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/c 2 =[  0  0 ] = T 2 L -2

“...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) Galileo

Fitzgerald L = L 0 (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 “ 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) La Science et l'Hypothèse (1902)

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 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.Space and Time [1] 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- v 2 n) 1/2 t = ( t – vnx) / (1- v 2 n) 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/v 2 Fixing the sign of n Energy of a free body: E = m/n(1- v 2 n) 1/2 E/m > 0  n>0 1- v 2 n > 0  v 2 <1/n  v 2 max = V 2 V = c = a universal maximum velocity

Limiting Velocity Galilean coordinates: ds 2 = c 2 dt 2 – dx 2 – dy 2 – dz 2 = η μν dX μ dX ν η μν = (1,-1,-1,-1), Minkowski metric tensor in g.c.X μ Arbitrary coordinates: x μ = x μ (X α ) ds 2 =   (x)dx  dx,   (x) = C  X α X β Time-like: ds 2 >0  ds 2 = c 2 dT 2 Space-like: ds 2 <0  ds 2 = - dl 2 = - dx 2 -dy 2 -dz 2 Isotropic or light-like: ds 2 = 0 ds 2 = c 2 [   00 dt +  oi dx i /(c   00 )]2- - [-  ik +  0i  0k /  00 ] dx i dx k = c 2 d  2 – dl 2 The true content of theory is:

ds 2 = c 2 d  2 – dl 2 =0 Physical velocity of light c = dl/d  Coordinate velocity of light ḉ = [  00 dt / (   00 -  0i e i ) ]c 0 < ḉ <  For massive particles ds 2 > 0  Causality V = dl/d  < V max = 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.