# How general relativity gets to an inverse-square law

I understand that a general interpretation of the $1/r^2$ interactions is that virtual particles are exchanged, and to conserve their flux through spheres of different radii, one must assume the inverse-square law. This fundamentally relies on the 3D nature of space.

General relativity does not suppose that zero-mass particles exchanged. What is the interpretation, in GR, of the $1/r^2$ law for gravity? Is it come sort of flux that is conserved as well? Is it a postulate?

Note that I am not really interested in a complete derivation (I don't know GR enough). A physical interpretation would be better.

Related question: Is Newton's Law of Gravity consistent with General Relativity?

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I believe it is the case that the $1/r^2$ law depends not only on the 3D nature of space but also on the flatness of that space. In GR, space (and spacetime) are not, in general, flat. See: math.ucr.edu/home/baez/einstein/node6a.html –  Alfred Centauri Jun 14 '13 at 21:41
Here I always wonder about the role of Einstein-Hilbert tensor, which is always a 2-index tensor, for any number of dimensions. –  arivero Jun 14 '13 at 22:13

I understand that a general interpretation of the $1/r^2$ interactions is that virtual particles are exchanged [...] General relativity does not suppose that zero-mass particles exchanged.

You don't need quantum field theory for this. In a purely classical field theory, we have field lines, and the field lines from a spherically symmetric source should radiate outward along straight lines. In a frame where the source is at rest, we expect by symmetry that the field lines are uniformly distributed in all directions. The strength of the field is proportional to the density of the lines, which falls off like $1/r^2$ in a three-dimensional space.

This whole description is complicated by the polarization of the field. Gravitational fields have complicated polarization modes. Nevertheless, the $1/r^2$ result is unaffected.

Finally, we have an issue unique to GR, which is that the field is the metric, and this means that the field itself affects the measuring tools that we use to measure things like $r$, the field, and the area of a surface through which we're counting the number of field lines that penetrate. These are all strong-field issues, so for large $r$, they don't affect the $1/r^2$ argument.

Is it a postulate?

No. In the standard formulation of GR, the main postulate is the Einstein field equations. From it, we can prove Birkhoff's theorem, which says that the Schwarzschild metric is the external field of a static, spherically symmetric source. The weak-field limit of the Schwarzschild metric corresponds to a $1/r^2$ field.

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So, if I understand your answer correctly, GR has the same interpretation as classical mechanics, i.e. a conservation of field lines? If yes, then a corollary to my question would be: is the conservation (or continuity) of field lines fundamentally required as a postulate? –  fffred Jun 14 '13 at 22:06
@fffred: is the conservation (or continuity) of field lines fundamentally required as a postulate? No, it's not logically independent of the field equations. The logical structure is basically the same as in E&M, where the field equations (Maxwell's equations) are sufficient to prove the validity of the usual rules for field lines. –  Ben Crowell Jun 14 '13 at 22:54
"The main postulate is the EFE". When I read Einstein's original paper, yes, this is what he did. But in Hilbert's paper, which is the better way of doing GR, I've heard (but have not read his actual paper) that he derived it from the EH action. And the approaches turned out to be equivalent! –  Dimensio1n0 Jun 30 '13 at 10:23

I found many explanations for this type of questions

http://settheory.net/cosmology http://settheory.net/general-relativity It's better than "The Meaning of Einstein's Equation" (John Baez).

In particular - It is directly applied to an important example (universal expansion) - The expression is simpler (relating 1 component of the energy tensor to 3 components of the Riemann tensor) - The relation between energy and curvature is not only expressed but also justified - Both (diagonal) space and time components of the relation are expressed and justified, resulting in showing their similarity "like a coincidence".

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## protected by Qmechanic♦Jul 22 '14 at 16:50

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