Inertial observers explanation for unruh thermalization

[T S Bailey]

Imagine two observers, one accelerating and one inertial. The accelerating observer detects a thermal bath whereas the inertial observer detects a vacuum. I would imagine that this hot gas would thermalize the accelerating observers spaceship, but how does the inertial observer reconcile this? They saw no such gas, so what is the origin of the energy?

 

[Geofleur]

Why do you say that the inertial observer would not see the thermal bath?

 

[T S Bailey]

Why do you say that the inertial observer would not see the thermal bath?

Because in quantum mechanics energy states are defined by a hamiltonian which is dependent on a time coordinate. But in special relativity Lorentz transformations only work for inertial frames, there is no way to transform to an accelerated frame. Since they don't share a time coordinate the observed energy for the vacuum will be different for an accelerating observer, ie the hamiltonian for each frame is different and so different quantum states are observed.

 

[Geofleur]

There is, in fact, a way to use special relativity to study accelerated frames. Choose an inertial frame that is instantaneously traveling at the same velocity as the spaceship. There will then be a Lorentz transformation relating time in the first inertial frame to that in the momentarily co-moving inertial frame.

 

[T S Bailey]

There is, in fact, a way to use special relativity to study accelerated frames. Choose an inertial frame that is instantaneously traveling at the same velocity as the spaceship. There will then be a Lorentz transformation relating time in the first inertial frame to that in the momentarily co-moving inertial frame.

It sounds like you're talking about Rindler coordinates which has the same effect. In this case I believe the problem is that, since a greatly accelerating observer has less available information (their light cone has diminished), there has been a corresponding increase in entropy. This increased entropy is then invoked as the reason for the thermal bath.

 

[Geofleur]

It seems that you are describing the Unruh effect (https://en.wikipedia.org/wiki/Unruh_effect), which I had not heard of before! Ah, I see it in the question title... erg. Maybe someone else can shed some light on this - I'm still learning QFT.

 

[SlowThinker]

Did you read the very first thread that comes up when searching PF for "Unruh"? It's fairly enlightening.
The general idea is that the inertial observer sees a particle being emitted from the accelerated detector, while the accelerated detector hits an antiparticle.
The Unruh effect is certainly a fascinating subject, with many interesting aspects, most notably what does a "particle" even mean.

 

[T S Bailey]

Did you read the very first thread that comes up when searching PF for "Unruh"? It's fairly enlightening.
The general idea is that the inertial observer sees a particle being emitted from the accelerated detector, while the accelerated detector hits an antiparticle.
The Unruh effect is certainly a fascinating subject, with many interesting aspects, most notably what does a "particle" even mean.

I will check it out.

 

[PeterDonis]

I would imagine that this hot gas would thermalize the accelerating observers spaceship, but how does the inertial observer reconcile this?

Suppose the accelerating detector detects a particle from the hot gas. This means that the detector undergoes a state transition: heuristically, it gains a quantum of energy from the quantum field (by absorbing the detected particle). But that means the quantum field must also undergo a state transition; heuristically, it must lose a quantum of energy corresponding to the energy absorbed by the detector.

To an inertial observer, this process will look like the detector is emitting a particle into the vacuum. In other words, to an inertial observer, the detector will lose a quantum of energy (but "energy" is now defined according to the inertial observer's time translation symmetry), and the field will gain one. So the quantum field will no longer be a vacuum; it will have a particle in it, emitted by the detector.

The thermalization of the accelerating detector by the hot gas will just be the total effect of a lot of these processes; assuming that the detector starts out at zero temperature, according to the accelerated observer, it will gradually heat up by absorbing particles from the hot gas, and the gas will cool down as it loses particles to the detector. They will eventually equilibrate at some intermediate state where the detector has absorbed a bunch of particles, and the quantum field has changed state to one containing that many fewer particles.

To an inertial observer, this process will look like the detector is cooling down--losing energy by emitting particles--while the quantum field is heating up--gaining energy by changing state from the vacuum to one containing a bunch of particles.

 

[PeterDonis]

They saw no such gas, so what is the origin of the energy?

Ultimately, it's coming from the energy that is causing the detector to accelerate.