thermodynamics - Are information conservation and energy conservation related?

as evident from the title, are both, conservation of energy and conservation of information two sides of the same coin??

Is there something more to the hypothesis of hawking's radiation other than the fact that information cannot be lost? or can I say energy cannot be lost??

Answer

First of all I do not think that conservation of information is an established statement. It seems to be an open problem still as far as black holes go.

Even if true, it is a different type of conservation, analogous to the unitarity requirements of a system of functions or phase space considerations.

From the conclusions of a paper by Hawking :

In this paper, I have argued that quantum gravity is unitary and information is preserved in black hole formation and evaporation. I assume the evolution is given by a Euclidean path integral over metrics of all topologies. The integral over topologically trivial metrics can be done by dividing the time interval into thin slices and using a linear interpolation to the metric in each slice. The integral over each slice will be unitary and so the whole path integral will be unitary. On the other hand, the path integral over topologically non trivial metrics will lose information and will be asymptotically independent of its initial conditions. Thus the total path integral will be unitary and quantum mechanics is safe.

How does information get out of a black hole? My work with Hartle[8] showed the radiation could be thought of as tunnelling out from inside the black hole. It was therefore not unreasonable to suppose that it could carry information out of the black hole. This explains how a black hole can form and then give out the information about what is inside it while remaining topologically trivial. There is no baby universe branching off, as I once thought. The information remains firmly in our universe. I’m sorry to disappoint science fiction fans, but if information is preserved, there is no possibility of using black holes to travel to other universes. If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized. It is like burning an encyclopedia. Information is not lost, if one keeps the smoke and the ashes. But it is difficult to read. In practice, it would be too difficult to re-build a macroscopic object like an encyclopedia that fell inside a black hole from information in the radiation, but the information preserving result is important for microscopic processes involving virtual black holes. If these had not been unitary, there would have been observable effects, like the decay of baryons.

Energy is a conserved quantity because of Noether's theorem: wherever it holds, energy is conserved. In extreme General Relativity scenaria energy itself loses its meaning, whereas phase space and unitarity may hold and if Hawking is correct, information is conserved.

So energy conservation and possible conservation of information are two unconnected effects.