Abstract
I introduce a covariant four-vector \({\mathcal {G}^a[v]}\), which can be interpreted as the momentum density attributed to the spacetime geometry by an observer with velocity \(v^a\), and describe its properties: (a) Demanding that the total momentum of matter plus geometry is conserved for all observers, leads to the gravitational field equations. Thus, how matter curves spacetime is entirely determined by this principle of momentum conservation. (b) The \({\mathcal {G}^a[v]}\) can be related to the gravitational Lagrangian in a manner similar to the usual definition of Hamiltonian in, say, classical mechanics. (c) Geodesic observers in a spacetime will find that the conserved total momentum vanishes on-shell. (d) The on-shell, conserved, total energy in a region of space, as measured by comoving observers, will be equal to the total heat energy of the boundary surface. (e) The off-shell gravitational energy in a region will be the sum of the ADM energy in the bulk plus the thermal energy of the boundary. These results suggest that \({\mathcal {G}^a[v]}\) can be a useful physical quantity to probe the gravitational theories.
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Notes
The signature is \((-,+,+,+)\). We use units with \(\hbar = c= (16\pi G) =1\) so that Einstein’s equations become \(G^a_b = (1/2)T^a_b\). Latin letters run through \(0-3\) and Greek letters run through \(1-3\).
This definition will lead to Einstein’s theory. There is a natural extension of all the results in this paper to Lanczos-Lovelock models which will be presented elsewhere.
This is indeed the off-shell, identically conserved, Noether current associated with \(q^a\). I stress that it can be obtained purely from a differential geometric identity without mentioning the action principle for gravity or any diffeomorphism invariance! [3]. In normal units, the left hand side should be multiplied by \(16\pi G\) which we have set to unity.
The fact that \(v^a v_a = -1\) does not affect the argument. One can see this more formally by writing Eq. (5) in a local inertial frame near the origin (with \(\nabla _a v^b =\partial _a v^b\)) and taking \(v^a=q^a/(-q_iq^i)^{1/2}\) where \(q^a\) is an arbitrary timelike vector. Using the Taylor series expansion \(q^b(x) = q^b(0) + M^b_{c}(0) x^c+\mathcal {O}(x^2)\), it is easy to see that we have sufficient freedom in the choice of \( q^b(0), M^b_{c}(0)\) to validate the above argument.
The Noether current \(J^a[q]\) is invariant under the ‘gauge transformation’ \(q^a\rightarrow q^a+\partial ^af\). In fact all the relevant algebra is identical to electrodynamics in curved spacetime with \(q^i\Leftrightarrow A^i, J^{mn}\Leftrightarrow F^{mn}, J^a\Leftrightarrow j^a\) where \(j^a\) is an electromagnetic current sourcing \(A^i\). Any vector \(q^i\) which satisfies source-free Maxwell’s equations in a given metric will have zero Noether current. Further, using the gauge freedom we can always set \(\nabla _iq^i=0\) and for such vector fields Eq. (5) will give the trace-free Einstein’s equations, which will lead to Einstein’s equations with the cosmological constant arising as an integration constant. I hope to revisit this idea in a later work.
It does not seem to have been widely appreciated that Noether current \(J^a[q]\) associated with an arbitrary vector field \(q^a\) is, in general, non-zero in the flat spacetime—which, incidentally, is yet another reason not to link it to diffeomorphism invariance of the Hilbert action. So the \({\mathcal {G}^a[q]}={\mathcal {P}^a[q]}\) attributed to a flat spacetime by, say, accelerated observers can be nonzero. This is a feature and not a bug; and Eq. (9) relates it to the thermal effects seen even in the flat spacetime.
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Acknowledgments
I thank Sumanta Chakraborty, S. Date and D. Kothawala for comments on an earlier draft. My work is partially supported by the J.C. Bose research Grant of the Department of Science and Technology, Government of India.
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Padmanabhan, T. Momentum density of spacetime and the gravitational dynamics. Gen Relativ Gravit 48, 4 (2016). https://doi.org/10.1007/s10714-015-1996-z
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DOI: https://doi.org/10.1007/s10714-015-1996-z