Acausal Preacceleration in
Classical Electromagnetism
After a while, physicists adjust to
the weirdness of quantum mechanics, but there are some things that are just too
weird to be accepted even there. For
example, most physicists reject David Bohm's interpretation of quantum
mechanics because the future would influence the present (and the
past)-creating problems for interpreting experiments, at least (not to mention
relativity and being able to propagate changes faster than the speed of light).
On the whole, though, physicists generally accept the intuition that causes
come before effects- even in relativity is careful to preserve this.
However, even this fundamental
concept faces a challenge in physics, and surprisingly not from quantum
mechanics. Instead, this confronted in
classical electrodynamics (and the problem stays around in relativistic
electrodynamics too). When a point
particle accelerates, it radiates off energy- which corresponds to a loss in
kinetic energy, as though a drag force was acting on it. An accelerating particle then exerts a force
on itself, through its fields acting on itself.
The Abraham-Lorentz formula describes this force (relativistically, it's
the Abraham-Lorentz-Dirac force). Unlike
most forces one runs into, this one is proportional to the derivative of
acceleration, which means that acceleration is proportional to its own
derivative, or that acceleration is an exponential. Since we don't observe charges spontaneously
accelerating exponentially in time, the best solution is to set the constant in
front (from integration) equal to zero.
This, though, creates another
problem- forces that are applied to the particle will affect the particle's
motion BEFORE they are applied. For example,
let it be t=0 sec now. At t=10 sec, I'm
going to apply a force F1 on the particle. However, to describe the particle's
motion at t=5 sec, I need to use the fact that I'm going to apply F1 at t=10
seconds- even though I haven't applied it yet!
Some have suggested that because the
issue only arises for point particles- so hopefully it points to a failure of
classical physics because of interference of quantum mechanics. In quantum electrodynamics the problems don't
quite go away, though. These self-fields
still create infinities in the mathematics of QED, which again aren’t observed
physically, so physicists simply remove them by renormalizing the wavefunction-
forcing them to go away with a little bit of handwaving. As for the future interfering with the
present and future, in quantum electrodynamics signals propagating backwards in
time are interpretted as antimatter.
Hence something that looks like an electron going backwards in time is
taken to be a positron moving forward in time.
However, to me, it seems as though this does not really solve the
problem of the future influencing the past, only gives us a different way of
describing it through antiparticles that, while at least moving the right
direction in time, somehow mysteriously conspire to give the exact influence of
the future (and as yet unapplied) force.
The first problem with this,
classically, is that relativity relies on information being transported a
speeds slower than c- all changes that are “meaningful” must move slower than
c. (An example of a non-meaningful
“change” is the collapse of a wavefunction, which we cannot actually control;
it collapses randomly and instantaneously over the whole spatial extent.) However, this is a meaningful change- we can
(presumably) control what force we exert, and if it is propagating faster than
c, it must be going faster than the speed of light. QED, with the antiparticles, will get around
this, because the antiparticles are traveling forward in time slower than c,
although how they manage to know the strength of the force that will be applied
is disconcerting.
However, F. Rohrlich’s article in
American Journal of Physics points to a solution of the Lorentz-Abraham-Dirac
equation found by Yaghjian. He has a
whole book on the solution, which I haven’t had a chance to read yet. He imposes asymptotic conditions to keep the
acceleration from exponentially increasing spontaneously, like described above
(only for the relativistic equation). He
discovered that the expansions involved in deriving these equations require an
adiabatic approximation, which does apply when the forces change
instantenously, and these solutions do not have any preacceleration. Problem solved! Causality is rescued!