# Inverse square law in gravitation

## Main Question or Discussion Point

Help!

Has anybody made a case as to why the inverse square law should apply to gravitation, a case that is based on pure reasoning, instead of empirical evidence? I have been trying to find such arguments, but no luck so far.

Janein

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Orodruin
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Physics, unlike mathematics, is an empirical science. Theories are based on observation and the best thing you can hope is to find a good description at as basic level as possible. You cannot do physics based on logic alone, you will always need experimental guidance and verifications. The ancient Greeks tried doing science using reasoning only and got a lot of things wrong.

tech99
Gold Member
Help!

Has anybody made a case as to why the inverse square law should apply to gravitation, a case that is based on pure reasoning, instead of empirical evidence? I have been trying to find such arguments, but no luck so far.

Janein
You are right to ask this. It is easy to just assume that the inverse square law applies to all sorts of phenomena, but it is not always true. For instance, the electric field associated with EM radiation follows an inverse law.

Bandersnatch
I've always found the following argument satisfactory:

Let's say that any mass produces some 'interaction' proportional to its magnitude, which when encountered by another mass produces acceleration proportional to the magnitude of the interaction.

Imagine you've got a one-dimensional space, i.e. a line. In this space, there exists a massive point ##M##, from which interaction propagates in all possible directions. In the one-dimensional space this means the interaction total 'produced' by the point must be split in two, shared between the two possible directions. So if we were to write an equation for the force felt by a test particle ##m## in this 1D space, it'd look like ##F=AmM/2##, where ##A## is some constant. We'd probably want to fold the '2' into the constant, so we'd end up with ##F=BmM##. The force in one dimension is independent of distance from the source.

Now, let's add another dimension. A 2D space is a plane. A massive point on a plane emanates its interaction in all possible directions, which in a 2D space means that it has to be shared between all points of a circle surrounding the point. The circumference of a circle is given by ##2πR##. ##R## is the distance from the massive point. Again, all of the interaction 'produced' by point ##M## must be shared between all points on the circle. We end up with the equation of gravitational force ##F=AmM/2πR##. Combining all constants together, we get ##F=CmM/R## - the force in two dimensions falls linearly with distance.

In three dimensions, the interaction produced by the central point is spread out and shared by points surrounding the point, i.e. on a sphere. Surface of a sphere is given by ##4πR^2##. The equation of force looks like ##F=AmM/4πR^2##. Combining the constants we get: ##F=DmM/R^2##. I.e., in 3D the force from ##M## on a test particle ##m## falls with the square of the distance.
As a final touch, we rename the constant ##D## to ##G##, because it's the initial letter of the word 'gravity'.

Physics, unlike mathematics, is an empirical science. Theories are based on observation and the best thing you can hope is to find a good description at as basic level as possible. You cannot do physics based on logic alone, you will always need experimental guidance and verifications. The ancient Greeks tried doing science using reasoning only and got a lot of things wrong.
Thank you Orodruin

I can't argue with that. After all, we are trying to understand what we observe. But we generally do attempt to understand and generalize what we observe and want to be able to say: "See that is why."

Somebody must have tried that?

Janein

Orodruin
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Homework Helper
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Thank you Orodruin

I can't argue with that. After all, we are trying to understand what we observe. But we generally do attempt to understand and generalize what we observe and want to be able to say: "See that is why."

Somebody must have tried that?

Janein
You can always try to go deeper and find a more fundamental description. The question is whether or not this will be successful. I mean, I could say it is because the Newtonian limit of the Einstein field equations turn out to be the Poisson equation, whose fundamental solution goes as 1/r^2, but that then just begs the question "has someone tried to find the Einstein field equations by pure reasoning?"

OK, pure reason may be too much to ask. I guess I better settle for, say "rationalization attempts" instead. I really wonder what people think about that.
J.

I've always found the following argument satisfactory:

Let's say that any mass produces some 'interaction' proportional to its magnitude, which when encountered by another mass produces acceleration proportional to the magnitude of the interaction.

Imagine you've got a one-dimensional space, i.e. a line. In this space, there exists a massive point ##M##, from which interaction propagates in all possible directions. In the one-dimensional space this means the interaction total 'produced' by the point must be split in two, shared between the two possible directions. So if we were to write an equation for the force felt by a test particle ##m## in this 1D space, it'd look like ##F=AmM/2##, where ##A## is some constant. We'd probably want to fold the '2' into the constant, so we'd end up with ##F=BmM##. The force in one dimension is independent of distance from the source.

Now, let's add another dimension. A 2D space is a plane. A massive point on a plane emanates its interaction in all possible directions, which in a 2D space means that it has to be shared between all points of a circle surrounding the point. The circumference of a circle is given by ##2πR##. ##R## is the distance from the massive point. Again, all of the interaction 'produced' by point ##M## must be shared between all points on the circle. We end up with the equation of gravitational force ##F=AmM/2πR##. Combining all constants together, we get ##F=CmM/R## - the force in two dimensions falls linearly with distance.

In three dimensions, the interaction produced by the central point is spread out and shared by points surrounding the point, i.e. on a sphere. Surface of a sphere is given by ##4πR^2##. The equation of force looks like ##F=AmM/4πR^2##. Combining the constants we get: ##F=DmM/R^2##. I.e., in 3D the force from ##M## on a test particle ##m## falls with the square of the distance.
As a final touch, we rename the constant ##D## to ##G##, because it's the initial letter of the word 'gravity'.
Thanks Bandersnatch,

That is the kind of stuff I was looking for.

Janein

if we were to write an equation for the force felt by a test particle mmm in this 1D space, it'd look like F=AmM/2
That is an assumption about how the world operates, upon which reason is applied. This argument is therefore not "pure reason," as nice as it is. Pure reason is far too high a standard for physical theory. All statements of physical law are based upon reason applied to results of experiment/observation, at minimum.

Orodruin
Staff Emeritus