We discuss on the problem of the electromagnetic radiation from the accelerated charged particle and the back reaction of that. The free-falling charged particle radiates the electromagnetic wave. The charged particle on the surface of the earth does not radiate the electromagnetic wave. The existence or the non-existence of the electromagnetic radiation from the charged particle and the back reaction of that is independent of the observer, which is consistent with the energy conservation. A paradox comes from combining this phenomenon with the equivalence principle in the theory of the general relativity. We consider the Hawking effect in the context of this paradox. We give our resolution on this paradox.

The problem of the electromagnetic radiation from the accelerated charged particle and the back reaction of that has a long history [

A paradox in connection with this problem is known from old days [

If we combine this problem to the equivalence principle in the theory of general relativity, this paradox becomes more complicated, and many authors proposed various resolutions to this paradox. We implicitly have given our resolution to this paradox in the previous papers [

An accelerating charged particle radiates the electromagnetic wave. Then this accelerating charged particle loses the energy as the back reaction of the electromagnetic radiation, which is given by

We consider the case that the +

By using

In the standard treatment, we make the back reaction term to be smoothing by taking the average in the form

In the situation of

If we consider the case of

But

Then we try to solve the original

If we take the limit of

In the case of

While, the general solution of

Though it is not the physical solution, if we take the–branch solution of the form

In the case of |

First, we consider the equation of motion under the gravitational force in addition to the constant electric force. In the case of

The relative position

Then the gravitational force is eliminated by taking the relative position, but the electric force and the back reaction of the electromagnetic radiation is not eliminated. This means that the free-falling observer

The existence of the electromagnetic radiation and back reaction of that is independent whether the acceleration comes from the electric force and/or the gravitational force. This is consistent with the energy conservation. Further, the existence of the electromagnetic radiation and the back reaction of that is independent of the observer.

Next, we consider the case that the charged particle is on the surface of the earth under the electric and the gravitational forces. The equation of motion is given by

The equation of motion of the observer is given by

The physically meaningful solution is given by

Because there is the term

The equation of motion of the charged particle

The equation of motion of the observer, who is forcibly accelerated by a rocket2, is given by

The physically meaningful solutions are

If we denote the relative position of the charged particle at rocket1 and the observer at rocker2 as

Then the observer at rocket2 observe the electromagnetic radiation from the charged particle at rocket2 and the back reaction of that, which is supplemented by the relative additional acceleration caused by relative additional force

It has a long history of a paradox on the electromagnetic radiation from the accelerated charged particle in the context of the general coordinate transformation [

The Maxwell equation is written in the following covariant form

If we express the electromagnetic current in the form of the field quantity by using the Dirac field

In order to explain the electromagnetic radiation from the charged particle, we must use the expression of the electromagnetic current in the form of the particle quantity

Pauli used the current in the form of the field

If the metric is regular, we can always make the metric to be locally flat by the general coordinate transformation. The frame of this locally flat metric is called the local inertial frame. In the locally flat metric, there is no gravitational force, so that the gravitational force and the acceleration by the general coordinate transformation is equivalent, which is called the equivalence principle.

But the terminology of the local inertial frame is misleading. Though the gravitational force is locally eliminated in this local inertial frame, the electromagnetic force is not locally eliminated, because the electromagnetic force does not satisfy the equivalence principle. If there are some various charged particles, we cannot locally eliminate the electric force for all charged particles by the general coordinate transformation. While, if there are some massive particles, we can locally eliminate the gravitational force for all massive particles by the general coordinate transformation, which we denote

Under the general coordinate transformation, i) we cannot eliminate the electromagnetic radiation from the charged particle, ii) we cannot eliminate the electric force, iii) physical meaning of the space and the time is not guaranteed except under the Lorentz transformation.

We discuss on the problem of the electromagnetic radiation from the accelerated charged particle and the back reaction of that. The free-falling charged particle radiates the electromagnetic radiation, which is observed by the free-falling observer. The charged particle on the surface of the earth does not radiate the electromagnetic radiation. The existence or non-existence of the electromagnetic radiation from the charged particle does not depend on the observer.

The paradox comes from the misleading terminology of the local inertial frame, which is realized by the general coordinate transformation. The general coordinate transformation is the formal transformation but it is not the physical transformation in a sense that the physical meaning of the space and the time is not guaranteed. Though the gravitational force is eliminated in the local inertial frame, it is not the physical inertial frame but only the formal inertial frame because we cannot eliminate the electric force and we cannot eliminate the electromagnetic radiation from the accelerated charged particle.

As the result of our resolution of the paradox, we conclude that the Hawking effect [

This phenomenon is similar to the following paradox. In the hydrogen atom, though the electron is accelerating, the electron in the ground state does not cause the electromagnetic radiation. The resolution of this paradox is the followings. The electron has the possibility to radiate the electromagnetic wave, but as there is no orbit to transit, the electron cannot lose the energy. Then the electron cannot radiate the electromagnetic wave because of the energy conservation. The energy conservation law takes precedence over the electromagnetic radiation.