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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.

Introduction

The problem of the electromagnetic radiation from the accelerated charged particle and the back reaction of that has a long history [1]–[6]. The standard equation, which treats the electromagnetic radiation from the accelerated charged particle and the back reaction of that, is known as the Lorentz-Abraham-Dirac equation [7]–[9]. Even now, there are still some difficulties in this problem such as the existence of the run-away solution [2], [3] and the pre-acceleration phenomenon [10].

A paradox in connection with this problem is known from old days [1], [3]–[6]. Pauli and Feynman concluded that there is no electromagnetic radiation from the constantly accelerating charged particle, and they concluded that the electromagnetic radiation occurs when the acceleration of the charged particle changes with time [1], [3]. Rohlich concluded that the electromagnetic radiation occurs from the charged particle even if the acceleration of the charged particle stays constant [4]–[6].

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 [11]–[13].

Back Reaction of the Electromagnetic Radiation under the Electric Force

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 (1)dWdt=e26πϵ0c3(d2q(t)dt2)2.

We consider the case that the +e charged particle is at the position (0, 0, q(t)), which is accelerated by the constant electric field in z-direction of the form E = (0, 0, −E). The energy conservation of the charged particle with the back reaction of the electromagnetic radiation is given in the form (2)ddt(12mq˙(t)2+eEq(t))+e26πϵ0c3q¨(t)2=0,which gives (3)mq˙(t)q¨(t)+eEq˙(t)+e26πϵ0c3q¨(t)2=0.

By using v(t)=q˙(t), (3) becomes in the form (4)mv(t)v˙(t)+eEv(t)+e26πϵ0c3v˙(t)2=0.

In the standard treatment, we make the back reaction term to be smoothing by taking the average in the form (5)e26πϵ0c3v˙(t)2→e26πϵ0c31t∫0tdt′v˙(t′)2=e26πϵ0c31t∫0tdt′[ddt′(v(t′)v˙(t′))−v(t′)v¨(t′)]=e26πϵ0c3v(t′)v˙(t′)|t′=0t′=t−e26πϵ0c31t∫0tdt′v(t′)v¨(t′).

In the situation of v(t′)v˙(t′)|t′=0t′=t=0, (4) effectively becomes in the form (6)mv(t)v˙(t)+eEv(t)−e26πϵ0c3v(t)v¨(t)=0.

If we consider the case of v(t) ≠ 0, (6) becomes in the form (7)mv˙(t)+eE−e26πϵ0c3v¨(t)=0.

But (7) contradicts with (4). Equation (7) has the special solution v˙(t)=−eE/m=(const.) but it is not the solution of (4).

Then we try to solve the original (4) instead of (7) to know the back reaction of the electromagnetic radiation. For v(t)≠0, we can rewrite (4) in the form (8)mv˙(t)=−eE−e26πϵ0c3v˙(t)2v(t),and FBack=−(e2/6πϵ0c3)(v˙(t)2/v(t)) can be considered to be the force of the back reaction. We rewrite (4) in the form (9)av˙(t)2+v(t)v˙(t)+bv(t)=0,where a=(e2/6πϵ0mc3), b = eE/m. This gives (10)dv(t)dt=−v(t)±v(t)2−4abv(t)2a.

If we take the limit of e→0, which is the case of no action bv(t)→0, we must take the solution of no back reaction av˙(t)2→0. Then we must take + branch solution in the form (11)dv(t)dt=−v(t)+v(t)2−4abv(t)2a.Thus we obtain (12)dt2a=dv(t)v(t)2−4abv(t)−v(t)=−14abv(t)(v(t)2−4abv(t)+v(t))dv(t)=−14ab(v(t)2−4abv(t)v(t)+1)dv(t).Hence we obtain the solution of the form (13)v(t)+v(t)2−4abv(t)−2ablog⁡|v(t)−2ab+v(t)2−4abv(t)|=−2bt+C1.

In the case of t→∞, we have v(t)→−∞, and we obtain −bt≈v(t)−ablog⁡|v(t)|, which gives v(t)≈−bt+ablog⁡|v(t)|≈−bt+ablog⁡t, (t→∞). While, if there is no back reaction, we have v(t)=−bt+C1. Therefore, the term ablog⁡t, (t→∞) is the back reaction for the velocity v(t).

While, the general solution of (7) is given by v(t)=−bt+C1et/a, which contains the run-away term C1et/a. For the physically meaningful solution of (7), we must put C1=0. Because if we put C1≠0, the reaction exists even if the action does not exist by putting b = 0 in v(t)=−bt+C1et/a, which is unphysical. In this case, we obtain the solution v(t) = −bt of (7) but it is not the solution of the original equation of (4).

Though it is not the physical solution, if we take the–branch solution of the form (14)dv(t)dt=−v(t)−v(t)2−4abv(t)2a,we obtain (15)−dt2a=−14ab(v(t)2−4abv(t)v(t)−1)dv(t),and we obtain the solution (16)v(t)−v(t)2−4abv(t)+2ablog⁡|v(t)−2ab+v(t)2−4abv(t)|=−2bt+C1.

In the case of |ab| ≪ v(t), we have −alog⁡|v(t)|≈t, (t→∞), that is, v(t)≈C1e−t/a, (t→∞). The solution of this case can be seen by taking limit b→0 in (9). This solution is different from the standard run-away solution of (7) in the form v(t)=C1et/a.

