American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  

http://asrjetsjournal.org/  
 

A Static Solution to Einstein’s Field Equations for a 

Spherical Distribution of Electrically Counterpoised Dust 

Sashinka Wimaladharmaa*, Nalin de Silvab 

aDepartment of Mathematics, University of Kelaniya, 11600, Sri Lanka. 
bNo.109/1, Railway Avenue, Maharagama , 10280, Sri Lanka. 

aEmail: wimaladharma@kln.ac.lk 
bEmail: nalink2003@yahoo.com 

 

 

 

Abstract 

The work on Einstein-Maxwell equations for a sphere composed of a special kind of matter distribution called 

electrically counterpoised dust (ECD) with constant density has been discussed. The better understanding of the 

electrically counterpoised dust model in detail will helps scientists to understand the universe in better way. 

Along this line, the understanding of static solutions for the electrically counterpoised dust will be a crucial 

factor. To understand the static solutions for the electrically counterpoised dust, static solutions for the Einstein-

Maxwell equations have been found. In particular, static solutions for a sphere composed of electrically 

counterpoised dust matter using standard boundary conditions tested. In addition to that the expressions for the 

redshift moving along a radial geodesic have been obtained. 

Keywords: Einstein field equations; electrically counterpoised dust. 

1. Introduction  

In a distribution of electrically counterpoised dust, there are no resultant force between each particle since all the 

forces between particles in such a distribution have been balanced. Wickramasuriya [1] considered a spherical 

distribution of electrically counterpoised dust (ECD) and proved that Einstein’s field equations can be written in 

a simple form for such a distribution. Using this form, a static solution to Einstein’s field equations has been 

found with the application of standard boundary conditions. 

------------------------------------------------------------------------ 

* Corresponding author.  

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

The red shift of a pulse of light which is emitted at a point inside the sphere as observed by an observer who is 

at a large distance in the exterior region is also calculated. 

2. Material and the Method  

Wickramasuriya [1] consider a spherically metric in the form 

𝒅𝒅𝒅𝒅𝟐𝟐 = 𝒆𝒆𝟐𝟐𝑼𝑼𝒄𝒄𝟐𝟐𝒅𝒅𝒅𝒅𝟐𝟐 − 𝒆𝒆−𝟐𝟐𝑼𝑼(𝒅𝒅𝒅𝒅𝟐𝟐 + 𝒅𝒅𝟐𝟐𝒅𝒅𝒅𝒅𝟐𝟐 + 𝒅𝒅𝟐𝟐 𝐬𝐬𝐬𝐬𝐬𝐬𝟐𝟐 𝒅𝒅𝒅𝒅𝒅𝒅𝟐𝟐)              (1) 

and proved that Einstein’s field equations for a spherically symmetric distribution of electrically counterpoised 

dust (ECD) can be written in the form  

𝟏𝟏
𝒅𝒅𝟐𝟐

𝒅𝒅
𝒅𝒅𝒅𝒅
�𝒅𝒅𝟐𝟐 𝒅𝒅𝒆𝒆

−𝑼𝑼

𝒅𝒅𝒅𝒅
� = −𝟒𝟒𝝅𝝅𝝅𝝅𝒆𝒆−𝟑𝟑𝑼𝑼                                                            (2) 

where  𝝅𝝅 is the density of the sphere. 

Using the transformations 𝒚𝒚 = 𝒆𝒆−𝑼𝑼 and 𝒅𝒅 = 𝒍𝒍𝒍𝒍, equation (2) can be written in the form  

𝟏𝟏
𝒍𝒍𝟐𝟐

𝒅𝒅
𝒅𝒅𝒍𝒍
�𝒍𝒍𝟐𝟐 𝒅𝒅𝒚𝒚

𝒅𝒅𝒍𝒍
� = −𝟒𝟒𝝅𝝅𝝅𝝅𝒍𝒍𝟐𝟐𝒚𝒚𝟑𝟑                                                             (3) 

where  𝒚𝒚 = 𝒚𝒚(𝒍𝒍)  and  𝒚𝒚′ = 𝒅𝒅𝒚𝒚
𝒅𝒅𝒍𝒍

. 

