Why doesn't an electron sitting on the Earth radiate?

[craigthone]

I have some puzzles when I combine the Equivalence principle and Unruh's effect.
Equivalence principle says that acceleration is equivalent to a uniformly gravitational field, or locally equivalent to gravitational field. Then the electron sitting on Earth should have Unruh's effect, i.e. radiate. What is wrong with the argument?

 

[Dale]

Don’t forget to include the detector in your setup. Is it free falling or co-accelerating?

 

[craigthone]

Sorry, the co-accelerating detector. The observer sits on the surface also.

 

[PeterDonis]

Equivalence principle says that acceleration is equivalent to a uniformly gravitational field, or locally equivalent to gravitational field.

Not quite. It says that accelerating in flat spacetime is locally equivalent to being at rest in a uniform gravitational field.

Then the electron sitting on Earth should have Unruh's effect, i.e. radiate.

No, that's not what the Unruh effect says. The Unruh effect says that an accelerating detector should detect radiation from the quantum vacuum. However, the intensity of the radiation for a 1 g acceleration is way, way too small to actually detect.

 

[craigthone]

No, that's not what the Unruh effect says. The Unruh effect says that an accelerating detector should detect radiation from the quantum vacuum. However, the intensity of the radiation for a 1 g acceleration is way, way too small to actually detect.

Forget the electron, do observers (detector) sitting near Earth see Unruh's radiation? I know the answer is no, but I do not have good explanations. Maybe I should ask in flat space for accelerating observer what is the quantum state corresponding to vacuum for static observer in the earth?

 

[haushofer]

An accelerating observer observes an event horizon, like an observer at rest near a BH. An observer on Earth doesn't.

 

[Khashishi]

There's some discussion on this here.
https://physics.stackexchange.com/questions/2696/do-all-massive-bodies-emit-hawking-radiation
I think the consensus is no, you don't get Unruh radiation because the Unruh radiation is perceived as coming from the Rindler horizon. But Earth has no horizon. The equivalence principle only applies locally - an accelerating frame looks like gravity. But an accelerating frame is equivalent to a constant gravitational field which extends to infinity. There is a Rindler horizon, which is some distance behind the accelerating frame where not even light can catch up with the center of the frame. But Earth's gravity isn't a constant gravitational field, and there isn't a horizon.

 

[PeterDonis]

Can I say that "being at rest in a uniform gravitational field is locally equivalent to accelerating in flat spacetime, free falling in a uniform gravitational field is locally equivalent to nonacceleration in flat spacetime ?"

Yes.

do observers (detector) sitting near Earth see Unruh's radiation? I know the answer is no, but I do not have good explanations.

I already answered this. We don't see Unruh radiation on Earth because it's way too faint for us to detect.

An accelerating observer observes an event horizon, like an observer at rest near a BH.

The Rindler horizon is not an event horizon; it's not the boundary of a region that can't send light signals to future null infinity. There are some local similarities between the two, but they're not the same.

[Edit: removed final statement, not correct.]

 

[PeterDonis]

I think the consensus is no, you don't get Unruh radiation because the Unruh radiation is perceived as coming from the Rindler horizon.

I know this is a common heuristic, but I'm not sure it's actually true. The actual derivation of the Unruh effect does not rely on the presence of the Rindler horizon. It relies on the fact that, in quantum field theory, which state is the "vacuum" depends on your state of motion. In particular, if the quantum field in flat spacetime is in a vacuum state with respect to inertial observers, that same state will not be a vacuum state with respect to an accelerated observer; it will be a thermal state at the Unruh temperature, and therefore the accelerated observer will observe radiation at that temperature. That is true right at the location of the accelerated observer; there's no need to reach out to the Rindler horizon.

 

[sweet springs]

OP 's concern is radiation from the event horizon, but as a layman I have a more primitive question of radiation from the charge itself.
Dose a charge sitting on the floor under gravitation of the Earth or accelerating rocket emit light by radiation? I do not think so. I would like to know the condition of radiation more in detail.

 

[Dale]

Sorry, the co-accelerating detector. The observer sits on the surface also.

So am accelerating electron (in an elevator) will not produce any radiation that can be detected by a co-accelerating detector. By the equivalence principle the same is what we expect for the charge and detector at rest in gravity.

