Journal of Modern Physics, 2011, 2, 370-373
doi:10.4236/jmp.2011.25045 Published Online May 2011 (http://www.SciRP.org/journal/jmp)
Using of the Generalized Special Relativity in Deriving the
Equation of the Gravitational Red-Shift
Mahmoud Hamid Mahmoud Hilo1,2
1Department of Physics, Faculty of Education, Al-Zaiem Al-Azhari University, Omdurman, Sudan
2Department of Physics, Faculty of Science and Arts at Al-Rass, Qassim University, KSA
Received January 9, 2011; revised March 10, 2011; accepted March 21, 2011
Abstract
In this work we present a study of a new method to prove the equation of the gravitational red shift of spec-
tral lines. That’s according to the generalized special relativity theory. The equation of the gravitational red
shift of spectral lines has been studied in many different works, using different methods depending on the
Newtonian mechanics, and other theories. Although attention was drawn to the fact that the well-known ex-
pression of the gravitational Red-Shift of spectral lines may be derived with no recourse to the general rela-
tivity theory! In this study a unique derivation has been done using the Generalized Special Relativity (GSR)
and the same result obtained.
Keywords: Generalized, Red-Shift, GSR, Approximations, Gravitational
1. Introduction
In physics, light or other forms of electromagnetic radia-
tions of a certain wavelength originating from a source
placed in a region of strong gravitational field (and
which could be said to have climbed “uphill” out of the
gravity well) will be found to be longer wavelength when
received by observer in a region of weak gravitational
field [1]. If we apply to optical wave-lengths this mani-
fests itself as a change in the color of the light, the wave-
length is shifted towards the red (making it less energetic,
longer in wavelength, and lower in frequency) part of the
spectrum. This effect is called the gravitational red shift,
and the other spectral lines found in the light, will also be
shifted towards the longer wavelength, or red end of the
spectrum. This shift can be observed along the entire
electromagnetic spectrum [2].
In all basic studies involved in the theory of general
relativity, attention was drawn to the three main prob-
lems related to it, those are Well-Known advance of the
perihelion of the planet Mercury, the Gravitational De-
flection of Light Rays and the Gravitational Red-Shift of
Spectral Lines. The gravitational red-shift discussed within
the emitted rays from a particle that located in the field
of another rest particle due to a spherical symmetry (such
as Solar field), the atoms that compose the gases edges of
a rest star forms light sources in the star field, and ac-
cording to this information the Gravitational Red-Shift
obtained [3]. This study introduces a new method to ob-
tain the same result of the Gravitational Red-Shift using
the Generalized Special Relativity theory, (GSR) by
adopting the approximation of the gravitational potential.
2. Objective
Objective of this work is to prove the equation of the gra-
vitational redshift as well as to test the theory of the gen-
eralized special relativity.
3. Generlized Special Relativity (GSR)
Theory
The Generalized Special Relativity theory is a new form
of the special relativity theory that adopts the gravita-
tional potential, and it gives the formula of relative mass
to be as follows [4]:
00 0
2
00 2
v
g
m
m
gc


(1)
where, 00 2
2
1gc

 and  denotes the gravitational po-
tential, or the field in which the mass is measured.
The derivation of the mass Equation 1 using the Gen-

M. H. M. HILO371
eralized Special Relativity (GSR) can be found as follows:
In the Special Relativity (SR), the time, length and
mass can be obtained in any moving frame by either
multiplying or dividing their values in the rest frame by a
factor  :
2
2
1v
c
  (2)
where c is the speed of light, and v is the velocity of the
particle.
To see how gravity affects these quantities it is a con-
venient to re-express  in terms of the proper time, [5]:
22
ddcx  gdx



(3)
where
g


is the metric tensor, and,

and

de-
notes the contravariant (covariant) vectors.
Which is a common language to both Special Relativ-
ity (SR) and General Relativity (GR). We know that in
special relativity (SR) Equation 3 reduces to [6]:
22 220
dddd,
ii
cctxxx   .ct (4)
where i denotes the particle position (covariant) vector
according to Lorentz covariance.
Then
2
22
d1dd
11
ddd
ii
xx v
ttt cc


 
(5)
Thus we can easily generalize  to include the effect of
gravitation by using Equation 3 and by adopting the
weak field approximation where [6]:
11 22 33002
2
1, 1gg ggc

  (6)
Then

becomes
2
00 00
2
d1dd
ddd
ii
2
x
x
gg
ttt c

   
v
c
(7)
When the effect of motion only is considered, the ex-
pression for time in special relativity (SR) takes the form
[5].
0
2
2
d
d
1
t
t
v
c


(8)
where the subscript 0 stands for the quantity measured in
the rest frame. While gravity only affects time, its ex-
pression is given by [6].
0
00
d
dt t
g
 (9)
In view of Equations 7 and 8, the expression:
0
d
dt t

 (10)
can be generalized to recognize the effect of motion as
well as gravity on time, to get:
0
2
00 2
d
dt t
v
gc


(11)
The same result can be obtained for the volume where
the effect of motion and gravity respectively gives [6]:
2
02
1v VV c  (12)
000
VgVgV0 (13)
The generalization can be done by utilizing Equation 7
to find that:
2
000
2
v
VV gV
c

0


(14)
To generalize the concept of mass to include the effect
of gravitation we use the expression for the Hamiltonian
in general relativity, i.e. [6]:
222
0
200 00
0000 00000
22
0
d
d
cm
x
HcgT ggg
V



c



 