Now choose  𝒍𝒍 such that  𝟒𝟒𝝅𝝅𝝅𝝅𝒍𝒍𝟐𝟐 = 𝟏𝟏. Then (3) reduces to 

𝟏𝟏
𝒍𝒍𝟐𝟐

𝒅𝒅
𝒅𝒅𝒍𝒍
�𝒍𝒍𝟐𝟐 𝒅𝒅𝒚𝒚

𝒅𝒅𝒍𝒍
� = −𝒚𝒚𝟑𝟑                                                                 (4) 

Equation (4) is in the form of Lane Emden equation, the solution of which is 𝒚𝒚, which we write as 𝒅𝒅(𝒍𝒍) since 

the solution of  𝟏𝟏
𝒍𝒍𝟐𝟐

𝒅𝒅
𝒅𝒅𝒍𝒍
�𝒍𝒍𝟐𝟐 𝒅𝒅𝒚𝒚

𝒅𝒅𝒍𝒍
� = −𝒚𝒚𝟑𝟑 is usually written as 𝒅𝒅(𝒍𝒍), the so called Lane Emden function.  

Lane Emden function has been expressed as a power series in even powers of 𝒍𝒍 .The power series converges 

slowly, and it is more advantageous to obtain the solution in terms of a table of values according to 

Dharmawardane [2]. 

He has plotted the graph of 𝒅𝒅�𝒅𝒅
𝒍𝒍
� and  𝒅𝒅�𝒅𝒅

𝒍𝒍
� + 𝒅𝒅

𝒍𝒍
𝒅𝒅′ �𝒅𝒅

𝒍𝒍
�  against 𝒅𝒅

𝒍𝒍
 which is reproduced in Figure 1, where 𝒍𝒍 = 𝒅𝒅

𝒍𝒍
   

Now as it has been shown in Wimaladharma [3] that  𝑒𝑒𝑈𝑈(𝑅𝑅) = 1 ± Φ(𝑅𝑅) , where  Φ(𝑅𝑅)  is the Newtonian 

potential divided by 𝑐𝑐2, where 𝑐𝑐 is the velocity of light.  

Assuming that when 𝑅𝑅 = 0, Φ  and 𝜕𝜕Φ 
𝜕𝜕𝑅𝑅

 to be equal to zero, due to physical conditions, it can be proved that  

𝑦𝑦(0) = 1  and 𝑦𝑦′(0) = 0. 

𝑦𝑦 

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

Therefore the initial of the equation (3) are the same as the initial conditions of Lane Emden equation. 

As an illustration we have, for a few values of  𝑥𝑥 , the values of the Emden function 𝜃𝜃(𝑥𝑥)  given by 

Dharmawardane [2].  

      

 

Figure 1: the graph of 𝜃𝜃 �𝑅𝑅
𝑙𝑙
� and  𝜃𝜃 �𝑅𝑅

𝑙𝑙
� + 𝑅𝑅

𝑙𝑙
𝜃𝜃′ �𝑅𝑅

𝑙𝑙
�  against 𝑅𝑅

𝑙𝑙
 

In Figure 2 a graph of   �−�𝑅𝑅
𝑙𝑙
�
2
𝜃𝜃′ �𝑅𝑅

𝑙𝑙
��  is reproduced against  𝑅𝑅

𝑙𝑙 . 

𝑅𝑅
𝑙𝑙  

Figure 2: the graph of �−�
𝑅𝑅
𝑙𝑙
�
2
𝜃𝜃′ �𝑅𝑅

𝑙𝑙
�� 

Since 𝜃𝜃(𝑥𝑥) = 𝑦𝑦(𝑥𝑥), the interior solution for the above distribution of electrically counterpoised dust (ECD) can 

be written as 𝑒𝑒−𝑈𝑈 = 𝜃𝜃 �𝑅𝑅
𝑙𝑙
�, where 𝑥𝑥 = 𝑅𝑅

𝑙𝑙
. 

0 0.5 1 1.5 2
0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 

0 

0.2 

0.6 

0.8 

1 
( )Ry θ= 






=

l
Ry θ  

𝑦𝑦 

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

Therefore the metric for the interior region comprising electrically counterpoised dust (ECD) can be written in 

the form 

Table 1: the values of the Emden function 𝜃𝜃(𝑥𝑥) 

𝑥𝑥 𝜃𝜃(𝑥𝑥) 
0.0 1.000000 
1.0 0.855058 
2.0 0.582851 
3.0 0.359227 
4.0 0.209282 
5.0 0.11082 
6.0 0.043738 
7.0 -0.0043122 

 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑𝜃𝜃2 + 𝑅𝑅2 sin2 𝜃𝜃 𝑑𝑑𝑑𝑑2)  ,𝑅𝑅 ≤ 𝑎𝑎        (5) 

where  𝑎𝑎 is the radius of the sphere. 