 

[craigthone]

Yes.
I already answered this. We don't see Unruh radiation on Earth because it's way too faint for us to detect.

I do not think so.
From QFT in Rinler space, the argument of Unruh's effect takes a special state $\rho=\exp(-2\pi H)$ where $H$ is the Rindler Hamiltonian and the Minkowski boost, this state $\rho=\exp(-2\pi H)$ is Minkowski vacuum. While for quantum fields on the earth, expanding around the surface, we do not have such state, and no compactification of Euclidean time and no temperature. The picture is totally different from near black hole horizon case. So there is no Unruh radiation rather than to weak to detect.

 

[stevendaryl]

This picture tries to give an intuitive reason that an accelerated charge radiates. Imagine that you have a charged particle moving along the x-axis at a constant speed. The electric field associated with that situation points toward the moving particle. Now, at some point, the particle abruptly halts. Nearby the particle, the electric field points toward the particle. But the information that the particle's motion has changed hasn't reached far-away places, so the electric field continues to point toward where the particle would have been, if it kept traveling at the same speed. Because electric fields have to be continuous in vacuum, there must be a transition region, spreading out spherically at the speed of light, where the electric field rapidly changes from one configuration to the other. This region of rapidly changing electric (and magnetic, since that changes, too) fields is a spherical burst of electromagnetic radiation. This is shown in the following picture:

To the extent that this intuitive picture is accurate, I think we can conclude that there is no radiation as viewed from a reference frame in which the situation is static. If the electric field far away from the charge isn't changing with time, then there shouldn't be a need for the radiation, which sort of acts like a "correction" to the field that propagates at the speed of light.

The interesting case is a charged particle undergoing constant proper acceleration. From the point of view of a Rindler observer, the situation is static, so there should be no radiation. But from the point of view of an inertial observer, the situation is not static, and it's also not the constant-velocity case. I'm not sure what the electric field looks like to the inertial observer.

A charge at rest on a planet shouldn't radiate, according to this heuristic, because the distant electric field is unchanging (in the frame in which the situation is static, which is someone at rest in the Schwarzschild coordinate system).

 

[stevendaryl]

This picture tries to give an intuitive reason that an accelerated charge radiates. Imagine that you have a charged particle moving along the x-axis at a constant speed. The electric field associated with that situation points toward the moving particle. Now, at some point, the particle abruptly halts. Nearby the particle, the electric field points toward the particle. But the information that the particle's motion has changed hasn't reached far-away places, so the electric field continues to point toward where the particle would have been, if it kept traveling at the same speed. Because electric fields have to be continuous in vacuum, there must be a transition region, spreading out spherically at the speed of light, where the electric field rapidly changes from one configuration to the other. This region of rapidly changing electric (and magnetic, since that changes, too) fields is a spherical burst of electromagnetic radiation. This is shown in the following picture:
View attachment 225442

To the extent that this intuitive picture is accurate, I think we can conclude that there is no radiation as viewed from a reference frame in which the situation is static. If the electric field far away from the charge isn't changing with time, then there shouldn't be a need for the radiation, which sort of acts like a "correction" to the field that propagates at the speed of light.

The interesting case is a charged particle undergoing constant proper acceleration. From the point of view of a Rindler observer, the situation is static, so there should be no radiation. But from the point of view of an inertial observer, the situation is not static, and it's also not the constant-velocity case. I'm not sure what the electric field looks like to the inertial observer.

A charge at rest on a planet shouldn't radiate, according to this heuristic, because the distant electric field is unchanging (in the frame in which the situation is static, which is someone at rest in the Schwarzschild coordinate system).

I have to caution readers that I'm not at all claiming to be an expert in this. I've just given an intuitive heuristic that may reflect a misunderstanding on my part.

 

[Vanadium 50]

This thread - as is the case for this topic, as well as it's evil twin, radiation of a charge in free fall - is a bit of a mess. But before I respond to the mention of my name, is this really at the A-level? It seems not to be. I am not going to spend a lot of time on writing an A-level answer only to get a reply like "What is a multipole?" or "What 16 components?"

