Relativity and Light
Vladimir A. Petrov,
Institute for High Energy Physics , NRC KI, Protvino
Dedicated to The Year of Light...
Maxwell
Electrodynamical constant c
1/c2 =[00] = T2 L-2
Galileo
“...have the ship proceed with any speed you like, so long as the motion is
uniform and not fluctuating this way and that. You will discover not the least
change in all the effects named, nor could you tell from any of them whether
the ship was moving or standing still “ (1632)
Fitzgerald
L = L0 (1-(v/c)2)1/2
Larmor
“. . . individual electrons describe corresponding
parts of their orbits in times shorter for the (rest)
system ...“ (1897)
Lorentz
Ether does exist but we can’t detect it...
Electromagnetic phenomena in a system moving with
any velocity smaller than that of light (1904)
Poincaré
La Science et l'Hypothèse (1902)
“The laws of physical phenomena must be the same
for a fixed observer and for an observer in rectilinear
and uniform motion so that we have no possibility of
perceiving whether or not we are dragged in such a
motion. (Plenary talk at St.-Louis, 1904)
Einstein
Zur Elektrodynamik bewegter Körper (1905)
Minkowski
“...space for itself, and time for itself shall completely
reduce to a mere shadow, and only some sort of union
of the two shall preserve independence.” (1909)
Ignatowski
1875 - 1942
Should we postulate the constancy
of the light velocity in addition to the relativity principle ?
Some general remarks on the relativity principle.
W. V. IGNATOWSKY
When EINSTEIN introduced the relativity principle some time ago, he simultaneously assumed that
the speed of light shall be a universal constant, i.e. it maintains the same value in all coordinate
systems. Also MINKOWSKI started from the invariant in his investigations, although it is to be
concluded from his lecture "Space and Time"[1], that he attributed to the meaning of a universal
space-time constant rather than that of the speed of light.
Now I've asked myself the question, at which relations or transformation equations one arrives when
only the relativity principle is placed at the top of the investigation, and whether the LORENTZian
transformation equations are the only ones at all, that satisfy the relativity principle.
In order to answer this question, we again repeat what is given to us by the relativity principle per se.
If we have two coordinate systems and , being in translatory motion with respect to each other, then
the relativity principle says that both systems can be seen as equally valid, i.e. any of them can be
seen as at rest and the other one as in motion. In other words: we cannot determine absolute
motion.
However, if and are equally valid, and if we can express in system any physical quantity by a function of
parameters , i.e. by writing
Ignatowski’s Transformations
x = ( x – vt ) / (1- v2 n)1/2
t = ( t – vnx) / (1- v2n)1/2
Follow from the homogeneity and isotropy of
space and time, and relativity principle with a
group character of the space-time
transformations.
n is a “world” constant apriori unknown both in
magnitude and sign.
-  n  1/v2
Fixing the sign of n
Energy of a free body:
2
1/2
E = m/n(1- v n)
E/m > 0  n>0
1- v2n > 0  v2<1/n  v2max = V2
V = c = a universal maximum velocity
Limiting Velocity
Galilean coordinates:
ds2 = c2dt2 – dx2 – dy2 – dz2 = ημνdXμdXν
ημν = (1,-1,-1,-1), Minkowski metric tensor in g.c.Xμ
Arbitrary coordinates: xμ = xμ (Xα)
ds2 = (x)dxdx,
 (x) = CXαXβ
Time-like: ds2>0  ds2 = c2dT2
Space-like: ds2<0  ds2 = - dl2 = - dx2-dy2-dz2
Isotropic or light-like: ds2 = 0
ds2 = c2[ 00 dt + oidxi/(c  00 )]2- [- ik + 0i 0k/ 00] dxidxk = c2d2 – dl2
ds2 = c2d2 – dl2=0
Physical velocity of light
c = dl/d
Coordinate velocity of light
ḉ = [00 dt / ( 00 - 0i ei) ]c
0<ḉ<
For massive particles ds2 > 0  Causality
V = dl/d < Vmax = c
The existence of the limiting velocity for massive
bodies is a consequence of the pseudo-Euclidean
geometry of Minkowski space-time.
The very essence of the relativity theory is that we live
in Minkowski space with pseudo-Euclidean geometry.
This makes relativity universal and valid for any kind of
interaction.
Heuristic role of light in this story is beyond any doubt.