Re: [Phys-L] acceleration of light to c

[John Denker <jsd@av8n.com>]

On 08/27/2013 10:38 AM, Bill Nettles wrote:
> Talking about electromagnetic radiation and its constant speed in
> class (college physical science), and a student asks, "Does light
> accelerate [using the term in the sense of increasing speed, NOT GR
> reference frames around massive objects] to that speed?"

I assume we are talking about speed, velocity, and acceleration
in 3-dimensional Euclidean space, in some chosen reference frame.
I mention this because due to a snafu in the terminology, the 
definition of velocity in 3D space (dx/dt) is different from the 
definition in spacetime (dx/dτ).  A light wave doesn't have any 
proper time τ, so it doesn't even have a 4-velocity, much less 
a 4-acceleration ... although it does have a perfectly well 
defined 4-momentum.
  http://www.av8n.com/physics/spacetime-welcome.htm

For photons (i.e. wave packets), the scalar acceleration is zero.
The speed is c.  The group velocity is c and the phase velocity is
c in every frame.  You can change the direction and/or magnitude 
of the momentum, but the speed remains c.

The same is true for any fast-moving particle.  Whenever the
total energy is large compared to the rest energy, the speed
is very nearly c (for a fast-moving massive particle) or exactly
c (for a massless particle).  Small changes in the momentum don't 
change the speed.

Note that the scalar acceleration (change in speed) is distinct
from the vector acceleration (change in velocity) for reasons
having nothing to do with general relativity.
  http://www.av8n.com/physics/acceleration.htm

> I'm okay with thinking about changes in an EM field propagating
> without increasing speed from 0 to c, and I can handle a wave being
> created instantaneously with a certain speed, because a wave is a
> propagation of change, but how do we handle talking about photons
> appearing instantaneously  with speed c?  Is it because a photon is
> merely a local manifestation of the field? Are there photons in a
> non-zero but static field?  [b]

I don't know what it means to speak of a wave being "created 
instantaneously".  The wave is created bit by bit, such that
each tiny bit is created at a particular time ... but creating 
the wave /as a whole/ takes a long time.  This is equally true 
for electromagnetic waves and waves on a string et cetera.

Suppose the string (the field) is initially at rest and the source
is initially at rest.  Then at time t=0 the source starts wiggling.
A wave starts coming out from the source.  At some later time t,
the string is still completely at rest at all distances more than
ct away from the source.

For light waves, no part of the wave is accelerating at all.  For
transverse waves on a string, no part of the wave is accelerating
in the direction of motion.

I also don't know what it means to talk about "photons" in this
context.  There are two main definitions of what "photon" means,
and AFAICT neither applies in this situation.

 *) In the context of an electrical harmonic oscillator, such as one 
 standing-wave mode of the electromagnetic field, we say that the
  Nth energy eigenstate has N “photons” in it. The operator a^† a 
  is the photon-number operator. These photons do not propagate at
  the speed of light; indeed they do not propagate at all. 

 *) In the context of a propagating wave, a “photon” is a wavepacket, 
  typically a Gaussian wavepacket, with some not-too-large spread in
  position and also some not-too-large spread in momentum.  During
  the emission process, on timescales short compared to the size
  of the wavepacket, we do not (yet) have a well-formed photon.
  Certainly the wavepacket is not "created instantaneously".

================

As always, a big part of the task is to figure out where the student 
is coming from.  One hypothesis to consider is that the student may
have an unduly crude notion of a photon as a "particle", such that
the photon supposedly exists in advance and is emitted by an atom the 
way a bullet is shot from a gun.  In reality, it doesn't work that way.

As a very general rule, the wave-mechanical description is safer and
better.  Anything you think of as a "particle" can be described in
terms of waves, i.e. a wave packet.  The converse is not true;  in 
general it's hard to describe a wave in terms of any naïve classical 
notion of "particles".

If that's not where the student is coming from, please clarify the
question.