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Advanced Series on Theoretical Physical Science Volume
Extended relativit)' « ® r o
10
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A Broader View of Relativity
Taiji relativity
General Implications of Lorentz and Poincare Invariance 2nd Edition
Jong-Ping Hsu Leonardo Hsu
A Broader View of Relativity General Implications of Lorentz and Poincare Invariance 2nd Edition
ADVANCED SERIES ON THEORETICAL PHYSICAL SCIENCE A Collaboration between World Scientific and Institute of Theoretical Physics Series Editors: Dai Yuan-Ben, Hao Bai-Lin, Su Zhao-Bin (Institute of Theoretical Physics Academia Sinica)
Published Vol. 1: Yang-Baxter Equation and Quantum Enveloping Algebras (Zhong-Qi Ma) Vol. 2: Geometric Methods in the Elastic Theory of Membrane in Liquid Crystal Phases (Ouyang Zhong-Can, Xie Yu-Zhang & Liu Ji-Xing) Vol. 4: Special Relativity and Its Experimental Foundation (Yuan Zhong Zhang) Vol. 6: Differential Geometry for Physicists (Bo-Yu Hou & Bo-Yuan Hou) Vol. 7: Einstein's Relativity and Beyond (Jong-Ping Hsu) Vol. 8: Lorentz and Poincare Invariance: 100 Years of Relativity (J.-P. Hsu & Y.-Z. Zhang) Vol. 9: 100 Years of Gravity and Accelerated Frames: The Deepest Insights of Einstein and Yang-Mills (J.-P. Hsu & D. Fine) Vol. 10: A Broader View of Relativity: General Implications of Lorentz and Poincare Invariance (J.-P. Hsu&L Hsu)
Advanced Series on Theoretical Physical Science ® « g t ® * « 6 ® < 8=""> 8 ® ® ®
r UCIIJ^I 1
tff% S I #*i /c, k x , ky, k z ) between two moving frames, F'=F'(V) and F"=F"(U), are given by
k"x = P T P
.
k"y=pk' y .
k"z = pk'z>
k"° = p K - p K x ,(5.10)
2
vi-p' 2
Vi-p'
where (k x , k y , k£) = (k' 1 , k' 2 , k' 3 ) and
p.m, K(V)
v.
u v
-
l-UV/c2
,
p.=v:.
(511)
c
Suppose one performs the experiment in the F'(V) frame and that the atoms are at rest in F"(U) so that kx = 2itA" and k"° = 2itv"/c. In practice, one cannot know the wavelength and frequency X" and v" of the light emitted by the atoms as measured by observers in the F"(U) frame. One can only compare the shifted X' and v' with the unshifted quantities X'0 and v'0 associated with the same kind of atoms at rest in the laboratory frame F'(V). Since F'(V) and F"(U) are not completely equivalent, as shown in (5.5), the wavelength and the frequency of light emitted by atoms at rest in F' and measured by observers in F' are not the same as those emitted by the same kind of atoms at rest in F" and measured in F". Thus, one does not have the usual relations X" = X'0 and v" = v'0 of special relativity. Rather, one has in general
The Contributions ofPoincare
49
k^o K(V)
K(U)
and
( m "/c) K(U)
=
K/c) K(V)
(512)
for the contravariant wave 4-vector k,(i = (k ,k ,k ,k' 3 ) because of the metric tensor in (5.5) for the two frames F'(V) and F"(U). Thus, one obtains
1
1
X"K(U)
X.'0K(V)
and
_ ^ = J^S_. K(U) K(V)
(5.13)
It follows from (5.10) and (5.13) that the observable Doppler shifts in the moving frame F'(V) are given b y ; H
± ^ ± v
°
V
Vo-v'-^Sl,
VI-P'2'
(5.14)
VI-P' 2 '
which are exactly the same as those in special relativity. Therefore, we see that the absolute velocities V and U cannot be determined individually by the Doppler shift experiment. The best one can do is to determine the "relative velocity" p' between F'(V) and F"(U), as shown by (5