 

[craigthone]

This thread - as is the case for this topic, as well as it's evil twin, radiation of a charge in free fall - is a bit of a mess. But before I respond to the mention of my name, is this really at the A-level? It seems not to be. I am not going to spend a lot of time on writing an A-level answer only to get a reply like "What is a multipole?" or "What 16 components?"

We need A level answers. It is better to have some equations rather than something else. Thanks in advance.

 

[PeterDonis]

From QFT in Rinler space, the argument of Unruh's effect takes a special state $\rho=\exp(-2\pi H)$ where $H$ is the Rindler Hamiltonian and the Minkowski boost, this state $\rho=\exp(-2\pi H)$ is Minkowski vacuum. While for quantum fields on the earth, expanding around the surface, we do not have such state

Technically, you're correct, an observer standing on Earth's surface is not in vacuum, there's air around him and ground below him. But imagine, for example, someone standing on an airless planet (or on the Moon). There is vacuum around that person, and they are accelerated, so they should in principle observe Unruh radiation. That's all I was trying to say.

 

[Vanadium 50]

Then the electron sitting on Earth should have Unruh's effect, i.e. radiate. What is wrong with the argument?

It's confusing two things. Are you saying this electron should radiate because it is charged? Or are you saying it should radiate because of the Unruh effect, which has nothing to do with charge? Until we get the question straight, it's unlikely we will be able to provide a good answer.

 

[craigthone]

It's confusing two things. Are you saying this electron should radiate because it is charged? Or are you saying it should radiate because of the Unruh effect, which has nothing to do with charge? Until we get the question straight, it's unlikely we will be able to provide a good answer.

In the process of discussing, I think the first question (this electron should radiate because it is charged) is answered by Dale using equivalence principle correctly.

The second question is Forget the electron, do observers (detector) sitting near Earth see Unruh's radiation? I know the answer is no, but I do not have clear argument.

[Moderator's note: Question about EP deleted; it was already asked in a previous post that has been moved to a new thread.]

 

[Vanadium 50]

The heuristic is that Unruh radition is a consequence of applying the Equivalence Principle to Hawking radiation. If you go the other way, you get Hawking radiation back - i.e. the geometry of a black hole, not the geometry of a planet.

If you are wondering why BH's emit Hawking radiation and planets don't, the easiest way to see this is conservation of energy. A BH can get energy by shrinking its horizon. Since a planet doesn't have a horizon, there's nowhere for the energy to come from.

 

[Dale]

The heuristic is that Unruh radition is a consequence of applying the Equivalence Principle to Hawking radiation. If you go the other way, you get Hawking radiation back

Very well explained!

 

[PeterDonis]

The heuristic is that Unruh radition is a consequence of applying the Equivalence Principle to Hawking radiation.

I would put this the other way around: Hawking radiation is (heuristically) the result of applying the EP to Unruh radiation. Unruh radiation can be derived rigorously using QFT in flat spacetime. Hawking radiation, if you want to try to derive if from first principles, has to be derived using a semi-classical approximation--QFT in a fixed curved background spacetime--which is not rigorous, and nobody AFAIK has figured out how to make it rigorous.

If you are wondering why BH's emit Hawking radiation and planets don't, the easiest way to see this is conservation of energy. A BH can get energy by shrinking its horizon. Since a planet doesn't have a horizon, there's nowhere for the energy to come from.

First, I think you mean "A BH can lose energy by shrinking its horizon".

Second, a planet has lots of other ways of losing energy, so I don't think "not being able to lose energy" is a good heuristic for why planets don't emit Hawking radiation. I think a much better heuristic (which is based, not on Hawking's original argument, but on how our understanding has developed in the decades since he first published his result) is that Hawking radiation is associated with trapped surfaces, not event horizons. For a black hole, the two coincide (or almost coincide, since a real hole will be accreting matter), but the difference is that the trapped surface is a local phenomenon--radially outgoing light does not move outward--so it's much easier to see how an object can "tell" whether it's "allowed" to emit Hawking radiation (a planet has no trapped surface, so no Hawking radiation, even though a planet can radiate in lots of other ways).

 

[Vanadium 50]

First, I think you mean "A BH can lose energy by shrinking its horizon".

I mean it can transfer energy from the horizon to the radiation field. A planet cannot do this.

