Relativity kinetic/potential energy question

Suppose I have a box floating out in space with two masses inside. I'm outside the box with a sensitive gravity meter, measuring the gravity coming from the entire box system. I start recording the gravity from the box when the two masses inside the box are some distance apart. The masses accelerate toward each other due to gravity until they collide elastically in the center of the box and stop. What would a plot of gravity vs. time look like during that process?

 

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Constant in both direction and magnitude.

Don't ask me why

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My understanding was that gravity in relativity is affected by kinetic energy, but not gravitational potential energy...which would lead me to suspect that the gravity should increase as the masses accelerate towards each other, then abruptly decrease when they collide and stop. But I would like to hear from someone who knows more about relativity than me.

 

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Gravitation depends on rest mass only, not relativistic mass.

 

What is the purpose of the box?

If the masses collide head-on, you would not detect any change, if the box is small. If the masses "inspiral" and collide, you would detect gravitational radiation. Numerical relativity researchers simulate collisions of massive objects such as black holes and neuron stars, and gravitational wave detectors such as LIGO, GEO and VIRGO are expected to detect them if something like those happen in our neighborhood.

 

Okay, it looks like I was simply mistaken in my belief that relativistic kinetic energy affects gravitation. So apparently, for example, a planet that was spinning wouldn't have any more gravity than an identical planet that wasn't spinning (because the extra relativistic mass of the spinning planet doesn't matter)?

 

In some sense you can say that relativistic kinetic energy affects gravitation, but it is more complicated than that. In general gravity does not just depend on one number, say, called "mass"; rather it depends on a collection of numbers called mass-energy tensor.

 

So does kinetic energy affect the mass-energy tensor?

Edit: Anyway, am I correct in my new-found belief that the box's gravitational pull won't be any stronger just because it the things in it have more kinetic energy?

 

I wonder can we take this further and assume that the combined relativistic kinetic energy of the massive particles of an object at rest contribute greatly to that objects gravitational pull?

If so, can temperature affect gravity

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Space-time curvature carries energy, via such things as the Bondi mass of a system. Thus if you change the space-time curvature you might change the energy or via versa.

Suppose you have two objects of mass, in that they, individually, product a space-time metric which isn't flat space-time. If you put them both into the same space-time region you have a space-time formed by their overlapping gravitational field. They have potential energy due to one anothers fields and the space-time have some kind of curvature to it. Suppose you move one of them, changing the potential energy between them. The space-time has a new configuration. It must do or else they wouldn't have contributions to gravity. You could argue that change in curvature is due in some way to the change in potential energy. Gravitational fields in GR do not add linearly, you can't plonk a black hole down next to another black hole and get the resultant space-time metric by just adding the two seperate black hole metrices, the actual metric is analytically a nightmare (or impossible) to find and you could view the additional, non-trivial, curvature due to this mixing of fields as somehow a space-time measurement of potential energy, as it only occurs when you have more than one object and the metrics not add nicely.

Don't take that as coming from anything other than a bit of intuition and experience, that's not something I got out of a textbook or a paper so could be utter nonsense.

 

Do they collide elastically or stop? You can't have both.

 

He must have meant a perfectly inelastic collision (with a zero coefficient of restitution).

 

As temur pointed out, kinetic energy must be accounted for in the relativistic mass-energy tensor, which in turn is used to determine the resulting spacetime curvature. So yes, kinetic energy does have an effect on the spacetime curvature. Problem is, as other are pointing out, the relativistic solution to this question would be an utter nightmare to solve analytically (there are ways of approximating the answer to arbitrary accuracy on a computer though). The kinetic energy cannot be treated the same as an ordinary rest mass, and doing the problem in GR is vastly more complicated than the relatively simple and straight-forward Newtonian approach even for two stationary bodies.

 

So is it true or not that the apparent gravity of the box would increase as the masses inside it accelerate due to their gravitational potential energy being converted into kinetic energy?

 

That's a damn good question, and to be honest I don't know the answer offhand, and I think it goes beyond the level of sophistication we used in my course. If anyone here understands concepts like the gravitomagnetic effect which I think is related to your question, they could probably give you a good answer. GR does allow for certain conserved quantities, and there's a notion of energy conservation in the sense that a system can be disturbed and then returned to its original configuration without any net gain or loss of energy. However, I don't think GR contains a concept of gravitational potential in the way that Newton used it.

 

This is tricky because you have to look at the geometry of space time instead of simple intrinsic properties. It would seem that potential energy is not an intrinsic property and so would not affect energy; however the very arrangement of the box and the two masses inside may somehow reflect the stored energy and hence no change in gravitation as the potential was already there.


I thought I would attach this to the discussion-

Different wavelength photons have different energies, right? This means that different wavelength photons should have different affects on space time. This would mean that photons have some kind of effect on their own speed - much like massive bodies have effects on photons.

Furthermore, the more energetic the photon the smaller the wave-packet (??) and so more localized the effect of the mass of the photon (via e=mc^2) .. I'm not sure if my understanding of the photon is sufficient, but there should certainly be some kind of self-affecting force . It can be certainly said that the more energetic photon will bend space time more than a less energetic photon. Whether this effect is even observable (perhaps it is so insignificant that the effect would register below Planck length) is another matter.

 

Speed? Photons, of all energies, move at the same speed. More energy just means more momentum, but their 4-momentum is always null.

 

Right, rest mass may be zero but traveling at the speed of light the momentum of a photon is expressed as h/wavelength.

If they distort space time in proportion to their energies then there will be differences in speed.. but to what extent?

I'm not sure how one localizes a wave.. what is the volume of a photon? Is it something like a wave-packet or what? The higher the energy of the photon the smaller this volume will be, and this would perhaps mean that the distortion on space time is greater. How does this even work? If you have a radio receiver do the photons register as sharp clicks or do they get absorbed as the entire wave-length passes?

 

In classical setting, you should consider the Einstein-Maxwell equation in vacuum, with initial data for the gravitational part equal to Schwarzschild and the electromagnetic part equal to a plane wave (or a localized wave packet). You somehow have to find a way to put this EM wave in the asymptotic (i.e., flat) region of the Schwarzschild spacetime. In general you need to use numerical simulation but if you assume the EM part is small then you should be able to get a perturbation theory in the strength of the EM part. I feel this must have been done already but I don't know the literature on this topic.

 

Yes. Its not like this effect would stimulate light to go faster than C, it would just make light travel slower than C as it already does in substances of refractive index > 1.

The space-time distortion would really be negligible; however it could possibly get exponentially stronger with ultra-high energy photons due to greater mass and smaller volume.