Now for the exterior vacuum region, the matter density 𝜌𝜌 is equal to zero since there is no matter present there. 

Therefore, for the exterior vacuum region (2) becomes 

1
𝑅𝑅2

𝑑𝑑
𝑑𝑑𝑅𝑅
�𝑅𝑅2 𝑑𝑑𝑒𝑒

−𝑈𝑈

𝑑𝑑𝑅𝑅
� = 0

                                       
(6) 

The solution of the differential equation (6) takes the form 

𝑒𝑒−𝑈𝑈 = 𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅

                                               
 
(7)

 

where 𝐴𝐴1 is a constant. 

Therefore the corresponding exterior metric can be written as 

𝑑𝑑𝑑𝑑2 = 𝑒𝑒2𝑈𝑈𝑐𝑐2𝑑𝑑𝑑𝑑2 − 𝑒𝑒−2𝑈𝑈(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑𝜃𝜃2 + 𝑅𝑅2 sin2 𝜃𝜃 𝑑𝑑𝑑𝑑2)                                   (8) 

where 𝑒𝑒−𝑈𝑈 = 𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅

  , 𝑅𝑅 > 𝑎𝑎. 

Now, from (5) and (8) the metric for the whole distribution of matter can be written as  

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

𝑑𝑑𝑑𝑑2 =
1

�𝜃𝜃 �𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐

2𝑑𝑑𝑑𝑑2 − �𝜃𝜃 �
𝑅𝑅
𝑙𝑙
��

2

(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑Ω2)  ,    𝑅𝑅 ≤ 𝑎𝑎 

𝑑𝑑𝑑𝑑2 = 1

�𝐴𝐴1+
𝐴𝐴2
𝑅𝑅 �

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅
�
2

(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑Ω2)        𝑎𝑎 < 𝑅𝑅                                      (9) 

where  𝑑𝑑Ω2 = 𝑑𝑑𝜃𝜃2 + 𝑅𝑅2 sin2 𝜃𝜃 𝑑𝑑𝑑𝑑2. 

Here we apply the standard boundary conditions, known as Lichernowicz’s  boundary conditions which says that 

metric coefficients and their derivatives are continuous at the boundary , 𝑅𝑅 = 𝑎𝑎 . 

Then we obtain 

1

𝜃𝜃�𝑎𝑎𝑙𝑙�
= 1

�𝐴𝐴1+
𝐴𝐴2
𝑎𝑎 �

                 (10) 

− 1

�𝜃𝜃�𝑎𝑎𝑙𝑙��
3 �

1
𝑙𝑙
𝜃𝜃′�

𝑎𝑎
𝑙𝑙�� = − 1

�𝐴𝐴1+
𝐴𝐴2
𝑎𝑎 �

3 �−
𝐴𝐴2
𝑎𝑎2
�

       

 (11) 

Substituting (10) in (11), we obtain  

𝐴𝐴2 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�.

                                                 
(12) 

Using the fact that  𝐴𝐴2 = 𝑚𝑚  according to Wickramasuriya [1], the value of the mass of the sphere of electrically 

counterpoised dust is found as 

𝑚𝑚 = 𝐴𝐴2 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�
                                                                                            (13) 

Substitution of (13) in (10) gives
 

𝐴𝐴1 = 𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
 .                                                     (14) 

Now the metric for the entire spherical distribution takes the form 

𝑑𝑑𝑑𝑑2 =
1

�𝜃𝜃 �𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐

2𝑑𝑑𝑑𝑑2 − �𝜃𝜃 �
𝑅𝑅
𝑙𝑙
��

2

(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑Ω2)  ,                               0 ≤ 𝑅𝑅 ≤ 𝑎𝑎 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅�

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
+ 𝑚𝑚

𝑅𝑅
�
2

(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑Ω2)                  𝑎𝑎 < 𝑅𝑅                                (15)     

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

According to the Figure 1,  𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
= 𝜃𝜃 �𝑎𝑎

𝑙𝑙
� + 𝑎𝑎

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� = 1  only when  𝑎𝑎

𝑙𝑙
= 0  which implies that 

 𝑎𝑎 = 0 and  𝑚𝑚 = 0. 