Second, a planet has lots of other ways of losing energy

And the PF quibbling begins!

You know what I meant. Yes, you can make this more complicated. No, that doesn't help with the understanding.

I knew that getting involved in this morass of a thread was a mistake. Lesson learned.

 

[PeterDonis]

do observers (detector) sitting near Earth see Unruh's radiation? I know the answer is no

I'm not sure the answer is no. See my post #10. In practice, Unruh radiation for an object accelerating at 1 g is many, many orders of magnitude too weak for us to detect, so there's no way of testing the question experimentally.

 

[PeterDonis]

Moderator's note: Posts about the equivalence principle have been moved to a new thread:

https://www.physicsforums.com/threads/local-meaning-of-equivalence-principle.947092/

Please direct further discussion of the EP there.

 

[pervect]

I'm regarding this as a "warm up" problem, but maybe it belongs in another thread. Let's take a case of accelerating a small, dense charge , initially at rest in flat space-time, with a constant acceleration for some finite period of time.

What is the radiation reaction force needed to accelerate the charge, how do we calculate it? Is there a natural limit from "small and dense" to "point charge", and how do we take this limit?

My understanding is basically that this isn't really very straightforwards, but I don't see how we can get very far with the more complicated questions if we don't understand this one.

I suppose we'd get the field from the retarded potential, that's about as far as my thinking has gone, and that doesn't really address the question of the "point charge limit".

 

[haushofer]

The Rindler horizon is not an event horizon; it's not the boundary of a region that can't send light signals to future null infinity. There are some local similarities between the two, but they're not the same.

I know this is a common heuristic, but I'm not sure it's actually true. The actual derivation of the Unruh effect does not rely on the presence of the Rindler horizon. It relies on the fact that, in quantum field theory, which state is the "vacuum" depends on your state of motion. In particular, if the quantum field in flat spacetime is in a vacuum state with respect to inertial observers, that same state will not be a vacuum state with respect to an accelerated observer; it will be a thermal state at the Unruh temperature, and therefore the accelerated observer will observe radiation at that temperature. That is true right at the location of the accelerated observer; there's no need to reach out to the Rindler horizon.

I'm also doubting now. Unruh radiation follows from the Fourier expansion of the quantum field being only covariant for inertial observers. In that sense its derivation is similar to deriving inertial forces from the fact that Newton 's 2nd law is only covariant for inertial observers.

 

[stevendaryl]

I know this is a common heuristic, but I'm not sure it's actually true. The actual derivation of the Unruh effect does not rely on the presence of the Rindler horizon. It relies on the fact that, in quantum field theory, which state is the "vacuum" depends on your state of motion. In particular, if the quantum field in flat spacetime is in a vacuum state with respect to inertial observers, that same state will not be a vacuum state with respect to an accelerated observer; it will be a thermal state at the Unruh temperature, and therefore the accelerated observer will observe radiation at that temperature. That is true right at the location of the accelerated observer; there's no need to reach out to the Rindler horizon.

Well, the fact that the definition of vacuum is dependent on your state of motion doesn't distinguish between black holes and other gravitating bodies, and doesn't explain why Hawking radiation should appear to be coming from the event horizon, rather than everywhere. Why isn't the presence of curvature enough for Hawking radiation? I'm not questioning the mathematics that you're quoting, I'm just saying that that heuristic for explaining Hawking radiation doesn't really understand the main facts about Hawking radiation.

Hawking's original layman's explanation (which everyone says is so bad) in terms of virtual particles had the benefit of connecting the radiation to the event horizon: Radiation occurs when a negative-energy virtual particle falls into the event horizon and is trapped. That intuitively explains why black holes are relevant, and why you don't see the radiation everywhere that space is curved. So from the point of view of explaining things without getting into the details of the equations, it seems to me that a better intuitive explanation should at least have this feature.

 

[pervect]

My understanding is not good, but if we uniformly accelerating a charge in flat space-time we say that it radiates energy. This energy has to come from somewhere - it comes from the radiation reaction force needed to accelerate the charge.

My understanding (which may be wrong) is that in the accelerated frame it appears as if this energy is not radiated, but absorbed. I don't quite understand the calculations which are allegedly supposed to say this , the bogoiubov transformations. This sort of sign reversal doesn't seem to me to be possible if we view the radiated energy as being carried away by time-like worldlines, the only thing I can assume is that the model of energy we are using has the energy being carried away by space-like worldlines.

