Apparent paradox: e=mc^2 violates conservation of energy?

[Codebender]

Take two uncharged particles (e.g., neutrons) at rest with respect to one another in empty space. Now apply 1 Joule of energy to accelerating them apart. They will eventually come again to rest with 1 Joule of gravitational potential and accelerate back together due to gravity. Their kinetic energy when they return to their original position will be the same 1 Joule that we originally gave them.

According to Special Relativity, energy and mass are equivalent and transmutable, so now instead of accelerating them apart, convert them each to mc² worth of energy (a hard gamma ray), and allow it to depart in opposite directions at the speed of light. Some arbitrary time later, convert each gamma ray back into a neutron. The two particles will then have gravitational potential that apparently came from nowhere, and will accelerate together yielding kinetic energy that they apparently should not have.

The potential solutions I've come up with so far are:

1. A limit on the conversion efficiency such that the "extra" energy must be put in as part of the mass/energy conversions, but that doesn't really make sense.

2. The energy of the photons somehow dissipates over distance at the same rate the gravitational potential increases. But this would make it impossible for us to see stars that are billions of light-years away.

 

[Privalov]

Now apply 1 Joule of energy to accelerating them apart. They will eventually come again to rest with 1 Joule of gravitational potential

It’s not really possible, since gravitational attraction of two neutrons is too small – they will run away to infinity. However, let’s assume we use smaller portion of energy, that 1 Joule.

Some arbitrary time later, convert each gamma ray back into a neutron.

It will not be possible. Photons will lose small fraction of their energy – what will make it not enough to produce a neutron.

2. The energy of the photons somehow dissipates over distance at the same rate the gravitational potential increases.

Exactly.

But this would make it impossible for us to see stars that are billions of light-years away.

This effect is not strong enough for that. It can only slightly shift the spectrum line of massive stars – so we can detect it and conclude General Relativity is correct; but that’s pretty much it. We can’t see this minor color change with naked eye.

 

[Codebender]

gravitational attraction of two neutrons is too small

Yes, of course. Consider and erg instead of a Joule or perhaps a larger object than a single particle. I was only trying to avoid electromagnetic or other effects that might confuse the issue.

Photons will lose small fraction of their energy – what will make it not enough to produce a neutron.

I'm assuming empty space here, so without the intervention of the few atoms of hydrogen that pervade intergalactic space, where does the energy go? It has to go somewhere or conservation remains violated. By what mechanism can the wavelength of a photon change while it travels through empty space?

And even so, it doesn't solve the problem. Let's exaggerate that half of the energy is dissipated, so that instead of the original objects I can only make an object half as massive out of the energy that arrives. (e.g., a golf ball instead of a baseball). These smaller objects still have gravitational potential that is unexplained. In order to retain conservation, the light would have to immediately be attenuated to nothing.

This effect is not strong enough for that. It can only slightly shift the spectrum line of massive stars

This sounds like doppler red-shift to me, related to relative velocity. I can't find any reference at all about changes in spectra over distance. Can you point me in the right direction, or tell me the proper name of this effect?

 

[Privalov]

I'm assuming empty space here, so without the intervention of the few atoms of hydrogen that pervade intergalactic space, where does the energy go?

You derived the resolution of this paradox yourself – the missing energy goes to the gravitational potential energy of the photons.

As particles move apart, against the force of each other’s gravity, their frequency and their energy decrease. At the same time, their gravitational potential energy increase by the same amount.

By what mechanism can the wavelength of a photon change while it travels through empty space?

I’m unaware of existence of any definitive answer to that question (other than “laws of General Relativity dictate that”).

Some researchers speculate gravity acts through exchange of virtual particles – gravitons. There is no completed theory at this moment though.

Let's exaggerate that half of the energy is dissipated, so that instead of the original objects I can only make an object half as massive out of the energy that arrives.

In order to be able to cut outgoing energy in half, baseball must be almost as massive, as a black hole. Let’s assume this theoretical possibility and continue.

These smaller objects still have gravitational potential that is unexplained.

It’s perfectly explained. Half of object mass were converted to object’s potential energy. (Although, we need an object as large as a black hole, in order to generate gravitational field strong enough to allow these kinds of potentials).

In order to retain conservation, the light would have to immediately be attenuated to nothing.

In fact, light will lose half of its energy. This is how you defined the size of a baseball: whatever energy is trying to escape will be cut in half.

This sounds like Doppler red-shift to me

Similar outcome; completely different cause. Doppler red-shift is caused by the fact distant galaxies are moving away from us. Obviously, all kinds of waves we can receive from there will have smaller frequency (comparing to frequency, measured by an observer at that distant galaxy).

I can't find any reference at all about changes in spectra over distance.

It’s not a change in spectra over distance; it’s change in spectra of a light, escaping massive object (like a star).

Can you point me in the right direction, or tell me the proper name of this effect?

It’s called gravitational redshift.

http://en.wikipedia.org/wiki/Gravitational_redshift

 

[Codebender]

I think I understand, thank you for your patience. The gravitational red/blue-shift explains it, somehow I had never come across that phenomenon.

I'm still missing something, though:

As particles move apart, against the force of each other’s gravity, their frequency and their energy decrease. At the same time, their gravitational potential energy increase by the same amount.

Photons have zero rest mass, and therefore do not generate any gravitational force, right? If I recall correctly, they bend around masses because space is distorted by the gravity of that mass, not because they interact with it gravitationally. Thus at no point does the photon every have any gravitational potential. So as the photon loses energy it's as if the energy has to be "stored" somewhere until the light becomes matter again, and the energy lost during transit out of a gravity gradient (possibly very far away) suddenly becomes gravitational potential.

 

[Privalov]

Photons have zero rest mass, and therefore do not generate any gravitational force, right?

No.

Wikipedia says: "Since photons contribute to the stress-energy tensor, they exert a gravitational attraction on other objects"

http://en.wikipedia.org/wiki/Photon

Thus at no point does the photon every have any gravitational potential.

Photon has a miniscule gravitational potential, which can be ignored at most cases (but not in this particular problem).