Thus the metric is Lorentzian at large distances only if a=0 and m=0, which makes the distribution devoid of 

matter. It is the flat space time without matter which is of no interest.  

3. Results 

3.1 The Red shift of a pulse of light 

Here we consider the red shift of a pulse of light which is emitted at a point inside of the sphere as observed by 

an observer who is at a large distance away in the exterior region using expressions for null geodesics in General 

Relativity. 

Consider a pulse of light with front emitted at 𝑅𝑅 = 𝑅𝑅𝑒𝑒 at 𝑑𝑑 = 𝑑𝑑𝑒𝑒 with frequency 𝜈𝜈𝑒𝑒  inside the sphere and an 

observer at 𝑅𝑅 = 𝑎𝑎,  just inside the surface of the sphere receiving it at 𝑑𝑑 = 𝑑𝑑𝑒𝑒′ with frequency  𝜈𝜈𝑒𝑒′. 

Now consider the radial null geodesics within the sphere. 

0 =
1

�𝜃𝜃 �𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐

2𝑑𝑑𝑑𝑑2 − �𝜃𝜃 �
𝑅𝑅
𝑙𝑙
��

2

𝑑𝑑𝑅𝑅2  

 

from which  we obtain  

𝑑𝑑𝑅𝑅
𝑑𝑑𝑑𝑑

= 𝑐𝑐

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2  .                                                                                                         (16) 

(The plus sign is taken since 𝑅𝑅 increases with time 𝑑𝑑 as the photons are going away from the centre of the 

sphere) 

Now integration of (16) gives 

∫ 𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒′
𝑑𝑑𝑒𝑒

= ∫ �𝜃𝜃 �𝑅𝑅
𝑙𝑙
��

2
𝑑𝑑𝑅𝑅𝑅𝑅𝑒𝑒′

𝑅𝑅𝑒𝑒
                                                                                 (17) 

Now consider the rear of the pulse emitted at 𝑅𝑅 = 𝑅𝑅𝑒𝑒 at  𝑑𝑑 = 𝑑𝑑𝑒𝑒 + Δ𝑑𝑑𝑒𝑒 with frequency 𝜈𝜈𝑒𝑒  . 

Let us assume that the rear of the pulse is observed at 𝑅𝑅 = 𝑎𝑎, the boundary of the sphere at 𝑑𝑑 = 𝑑𝑑𝑒𝑒′ + Δ𝑑𝑑𝑒𝑒′ with 

frequency  𝜈𝜈𝑒𝑒′. 

Integration of (17) for the rear of the pulse gives  

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

∫ 𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒′+Δ𝑑𝑑𝑒𝑒′ 
𝑑𝑑𝑒𝑒+Δ𝑑𝑑𝑒𝑒 = ∫ �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
𝑑𝑑𝑅𝑅𝑅𝑅𝑒𝑒′

𝑅𝑅𝑒𝑒
                                                     (18) 

  From  (17) and (18) we obtain 

Δ𝑑𝑑𝑒𝑒 = Δ𝑑𝑑𝑒𝑒′                                                  (19) 

Now from the metric (15) the proper time intervals of the two observers corresponding to Δ𝑑𝑑𝑒𝑒  and Δ𝑑𝑑𝑒𝑒′ are 

given by 

Δ𝜏𝜏𝑒𝑒 = � 1

𝜃𝜃�𝑅𝑅𝑒𝑒𝑙𝑙 �
� Δ𝑑𝑑𝑒𝑒   and  Δ𝜏𝜏𝑒𝑒′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�
� Δ𝑑𝑑𝑒𝑒′                                        (20) 

where Δ𝜏𝜏𝑒𝑒 and Δ𝜏𝜏𝑒𝑒′ are the proper time intervals of the two observers at 𝑅𝑅 = 𝑅𝑅𝑒𝑒  and 𝑅𝑅 = 𝑎𝑎,  just inside the 

surface of the sphere, respectively, emitting and receiving the pulse. 

Since the number of cycles of the pulse remain the same at emission and  observation, we have 

𝜈𝜈𝑒𝑒Δ𝜏𝜏𝑒𝑒 = 𝜈𝜈𝑒𝑒′  Δ𝜏𝜏𝑒𝑒′                                                         (21) 

where 𝜈𝜈𝑒𝑒 and 𝜈𝜈𝑒𝑒′  are the emitted and observed frequencies respectively.  