Exactly what model of energy are we using? It's not clear, but I assume it has energy as a conserved quantity, and that it's related to the Killing vectors of whichever space-time we're using (the Rindler space-time for the accelerating charge).

With this definition of energy, we can't have radiated energy from the passive system of a charge suspended on a pedestal in a Schwarzschild space-time. The pedestal isn't doing any work on the charge.

 

[Jonathan Scott]

There's a plausible-looking Wikipedia entry on this subject, with various interesting references:
Paradox of radiation of charged particles in a gravitational field

 

[PeterDonis]

the fact that the definition of vacuum is dependent on your state of motion doesn't distinguish between black holes and other gravitating bodies

You're right, it doesn't. Hawking radiation is not the same thing as Unruh radiation. The equivalence principle analogy between them only holds locally for an observer that is accelerated close to the horizon, i.e., an observer whose Rindler horizon coincides with the black hole's event horizon. It does not constitute a complete explanation of why black holes radiate.

In the post you quoted, I was not making any claims about Hawking radiation; I was only talking about Unruh radiation. I did refer to the analogy between Hawking radiation and Unruh radiation in post #22. As I said in that post, the way our understanding of Hawking radiation has developed, it seems likely that the key requirement for Hawking radiation is the presence of a trapped surface, not an event horizon. The presence of a trapped surface is detectable locally, so this view offers a local explanation of why a black hole emits Hawking radiation but other gravitating bodies don't.

 

[stevendaryl]

You're right, it doesn't. Hawking radiation is not the same thing as Unruh radiation. The equivalence principle analogy between them only holds locally for an observer that is accelerated close to the horizon, i.e., an observer whose Rindler horizon coincides with the black hole's event horizon. It does not constitute a complete explanation of why black holes radiate.

In the post you quoted, I was not making any claims about Hawking radiation; I was only talking about Unruh radiation. I did refer to the analogy between Hawking radiation and Unruh radiation in post #22. As I said in that post, the way our understanding of Hawking radiation has developed, it seems likely that the key requirement for Hawking radiation is the presence of a trapped surface, not an event horizon. The presence of a trapped surface is detectable locally, so this view offers a local explanation of why a black hole emits Hawking radiation but other gravitating bodies don't.

It's a little puzzling why the trapped surface is relevant in the Hawking case, but not the Rindler case, if they are both manifestations of doing QFT in a noninertial frame.

 

[PeterDonis]

It's a little puzzling why the trapped surface is relevant in the Hawking case, but not the Rindler case

There is no trapped surface in the Rindler case.

if they are both manifestations of doing QFT in a noninertial frame.

A trapped surface has nothing to do with QFT; it appears at the classical spacetime geometry level.

 

[stevendaryl]

There is no trapped surface in the Rindler case.

That's what I meant---if a trapped surface is necessary for Hawking radiation, then why is there radiation in the Rindler case, which doesn't have a trapped surface?

A trapped surface has nothing to do with QFT; it appears at the classical spacetime geometry level.

I know that. I meant the radiation in both cases is a manifestation of doing QFT in a noninertial frame.

 

[PeterDonis]

if a trapped surface is necessary for Hawking radiation, then why is there radiation in the Rindler case, which doesn't have a trapped surface?

Because Hawking radiation and Unruh radiation aren't the same thing. There just happens to be a mathematical analogy involved in one way of looking at both of them. But that way of looking at them, which ties both to the presence of a horizon, is not the only way of looking at them.

I've suggested that Hawking radiation might be better viewed as being produced by trapped surfaces. (My understanding is that this is part of most physicists' current view of Hawking radiation, but I am not familiar with all of the twists and turns in the literature.) Earlier in the thread, I also suggested that Unruh radiation might be better viewed as being produced by acceleration. On these views, there is no necessary connection between the two phenomena. The analogy between them, on these views, is due to the coincidence that both trapped surfaces and acceleration are associated with horizons, and the horizons happen to look the same to accelerated observers close enough to them, which is just a coincidence and does not point to any deep similarity between them.