In the previous post we arrived at conclusions that will enable us to address the basic paradox that started this series of posts, and is restated as Issue 3 below. I will be introducing some mathematics here is order to make the line of reasoning clearer for those who know something of this topic but for those readers for whom this is unfamiliar, I would encourage you to not sweat the details but simply read the text between the equations and I think (hope!) you will get the gist.
Issue 3: If a detector D on Earth detects radiation from a falling charge (Scenario 3), that implies that the charge loses some energy in the form of radiation and thus one would expect that it will fall more slowly than a neutral particle, thus violating Postulate #1 and the Principle of Equivalence that says that all objects falling freely in a gravitational field will fall at the same rate. If it still falls at the same time as the neutral particle and thus does not violate Postulate #1, that must mean that its kinetic energy is unaffected by the emission of radiation. Does that not violate the law of conservation of energy? We seem to have arrived at two irreconcilable and unpalatable options.
To analyze more carefully what is happening with Issue 3, we need to have an equation of motion for the charged particle and here we run into a fundamental difficulty. This is because of what I alluded to in post #6 and that is the non-linearity involved with gravity, where mass and energy affect the curvature of the space, and it is difficult to formulate physics in such a curved space. While we have Maxwell’s equations to calculate the electric and magnetic fields due to a distribution of charges and currents, and we have the Lorentz force to provide the force on a charge due to those fields, and we have equations for the radiation emitted by an accelerating charge, these are all obtained for inertial frames in a ‘flat’ (Minkowski) space. i.e., a space in the absence of gravitational fields.
But the charged particle’s mass and electromagnetic fields change the curvature of space which in turn affects the motion of the charge. To fully describe the motion of a charged particle in a gravitational field, we have to deal with such a curved space but we do not have a unified theory of gravity and electromagnetism that would give us a Lagrangian from which we could obtain the equation of motion. In the absence of such a complete theoretical understanding of what the equation of motion should be for a radiating electric charge that is falling in a gravitational field, what we are forced to do is to patch together things that are obtained in flat space, choose an equation of motion that we think is plausible and might work the best, and see where it leads us. Since the charged particle’s mass and electromagnetic fields are small, their gravitational effects could also be expected to be small, making this a good approximation.
There are various candidate theories for the equation of motion of the charge and Fulton and Rohrlich (Annals of Physics 9, 499-517, 1960) choose to use what they refer to as the Abraham-Dirac equation (also called the Lorentz-Dirac or the Abraham-Lorentz-Dirac equation) given by:
F𝜇 + 𝜞𝜇 = ma𝜇
This equation takes the form of the relativistic generalization of Newton’s second law of motion F = ma, written in four-vector notation instead of the more familiar three-vectors of space, where the superscript is the component of the four-vectors. m is the mass of the charge, a𝜇 is its four-acceleration given by a𝜇 = dv𝜇/d𝜏 where v𝜇 is its four-velocity and 𝜏 is the proper time, F𝜇 is the external force acting on the charge, and 𝜞𝜇 is what is referred to as the radiation reaction force. In the case of the falling charge, F𝜇 is the force of gravity on it. The radiation reaction force is given by
𝜞𝜇 = (2e2/3c3)(da𝜇/d𝜏 – (1/c2)v𝜇a𝝀a𝝀)
This radiation reaction force arises as a basic consequence of the law of conservation of momentum that says that if a body sends out something that has momentum, then the body must recoil with that same amount of momentum in order to maintain the same total momentum of the whole system. This is a common phenomenon. This is the cause of the recoil one feels when one fires a gun. It is also used in rocket propulsion when course changes are required in outer space. By burning fuel and expelling it in one direction, the rocket recoils in the opposite reaction. So when an accelerating charged particle emits radiation, then since radiation carries energy and momentum, the particle will experience a reaction force in the opposite reaction, hence the label radiation reaction force. This will try to slow it down, and thus decrease the magnitude of its acceleration. For a uniformly accelerating charge, 𝜞𝜇 = 0.
Fulton and Rohrlich say that while we are not obliged to accept this equation of motion, if we do accept it we are obliged to accept the consequences that follow from it, which they then proceed to work out. They then show that the fact that the radiation reaction force is zero does not imply that there is no radiation. By looking at the zeroth component of the above equation of motion, they obtain the result:
dT/dt – (dQ/dt – ℛ) = dW/dt
where T is the particle’s kinetic energy and W is the work done by the external force (in this case the gravitational field). The term in parentheses represents the net rate of work done by the radiation reaction force, where ℛ = 2e2g2/3c3 is the Larmor formula found in textbooks for the rate of radiation energy emitted by a charge e having an acceleration g. Q = 2e2ao/3c2 where ao = 𝛾v·a is the zeroth component of the covariant four-acceleration of the charge a𝜇 and 𝛾 is the usual Lorentz factor. ℛ is always positive and in this situation Q is also positive since the velocity and acceleration are in the same direction, making ao a positive number. But dQ/dt can be either positive or negative. (Note that the Q used here does not stand for the charge of the particle as it did in earlier posts.)
If dQ/dt = 0, then dT/dt = dW/dt – ℛ. Since ℛ is positive, the kinetic energy of the falling charge gets reduced because of the radiation it emits and it will arrive at the bottom after the neutral particle, violating Postulate #1. It is the presence of the dQ/dt term that enables the preservation of the Principle of Equivalence even in the presence of ℛ, because it is possible for the (dQ/dt – ℛ) term to be zero even if ℛ is not zero provided that ℛ = dQ/dt. We see that in that case, the radiated energy ℛ comes, not at the expense of the particle’s kinetic energy, but at the expense of Q. So when the charge accelerates as it falls, the value of Q increases as its speed increases but it is that increase in energy that is carried away by the radiation. As a practical matter, according to ℛ, the energy radiated by a single accelerating charge is so infinitesimally small that measuring it is next to impossible. In one second, the energy radiated by an electron falling in the Earth’s gravitational field is roughly 10-50 times the mass of the electron.
So to summarize, since dQ/dt – ℛ = 0, we get that 𝜞𝜇 = 0 and dT/dt = dW/dt. Hence the particle’s acceleration is unaffected by the radiation and all the work done by the external gravitational force goes into the particle’s kinetic energy so that there is no loss in kinetic energy due to the charge radiating. Thus the charged particle falls just like a neutral particle even though it is radiating energy.
So what is this mysterious Q that seems to have saved the day by enabling us to escape the paradox raised in Issue 3? To understand it takes us into another subtlety, and that is the mass of the electron, which will be examined in the next post.
(To be continued.)