Energy conservation in relativity

[Jandell254]

I have a question with regards to the increase in mass of an an accelerated object. I actually posted this question yesterday but couldn't find it, so maybe it didn't get through...

Anyway, consider 2 large masses very far away from each other. This system has a certain gravitational potential energy.

They will accelerate towards each other due to gravity. Say they reach about 60% light speed by the time they hit each other. According to relativity, their masses will be increased (due to their high velocity), along with the attractive force between them. They will collide and convert their energy into heat.

If the masses are now pushed away from each other, to their original positions, at a slower velocity (say 1000 m/s), won't this take less energy than they had when they crashed together since they will now be back to their normal masses, as their masses won't be increased due to their slow speed. The attractive force between them will thus also be less.

This example makes it seem that energy has been created from nothing due to the mass increase of the objects. Am I missing out something obvious?

 

[Hurkyl]

Well, the main problem is that you're mashing together facts from different physical theories, and so you can't really expect the result to make any sense.

For example, Newtonian gravitation and special relativistic mechanics simply don't fit together, and I think you might be blending together relativistic and non-relativistic mechanics.

 

[Jonathan Scott]

I have a question with regards to the increase in mass of an an accelerated object. I actually posted this question yesterday but couldn't find it, so maybe it didn't get through...

Anyway, consider 2 large masses very far away from each other. This system has a certain gravitational potential energy.

They will accelerate towards each other due to gravity. Say they reach about 60% light speed by the time they hit each other. According to relativity, their masses will be increased (due to their high velocity), along with the attractive force between them. They will collide and convert their energy into heat.

If the masses are now pushed away from each other, to their original positions, at a slower velocity (say 1000 m/s), won't this take less energy than they had when they crashed together since they will now be back to their normal masses, as their masses won't be increased due to their slow speed. The attractive force between them will thus also be less.

This example makes it seem that energy has been created from nothing due to the mass increase of the objects. Am I missing out something obvious?

It all depends where the heat goes. When the masses collide, their kinetic energy (derived from the potential energy of the original system) goes into heat. If that heat energy remains in the masses, it will mean that the rest energies of the final masses (including their internal heat energy) will be the same as the total energy of the masses just before they collided including their heat energy. If it's radiated away, you've lost that much energy.

 

[Dale]

Hi Jandell, welcome to PF

I have a question with regards to the increase in mass of an an accelerated object.

Most modern scientists don't use the concept of relativistic mass. It is not terribly useful, because it is simply another name for energy. Instead, when most modern scientists use the word "mass" they refer to the "rest mass" which is frame invariant.

According to relativity, their masses will be increased (due to their high velocity), along with the attractive force between them. They will collide and convert their energy into heat.

The invariant masses of the objects will not be increased. Nor will the energy of the system. You will have some PE -> KE -> heat, but since no external force is acting on the system the total energy will be constant.

If the masses are now pushed away from each other, to their original positions, at a slower velocity (say 1000 m/s), won't this take less energy than they had when they crashed together since they will now be back to their normal masses, as their masses won't be increased due to their slow speed.

No, it will take the same amount of energy, that is what it means to have a conservative potential. It takes the same amount of energy to get from one point to another regardless of the speed or path taken.

The attractive force between them will thus also be less.

Don't forget, even if you want to use the concept of relativistic mass, neither relativistic nor rest mass is the source of gravity in GR. In GR gravity is much more complicated than that. The source of gravity is the stress energy tensor of the Einstein field equations. This tensor has 16 terms, only 1 of which could be called mass.

 

[Naty1]

I think Dalespam's post looks good after a quick read...there's an awful lot embodied in your post question...this does not modify his explanations:

It might be easier for you to see what's happening if you just assume a mass comes whizzing in from a great distance, speed increases, it bypasses the other mass, maybe because of initial velocity, and keeps going now with decreasing speed...when it gets way back out the same distance on the other side, it's back to initial conditions...as I think Dalespam implies. If you assume heat or friction, energy might be manifested a bit differently.

If it did work, we could generate power that way!

In general, as you bring two masses closer together from infinity, zero gravitational potential energy at infinity becomes more negative, is converted to kinetic energy...positive work is then required to move the objects apart...

 

[Jandell254]

The invariant masses of the objects will not be increased. Nor will the energy of the system. You will have some PE -> KE -> heat, but since no external force is acting on the system the total energy will be constant..

However, as I understand it, while the masses are moving at 60% light speed, they will see each other as being heavier than their original, invariant rest masses. So would a stationary observer. Whether this mass comes from their rest mass or kinetic energy doesn't matter, it still increases the force acting between them, and if mass is being "added" it is no longer a closed system (although relativity probably has some explanation for why it stays a closed system...).

No, it will take the same amount of energy, that is what it means to have a conservative potential. It takes the same amount of energy to get from one point to another regardless of the speed or path taken..

It will take me the same amount of energy to lift an apple 1m higher above the earth, whatever route I take. This I understand according to classical physics. But if the mass of the apple varies with the velocity at which I move it, this rule of conservation of energy no longer applies. If it weighs 2N if I lift it straight up at 1m/s, and 10N when I drop it straight down at 2m/s, I could get more energy out than I put in.

Don't forget, even if you want to use the concept of relativistic mass, neither relativistic nor rest mass is the source of gravity in GR. In GR gravity is much more complicated than that. The source of gravity is the stress energy tensor of the Einstein field equations. This tensor has 16 terms, only 1 of which could be called mass.

I don't know the maths of relativity, but is there anything in the theory that cancels out the gravitational potential energy introduced by the introduction of "new" mass due to the velocity increase?
Like Hurkyll said, the main problem is that normal physics doesn't always apply in relativity, so I wouldn't know how to work out the total energy of the beginning and end states of the masses.

To make the problem simpler you could just consider a single mass, falling towards a much heavier, stationary mass at near light speed. the falling mass will weigh a lot and will have a very high velocity when it hits, therefore will have a lot of energy. When you lift it back to its original position at a slower velocity, it will be much lighter and this will thus take less energy, even considering the fact that it will take much longer to move it back to its original position.

Naty said that if the 2 masses in my original post went past each other, they would end up back where they where originally, just in opposite places. This makes sense in relativity and normal physics. The problem comes in when the masses collide, and no longer use their kinetic energy to escape from each others gravitational field.

 

[Dale]

Whether this mass comes from their rest mass or kinetic energy doesn't matter, it still increases the force acting between them,

The part in bold is incorrect. What you are calling "mass" is a specific deprecated concept known as "relativistic mass". There is no theory under which the source of gravity is relativistic mass. In Newtonian physics the source of gravity is rest mass (since there is no relativistic mass in Newtonian physics), and in General Relativity the source of gravity is the stress energy tensor.