure-gravity extension is a real but bounded bonus.
Why Pith says this
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Referee report
The paper
The paper claims
A surface-term derivation of the Einstein equations is shown to work for pure gravity but to fail once matter is included.
The reading
What carries the argument
The central object is the split of a gravitational Lagrangian density $\mathcal{L}_{\rm grav}$ into a bulk part $\delta \mathcal{L}_B = \sqrt{-g} B_{\mu\nu} \delta g^{\mu\nu}$ and a surface part $\delta \mathcal{L}_S$ that is a total divergence. The argument also relies on the class of active diffeomorphisms generated by a vector $\zeta^\mu$ that is tangential to the boundary ($\zeta^\mu n_\mu = 0$) and satisfies the Killing equation on the boundary, so that the metric is unchanged there; local Rindler horizons allow this to be applied at every spacetime point. The decisive identity is the invariance of the matter action: for any generally covariant matter Lagrangian density, $\delta_\zeta S_M = -\int_{\partial U} d^3x \sqrt{|h|} L_M \zeta^\mu n_\mu = 0$ when $\zeta^\mu n_\mu = 0$, whereas the earlier derivation used a bulk expression $T^{\mu\nu} \nabla_\mu \zeta_\nu$ and neglected matter boundary terms.
Extended reading notes
Core claim
The central claim is that $\delta_\zeta S_M = 0$ for any boundary-preserving diffeomorphism (their Eq. (25)), so the surface-term derivation of the Einstein equations is invalid in the presence of matter, contrary to the earlier derivation in refs. [6,7,10]. A careful treatment of the matter variation keeps surface terms coming from the matter fields themselves; these cancel with the stress-energy bulk term, leaving the matter action invariant. In pure gravity, in contrast, the invariance of either the bulk or the surface gravitational action under diffeomorphisms that are tangential to a local Rindler horizon yields the generalized contracted Bianchi identity and the vacuum field equations $B_{\mu\nu} = \Lambda g_{\mu\nu}$, with the cosmological-type constant left undetermined by the method. The authors conclude that matter cannot be treated on the same footing as the surface part of the gravitational action, and that the earlier holographic interpretation overreaches, since the argument needs local Rindler horizons at every spacetime point.
Load-bearing premise
The negative result assumes the transformation genuinely moves the matter fields, so the matter's own surface terms cannot be dropped; if instead one treats the transformation as applying only to the metric, the field equations with matter can still be obtained and the disagreement becomes a matter of interpretation.
Editorial extensions
If this is right
- The vacuum Einstein equations, up to an undetermined cosmological-type constant, can be obtained from boundary-term invariance alone for any second-order gravitational Lagrangian, not just Einstein–Hilbert.
- The procedure cannot produce the Einstein equations with sources; matter fields must enter through the standard bulk variation of the action.
- Holographic-style claims that gravitational dynamics are encoded entirely on null boundaries lose support, because the derivation works only for pure gravity and requires a local Rindler horizon at each point.
- A revised version that demotes the diffeomorphism to a restricted metric variation can recover field equations with matter, but only by abandoning the interpretation of the result as following from diffeomorphism invariance.
Reading between the lines
- A natural extension would be to test scalar-tensor or $f(R)$ theories to see which classes of modified gravity retain the pure-gravity surface-term derivation and which break it even without matter.
- The same boundary-term obstruction may reappear in horizon-thermodynamics derivations of the Einstein equations, which share the local Rindler horizon input and often ignore matter surface terms.
- The failure suggests that only matter actions that are not strict generally covariant scalars could rescue the surface-term route, which would tie the approach to exotic matter couplings rather than standard fields.
Original abstract
As is well known, in order for the Einstein--Hilbert action to have a well defined variation, and therefore to be used for deriving field equation through the stationary action principle, it has to be amended by the addition of a suitable boundary term. It has recently been claimed that, if one constructs an action by adding this term to the matter action, the Einstein field equations can be derived by requiring this action to be invariant under active transformations which are normal to a null boundary. In this paper we re-examine this approach both for the case of pure gravity and in the presence of matter. We show that in the first case this procedure holds for more general actions than the Einstein-Hilbert one and trace the basis of this remarkable attribute. However, it is also pointed out the when matter is rigorously considered the approach breaks down. The reasons for that are thoroughly discussed.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
As is well known, in order for the Einstein--Hilbert action to have a well defined variation, and therefore to be used for deriving field equation through the stationary action principle, it has to be amended by the addition of a suitable boundary term. It has recently been claimed that, if one constructs an action by adding this term to the matter action, the Einstein field equations can be derived by requiring this action to be invariant under active transformations which are normal to a null boundary. In this paper we re-examine this approach both for the case of pure gravity and in the presence of matter. We show that in the first case this procedure holds for more general actions than the Einstein-Hilbert one and trace the basis of this remarkable attribute. However, it is also pointed out the when matter is rigorously considered the approach breaks down. The reasons for that are thoroughly discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | As is well known, in order for the Einstein--Hilbert action to have a well defined variation, and therefore to be used for deriving field equation through the stationary action principle, it has to be amended by the addition of a suitable boundary term. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | It has recently been claimed that, if one constructs an action by adding this term to the matter action, the Einstein field equations can be derived by requiring this action to be invariant under active transformations which are normal to a null boundary. | background 0.99 | 0.04 | 0.08 | asserted 0.72 |
| s2 | In this paper we re-examine this approach both for the case of pure gravity and in the presence of matter. | contribution 0.99 | 0.78 | 0.04 | asserted 0.81 |
| s3 | We show that in the first case this procedure holds for more general actions than the Einstein-Hilbert one and trace the basis of this remarkable attribute. | result 0.95 | 0.94 | 0.17 | asserted 1.00 |
| s4 | However, it is also pointed out the when matter is rigorously considered the approach breaks down. | result 0.91 | 0.88 | 0.54 | asserted 1.00 |
| s5 | The reasons for that are thoroughly discussed. | contribution 0.82 | 0.59 | 0.06 | unclear 0.58 |
What was examined, and what was not
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References
Discussion
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