Does the Unruh Effect Contradict Energy Principles in Flat Spacetime?

[Adam Lewis]

I have a question about the Unruh effect. For the spacetime to be flat, it must contain no energy. But the Rindler observer is supposed to be eternally accelerating, which would require infinite energy. Is there an obvious resolution to this apparent contradiction?

 

[Mentz114]

The Rindler observer does not need to be eternally accelerating. We are all Rindler observers if we are accelerating at a constant acceleration, and cease to be when we are not.

The energy for the acceleration comes from fuel, fed to the engines or mechanisms that make the acceleration.

Or perhaps you meant something different ?

 

[PeterDonis]

For the spacetime to be flat, it must contain no energy.

Actually, part of what makes the Unruh effect work is that "no energy" is frame-dependent. The concept of "energy" (at least as it's being used here) requires a notion of "time translations" (because of Noether's theorem), and the Unruh effect arises from the observation that a state which has zero energy with respect to the time translations of inertial observers will *not* have zero energy with respect to the time translations of accelerated observers.

This is relevant because it means that the Unruh effect is not limited to flat spacetime; it works in any spacetime which has multiple possible notions of "time translations". For example, a similar effect is observed in the spacetime surrounding a black hole: a state which has zero energy with respect to inertial observers freely falling into the hole does *not* have zero energy with respect to accelerated observers who are "hovering" at a constant height above the hole's horizon. (This also comes into play in explaining the Hawking effect, why black holes radiate when quantum effects are included.)

But the Rindler observer is supposed to be eternally accelerating, which would require infinite energy. Is there an obvious resolution to this apparent contradiction?

The "Rindler observer" in question, in the standard derivation of the Unruh effect, is a "test observer" who is idealized as having negligible rest mass and expending negligible energy to accelerate even for an indefinite period. Yes, this is unphysical, but it's only supposed to be an idealization. If you want to be more precise, you could say that the worldlines defined by the idealized "Rindler observers" can be picked out even if no real observer actually follows them for all time.

In a "real" scenario where the Unruh effect was observed, the accelerated observer would not be accelerating indefinitely; he would start accelerating at some finite time and stop accelerating at some later finite time. He would still observe the effect during the period when he was accelerating (but not when he was inertial).

 

[Adam Lewis]

I mean, the energy in the fuel should curve the spacetime, and that the situation being described is therefore not a solution to the field equations. By energy I mean the whole stress-energy tensor, which admittedly is not standard usage. (this was posted before PeterDonis' reply)

 

[Adam Lewis]

Actually, part of what makes the Unruh effect work is that "no energy" is frame-dependent. The concept of "energy" (at least as it's being used here) requires a notion of "time translations" (because of Noether's theorem), and the Unruh effect arises from the observation that a state which has zero energy with respect to the time translations of inertial observers will *not* have zero energy with respect to the time translations of accelerated observers.

This is relevant because it means that the Unruh effect is not limited to flat spacetime; it works in any spacetime which has multiple possible notions of "time translations".

That's interesting. Do you know what's supposed to happen for non-inertial observers in spacetimes with no timelike Killing vector? What does it mean more formally to have 'multiple possible notions of time translations'?

For example, a similar effect is observed in the spacetime surrounding a black hole: a state which has zero energy with respect to inertial observers freely falling into the hole does *not* have zero energy with respect to accelerated observers who are "hovering" at a constant height above the hole's horizon. (This also comes into play in explaining the Hawking effect, why black holes radiate when quantum effects are included.)

I mean, the stress-energy tensor should be zero since both of these are vacuum solutions. This seems to me to cause some tension with the assumption that the stress-energy associated with the detector is negligible, since any perturbation to the stress-energy will overwhelm 0. I agree you can still pick out the worldlines without actually following them, but doing so wouldn't necessarily bear much upon reality. If someone has worked this out for like a dust solution or something I'd feel a bit better.

In a "real" scenario where the Unruh effect was observed, the accelerated observer would not be accelerating indefinitely; he would start accelerating at some finite time and stop accelerating at some later finite time. He would still observe the effect during the period when he was accelerating (but not when he was inertial).

This would also make me a bit more comfortable; are you aware of a source that demonstrates it? I had thought the calculation made use of the asymptotic behaviour of the accelerated observer in order to e.g. build the Fock space.

Thanks for the reply, and sorry to drain your time; feel free to respond to none or only part of this.

 

[PeterDonis]

Do you know what's supposed to happen for non-inertial observers in spacetimes with no timelike Killing vector?

Not sure. Locally the reasoning that leads to the prediction of the Unruh effect should still hold true--if the quantum field is in a state that looks like vacuum (zero energy) to an inertial observer at some particular event, then an accelerating observer at that event should see the state as having non-zero energy. But globally, which state was "vacuum" to inertial observers might change. I haven't seen any real treatment of this case in the literature (not that my knowledge of the literature is very extensive).

What does it mean more formally to have 'multiple possible notions of time translations'?

Strictly speaking, a "time translation" is a Killing vector field with timelike orbits. To apply Noether's theorem you need a Killing vector field. So strictly speaking, the vector field of infalling inertial observers in Schwarzschild spacetime (around a black hole) is *not* a "time translation" because it's not a Killing vector field. However, for the purposes of the Unruh effect, any vector field with timelike orbits for which the "vacuum" state of the field is constant along the orbits will do, and I'm pretty sure that holds for infalling observers in Schwarzschild spacetime.

I mean, the stress-energy tensor should be zero since both of these are vacuum solutions.

Classically speaking, yes, but remember that the Unruh effect is a quantum effect. See below.

This seems to me to cause some tension with the assumption that the stress-energy associated with the detector is negligible, since any perturbation to the stress-energy will overwhelm 0.

Again, classically, yes, that's true, but once you're dealing with quantum fields the whole business of "zero" stress-energy is problematic. After all, formally, the energy in the quantum field even in the vacuum state is infinity, because you can have vacuum fluctuations in modes with arbitrarily high frequency (provided the fluctuations last a short enough time).

I agree you can still pick out the worldlines without actually following them, but doing so wouldn't necessarily bear much upon reality. If someone has worked this out for like a dust solution or something I'd feel a bit better.

I don't know that anybody has, but I know that you can do quantum field theory in curved spacetime, for example in Schwarzschild spacetime, and still have the Unruh effect come out. Even in a more realistic spacetime, like the Oppenheimer-Snyder solution matching a collapsing FRW dust to a Schwarzschild exterior, this would still cover anywhere in the exterior region, where there are certainly inertial infalling worldlines and accelerated "hovering" worldlines in vacuum. But I haven't seen a derivation of the Unruh effect for something like the interior, where the SET is nonzero throughout the region; I suspect that hasn't been considered because the whole point of the effect is that the quantum field is in the vacuum state with respect to the inertial observer, which certainly wouldn't be true for an inertial "comoving" observer inside the collapsing dust.

This would also make me a bit more comfortable; are you aware of a source that demonstrates it? I had thought the calculation made use of the asymptotic behaviour of the accelerated observer in order to e.g. build the Fock space.

I think Wald, in his book Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, talks about this, but I'd have to dig out my copy to be sure. Basically I believe you would use the same sort of reasoning that is used to justify using asymptotic fields in standard scattering theory--the incoming and outgoing particle modes aren't really at "past infinity" and "future infinity", but the timescales are so long compared to the interaction timescales that it doesn't matter, the answer you get by taking the limit as t goes to infinity is a good enough approximation to the answer at t = some finite time which is many orders of magnitude larger than the interaction time.