# The origin of the inverse square law

Where does the inverse square law of gravitation come from? Is it just inserted by hand into general relativity? Or can it be deduced from just Einstein's postulates that in the nonrelativistic and weak gravity limit, the inverse square law just naturally arises?

Any help would be greatly appreciated.

tom.stoer
In GR the inverse square law (plus corrections) can be derived as the Newtonian limit (plus corrections like an 1/r³ term). This is standard textbook physics. For weak fields / nearly flat spacetime one finds something like

$$\Delta \phi \sim \rho$$

describing a potential with a gravitating mass density as its source. The inverse of the Laplacian is just an integral operator with the 1/r kernel (the same as for the Coulomb law).

$$\phi \sim \Delta^{-1} \rho \to \int d^3r^\prime \frac{\rho(\vec{r}^\prime)}{|\vec{r}^\prime-\vec{r}|}$$

This reasoning works in higher dimensions as well. For d=3, 4, ... one finds 1/rd-2.

Where does the inverse square law of gravitation come from?
Nobody knows. My best guess is that it must be related to the 3-dimensionality of space.

Is it just inserted by hand into general relativity?
Pretty much, Einstein knew that for the weak field Newton's law was correct so he started there.

Or can it be deduced from just Einstein's postulates that in the nonrelativistic and weak gravity limit, the inverse square law just naturally arises?
No it can't.

HallsofIvy
Homework Helper
First, the "inverse square law" does not come from relativity. It was, I believe, Newton, that first deduced that the strength of gravity falls off as the inverse square. He deduced that from the fact that planet's orbits are ellipses. Relativity, in fact, gives a quite different picture of gravity that is NOT always "inverse square".

I agree with Passionflower that it is due to the 3 dimensionality of space. My high school physics teacher had what he called a "butter gun". It was a squirt gun with four dowels coming from the barrel. You could put one or more pieces of "toast" in the dowels and squirt "butter" on them. The point was that twices as far from the gun, the dowels, being linear, encompasses 4 times the area- you could put four times as many pieces of toast to put the same amount of butter on. The thickness of butter on the toast decreased in and "inverse square law". The surface area of a sphere increases as the square of the radius. A fixed quantity, spreading out in all directions in space is spread out over that "4 times" surface area and so its density at each point obeys an "inverse square" law.

tom.stoer
First, the "inverse square law" does not come from relativity. It was, I believe, Newton, that first deduced that the strength of gravity falls off as the inverse square. He deduced that from the fact that planet's orbits are ellipses. Relativity, in fact, gives a quite different picture of gravity that is NOT always "inverse square".
Sorry, but I can't agree.

I never claimed that the inverse square law was first derived from GR. Of course it has been discussed already by Newton, but that does not mean that it cannot be re-derived from GR.

Newton did not deduce this law from planet's orbits. You cannot derive a law based on experimental phenomena using deductive reasoning, it is inductive reasoning (Aristotle); sorry about that.

Of course relativity gives a different picture of gravity which is not always "inverse square"; it is not even a central potential in general. But in certain scenarios which are relevant for planetary motion the inverse square law can be deduced from GR in a certain approximation. Of course this is an approximation only, I know that, and I stated that clearly, but in many cases it's sufficient. Newtonian gravity with its inverse square law is an approximation to GR; that's what I am saying.

The OP asked if it (the inverse square law) is just inserted by hand into general relativity? This is not the case; it follows from GR using a certain approximation.

Last edited:
DaveC426913
Gold Member
I agree with Passionflower that it is due to the 3 dimensionality of space. My high school physics teacher had what he called a "butter gun". It was a squirt gun with four dowels coming from the barrel. You could put one or more pieces of "toast" in the dowels and squirt "butter" on them. The point was that twices as far from the gun, the dowels, being linear, encompasses 4 times the area- you could put four times as many pieces of toast to put the same amount of butter on. The thickness of butter on the toast decreased in and "inverse square law". The surface area of a sphere increases as the square of the radius. A fixed quantity, spreading out in all directions in space is spread out over that "4 times" surface area and so its density at each point obeys an "inverse square" law.
As HoI says, the inverse square law is a very general principle involving the increase in area of a section of a spherical shell as a function of radius. It's simple 3D geometry that applies just as easily to other things such as intensity of light. Double the distance, intensity drops by 4.

The 'surface area of a 2-sphere' item is cool, however I don't know if we can say this is 'where it comes from' why should gravity be related to the surface area of a 2-sphere, just because intensity falls off that way doesn't mean a force should, in fact not all forces do. So what's special about gravity. To summarize that above (correct me if I'm mistaken) Newton worked out the details from available information and Einstein found a more general theory of gravity which reduces to the inverse square law in certain limits and there is currently work to find yet another 'more general' theory which will again reduce. I think there are several answers to slightly different questions here, chronological and theoretical.

tom.stoer
To summarize that above (correct me if I'm mistaken) Newton worked out the details from available information and Einstein found a more general theory of gravity which reduces to the inverse square law in certain limits and ...
Agreed!

The 'surface area of a 2-sphere' item is cool, however I don't know if we can say this is 'where it comes from' why should gravity be related to the surface area of a 2-sphere, just because intensity falls off that way doesn't mean a force should, in fact not all forces do.
Exactly!

Clearly 3-dimensionality of space is a clue and perhaps the holographic principle will eventually fit into this as well. But that is all speculative, so would I prefer a 'we don't know'.

fzero
Homework Helper
Gold Member
The post in question states that the inverse square law cannot be derived from GR. Whether it should be deleted or not, it is certainly incorrect.

tom.stoer
Exactly!

