# The gravitational inverse square law and general relativity

I know this has been discussed previously, but I am wondering if the equations of general relativity independently reduce to an inverse square law at low speeds or if the formulas were created in a way that they would (chicken and egg question). (boundary condition).

One answer mentioned how Einstein's field equations can prove Birkoff's Theorem which probably answers my question, but could someone give a clarifying comment.

As many times as it has been explained, it still seems to me that general relativity says that gravity is not a force in the "true" sense of the word. On the other hand, an inverse square law suggests a "force" carried by particles (gravitons) due strictly to geometric considerations.

• This is a question I've asked myself several times before and I still want to know the answer to. The most I know about this is that Einstein's equations contain Newton's gravitational constant due to the classical limit, but never have I seen a full explanation on if GR happens to reduce to an inverse square or if it was made so that it would reduce to Newton's Law. Great question. – João Vítor G. Lima Nov 30 '18 at 4:23
• this might deal with your question. i might think this would be in a General Relativity text for undergraduate physics majors. – robert bristow-johnson Nov 30 '18 at 4:55
• Discussed previously where? Which answer? – Qmechanic Oct 7 '19 at 18:18

## 1 Answer

If general relativity didn't reproduce Newton's model of gravity (the inverse square law) in the low-speed weak-gravity approximation, then general relativity would have been ruled out, because we already know that Newton's model is an excellent approximation under those conditions. In this sense, the inverse square law played an important role in testing general relativity.

However, in hindsight, general relativity now stands on its own as our fundamental model of gravity, and the inverse square law is better regarded as one of the many correct predictions of general relativity. So there is no chicken-and-egg problem. This is a common theme in physics: the principles that were once regarded as fundamental are later regarded as mere approximations to something deeper. A new foundation is adopted, and the original foundation becomes a prediction instead.

The structure of general relativity can actually be deduced from a few simple principles that don't explicitly refer to any kind of inverse square law (this is supposed to be a non-technical rendition of Lovelock's theorem):

• The action principle. Loosely translated, this says that if thing A influences thing B, then thing B must also influence thing A in a related way.

• The list of "things" in the universe should include the metric field.

• Diffeomorphism invariance. Loosely translated, this means that if we take all the "stuff" in the universe and distort it all together in some smooth but otherwise arbitrary way, then we haven't actually changed anything at all (because we distorted all the stuff the same way, and the only things that matter are stuff compared to other stuff).

• The "action" in the action principle shouldn't involve any more than second-derivatives. (No third-derivatives, for example.) This, in turn, can be inferred using the idea that general relativity itself is probably just an approximation to something deeper that current experiments are unable to resolve. This is discussed in "Introduction to the Effective Field Theory Description of Gravity", https://arxiv.org/abs/gr-qc/9512024.

...it still seems to me that general relativity says that gravity is not a force in the "true" sense of the word. On the other hand, an inverse square law suggests a "force" carried by particles (gravitons) due strictly to geometric considerations.

The way we translate mathematically-formulated principles into words is, at least to some degree, a matter of taste. In classical general relativity, gravity is mediated by a field (the metric field) that both influences and is influenced by everything else, in accordance with the action principle. When we say that gravity is not a force in the "true" sense of the word, we are alluding to the fact that this same metric field is what we use to define geometry and proper time, which are things that we usually think of as prerequisites for doing any kind of physics at all. But we can also think of things the other way around: geometry and proper time are useful concepts because of the characteristics of this particular field and the way it interacts with everything else. Whether we use the "geometry-first" language or the "geometry-second" language, the important principles underlying classical general relativity are principles like those listed above.

Here's a concise review of how Newton's law of gravity is derived from general relativity: "Normalization conventions for Newton’s constant and the Planck scale in arbitrary spacetime dimension" (https://arxiv.org/abs/gr-qc/0609060). This paper highlights the fact that the inverse-square law result is specific to four-dimensional spacetime.

Regarding gravitons, a proper understanding of gravity in those terms requires a theory that reconciles general relativity with quantum theory (beyond the context of the low-resolution approximation that is used in the "Effective Field Theory" paper cited above). Exactly what such a theory has to say about the "right" way to think about gravity is an interesting question that I'm not qualified to answer, but here is a related (technical) post:

String theory and one idea of “quantum structure of spacetime”