A Comment on the Invariance of the Speed of Light Harihar Behera

A Comment on the Invariance of the Speed of Light
Harihar Behera
Patapur, P.O.: Endal, Jajpur-755023, Orissa, India
E-mail : [email protected]
Abstract
The invariance of the speed of light in all inertial frames is shown
to be an inevitable consequence of the relativity principle of special
relativity contrary to the view held by Hsu and Hsu in taiji relativity
where the speed of light is no longer a universal constant.The present
approach is not only new but also much simpler than the existing approaches.
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The two postulates of special relativity, viz.,(i) the relativity principle
and (ii) the postulate on the speed of light, are the basics of a theory that
has been tremendously successful in describing a wide range of phenomena,
although there exist derivations of Lorentz transformations (LT) without the
postulate on the speed of light [1-8]. However, using the relativity principle
only, Hsu and Hsu [9] have developed a general theory, named taiji relativity, which has four dimensional symmetry and is consistent with experiments
although the speed of light is no longer a universal constant within its framework. In the development of taiji relativity the authors have considered the
usual case of two inertial frames F and F 0 and to simplify the discussion
defined the speed of light to be constant and isotropic in the frame F by synchronizing the clocks in that frame according to the usual method of special
relativity. The authors have made this definition in one frame only and could
just as easily have used the F frame, as all frames are equivalent, so this procedure does not select a preferred frame. As the principle of relativity itself
does not specify how the F and F 0 clocks should be related, the theory does
not tell us anything about the speed of light in any other frame, they say
[9-12]. But here in this communication, the invariance of the speed of light
in all inertial frames is shown to be a natural consequence of the relativity
principle contrary to the view held by the authors [9-12].
Consider two inertial frames F and F 0 which are in uniform relative motion. An event may be characterized by specifying the co-ordinates (x, y, z, t)
of the event in F and the event is characterized by the co-ordinates (x0 , y 0, z 0 , t0 )
in F 0 . Let us proceed to find a transformation between (x, y, z, t) and (x0 , y 0, z 0 , t0 )
without the light-speed postulate of special relativity. To simplify the algebra let the relative velocityv of F and F 0 be along a common x/x0 axis with
corresponding planes parallel. Also at the instant the origins 0 and 00 coincide, we let the clocks there to read t = 0 and t0 = 0 respectively. The
homogeneity of space and time in inertial frames requires that the transformations must be linear so that the simplest form they can take is
x0 = k(x − vt); y 0 = y; z 0 = z; t0 = lx + mt
(1)
In order to determine the values of the three co-efficients k, l, and m we shall
use the idea of the homogeneity and isotropy of space. Let us assume that
at the time t = 0 a spherical electromagnetic wave left the origin of F which
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coincides with the origin of F 0 at that moment. Let the speed of light be
assumed as c and c0 in F and F 0 respectively with c 6= c0 . We know that
the two frames F and F 0 are equivalent. Therefore the electromagnetic wave
propagates in all directions in each inertial frame as the space is homogeneous
and isotropic. By the equivalence of inertial frames (i.e. relativity principle),
the wave forms (fronts) of the electromagnetic wave in the two frames F and
F 0 must be equivalent or similar, otherwise it would be possible to determine
from the shape of the wave front which one i.e.F or F 0 is at rest or in uniform
motion from which the absolute nature of motion or rest may be inferred
contrary to the relativity principle. Again the wave front of a spherical
electromagnetic wave in an isotropic and homogeneous medium describes a
sphere whose radius expands with time at a rate equal to its speed in that
medium. Thus the equation of the spherical wave fronts in the two inertial
frames must take the forms:
x2 + y 2 + z 2 = c2 t2
02
02
02
(2)
02 02
x +y +z =c t
(3)
when the transformations (1) are substituted in (3), we get
(k 2 − l2 c02 )x2 + y 2 + z 2 − 2(lmc02 − k 2 v) = (m2 c02 − k 2 v 2 )t2
(4)
In order for the expression (4) to agree with (2), which represents the same
thing, we must have
(i)
(ii)
(iii)
k 2 − l2 c02 = 1,
m2 c02 − k 2 v 2 = c2 ,
lmc02 − k 2 v = 0
(5)
This set of three equations (5) when solved for k, l and m yield
k = (1 − v 2 /c2 )−1/2 ,
l = −(c/c0 )(v/c2 )(1 − v 2 /c2 )−1/2 ,
m = (c/c0 )(1 − v 2 /c2 )−1/2
(i)
(ii)
(iii)
(6)
Now in view of these values of k, l and m, the transformations (1) can be
represented in the following matrix form, viz.,





