This short note shows how re-writing the four-vector momentum of Special relativity in a five-vector form, and by introducing a reduced velocity M (M^2 = c^2 - v^2), the relativistic energy-momentum equation can effectively be written in a massless form in terms of velocities only, with a zero-norm, null inner product.
Symbols used in alphabetic order -------------------------------- c = speed of light E = relativistic energy, mass m, E = m*c^2 E0 = rest mass energy, E0 = m0*c^2 m = relativistic mass, speed v>=0 m0 = rest mass, speed v=0 M = reduced velocity, see text p = momentum magnitude of vector p3, p = sqrt(px^2+py^2+pz^2) P3 = momentum three-vector (px,py,pz), px = m*vx etc P4 = momentum four-vector, see text P5 = momentum five-vector, see text v = velocity magnitude V of vector v3, v=sqrt(vx^2+vy^2+vz^2) v3 = velocity three-vector (vx,vy,vz) v5 = velocity five-vector, see text
Four Vector Momentum --------------------
Traditionally, Special Relativity works in four-vectors and, in particular, using Matlab's vector/matrix notation, the momentum four-vector 'p4' is defined as the following column vector:
p4 = [E/c ; P3],
with its reciprocal, row-vector p4' defined as
p4' = [E/c -p3],
where the minus sign in the three vector momentum p3 is a consequence of the Minkowski metric.
The inner product (or 'contraction') of the two vectors is given by
(1) p4'*p4 = (E/c)^2 - p^2 = (E0/c)^2,
which is just the relativistic energy momentum equation, i.e. multiplying throughout by c^2 and rearranging gives
E^2 = (pc)^2 + E0^2,
and the four vector 'norm' is non-zero and equal to the rest mass energy E0 squared.
This is all standard relativity, now for the five-vector forms.
Five Vector Momentum --------------------
Defining a five-vector momentum p5 as
p5 = [E0/c ; p3 ; E/c]
and its reciprocal, row-vector p5' defined as
p5' = [E0/c p3 -E/c]
then the same inner product (1) now becomes the zero norm equation
(2) p5'*p5 = (E0/c)^2 + p^2 - (E/c)^2 = 0
There has been a rerrangement and sign change in the vectors p5, p5' compared with p4 and p4', but this is just notational convenience.
Alternatively stated, the norm of p5, denoted here by ||p5|| is zero, i.e.
p5'*p5 = ||p5|| = 0.
Such a zero norm is desirable because it is akin to obtaining zero interval, or null trajectories in the space of vectors (eigenvectors, see further), and related to the concept of massless, photon-like trajectories.
A velocity quantity, termed reduced velocity, symbol M, is defined as
(2.1) M^2 = c^2 - v^2
and using the gamma factor of relativity, symbol g here (not to be confused with the metric 'g' of GR),
g = 1/sqrt(1 - [v/c]^2)
then the ratio of the reduced velocity to the speed of light is nothing more than the inverse of g, i.e.
(3) M/c = 1/g.
and since the ratio of the rest mass (m0) to the relativistic mass (m) is also the inverse of g,
m0/m = 1/g,
then from (3) the reduced velocity can be defined in terms of the mass ratio (m0/m) as
(4) M/c = m0/m
Note that the reduced velocity is so-named because it is measured down from the speed of light, i.e. if v=0 then M=c, whereas the familiar lab velocity v is measured from zero upward. This may seem an ad-hoc definition but, given the invariance of the speed of light, it actually makes just as much sense (if not more) to use c as a velocity reference point, and not the zero-speed lab frame, which is, of course, relative!
Anyhow, by the usual relativistic definitions of E0 and E, plus momentum p,
E0 = m0*C^2, E = m*c^2, p = m*v,
then the inner product (2) can be written in terms of the masses as
m0^2*c^2 + m^2*v^2 - m^2*c^2 = 0.
Lastly, substituting for the rest mass from definition (4) of the reduced velocity M, the relativistic mass m cancels and the entire inner product (2) now reduces to the following Pythagorean triangle of velocites:
(5) M^2 + v^2 - c^2 = 0,
which is nothing more than the original definition (2.1) of M.
Lastly, defining the five-vector velocity v5 as
(5.1) v5 = [M ; v3 ; c],
and its reciprocal, row vector (co-vector) v5' as
v5' = [M v3 -c],
then the relativistic energy momentum equation (5) is written as the five-vector, zero norm inner product
(6) v5'*v5 = [M v -c] * [M ; v ; c] = 0.
Thus, by introducing a five-vector, velocity form of the four vector momentum, and a reduced velocity M, the entire relativistic energy momentum equation can be written purely in terms of velocities (M,v,c), as a zero-norm contraction of two five-vectors, with no mass terms.
What is a mass energy equation (1) in four-vector relativity with non-zero norm E0^2, is now a seemingly mass-free, zero-norm equation in 'URM5' (5x5 Unity Root Matrix Theory), where The five-vector velocity v5 is an eigenvector in URM5 and the inner product of it and its reciprocal p5' is an invariant of value zero, as is the speed of light c also invariant.
With no mass present, and a zero-norm, null trajectory, what was a particle with mass and finite, relativistic interval (not shown, see refs below), is now more akin to a massless, photon-like equation.
In fact, the relativistic energy momentum equation (5) is actually now a consequence of the standard result in matrix algebra that eigenvectors to different eigenvalues are orthogonal. Pure algebra, no physics.
Unity Root Matrix Theory (URMT) thus reduces the famous relativistic energy equation to the simple, legal combination of integers, that is 5-dimensional Pythagoras here - URMT is ultimately a discrete theory, hence integers.
But with eigenvectors also comes basis states, matrix operators, commutators etc. and paves the way for a treatment of QM and beyond.
Why should our current formulations of the laws of physics be so complicated when what appears in nature as very complicated phenomena, time and time again reveals to have a very simple explanation? Chaos, complexity, and life itself (just four DNA bases) are prime examples.
URMT:- What originated as a study in congruence relations and linear Diophantine equations, in particular with regard to their symmetry and invariants, appears to generate a surprisingly rich field of physical phenomena, and supports the author's premise that, at the smallest, Planck scale, nature reduces to some very simple rules, with its laws formulated as integer equations, and more the realm of number theory than physics. The laws of nature are thus reduced to the legal combinations of integers, which currently appear to be of quadratic degree, in particular hyperbolic and n-dimensional Pythagoras.