Energy Conservation and Expansion of Space: A Thought Experiment

[sten]

Here's a thought experiment with a few questions:

Space expands, probably at accelerating rate.

We can make a thought experiment like this: there's a star surrounded by billions of light-years of empty space (say 4300 Mpc radius, well beyond the size of what we see - it's a thought experiment). On the outer layer of our empty space are solar cells collecting the energy of the star at all wavelengths at 100% efficiency. At that distance Hubble's law states the run-away speed would be above c (assuming 70km/s/Mpc value of Hubble's constant and assuming the law is valid over such distances and that it's linear in nature)

So... what's the sum of the energy collected by all solar cells?

If light never reaches the solar cells then the total energy output of the star at that distance will be Zero... I suppose?

So the star has event horizon even though it's not a black hole?

And if the energy from the star was sent "somewhere" but never reached the solar cells then what happened to it?

 

[sten]

My own take on this experiment's interpretation is that the energy was not "lost", it was used to "fuel" the expansion (therefore the accelerating rate).

That would mean dark energy is not that dark in its origin and that we'd never see it.
Even further - energy and space will have to be equivalent somehow, similar to the way energy and mass are equivalent

 

[bcrowell]

So the star has event horizon even though it's not a black hole?

Yes. In fact, you don't need the star at all. Any observer at any position in space has a horizon in this cosmological model. There's nothing really unusual about this. Horizons are a very commonplace phenomenon in GR. For instance, any accelerating observer has a horizon.

And if the energy from the star was sent "somewhere" but never reached the solar cells then what happened to it?

It still exists. It just doesn't escape. (Also, GR doesn't really have global conservation laws, but that's probably beside the point.)

 

[sten]

If the outside "bubble" is unaware if a star exists in its center and only uses energy measurements to verify the existence of such a star then the conclusion is that no energy is output and therefore no star is present (it could be black hole if gravity can be measured). So to an observer located at the solar cells no energy is coming out of the bubble.
To this observer the bubble (or the part of it he can see) is growing faster and faster and acceleration takes energy (what we call "dark")
When I put these things together I see this: energy is released inside the bubble, energy doesn't leave the bubble, bubble grows at accelerating rate (which requires energy).

How would you define where does the energy reside if it exists somewhere?

 

[Ich]

There are some fine points.

Any observer at any position in space has a horizon in this cosmological model.

Obviously, you're referring to a cosmology with a cosmological constant. It is not clear that this is the OP's intention, and I don't think that one should leave the OP's presumption "empty space" uncommented then.

In a really empty universe, the solar cells are not behind a horizon. Without going into details: the light will reach them, albeit redshifted, and will increase their kinetic energy by the amount needed to conserve energy.
It is important in this context that energy conservation depends on the coordinates you use. My comment is based on standard coordinates, not cosmological ones.

 

[sten]

if you sum the red-shifted energy received by the solar cells your total would be less than the star's output as measured near its surface. where did the rest of the energy go?

 

[Ich]

where did the rest of the energy go?

To the kinetic energy of the solar cells. The light accelerates the cells away from the sun.
From the sun's point of view, where the light is not redshifted, the cells convert only part of its energy to electricity and heat (the redshifted part), while the rest increases the kinetic energy of the cells.