Re: [Phys-l] Absolute four-momentum of massless particles

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding:

> Bob Sciamanda writes 
>
> > Yes, but that approach simply substitutes the validity of Maxwell's 
> > Equations as the second postulate!"
>
> I suggest that they are included under "all laws of physics" in the
> first postulate! 
>
> (I love these conversations!  We can get deeper insights into both
> the physics itself and physics pedagogy by learning from the
> group's collective wisdom.)

I don't think putting all the laws of nature into a hodgepodge that one calls the first postulate is a good idea.  To my mind the postulates are supposed to be meta-laws that the observed laws of are to obey.  This means that we ought not assume Maxwell's equations a priori before we get the meta-laws down.  The way I see it Maxwell's equations *depend on* the Lorentz invariance of SR; they don't dictate that invariance *to* SR (contrary to the formulations of SR that one often sees that takes the constancy of the speed of light as the 2nd postulate of SR).

The way I see things is that SR's Lorentz transformations and the Lorentz invariance of the laws of nature (or a local version of such invariance if we want to accommodate a curved spacetime for GR later) depends on the following two postulates:

P1) The laws of physics (whatever they happen to be, and regardless of which particles may or may not be massless) are form-invariant under transformations among the equivalence class of (local) inertial reference frames.

P2) There is no such thing as instantaneous-interaction-at-a-distance.

The first immediate corollary of P2) is:

C1) A speed limit of causation must exist between pairs of fixed (locally placed) events that are separated in space.

The next stronger corollary based on both C1) *and* P1) is:

C2) The causal speed limit between events is a universal constant that has the same value for all pairs of (locally placed) events throughout all of spacetime and is independent of direction, inertial reference frame used, etc.  We, by convention, call this universal speed limit of causation c.

The first lemma, coming from P1) is:

L1) The transformations for transforming between the Cartesian coordinate spacetime intervals between pairs of (local) events of spacetime in one inertial frame to the corresponding Cartesian coordinates of those spacetime intervals in another Cartesian inertial reference frame are *linear* transformations (because nonlinearities in the transformations would cause spacetime inhomogeneities that are inconsistent with P1).

Using L1), C2), and P1) we then derive Theorem:

T1) The actual form of the transformation laws for (local) spacetime intervals among the equivalence class of inertial reference frames is the group of homogeneous Lorentz Transformations.  These transformations depend on a 6-parameter family of constants in 4-d spacetime (3 parameters for velocity boosts and 3 for orientation changes, i.e. rotations, in space).

Because of the linearity of the homogeneous LTs the (local differential) spacetime intervals can be integrated giving the form of the transformation laws between the actual (local) coordinates of individual events in one inertial coordinate system in terms of the corresponding spacetime coordinates of those events in another inertial Cartesian spacetime coordinate system.  This integration of the intervals generates a set of 4 integration constant offsets between the spacetime origins among the different inertial coordinate systems.  This leads us to:

T2) The actual form of the transformation rules between the Cartesian coordinates of the events of spacetime among the equivalence class of all inertial reference frames is the (10-parameter) Poincaré group of transformations.

All the usual special relativistic effects, like length contraction, time dilation, lack of absolute simultaneity and relative time ordering of spacelike separated events are all consequences of the structure of the equations in T1) (and T2)).

None of any of this involves anything about light or electromagnetism.  It happens at a meta level before the specific laws of physics are found.  When they are found they must be consistent with all the previously derived postulates, corollaries, lemmas, and theorems found so far.  To get the specific form of the various laws of nature in mechanics, electromagnetism, etc we need to supplement all of this stuff with a third meta-law/postulate.  That 3rd postulate is:

P3)  The guiding principle for finding the laws of physics, at least at the classical level, is Hamilton's Principle of least/stationary action.  In particular, the laws of nature are the Euler-Lagrange equations of stationarity of an appropriate action functional that is form invariant under the action of all the elements of the Poincaré Group in T2).

Also P3) is what tells us the formulae to use for the momentum, energy, angular momentum, etc. for particles and fields that appear in the laws of physics.

Specifically, Maxwell's equations of E&M come from P3) when we extremize the action w.r.t. to a gauge field having a local U(1) gauge symmetry.  Other laws of nature come out when other imposed symmetries for other interactions are included.

The usual SR formulae for the energy and momentum of a free particle with a mass m come out of P3) when the action is taken as a Lorentz scalar that is proportional to the elapsed proper time of the particle along its world line.  The proportionality constant ends up being -m*c^2.  One of the important results of this is that the speed of a particle of nonzero (real) mass is always less than the causal speed limit c with the energy and momentum magnitude of the particle diverging to infinity as the speed of the particle is increased toward c.  Another consequence is that a particle of mass m has a rest energy of m*c^2.  Another consequence is that if a particle has exactly zero mass its speed is always c no matter how large or small its energy & momentum magnitude may be.  All massless particles must always travel *at* the causal speed limit c (in all inertial reference frames).

As was discussed in this forum earlier this year, Hamilton's principle and the symmetries it obeys are not sufficient to uniquely determine the laws of nature.  But the possibilities can be further winnowed down (and sometimes even zeroed in on completely, i.e. in the case of a free particle) by making a further 4th postulate:

P4) The E-L equations found from P3) are to be no higher than 2nd order in their derivatives.

In any event we don't get to light & electromagnetism until *after* we work out the dynamical consequences of P3) and that comes well *after* we have derived the usual spacetime effects of SR.  In short, the constancy of the speed of light is a theorem rather than a postulate of SR.

David Bowman