Potential energy and GR, something doesn't add up

[tbitz]

Here is a thought experiment that doesn't seem to add up.

Suppose you have a unit of mass at rest on the surface of Earth that spontaneously transforms to photos all moving radially outward into space. At some distance (say 1Km) the photons revert back to the original mass but are now 1Km above the surface of the Earth at rest. During this process no external energy has been added.

Now the mass at 1Km above the Earth begins to fall. When it passes it's original location it now has the same mass as when it started but has this additional kinetic energy. Where did this additional energy come from?

Tony

 

[clj4]

sorry, my error

 

[masudr]

This is nothing to do with GR. When the mass spontaneously becomes photons, they would have to radiate in ALL directions in order to conserve momentum. That includes up and down. This should solve your problem.

 

[Garth]

Not really, the isotropic radiation could be reflected by an external parabolic miror to the point 1 km high. The back reaction on the mirror would explain the change in momentum.

The answer to the problem is the photons at 1 km would be gravitationally red shifted (GR) and therefore of less total energy than originally. When converted back into mass it would be less than the original mass by the loss of potential energy/c². When that fell back to Earth it would regain that loss of mass/energy as kinetic energy and theoretically could be brought to a halt and that KE changed into extra mass so it ends up the same mass as it began, (perfect efficiency assumed of course!)

I hope this helps.

Garth

 

[tbitz]

Not really, the isotropic radiation could be reflected by an external parabolic miror to the point 1 km high. The back reaction on the mirror would explain the change in momentum.

The answer to the problem is the photons at 1 km would be gravitationally red shifted (GR) and therefore of less total energy than originally. When converted back into mass it would be less than the original mass by the loss of potential energy/c². When that fell back to Earth it would regain that loss of mass/energy as kinetic energy and theoretically could be brought to a halt and that KE changed into extra mass so it ends up the same mass as it began, (perfect efficiency assumed of course!)

I hope this helps.

Garth

Thats a good explanation. Thanks. (I'll have to assume the math works).

On the bit about momentum, what if there is enough mass to create one photon at a given wavelength. It would travel out radially so no mirrors needed. The question now is the momentum was zero before the transformation, but non-zero after the single photon was created. How can that be?

Is the answer simply this cannot happen because it would violate conservation of momentum?

Tony

 

[masudr]

No process can create a photon in a vacuum - there has to be two photons. A single photon can be created near a massive particle, since the massive particle can recoil.

 

[pmb_phy]

Here is a thought experiment that doesn't seem to add up.

Suppose you have a unit of mass at rest on the surface of Earth that spontaneously transforms to photos all moving radially outward into space. At some distance (say 1Km) the photons revert back to the original mass but are now 1Km above the surface of the Earth at rest. During this process no external energy has been added.

Now the mass at 1Km above the Earth begins to fall. When it passes it's original location it now has the same mass as when it started but has this additional kinetic energy. Where did this additional energy come from?

Tony

In the given frame of reference the spacetime is static (g_uv = constant in time) then the 4-momentum of the original particle will separate into two photons, each with an energy (i.e. E = P₀) is a constant of motion. So the total energy of the two photons remains constant. Therefore when they return to the same location and once again form a single particle the energy and the mass will be the same as when the process started.

Pete