Using of the Generalized Special Relativity in Deriving the Equation of the Gravitational Red-Shift

doi:10.4236/jmp.2011.25045

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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

E-mail: [email protected]

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

00 2





(1)

where, 00 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)

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]:

ddcx



gdx







(3)

where





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,

cctxxx



 .ct (4)

where i denotes the particle position (covariant) vector

according to Lorentz covariance.

Then

d1dd

ddd

xx v

ttt





 

(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

1, 1gg ggc



  (6)

Then



becomes

00 00

d1dd

ddd

ttt





  

(7)

When the effect of motion only is considered, the ex-

pression for time in special relativity (SR) takes the form

[5].





(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].

 (9)

In view of Equations 7 and 8, the expression:



 (10)

can be generalized to recognize the effect of motion as

well as gravity on time, to get:

00 2





(11)

The same result can be obtained for the volume where

the effect of motion and gravity respectively gives [6]:

VV c



(12)

000

VgVgV0

(13)

The generalization can be done by utilizing Equation 7

to find that:

000

VV gV







(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

200 00

0000 00000

HcgT ggg











 





(15)

where H is Hamiltonian,



is the density, and is

energy tensor.

Using Equations 14 and 15 yields:

200 0

cVV



 (16)

Therefore:

00 0

00 2





(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

, then the relative mass according to (GSR)

is found to be:



















(18)

When the field is weak in the sense that:



 (19)

And when the speed v is very low such that:

 (20)

M. H. M. HILO

372

Equation 18 reduced to:

mmm







 















(21)

Using the identity



11 for

xnxx 

We can get that

1mm c















(22)

And, when the field is so strong such that

21 and 1



 (23)

Then Equation 18 reduced to

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]:

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:

1Emcmcc





 









(26)

The difference in particles or light energy presented as

[7].

22 22

00 0

1Emcmcmc mcm







  



 (27)

Then

000

Ehv

mc c





 

(28)



 (29)

As mentioned before,



denotes the potential field,

which can be given by







Substituting



in Equation 29, then the following

equation is obtained

vc rc





 (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

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

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Relativi

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