Paradox of a Charged Particle in Gravitational Field（原声文稿）

I have been asking physicists the same question for more than a year now, and all of them answered straight away without any hesitation.

But none of them actually realized how difficult this question is to answer.

And I have been asking everywhere, even at a conference about general theory of relativity.

The problem originates from the known result from electrodynamics, The Larmor formula, which says that if you have an electric charge under acceleration it will produce an electromagnetic radiation.

This phenomenon is very well known and it's widely used in current technologies, so we know that this formula is correct.

And one night we were enjoying a beer in a bar and have some fun and one of our students asked a simple question related to her master thesis, which was about possible ways how cosmic rays could be losing energy.

And she wanted to know if a charged particle could lose energy by moving close to a very massive body, which would curve its trajectory and due to this acceleration it would radiate some energy away according to this Larmor formula.

And this immediately sparked a discussion.

But not about this particular problem concerning the cosmic rays.

But whether a charged particle could be radiating according to this Larmor formula if it was accelerated by gravity in general, whether it's orbiting or just freely falling.

And since this discussion remained unresolved, I've been asking this question to all the physicists I met.

And the immediate answer to this question was, of course, it's a charge under acceleration And therefore, according to this Larmor formula, it must produce an electromagnetic radiation.

But knowing a thing or two about gravity, I didn't like the answer.

And this question shouldn't be dismissed so quickly.

So now, think about the consequences of this answer.

We know since Galileo Galilei that all objects fall down on Earth with equal acceleration.

But if you dropped a charged particle with a neutral particle, the charged particle would start to radiate, which costs energy, and therefore the charged particle would have to accelerate slower than the neutral particle.

This would be in contradiction to Galileo's findings.

But Galileo didn't say anything about charged particles, so who cares?

Much later, in 1915, Albert Einstein discovered the general theory of relativity.

And one of the core pillars of this theory is the equivalence principle, which is basically a generalization of these Galileo's findings, that all objects fall down in the same acceleration.

It states that locally there is no physical experiment that can distinguish between freefall and being out in space, as well as between being on the surface of a planet and accelerating the rocket.

But according to this principle, a charge that is freely falling should behave the same way as being out in space.

And therefore it shouldn't start to randomly accelerate as it would in a free -fold due to radiation.

A particle either radiates or don't radiate.

and that is something that everybody has to agree on and therefore if a falling charge would radiate while the charge in outer space would not then you could experimentally distinguish these two cases and therefore the equivalence principle would be broken.

So in order to save the equivalence principle we must say that falling charge would not radiate.

Since locally it is at rest while the surface of the planet is accelerating towards it.

And this makes sense, because the electric field is also subject to gravitational acceleration.

And therefore, for a falling charge, there is no relative acceleration between the particle and its field.

And what is even more interesting is that if there was a charge sitting on Earth, from the reference frame of the falling observer, the charge on the surface should radiate according to Larmor formula, due to non -zero acceleration.

Because according to equivalence principle, this should be indistinguishable from the charge being accelerated by rocket.

So is this correct resolution to the question?

I think it needs a bit more explanation. Electric field is also influenced by gravity.

And therefore, if a charge is in free fall, its field is in free -flow as well, resulting in zero relative acceleration between the field and the charge.

And the Larmor formula was derived in such a way that it was only the charge that is accelerated, and not the field.

And this relative acceleration creates a self -interaction of the charge with its field, resulting in radiation.

It is important to know that electric field around a charged particle is not an integral part of the particle, but it's something external to the particle.

And therefore the particle and its field can be influenced separately.

Imagine for example a charged particle moving in magnetic field.

There is magnetic force acting on this particle, making it to curve.

But this magnetic force only applies to the particle itself, but not its electric field.

This creates a relative acceleration between the particle and its field, causing it to radiate.

But for gravity, both the particle and the field are influenced by gravity.

And therefore, there is no relative acceleration.

If you had a charge supported on the surface of the planet, there would be a force acting on the particle against the gravity from the ground. But this force would not act on its field.

The field of the particle would fall through the surface, causing relative acceleration and therefore radiation.

Ok, now it starts to be pretty weird.

This tells me that if I was holding a charge here, it would radiate energy away because it's in relative acceleration between the charge and its field.

But there are a couple of problems with this view.

One of the problems is that while trying to make this animation as demonstrative as possible, Light is an electromagnetic radiation and therefore you also need a magnetic field to have a light.

According to Maxwell's equation, the magnetic field can arise only in a couple of ways.

You either have a source which would be a magnetic monopole or you need either a current or electric field intensity that changes with time.

But this setup has none of it.

The charge is stationary therefore no current and the charge is still the same distance away from us and therefore the intensity of electric field we measure doesn't change with time.