Clearly 3-dimensionality of space is a clue and perhaps the holographic principle will eventually fit into this as well. But that is all speculative, so would I prefer a 'we don't know'.
The holographic principle is speculative, GR isn't. You can derive the inverse square law (in the sense discussed above) from GR, but you can't derive GR from the holographic principle, can you? So you are not asking where the inverse square law comes from, but you seem to aks where GR comes from.

To expand on my earlier comment on this question:

"Or can it be deduced from just Einstein's postulates that in the nonrelativistic and weak gravity limit, the inverse square law just naturally arises?"

It is true that in the weak field limit Newton's law applies, however GR does not explain that at all, it is simply 'plugged' in, hence it cannot be deduced.

fzero
Homework Helper
Gold Member
To expand on my earlier comment on this question:

"Or can it be deduced from just Einstein's postulates that in the nonrelativistic and weak gravity limit, the inverse square law just naturally arises?"

It is true that in the weak field limit Newton's law applies, however GR does not explain that at all, it is simply 'plugged' in, hence it cannot be deduced.
The inverse square law behavior is actually deduced in the weak field limit of the Schwarzschild solution. The identification of the coefficient and hence the identification of the Schwarzschild radius with the mass of the black hole tends to be "plugged in" in most elementary discussions. However, it is not necessary to do so once you have developed advanced concepts such as the Komar mass.

tom.stoer
It is true that in the weak field limit Newton's law applies, however GR does not explain that at all, it is simply 'plugged' in, hence it cannot be deduced.
Sorry, but this is simply wrong.

The inverse square law behavior is actually deduced in the weak field limit of the Schwarzschild solution.
Agreed, thanks.

It is deduced from Newton's law but not from GR. In other words GR clears up nothing about Newton's law.

tom.stoer
Please check some GR textbooks. That's all I have to say.

Please check some GR textbooks. That's all I have to say.
Please go ahead and reference a textbook (chapter and page number) where GR shows why we have Newton's law in the weak field limit.

tom.stoer
Please check e.g. http://arxiv.org/PS_cache/gr-qc/pdf/9712/9712019v1.pdf which is available online. Misner, Thorne and Wheeler should be OK as well. And Landau & Lifschitz is a good reference, but I don't know if it's available in English.

You should tell us which assumption or derivation is incomplete or insufficient.

Please check e.g. http://arxiv.org/PS_cache/gr-qc/pdf/9712/9712019v1.pdf which is available online. Misner, Thorne and Wheeler should be OK as well. And Landau & Lifschitz is a good reference, but I don't know if it's available in English.

You should tell us which assumption or derivation is incomplete or insufficient.
Do you have the chapter and page number please where the claim is made?

Don't you think it is a little silly to reference MTW without giving a chapter and page number? I suppose you are aware how many pages this book has.

You made the claim that GR explains Newton's law, so you should give me the reference, I cannot prove a negative.

tom.stoer
In Carroll's online reference you can simple search for "the Newtonian limit".

In Carroll's online reference you can simple search for "the Newtonian limit".
In chapter 4 page 112 you can see how Newton's law is plugged in to make things work.

I quote:

But this is exactly (4.36), if we set κ = 8πG.
So our guess seems to have worked out. With the normalization fixed by comparison
with the Newtonian limit, we can present Einstein’s equations for general relativity:

Plugging in the Newtonian limit.

tom.stoer
Of course you have to fix the constant. You always do. In every theory. You write F=mg and fix g by experiment. You write F=qE and fix q by experiment, you write E=mc² and fix c by experiment. What would you suggest instead?

fzero
Homework Helper
Gold Member
In chapter 4 page 112 you can see how Newton's law is plugged in to make things work.

I quote:

But this is exactly (4.36), if we set κ = 8πG.
So our guess seems to have worked out. With the normalization fixed by comparison
with the Newtonian limit, we can present Einstein’s equations for general relativity:

Plugging in the Newtonian limit.
It's not quite plugging in the Newtonian limit, it's fixing a normalization. The weak field limit has the same functional form, with a coefficient in the GR result that is only determined by experiment. The "experiment" in this case is obtaining the Newtonian result in the correct limit.

There's another place you might look for a derivation of the inverse square law, which is (7.48) in the context of the Schwarzschild BH. This result does not even rely on the weak field limit, it is an exact result. You might still object that somewhere along the way the value of $$m$$ was also fixed by comparing to the Newtonian result. However, this was not necessary. One can fix the value of the Schwarzschild radius by computing the Komar mass http://en.wikipedia.org/wiki/Komar_mass without using any comparison with Newtonian physics. It is just much simpler and quite instructive to explain how the Newtonian approximation works.

Of course you have to fix the constant. You always do. In every theory. You write F=mg and fix g by experiment. You write F=qE and fix q by experiment, you write E=mc² and fix c by experiment. What would you suggest instead?
What I was saying in the first place that GR does not explain Newton's law.

By analogy, if someone were to ask 'does GR show that orbits are elliptical' the answer is 'no it does not', 'Newton's theory showed that orbits are elliptical but GR shows that orbits are not exactly elliptical'.

GR is built on top of Newton's assumptions which are assumed correct and of course we have no experimental data in the weak field that contradicts Newton but we obviously cannot prove that in the weak field limit Newton's law is exact either.

tom.stoer