x0
y0
z0
t0






 = A


x
y
z
t



,





where A = 
3
k
0
0
−kv/cc0
0
1
0
0

0 −kv
0
0 


1
0 
0 kc/c0
(7)
The inverse transformations for x, y, z, t, then given by





x
y
z
t






 = A−1 


x0
y0
z0
t0



,





where A−1 = 
k
0
0
kv/c2
0
1
0
0
0 kvc0 /c
0
0
1
0
0 kc0 /c





(8)
Hence the transformation for x becomes
x = k(x0 + vc0 t0 /c)
(9)
But relativity principle demands that the transformation for x must be given
by [13]:
x = k(x0 + vt0 )
(10)
Thus Eq. (9) contradicts Eq. (10). Such a contradiction arises because of
our assumption that c 6= c0 . Hence in the order for Eq. (9) to be in accord
with the relativity principle (so with Eq. (10)),we must have c = c0 .
From the above discussion, we may conclude that the postulation on the
speed of light in special relativity is an inevitable consequence of the relativity principle taken in conjunction with the idea of the homogeneity and
isotropy of space and the homogeneity of time in all inertial frames. The
present approach is physically distinct from and logically simpler than the
standard special relativity, as we make use of only one postulate. The idea
of constructing a relativity theory by using only the relativity principle has
also been discussed by Ritz, Tolman and Pauli [14], but the present approach
is not only new but also much simpler than the existing approaches and has
been given as a preliminary report in [15].
The author thanks Prof. N.Barik, Dept. of Phys. Utkal University,
Bhubaneswar, P.C.Naik, Dept. of Physics, D.D.College, Keonjhar and N.K.
Behera, Dept. of Chem. U.N. College, Soro for fruitful discussions and
suggestions.The author also acknoledges the help received from the Institute
of Physics,Bhubaneswar for using its Library and Computer Centre for this
work.
References
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[2] L.A. Pars, Philos. Mag. 42, 249 (1921).
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1977) p.51.
[4] V.Berzi and V. Gorini,“Reciprocity principle and Lorentz transformations”J. Math.Phys.10 1518-1524 (1969).
[5] A.R.Lee and T.M.Kalotas,“Loerntz transformation from the first postulate”, Am.J.Phys.43,434-437(1975).
[6] J.M.L´evy-Leblond,“One more derivation of the Lorentz transformation,” 44, 271-277 (1976).
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physics/0012011
[8] O. L. Lange,Comment on “Space-time exchange invariance:Special relativity as a symmetry principle” by J. H. Field [Am. J. phys. 69(5),569575 (2001)], Am. J. Phys.70(1) ,78-79 (2002).
[9] Jong-Ping Hsu and L. Hsu, A physical theory based solely on the first
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[11] J.P. Hsu, Phys. Lett. A, 97 (1983) 137; Nuovo Cimento B 74 (1983) 67.
[12] J.P. Hsu and C. Whan, Phys, Rev. A 38 (1988) 2248, Appendix.
[13] A.N. Matveev,Mechanics and Theory of Relativity, Eng. Translation(Mir. Publishers, Moscow, 1989) p. 101.
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[15] H. Behera,Bulletin of Orissa Physical Society, Vol. X (2002) 153.
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