Back Reaction of the Electromagnetic Radiation under the Gravitational Force

A Free-Falling Object

First, we consider the equation of motion under the gravitational force in addition to the constant electric force. In the case of q˙1(t)≠0, the equation of motion of the accelerated changed particle (0,0,q1(t)) is given by (17)mq¨1(t)+eE+mg+e26πϵ0c3q¨1(t)2q˙1(t)=0.Equation (17) is solved in the same way as (4) by replacing eE→eE+mg. While the equation of the accelerated observer (0,0,q2(t)) is given by (18)q¨2(t)+mg=0.

The relative position Q(t) between the charged particle q1(t) and the observer q2(t), is given by Q(t)=q1(t)−q2(t), and we obtain the equation of motion (19)mQ¨(t)+eE+e26πϵ0c3q¨1(t)2q˙1(t)=0.

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 q2(t) observe the electric force acting on the charged particle q1(t) and the electromagnetic radiation from the charged particle q1(t) and the back reaction of that.

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.

An Object Placed on the Surface of the Earth

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 (20)mq˙1(t)q¨1(t)+eEq˙1(t)+mgq˙1(t)+e26πϵ0c3q¨1(t)2=N1q˙1(t),where N1 is the normal force with N1=eE+mg. This gives the equation of motion in the form (21)mq˙1(t)q¨1(t)+e26πϵ0c3q¨1(t)2=0.Then the physically meaningful solution is given by q¨1(t)=0, which gives q1(t)=0 because we assume that the charged particle is fixed on the surface of the earth.

The equation of motion of the observer is given by (22)mq¨2(t)+mg=N2,where N2=mg and we obtain (23)mq¨2(t)=0.

The physically meaningful solution is given by q2(t)=0 because we assume that the observer is also fixed on the surface of the earth.

Because there is the term (e2/6πϵ0c3)q¨1(t)2 in (21), the charged particle has the possibility to lose the energy, but the charged particle is fixed on the surface of the earth, so that the charged particle cannot lose the energy. Because of the energy conservation, the electromagnetic radiation from the charged particle does not occur as the charged particle cannot lose the energy. The non-existence of the electromagnetic radiation is independent whether the electric force and/or gravitational forces act. This is consistent with the energy conservation. Further, the non-existence of electromagnetic radiation is independent of the observer.

An Object Being Accelerated by a Ricket

The equation of motion of the charged particle q1(t), which is forcibly accelerated by a rocket1, is given by (24)mq¨1(t)+e26πϵ0c3q¨1(t)2q1(t)=F1(t),where F1(t) is the forced acceleration by the rocket1, which is given by (25)F1(t)=mq¨0(t)+e26πϵ0c3q¨0(t)2q0(t), (26)q0(t)=12α2t2+C1t+C2.

The equation of motion of the observer, who is forcibly accelerated by a rocket2, is given by (27)mq¨2(t)=F2(t)=mq¨0(t)=mα.

The physically meaningful solutions are q1(t)=q0(t) and q2(t)=q0(t). The charged particle radiates the electromagnetic wave and receive the back reaction of that, which is supplemented by the additional acceleration caused by the additional force of F1(t). This is consistent with the energy conservation.

If we denote the relative position of the charged particle at rocket1 and the observer at rocker2 as Q(t)=q1(t)−q2(t), the equation of motion is given by (28)mQ¨(t)+e26πϵ0c3q¨1(t)2q1(t)=F1(t)−F2(t)=e26πϵ0c3q¨0(t)2q0(t).

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 F1(t)−F2(t). The existence of the electromagnetic radiation from the accelerated charged particle and the back reaction of that is independent of the observer.

Local Inertial Frame and the Radiation of the Electromagnetic Wave

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 [1]–[6]. We implicitly give our solution for this paradox [11], [12].

The Maxwell equation is written in the following covariant form (29)∇μfμν(x,t)=−μ0jν(x,t),  (30)∇μfμν(x,t)=∂μfμν(x,t)+Γαμμ(x,t)fαν(x,t)+Γαμν(x,t)fμα(x,t)−Γαμα(x,t)fμν(x,t).

If we express the electromagnetic current in the form of the field quantity by using the Dirac field (31)jμ(x,t)=eψ¯(x,t)γμψ((x,t),we can prove that the Maxwell equation covariantly transform under the general coordinate transformation. If we look at this fact the other way, we can say that we cannot explain the physical phenomenon of the electromagnetic radiation from the accelerating charged particle by the general coordinate transformation. In another words, we cannot eliminate the electromagnetic radiation from the accelerated charged particle by the general coordinate transformation. Thus the general coordinate transformation is not the physically meaningful transformation but only the formal transformation in the theory of general relativity.

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 (32)ρ(x,t)=eδ3(x−q(t)), j(x,t)=eq˙(t)δ3(x−q(t)). 

Pauli used the current in the form of the field ρ(x,t)=eδ3(x) and started from the expression

(33)E(x,t)=e4πϵ0x|x|3,and he concluded that the electromagnetic radiation from the constantly accelerated charged particle does not occurs [1]. But we must use the current of (32) to explain the electromagnetic radiation from the accelerated charged particle.

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 mI=mG for any massive particles.

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.

Summary and Discussions

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 [14] does not occur. There is an analogy between the Hawking effect, by putting the box of the vacuum on the surface of the surface, and the electromagnetic radiation, by putting the charged particle on the surface of the earth. There is the possibility to occur the Hawking effect for the box of the vacuum on the surface of the earth, just as there is the possibility to occur the electromagnetic radiation from the charged particle on the surface of the earth. But the box of the vacuum on the surface cannot lose the energy as the box is fixed on the surface of the earth, so that the Hawking effect cannot occur because of the energy conservation. This corresponds to the phenomenon that the charged particle on the surface of the earth cannot lose the energy as the charged particle is fixed on the surface of the earth, so that the electromagnetic radiation does not occur because of the energy conservation.

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.

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