(15)
where H is Hamiltonian,

is the density, and is
energy tensor.
00
T
Using Equations 14 and 15 yields:
2 2
200 0
g
mc mc
cVV   (16)
Therefore:
00 0
2
00 2
g
m
m
v
gc


(17)
which is the expression of mass in the presence of gravi-
tation.
In view of Equation 1 and when we substitute the
value of 00
g
, then the relative mass according to (GSR)
is found to be:
02
2
22
2
1
2
1
mc
m
v
cc









(18)
When the field is weak in the sense that:
2
21
c

 (19)
And when the speed v is very low such that:
2
21
v
c
 (20)

M. H. M. HILO 372
Equation 18 reduced to:
02
02
2
2
12
1
2
1
mc
mmm
c
c





 







1
(21)
Using the identity

11 for
n
xnxx 
We can get that
02
1mm c







(22)
And, when the field is so strong such that
2
22
21 and 1 v
cc

 (23)
Then Equation 18 reduced to
02
2
mm c

 (24)
4. Derivation of the Red Shift Equation
Einstein’s mass and energy equivalent relation, agrees
that the energy of a particle is given by [7]:
2
Emc hv (25)
where h is Planck’s constant and c is the speed of light,
substitute the mass m from Equation 22 in Equation 25,
one gets:
22
02
1Emcmcc


 




(26)
The difference in particles or light energy presented as
[7].
22 22
00 0
2
1Emcmcmc mcm
c0



  

 (27)
Then
0
2
000
m Ehv
Ehv mc c
2



 
(28)
Or
2
0
v
vc

 (29)
As mentioned before,

denotes the potential field,
which can be given by
M
G
r



Substituting
 in Equation 29, then the following
equation is obtained
22
0
vM
vc rc
G


 (30)
Equation 30 is found to be the
Gravitational Red-Shift of the spect
expecte
his paper are only con-
ion of the gravitational
d 24 are obtained to give the expression
e presence of gravitational field within the
gravitational red shift from the gener-
ativity (GSR) model stand point, was
po
same formula of the
ral lines which is the
d result.
5. Materials and Methods
Materials and methods applied in t
centrated on mathematical derivat
redshift equation. Starting with the equation of mass in
the sense of the generalized special relativity theory, we
found that it is possible to prove gravitational redshift
equation.
6. Results
Equations 17 an
of mass in th
special relativity theory, which means generalizing the
special relativity to include the gravitational potential.
Equation 30 is the same as the equation of the gravita-
tional red-shift, obtained by Equation 24 using Einstein's
mass and energy equivalent relation, and that presents
the generalized special relativity as an adequate theory in
proving the equations of famous physical phenomena.
7. Discussion
The expression of
alized special rel
derived in previous Section 4, the expression obtained in
Equation 30 is in complete agreement with that obtained
from general relativity in [7]. In one hand, that means
adopting of the weak field approximation in the general-
ized special relativity, leads to succeed in proving an
important theory in physics such as the gravitational red-
shift of the spectral lines, and on the other hand, it ex-
plains that the appearance of the weak field does not af-
fect the red shift negatively, but, oppositely the general-
ized special relativity succeeded again.
The result obtained in this study using Generalized
Special Relativity theory (GSR), and the result obtained
by Evans and Dunning [7], which, used the Newtonian
mechanics, agree that the (GSR) is another crucial theory
to examine one of the three important tests related to the
theory of the general relativity, those are Well-Known
advance of the perihelion of the planet Mercury, the
Gravitational Deflection of Light Rays and the Gravita-
tional Red-Shift of Spectral Lines [6,8,9].
When we used the approximation of g00, we depend
completely on the result found in [5], dealing with the
special relativity (SR) in the presence of gravitational
tential and not only on the general relativity (GR).

M. H. M. HILO 373
Expanding Universe,” Monthly Notices of the Royal As-
ciety, Vol. 388, No. 3, 2008, pp. 960-964.
365-2966.2008.13477.x
Retadration,
ty in Estimating the Proton (Nucleon) Mass to
r Calculus and
ation and Cosmology,” 3rd Edition,
ational
Cos-
Bazin and M. Schiffer, “Introduction to
8. References
[1] G. F. Lewis, et al. “Cosmological Radar Ranging in an R
tronomical So
doi:10.1111/j.1
[2] T. Koupelis and K. F. Kuhn, “In Quest of the Universe,”
5th Edition, Jones & Bartlett Publishers, Sudbury, 2007,
p. 557.
[3] O. D. Jefimenko, “Electromagnetic and mology,” 1st Edition, Oxford University Press, Oxford,
1934, p. 591.
[9] R. Adler, M.
Theory and Relativity,” 2nd Edition, 2005.
[4] M. H. M. Hilo, et al. “Using of the Generalized Special
Relativi
Explain the Mass Defect,” Natural Science, Vol. 3, No. 2,
pp. 141-144. doi:10.4236/ns.2011.32020
[5] F. L. Derek, “An Introduction to Tenso
elativity,” 3rd Edition, John Wiley & Sons Ltd., New
York, 1982, pp. 5-6.
[6] S. Weinberg, “Gravit
John Wiley & Sons Ltd., New York, 1972, p. 688.
[7] R. F. Evans and J. Dunning-Davies, “The Gravit
Red-Shift,” University of Hull, England, 2004, p. 3.
[8] R. C. Tolman, “Relativity, Thermodynamics and
General Relativity,” 2nd Edition, McGraw-Hill, New
York, 1975.