Re: Massless Particles

Oh No

Thus spake Peritas <galoislie@yahoo.com>
>Peritas wrote:
>> Given a massless particle (rest mass = 0), other than a photon or
>> graviton, what would be the relativistic explanation for why it can
>> travel faster than c? What aspect of electromagnetic radiation make
>> its speed of propagation in free space an upper bound on speed? I
>> suppose the answer lies with the derivation of the sqrt(1-v2/c2) term
>> in SR, but I could use some insight from those of you that have a
>> strong understanding of relativity and its derivation.
>>
>> Thanks
>
>Please allow me to correct what I'm asking here. I meant to ask,
>"...why it *can't* travel faster than c?" in the first sentence.
>
That's a better question, one I can answer without attempt to undo
confusion first - which generally leads to more confusion.

First of all, the limit on speed is really a limit on the transfer of
information, not on the speed of photons, the elementary particles which
make up light. In quantum field theory we find that a particle can be
created at one point and annihilated at a point outside the light cone,
but that no information can be transmitted this way and the process
cannot be observed.

To understand the reasons for a maximal speed we have to look at the way
in which space-time co-ordinates in physics are really defined. We have
to challenge the assumption that space-time co-ordinates are pre-
existent, and then measured, but actually think in terms of the
measurement providing the definition.

If we are to measure the time and distance of an event spacially
separation from ourselves, then information must travel between us and
the event. If we know the speed of information transfer, we can easily
determine the time and distance. But speed is defined in terms of times
and distances, so we have a paradox. We must know the co-ordinate of the
event to talk of the speed of information, but we must know the speed of
information to talk of the co-ordinate of the event.

To resolve the paradox we must find something fundamental, and base
everything else on it. If we do not believe in instantaneous action at a
distance, then we may say that there is always a maximum speed of
information, which we can call c=1, and base a co-ordinate system on
that.

From this assumption it is straight forward to use the methods of k-
calculus to find relationships between inertial coordinate systems (see
the ch2 of D'Inverno, Introducing Einstein's Relativity, or Bondi's
Relativity and Common Sense, or failing that http://xxx.lanl.gov/abs/phy
sics/9909048).

From that point one establishes the properties of vectors in inertial
coordinate systems, and finds that, corresponding to the Pythagorean
magnitude of a 3-vector, for space time a 4-vector (E,p) has a magnitude

m^2 = E^2 - p^2

where E is the time component of the 4-vector and p is a space 3-vector.
then the statement m^2=0 is equivalent to a statement that the vector is
directed in a light-like direction, representing something moving at the
speed of light

Regards

--
Charles Francis
substitute charles for NotI to email

Christophe de Dinechin

Oh No wrote:
> To understand the reasons for a maximal speed we have to look at the way
> in which space-time co-ordinates in physics are really defined. We have
> to challenge the assumption that space-time co-ordinates are pre-
> existent, and then measured, but actually think in terms of the
> measurement providing the definition.

I second that, but then, don't these measurements themselves rely on
electromagnetic waves?

> From that point one establishes the properties of vectors in inertial
> coordinate systems, and finds that, corresponding to the Pythagorean
> magnitude of a 3-vector, for space time a 4-vector (E,p) has a magnitude

An essential point which you did not make explicit is that Einstein
wanted laws of physics to be invariant by change of coordinates.
Therefore, acceptable entities in general relativity had to be tensors.

It so happens that what we called "mass" is a rank 0 tensor which
happens to be the norm of a rank 1 tensor containing of the
measurements we traditionally called "energy" and momentum. That's
where the relation below comes from.

> m^2 = E^2 - p^2
>
> where E is the time component of the 4-vector and p is a space 3-vector.
> then the statement m^2=0 is equivalent to a statement that the vector is
> directed in a light-like direction, representing something moving at the
> speed of light

Another important step: it's the contravariance relation between
energy-momentum and "space-time displacement" (dx,dy,dz,dt) that
implies that if one has zero norm, then so has the other. And that's
how you deduce that m=0 implies a speed-of-light displacement for the
corresponding entity.


Regards
Chistophe