# Gravity affecting gravity?

I was thinking about how a compressed spring weighs more than the same spring uncompressed, and it got me wondering about the earth and moon system.

Basically, if you separate the two and weigh them individually, the sum is not going to weigh as much as if you weighed them as a system because of the potential energy bound in the gravitational field. Well, this potential energy has mass too, right? Which means if we want to calculate completely how the earth and moon interact, we have to take a third mass into consideration.

However, this third mass (I'll just call it the field) interacts itself with the earth and moon, and this interaction is itself another potential (or simply modifies the field from what we assumed it was)!

It seems to me this recursive process goes unto infinity, so you can't really calculate the interacting system directly -- you have to iterate and converge onto the correct solution.

Is this right? Or am I way off base?

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The energy in the system doesn't "have" mass, it "is" mass. So you don't add in the mass of the energy, and then recalculate what the total energy of the new system in addition to the sum of the energy and mass of the old system would be.
Too wordy? Or does that make sense?

The energy in the system doesn't "have" mass, it "is" mass.

You know what I meant. I didn't think I had to state in every single post "mass is energy".

I'm approaching it from a calculating point of view, where you start with an initial inaccuracy and converge. How else would you calculate it?

Well, just from what you said there, you automatically know that you won't have a recursive process. I don't know what the potential energy of the gravitational relationship between the moon and the earth is, so as far as calculating goes....
But what I AM saying is that you take the total mass of the earth-moon system, find the potential energy of the relationship, and find the sum for the total mass(as you know), which probably will be several place values off from significant. That is it.
My only point is that, if you find the potential energy of THAT system, you are creating energy.

When it WOULD be significant, and maybe even large enough in some cases to have an effect, is if you introduce a third gravitational body.
If two black holes were orbitting each other, and you introduce a third black hole, the total mass of the two orbitting black holes may be enough to alter gravitational pull. Good luck with the three body problem though ;)

SteamKing
Staff Emeritus
Homework Helper
Compressed spring weighs more than uncompressed spring?!?

What strange new physics is this?

Steamking, a compressed spring has more energy than an uncompressed spring. E = mc^2, therefore the spring has more mass as well. Unless you were being sarcastic...in which case stop being sarcastic haha.

Keep in mind that this is not a discussion of proper mass, but relativistic mass, which I think would apply since we are talking about bodies in motion. I believe the same goes for the spring, it only weighs more when in motion...though I am not a physics doctorate and would gladly defer to one on that matter :)

SteamKing
Staff Emeritus
Homework Helper
A compressed spring is not in motion. It is compressed.

Hence my confusion: Does the compressed spring actually weigh more or does it have to be being compressed, or decompressing?
I know that a body with greater energy(i.e. motion or heat) will have greater relativistic mass. However, I can't remember if potential energy actually increases relativistic mass...especially considering the common use of the term "rest mass," which would seem to indicate that an object at rest has minimal mass. BLAH! There is so much to remember, and so so much to learn....

But what I AM saying is that you take the total mass of the earth-moon system, find the potential energy of the relationship, and find the sum for the total mass(as you know), which probably will be several place values off from significant. That is it.
My only point is that, if you find the potential energy of THAT system, you are creating energy.

I think you're missing my point a little bit. I'm saying the potential energy stored in the gravitational field itself is not a simple function of the masses of the earth, the moon and their distance apart. Because the potential energy of the field is itself mass, there is, for lack of a more semantically astute way of expressing it, a gravitational force to the gravitational field itself. You see where the recursive effect comes in? I'm not saying there's a series of events that happens or that energy is magically generated -- I'm just saying that to calculate the potential of the interaction requires some sort of recursive function.

The effects of course diminish very rapidly. But this isn't a practical question -- it's purely theoretical. When someone is trying to do relativistic calculations, you wouldn't tell them to just use Newton's law because the difference is insignificant at slow speeds. How would you ever learn the theory then?

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Hence my confusion: Does the compressed spring actually weigh more or does it have to be being compressed, or decompressing?
I know that a body with greater energy(i.e. motion or heat) will have greater relativistic mass. However, I can't remember if potential energy actually increases relativistic mass...especially considering the common use of the term "rest mass," which would seem to indicate that an object at rest has minimal mass. BLAH! There is so much to remember, and so so much to learn....

The spring simply needs to be compressed -- it does not have to be in motion. I was under the impression you knew more about this area of physics than me, but as this seems to not be the case, see this Wikipedia article for an introduction to the subject: http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence" [Broken]. It has the spring example, as well as a few others.

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Runner1, its hard to remember certain things sometimes, and I apologize. I'm not a doctor, and so don't claim to be an expert(I admittedly have struggled with some aspects of relativity in school), but I do have valuable knowledge to share with you. I do not need an introduction, but I appreciate your sincere gesture.
However, regarding your question on the recursive properties of gravity calculations....do you feel they have been answered or are you still confused?

