The origin of the inverse square law

In summary, the inverse square law of gravitation can be derived from general relativity as the Newtonian limit, with corrections for weak fields and nearly flat spacetime. This can be seen through the equation \Delta \phi \sim \rho, which describes the potential of a gravitating mass density, and the use of an integral operator with a 1/r kernel. However, this is an approximation and not always accurate, as relativity gives a different picture of gravity that is not always inverse square. Newton originally deduced the inverse square law from the fact that planet's orbits are ellipses, but it cannot be derived using deductive reasoning. The inverse square law can also be understood through the concept of surface area of a
  • #1
lugita15
1,554
15
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.

Thank You in Advance.
 
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  • #2
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

[tex]\Delta \phi \sim \rho[/tex]

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).

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

This reasoning works in higher dimensions as well. For d=3, 4, ... one finds 1/rd-2.
 
  • #3
lugita15 said:
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.

lugita15 said:
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.

lugita15 said:
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.
 
  • #4
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.
 
  • #5
HallsofIvy said:
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.
 
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  • #6
HallsofIvy said:
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.
 
  • #7
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.
 
  • #8
homology said:
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!
 
  • #9
homology said:
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'.
 
  • #10
HallsofIvy said:
Well, I've just undeleted it! I will pm ZapperZ about this.

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.
 
  • #11
Passionflower said:
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.
 
  • #12
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.
 
  • #13
Passionflower said:
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.
 
  • #14
Passionflower said:
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.

fzero said:
The inverse square law behavior is actually deduced in the weak field limit of the Schwarzschild solution.
Agreed, thanks.
 
  • #15
It is deduced from Newton's law but not from GR. In other words GR clears up nothing about Newton's law.
 
  • #16
Please check some GR textbooks. That's all I have to say.
 
  • #17
tom.stoer said:
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.
 
  • #18
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.
 
  • #19
tom.stoer said:
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.
 
  • #20
In Carroll's online reference you can simple search for "the Newtonian limit".
 
  • #21
tom.stoer said:
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 into 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.
 
  • #22
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?
 
  • #23
Passionflower said:
In chapter 4 page 112 you can see how Newton's law is plugged into 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 [tex]m[/tex] 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.
 
  • #24
tom.stoer said:
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.
 
  • #25
I think you do not fully appreciate what Newton's theory and GR do for us. They tell us that F(r) ~ -mM/r² (plus corrections for GR, e.g. ~ 1/r³). That's a prediction derived from Newtonian theory and GR (independently and beyond the Newtonian theory). What both theories cannot tell us is the specific value of the constant G. Again - GR is not build on top of Newtonian gravity, it only uses the same constant.

Once you write down the Einstein-Hilbert action the inverse square law (plus corrections) is fixed w/o any additional input in the limit of weak fields for mass densities with spherical symmetry. Nothing else is required but to fix G.

@fzero: You need the weak field limit for the Schwarzschild BH as you know that for large curvature the orbits are "deformed" (large precession, ergosphere, ..which is not known from a strict 1/r potential)
 
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1. What is the inverse square law?

The inverse square law is a mathematical principle that describes the relationship between the intensity of a physical quantity and the distance from its source. It states that the intensity of a physical quantity decreases proportionally to the square of the distance from its source.

2. Who discovered the inverse square law?

The inverse square law was first described by the English physicist Isaac Newton in the late 17th century. However, it was later refined and expanded upon by other scientists such as Joseph-Louis Lagrange and Pierre-Simon Laplace.

3. What is an example of the inverse square law?

One example of the inverse square law is the relationship between the intensity of light and its distance from a source. As the distance from a light source increases, the intensity of the light decreases according to the inverse square law.

4. How is the inverse square law used in science?

The inverse square law is used in various fields of science, such as physics, astronomy, and engineering. It is commonly applied to understand the behavior of physical phenomena such as gravity, light, and sound. It is also used in designing experiments and predicting the behavior of physical systems.

5. Are there any exceptions to the inverse square law?

Yes, there are some exceptions to the inverse square law. For example, in certain situations where the distance between objects is very small or the source is not isotropic (emits energy equally in all directions), the inverse square law does not apply. Additionally, in some cases, the inverse square law may be modified or replaced by other laws that better describe the behavior of a physical system.

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