
Fivevectors in STR
Posted:
May 17, 2013 9:57 AM


This short note shows how rewriting the fourvector momentum of Special relativity in a fivevector form, and by introducing a reduced velocity M (M^2 = c^2  v^2), the relativistic energymomentum equation can effectively be written in a massless form in terms of velocities only, with a zeronorm, 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 threevector (px,py,pz), px = m*vx etc P4 = momentum fourvector, see text P5 = momentum fivevector, see text v = velocity magnitude V of vector v3, v=sqrt(vx^2+vy^2+vz^2) v3 = velocity threevector (vx,vy,vz) v5 = velocity fivevector, see text
Four Vector Momentum 
Traditionally, Special Relativity works in fourvectors and, in particular, using Matlab's vector/matrix notation, the momentum fourvector 'p4' is defined as the following column vector:
p4 = [E/c ; P3],
with its reciprocal, rowvector 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 nonzero and equal to the rest mass energy E0 squared.
This is all standard relativity, now for the fivevector forms.
Five Vector Momentum 
Defining a fivevector momentum p5 as
p5 = [E0/c ; p3 ; E/c]
and its reciprocal, rowvector 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, photonlike 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 sonamed 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 adhoc 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 zerospeed 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 fivevector velocity v5 as
(5.1) v5 = [M ; v3 ; c],
and its reciprocal, row vector (covector) v5' as
v5' = [M v3 c],
then the relativistic energy momentum equation (5) is written as the fivevector, zero norm inner product
(6) v5'*v5 = [M v c] * [M ; v ; c] = 0.
Thus, by introducing a fivevector, 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 zeronorm contraction of two fivevectors, with no mass terms.
What is a mass energy equation (1) in fourvector relativity with nonzero norm E0^2, is now a seemingly massfree, zeronorm equation in 'URM5' (5x5 Unity Root Matrix Theory), where The fivevector 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 zeronorm, null trajectory, what was a particle with mass and finite, relativistic interval (not shown, see refs below), is now more akin to a massless, photonlike 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 5dimensional 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.
For the full story, see
http://www.urmt.org/urmt_str_paper.pdf
or, more generally,
http://www.urmt.org
Richard Miller Equations not words.
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 ndimensional Pythagoras.