Oh no offense was meant at all! I hope you didn't take it that way. I forget a lot myself ;)

Regarding my question, I'm really just wondering if I'm right, and if so, how one goes about doing these calculations. I'm kind of hoping an expert in this area will respond...

Yes I see what your saying, and I suppose you could call it recursive. The problem is that relativity is background independent. SO, when you calculate the masses of the Earth and Moon, your fine and dandy. Then, when you calculate the potential energy of the relationship between the two, your still fine and dandy, because it is the potential energy of one object relative to another, regardless of background. Agreed?
Now, next step: You find the mass of the energy in the system described above. Once you do this, you have the mass of the entire system, which could be floating off in oblivion and be the whole extent of the universe, and your calculations are, once again, fine and dandy.
In order to calculate the Gravity of the gravity, you must introduce another body. This extends your universe, and you can once again calculate the mass of gravity, HOWEVER this time you will calculate the mass of the gravity of the entire 3 body system, which I have never done and is no doubt entirely too complicated.
THIS process can go on and on and on until you've calculated the mass of the entire universe....What I'm trying to say is that, in your two body system, you run out of objects which can be used for relativistic calculations.

Sorry, got caught up for a bit. Long story short:

The POTENTIAL gravitational energy between two objects could be found by taking the negative gravitational constant(google it), multiplying it by (the masses of your two objects divided by the distance between them) and adding the constant of integration.

You have a constant of integration because the formula above is a simplified way of integrating the gravitational force. (In case your unfamiliar with calculus, which it seems like you are, this should maybe provide you some closure as to your feelings of a recursive problem, although I hope that you see the gravity doesn't actually compound.)

Then, naturally, you can convert the potential energy into mass using Einstein's equation :) Don't keep doing that though, haha.

In case your unfamiliar with calculus, which it seems like you are

How does it seem that I am unfamiliar with calculus? I got a 5 on Calc BC in high school, and A's in Calc II, Calc III/linear algebra, Physics I, Physics II, and Diff eq my past 4 years in college. My questions on PhysicsForums are more for conceptual understanding. The math follows pretty easily.

Does anyone else want to chime in on this thread or is it just us two that have an interest in this? For all I know, we could be exchanging all sorts of inaccuracies.

How does it seem that I am unfamiliar with calculus? I got a 5 on Calc BC in high school, and A's in Calc II, Calc III/linear algebra, Physics I, Physics II, and Diff eq my past 4 years in college. My questions on PhysicsForums are more for conceptual understanding. The math follows pretty easily.

Does anyone else want to chime in on this thread or is it just us two that have an interest in this? For all I know, we could be exchanging all sorts of inaccuracies.

Oh no! That's not at all what I meant to say! I apologize, I meant to say something more along the lines of "just IN CASE you aren't familiar with calculus, though it seems that you are...."

I just threw it in there on the odd chance that you were an enthusiast and not familiar with the math :( Sorry.

I am pretty sure potential energy does not have a "mass." Mass is mass, energy is energy, though they can be converted. The gravity that comes from the earth or the moon is a result of the mass. When two charges (say, electrons) are near each other, does their electric fields cause them to actually have a greater charge? There may be a force, which will cause them to move, but their charge certainly does not increase.

However I could be mistaking...

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Is this right?
General Relativity is in fact non-linear, unlike say, Maxwell equations.

I am pretty sure potential energy does not have a "mass." Mass is mass, energy is energy, though they can be converted....
However I could be mistaking...

Look up "rest mass" or "proper mass" and "relativistic mass."

I think that runner1 is aware that relativity is non linear. That would sure make things easy :)

Basically, if you separate the two and weigh them individually, the sum is not going to weigh as much as if you weighed them as a system because of the potential energy bound in the gravitational field.

This is backwards; gravitational potential energy is negative. It will be zero when the two objects are at infinite distance, and a finite negative value when they are close. In other words, positive work would have to be done to separate two gravitationally attractive objects.

Gravity IS non linear...it IS self interacting as suggested in the original post. That is what in part caused Einstein to take ten years developing GR....he did not initially know the proper
mathematics...anyway, that characteristic is taken into account
by the Einstein Field equations (tensors).

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This is backwards; gravitational potential energy is negative. It will be zero when the two objects are at infinite distance, and a finite negative value when they are close. In other words, positive work would have to be done to separate two gravitationally attractive objects.

Gravity IS non linear...it IS self interacting as suggested in the original post. That is what
in part caused Einstein to take ten years developing GR....he did not initially know the proper
mathematics...

But not self compiling....is the method I gave for calculating the potential energy not accurate? Integrating the gravitational force?

Hey runner, lets collaborate a little here. I've been suggesting the use of E = mc^2 to calculate the mass of energy. This is Einsteins equation for particles at REST. His equation for particles in MOTION is different. Since, in GR, when discussing gravity we're not discussing particles, do you have any thoughts on which we should be using to convert potential energy to mass?