# Radiation Reaction Force (or Self Force of charge particle)

Is it true that there are cases in which the radiation reaction force on an electron is nonzero even if its acceleration is constant? I am confused because the radiation reaction force is proportional to the time derivative of the acceleration.
Also when considering the radiation reaction force (or self-force) as a series, the first term, proportional to the acceleration goes to infinity for a point charge. In this case how would you treat a charged particle in free fall, i.e. what would be its equation of motion? And I am assuming that this particle should radiate as it is in free fall, since radiation is proportional to the square of the acceleration.

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Andrew Mason
Homework Helper
martinan said:
Is it true that there are cases in which the radiation reaction force on an electron is nonzero even if its acceleration is constant? I am confused because the radiation reaction force is proportional to the time derivative of the acceleration.
Also when considering the radiation reaction force (or self-force) as a series, the first term, proportional to the acceleration goes to infinity for a point charge. In this case how would you treat a charged particle in free fall, i.e. what would be its equation of motion? And I am assuming that this particle should radiate as it is in free fall, since radiation is proportional to the square of the acceleration.
Welcome to PF.
This is a difficult but very interesting issue. There seems to be a disagreement whether acceleration causes a charged particle to radiate.
See for example the discussion https://www.physicsforums.com/showthread.php?t=77786":

Gravitational acceleration will not cause a charge to radiate. It seems that charges radiate only when interacting with other electric fields. A charge will, of course, accelerate in such a situation. But can we say that it is the acceleration rather than the interaction with the other field that causes the radiation?

In my view, all we can say is that radiation is observed when a charge experiences a time dependent interaction with another electric field. There is no need to invent the radiation reaction force or the concept of self-interaction. The laws of physics seem to be the same without it.

AM

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What about the Larmor formula (or for v~c Lienard's generalization of the Larmor formula) which gives the total power radiated by an accelerating charge? It is proportional to a^2. This was derived using only the electric and magnetic field of the moving charge and did not consider the interaction with other fields.
As for the radiation reaction force, isn't it true that this 'self force' increases the the inertia of the charged particle and therefore a charged particle will experience less acceleration than a particle of the same mass subject to the same force?

Andrew Mason
Homework Helper
martinan said:
What about the Larmor formula (or for v~c Lienard's generalization of the Larmor formula) which gives the total power radiated by an accelerating charge? It is proportional to a^2. This was derived using only the electric and magnetic field of the moving charge and did not consider the interaction with other fields.
As for the radiation reaction force, isn't it true that this 'self force' increases the the inertia of the charged particle and therefore a charged particle will experience less acceleration than a particle of the same mass subject to the same force?
There is a relationship between acceleration and force, of course. The force is supplied, in the case of a charge, by the charge's interaction with another electrical field. So, although expressed in terms of acceleration, it can be just as easily expressed in terms of force.

My point is that you cannot just look at acceleration. Acceleration cannot occur without an external field interacting with the charge. The math may be equivalent, but the underlying physical explanation is not. It makes it difficult to explain, for example, why acceleration caused by gravity does not cause a charge to radiate.

AM

George Jones
Staff Emeritus
Gold Member
Andrew Mason said:
It makes it difficult to explain, for example, why acceleration caused by gravity does not cause a charge to radiate.

AM
But the acceleration of any freely falling object, including an electric charge, is zero, i.e.,

$$\mathbf{a} = \nabla_{\mathbf{u}} \mathbf{u} = 0.$$

Regards,
George

But the covariant acceleration is not equivalent to the regular classical time rate of change of the velolcity, which is the acceleration that the Larmor formula refers to. Isn't there a classical explanation?

George Jones
Staff Emeritus
Gold Member
martinan said:
But the covariant acceleration is not equivalent to the regular classical time rate of change of the velolcity, which is the acceleration that the Larmor formula refers to. Isn't there a classical explanation?
According to Jackson, the special relativistic generalization of the Larmor formula gives that the power radiated is proportional to the "length" of the charge's 4-acceleration, where "length" means with respect to the Minkowski space "inner product". So, no 4-acceleration, no radiation. I would think that this still holds true in general relativity, which means that freely falling electrons don't radiate.

If gravity isn't a force, then, as Andrew Mason says, only an electromagnetic field can accelerate an electron.

Regards,
George

Meir Achuz