So in this scenario where I hold this charge in my hand I should not see any electromagnetic waves.

But if I was freely falling I would see a current and the electric field intensity change with time and therefore I should see the magnetic component of the field and therefore radiation is allowed.

But whether something radiates or not, that should be an objective event.

So which one is correct?

Secondly, if the charge would radiate, we could harness its energy and have an infinite source of energy, so that the energy conservation law would be violated.

The problem is that we are working with uniformly accelerated frames.

And that's kinda tricky. For this we need to use the Rindler coordinates.

In this coordinates, an accelerated observer has something like event horizon, from which no signal can ever reach him, as long as he remains in accelerated motion.

According to this article, for a commuting observer, all the radiation happens beyond this horizon, and only electrostatic field appears in this region.

But freely falling observers would detect photons from the charge at the surface.

And it is an objective event which the commuting observer at the surface have to agree with.

But how can the observer on the ground make sense of this?

He would see the falling observer picking up photons without any reason.

And he would also see him falling slower than he should due to radiative pressure.

So where does the radiation come from if it's not from the charge?

Well, this comes from Anru Effect, which is basically a Hawking radiation from the Rindler horizon.

But this is far beyond what I wanted to talk about in this video.

I wanted to talk about whether a charge in a free fall would radiate or not. And the answer is yes.

Did I just confuse you?

What about the relative acceleration between the charge and its field?

The problem is that equivalence principle applies only locally.

And while the particle itself is small enough, so that we consider it to be local object, the field it generates spans to infinity.

This creates a different acceleration at different parts of the field, causing a back -reaction to the charge itself.

And therefore there is a small relative acceleration between the charge and its field.

But this acceleration would be much smaller than 9 .81 meters per second squared.

So, if the question was, would a following charge radiate according to Larmor formula, where we put g into the acceleration, gosh, I wish I could say no with 100 % certainty.

But this mystery is still unresolved.

And there are still papers being published to this day about this topic. But despite this, there are some agreements.

There is strong consensus that charge sitting on earth won't radiate according to commuting observer.

There is strong consensus that freely falling charge won't radiate according to freely falling observer.

There is also quite strong consensus that charge sitting on earth will radiate for freely falling observer.

But, for example, in this article, Signal claims that it would not radiate, but does a freely falling charge radiate from an observer on the surface? And this is the controversial one.

Most physicists that work in this field would say no due to equivalence principle. But there are also strong arguments for yes.

But in the past, there wasn't even consensus whether a uniformly accelerated charge radiates or not.

Even physicists like Born, Poli, Feynman thought it wouldn't and as I showed you signals still think this.

There is yet another consensus and it is the validity of the equivalence principle.

So people who are saying that the falling charge would radiate try to make sure that they won't break it.

Although it is still unclear whether equivalence principle should apply to the charged particles. But I want to clarify a couple of things. First, what is actually radiation?

The electromagnetic energy flux is given by a pointing vector, which has a clear definition as a vector product of electric and magnetic field.

I said that a falling observer would detect magnetic components of the field created by the charge sitting on the surface, Since it detects non -zero current or electric field intensity change with time. But you don't have to be accelerated to see this.

Even an inertial observer with non -zero velocity would detect a magnetic component, since it detects a current or electric field intensity to change with time.

So does a uniformly moving charge radiate?

And obviously no, because that would bring a lot of problems with the principle of relativity.

Despite having non -zero pointing vector, to say a charge is radiating, you have to make an integral over the closed surface around the charge to see whether an energy is flowing away.

But if you did this to an inertial observer, you would see the same flux coming in as flux coming out, leaving zero net flux from the charge.

So we can't say that the charge is releasing energy away.

To see energy falling out, you would have to see something like this.

But does this happen to a falling charge, or better to a charge in free space just because you have your rocket engine on? And where does the energy come from?

If this is the case, then the equivalence principle is in big danger.

But anyway, to have a definite answer, we have to do an experiment.

But in this case it's seriously difficult, considering the fact that if the Larmor formula did apply to a falling charge at 9 .81 meters per second squared, it would radiate just 3 .4 times 10 to the minus 33 electron volts per second, or around 5 times 10 to the minus 52 watts.

So you need to pile up 5 times 10 to the 52 unit charges to have a 1 watt of radiative power.

And considering that there is only 10 to the 50 atoms in the earth, you imagine that such experiment would be quite difficult to make on Earth.

Not to mention that if you piled up enough electrons from the Earth, then it would become charged and the gravitational force would be negligible compared to the electrostatic force caused by the electric field of the Earth. So what do you guys think? Because this question is really haunting me for a long time.

And I know that there are some experts in the comments so I would really like to hear your insight on this. Thanks for watching!