Potentials in General Relativity

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The role of the components of the metric tensor as gravitational potentials in general relativity is described quite well by D'Inverno [1] 

    In special relativity, in a coordinate system adapted to an inertial frame, namely Minkowski coordinates, the equation for a free test particle is 

If we use a non-inertial frame of reference, then this is equivalent to using a more general coordinate system. In this case, the equation becomes

where G^a_bc is the metric connection for g_ab, which, which is still a flat metric but not the Minkowski metric for h_ab. The addition terms involving G^a_bc which appear are precisely the inertial force terms we have encountered.  Then the principle of equivalence requires that the gravitational forces, as well as the inertial forces, should be given by an appropriate G^a_bc. In this case, we can no longer expect space-time to be flat, for otherwise there would be no distinction from the non-gravitational case. The simplest generalization is to keep G^a_bc as the metric connection, but now take it to be the metric connection of a non-flat metric. If we are to interpret G^a_bc as force terms, then it follows that we should regard gab, as potentials. The field equations of Newtonian gravitation consist of second-order partial differential equations in the gravitational potential F. In an analogous manner, we would expect general relativity to involve second-order partial differential equations in the potentials for g_ab. The remaining task which will allow us to build a relativistic theory of gravitation is to choose a likely set of partial differential equations.

 MTW also refer to g_mn as gravitational potentials at least at one point in their book [3]

Another example of potentials as used in general relativity can be found in Mould (Note: Mould uses the notation g₄₄ as g₀₀) [2]

  It is apparent from what has been said that the g₄₄ component of the metric tensor is associated with the Newtonian gravitational potential and that the component of the field equation governing g₄₄ is Poisson's equation relating the gravitational potential to the energy density T₄₄ in the field. .... It is possible to think of each component of the metric tensor g_mn as a separate potential of the gravitational field. Instead of just one potential and one Poisson equation as in Newton's theory, there are 10 of each in the Einstein theory corresponding to the 10 independent components of the symmetric tensor g_mn, and 10 governing equations contained in the field.

Nightingale and Foster have an entire section on this subject at in their GR book called Gravitational potential and the geodesic. [4] 

In Rindler’s latest general relativity text the author writes [5]

… the field equations must determine the whole metric; that is, the g_mn. And there are just ten of these. Once we lay an arbitrary coordinate system over spacetime, the ten functions g_mn should be uniquely determined. In fact, these ten tensor components g_mn are analog of the Newtonian scalar potential F (for which there is just one field equation) and of the 4-vector potential F_m of Maxwell’s theory (for which there are four field equations). This is why Einstein chose the symbol g (for gravitational potentials) to denote the metric.

Ohanian writes [6]

We have seen that, in gravitational theory, g_mn (or h_mn) plays a role analogous to that of the vector potential of electrodynamics.

Richard C. Tolman explains it as follows [7]

… we may expect that the relativistic analogue of Poisson’s equation will be a relation connecting all ten of the gravitational potentials g_mn with the distribution of matter and energy as given by the ten components of the energy-momentum tensor T₄₄.

References: 

[1] Introducing Einstein's Relativity, Ray D'Inverno, Clarendon Press-Oxford (1992), page 130.
[2] Basic Relativity, Richard A. Mould, Springer-Verlag (1994), page 321
[3] Gravitation, Misner, Thorne and Wheeler, W.H. Freedman & Co., page 434.
[4] A Short Course in General Relativity, J. Foster & J.D. Nightingale, Springer-Verlag (1994), page 90.
[5] Relativity; Special, General and Cosmological, Wolfgang Rindler, Oxford University Press (2001), page 223.
[6] Gravitation and Spacetime – Second Ed., Hans C. Ohanian & Remo Ruffini, W.W. Norton (1994), page 339.
[7] Relativity, Thermodynamics and Cosmology, Richard C. Tolman, Dover Pub. (1987), page 188.

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