Unruh effect and energy imparted to a detector

[asimov42]

Hi all,

One more question about the Unruh effect, i.e., that an accelerating observer will see a vacuum filled with particles in thermal equilibrium (a warm gas) where an inertial observer would see zone.

If I have a detector moving with constant acceleration, how would I go about determining the amount of (thermal) energy imparted to the detector over a period time (from the warm gas, in the accelerated frame)? For some temperature T, would the detector continue to heat up until reaching T (assuming the detector was colder than T initially), and then remain at T (in thermal equilibrium).

Essentially, if I understand correctly - from an inertial observer's perspective, a portion of the energy powering the acceleration of the detector will go towards heating the detector. I'd like to work out what this quantity (amount of energy) is, over varying amounts of time.

 

[mfb]

I would expect that you can use the usual formulas for blackbody radiation (you need the surface area and heat capacity of your object), but I am not sure.

 

[haael]

It's not about accelerating detectors. It's about accelerating frames.

Suppose you have a thermometer floating in vacuum. An inertial observer passing by it will see it recording zero temperature. An accelerating observer will see a nonzero temperature indication, no matter how strange it sounds.

The detector movement has nothing to do. Accelerating thermometer will still show zero temperature for an inertial observer, since it moves through vacuum and it has nothing to collide with. However, if you were moving along with the accelerating thermometer, then you will see nonzero temperature on it.

This all seems strange, but the quantum mechanics IS strange. In particular, with Unruh effect you could have something similar to black hole complementarity. Suppose you have a thermometer with a red and a green lamp. If it records zero, it blinks red. If it records nonzero, it blinks green. With this setup, inertial and non-inertial observers will see different macroscopic reality. Suppose you have an accelerating observer holding a thermometer and a second inertial one. Also suppose that the accelerating observer is smiling when he sees a green light and sad when he sees red. Then the accelerating observer will himself see a green light and smile, but the inertial one will see a sad accelerating guy watching a red light. You have two completely different histories depending on the reference frame. With black hole complementarity the paradox is somewhat hidden, since one observer dives into a black hole and can not ever argue with the second one on the objective event course. With Unruh effect it is worse, since the accelerating observer may turn back and meet the inertial one. On the other hand, you have something similar to the event horizon in the Unruh effect (the Rindler horizon), but I don't know whether it can be used to avoid the paradox like with the black hole complementarity.

That said, I am not certain if all this is true. This is my limited understanding of the topic.

 

[mfb]

Suppose you have a thermometer floating in vacuum. An inertial observer passing by it will see it recording zero temperature. An accelerating observer will see a nonzero temperature indication, no matter how strange it sounds.

No, you can design a thermometer to produce frame-invariant readings (a digital display for example). Every observer has to agree on the temperature the thermometer measures in its frame.

Unruh effect is the opposite: An accelerating thermometer will measure a non-zero temperature. And all observers agree on that.

since it moves through vacuum and it has nothing to collide with.

Virtual particles.

 

[asimov42]

Hmm, now I'm a bit confused - do all observers have to agree that the thermometer shows the *same* temperature?

E.g., let's say Alice is sitting in an inertial frame, and Bob is in the same (uniformly accelerating) frame as our thermometer. Alice looks at the thermometer as it goes by - does she read the same temperature that Bob reads?

Put another way: does the thermometer (or whatever we're using as a detector) become thermally excited in the inertial frame?

No, you can design a thermometer to produce frame-invariant readings (a digital display for example). Every observer has to agree on the temperature the thermometer measures in its frame.

Unruh effect is the opposite: An accelerating thermometer will measure a non-zero temperature. And all observers agree on that.


Virtual particles.

 

[mfb]

do all observers have to agree that the thermometer shows the *same* temperature?

Right. Just let it have a digital display - if you apply Lorentz transformations to a digital display, it will not change its numbers.
Observers can disagree about the temperature of a sample (it is not trivial to define temperature of quickly moving objects at all), but the thermometer has some specific way to measure temperature - and physics is the same, independent of the frame you work in. You can always switch your point of view to the frame of the thermometer or any other frame.

 

[asimov42]

Doesn't the Unruh effect then imply that we have to adjust any calculation involving accelerating masses to consider losses due to thermalization of the masses?

E.g. If I have two large masses placed some distance apart, gravitational attraction will accelerate them towards one another. Along the way, both should become 'warm' due to the Unruh effect? Granted the amount of heating will be very small unless the masses are large.

Not sure if I'm on the right track here?

Right. Just let it have a digital display - if you apply Lorentz transformations to a digital display, it will not change its numbers.
Observers can disagree about the temperature of a sample (it is not trivial to define temperature of quickly moving objects at all), but the thermometer has some specific way to measure temperature - and physics is the same, independent of the frame you work in. You can always switch your point of view to the frame of the thermometer or any other frame.

 

[mfb]

I am not sure if gravitational attraction gives acceleration in that sense - in GR, that trajectory would correspond to zero acceleration in spacetime.

Granted the amount of heating will be very small unless the masses are large.

Even more important, you would need extremely cool objects, otherwise they are so hot that you cannot see the effect at all.

 

[asimov42]

Here's a related thermodynamics question:

Consider a very cold object being continuously accelerated. As a result of the Unruh effect, the object 'sees' a thermal bath (i.e., particles in thermal equilibrium) at some temperature T (assume that the object is initially much colder than T). As the object moves, it will begin to warm up...

Since energy is constantly being redistributed in random ways in the bath, is it possible for the temperature of the object to be greater than T at some point in time? I.e., can the object, considered alone, ever be hotter than T?

Even more important, you would need extremely cool objects, otherwise they are so hot that you cannot see the effect at all.

 

[mfb]

I would expect that the unordered energy can exceed the expectation value for temperature T a bit - probably not enough to consider it as "different temperature".

 

[asimov42]

Thanks mfb. Just a clarification/question about gravity:

If a mass is moving under the influence of gravity, does the Unruh effect apply? I've looked at a couple of papers that mention both linear and circular acceleration (when discussing the effect). But, if we consider, e.g., a planet orbiting the sun, this would imply that the planet 'loses' a small amount of energy on each orbit, due to heating (i.e., the Unruh effect). This doesn't seem right...

I am not sure if gravitational attraction gives acceleration in that sense - in GR, that trajectory would correspond to zero acceleration in spacetime.