hellfire said:
What bothers me, specially in case of the Unruh radiation, is the following: You can get the result of an observer dependent horizon considering only the action of some field (a scalar field for example) in flat spacetime viewed by an accelerated observer. This observer will detect a thermal bath of particles. Calculations and qualitative discussions I am aware of stop at that point. However, a thermal bath of particles should create a gravitational field that should pertub the curvature making it non-vanishing. But, does this make sense? The inertial observer and the accelerated observer, both in the "same" spacetime, would measure different curvatures.
For now, I'm going largely to restrict my comments to this passage.
Most courses in quantum field theory and books like Peskin and Schroeder use inertial observers in a particular solution to Einstein's equation, Minkowski spacetime. In particular, the contribution from quantum fields to the stress-energy tensor in Einstein's equation is not taken into account. Stress-energy tensors of qunatum fields are considered, but they're not fed into Einstein's equation. This is a very useful approximation - quantum fields propagating in Minkowki spacetime without affecting the spacetime background as viewed from inertial frames.
This leads naturally to the idea that doing something similar usiing non-inertial frames and/or non-flat spacetimes might be interesting. Take an interesting solution to Einstein's equation, and consider quantum fields propagating through spacetime without affecting the spacetime background, i.e., don't take into account the contribution from quantum fields to the stress-energy tensor in Einstein's equation.
This results in a big payoff - the Unruh effect, Hawking radiation, and cosmological radiation. The payoff is large, but the effects themselves are usually very small.
After doing this, the "back-reaction" of the stress-energy tensor of the quantum fields on spacetime can be considered, but the methods needed to do this are often quite subtle and difficult. Generalizing renormalization of the expectation values of components of the stress-energy tensor from Minkowski spacetime to other spacetimes is not straightforward because the concept of particle, as formulated in QFT in Minkowski spacetime, often no longer applies. A field (as opposed to particle) interpretation rules! I may talk a little more about this in another post in this thread.
Your point about the consistency required between the views different observers take for the Rindler spacetime Unruh effect is a good one. In this Unruh effect, suppose the quantum field in is the vacuum of the inertial observer. Then, the expectation values of the components of the stress energy tensor for this observer are all zero. Since the stress-energy tensor is a tensor, the values of the components of the stress energy tensor are all zero in every coordinate system, including the coordinate system of the accelerated observer.
What is happening here? How does the non-accelerated observer feel a temperature? To explain this, I'll quote a passage from Birrell and Davies about how an idealized accelerated particle detector reponds to the inertial vacuum.
"The explanation comes from a consideration of the agency that brings about the acceleration of the detector in the first place. As the detector accelerates, its coupling to the field causes the emission of quanta, which produces a resistance against the accelerating force. The work done by the external force to overcome this resistance supplies the missing energy that feeds into the field via the quanta emitted from the detector, and also into the detector which simulaneously makes upward transitions. But as far as the detector is concerned, the net affect is the absorption of thermally distributed quanta."
Of the three effects, I have worked through (a few years ago) the Unruh effect in some detail, looked at some of the details for Hawking radiation from eternal Schwarzschild black holes, and hardly looked at all at cosmological radiation.
As I said before, cosmological radiation is not just associated with expanding universes the have positive acceleration. I think that the point that
John Baez makes is as follows.
As pervect noted, cosmological radition is presently very small - much smaller than the cosmic background radiation. As our universe expands, the scale factor (according to present models) will go to infinity, causing the CMB temperature to tend to zero. At some point in the very distant future, cosmological (Unruh) radiation will dominate the CMB radiation.
Regards,
George