Then the equations (20) and (21) give                                             

𝜈𝜈𝑒𝑒′ 

𝜈𝜈𝑒𝑒
= Δ𝜏𝜏𝑒𝑒

Δ𝜏𝜏𝑒𝑒′

   

=
𝜃𝜃�𝑎𝑎𝑙𝑙�

𝜃𝜃�𝑅𝑅𝑒𝑒𝑙𝑙 �
�Δ𝑑𝑑𝑒𝑒
Δ𝑑𝑑𝑒𝑒′

�

                         

(22) 

The equation (19) leads the equation (22) to

  

𝜈𝜈𝑒𝑒′ 

𝜈𝜈𝑒𝑒
=

𝜃𝜃�𝑎𝑎𝑙𝑙�

𝜃𝜃�𝑅𝑅𝑒𝑒𝑙𝑙 �
                                    (23) 

Now consider the moment at which the pulse of light passes through the boundary. 

Earlier in this section, we assumed that the observer at 𝑅𝑅 = 𝑎𝑎, at the boundary of the sphere but inside it 

observed the pulse at 𝑑𝑑 = 𝑑𝑑𝑒𝑒′  with frequency 𝜈𝜈𝑒𝑒′. Now our assumption is that an observer at 𝑅𝑅 = 𝑎𝑎 but outside 

of the sphere observed the pulse at 𝑑𝑑 = 𝑑𝑑0′ with frequency 𝜈𝜈0′. 

Then using the metrics in (15), the proper times of the two observers corresponding to Δ𝜏𝜏𝑒𝑒′ and Δ𝜏𝜏0′ can be 

written as  

Δ𝜏𝜏𝑒𝑒′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�
� Δ𝑑𝑑𝑒𝑒′      

 

and   Δ𝜏𝜏0′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑎𝑎
�Δ𝑑𝑑0′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�
� Δ𝑑𝑑0′            (24) 

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

where Δ𝜏𝜏0′  and Δ𝜏𝜏𝑒𝑒′  are the proper time intervals of the two observers at 𝑅𝑅 = 𝑎𝑎, inside the sphere and outside  

the sphere  respectively observing the pulse. 

Since the number of cycles in the pulse remain the same as it crosses the boundary, from the equation (24)  we 

obtain 

𝜐𝜐𝑒𝑒′
𝜐𝜐0′

= Δ𝜏𝜏0′
Δ𝜏𝜏𝑒𝑒′

= Δ𝑑𝑑0′
Δ𝑑𝑑𝑒𝑒′

                                                                     (25) 

According to the standard boundary conditions, the coordinate time intervals do not vary across the boundary.  

Then the equation (25) simplifies to  

𝜐𝜐𝑒𝑒′
𝜐𝜐0′

= 1
                                                

(26)
 

                             
 

Now assume that an observer at a large distance 𝑅𝑅 = 𝑅𝑅0, outside of the sphere observed the front of the pulse 

which is passed through the boundary at 𝑑𝑑 = 𝑑𝑑0′ outside the sphere at 𝑑𝑑 = 𝑑𝑑0, and the rear of the pulse which is 

passed through the boundary at  𝑑𝑑 = 𝑑𝑑0′ + Δ𝑑𝑑0′  outside the sphere,  at  𝑑𝑑 = 𝑑𝑑0 + Δ𝑑𝑑0 respectively with the 

frequency 𝜐𝜐0. 

Now we have to consider the exterior metric in (15) which is 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅�

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
+ 𝑚𝑚

𝑅𝑅
�
2

(𝑑𝑑𝑅𝑅2 + 𝑅𝑅2𝑑𝑑Ω2). 

Considering the null geodesics of the metric we obtain  

𝑐𝑐 𝑑𝑑𝑑𝑑 = �𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
+ 𝑚𝑚

𝑅𝑅
�
2
𝑑𝑑𝑅𝑅                                    (27) 

(Plus sign is taken since the null ray is away from the centre of the sphere.) 

Integration of (27) gives  

∫ 𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑0
𝑑𝑑0′

= ∫ �𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
+ 𝑚𝑚

𝑅𝑅
�
2
𝑑𝑑𝑅𝑅𝑅𝑅0

𝑅𝑅=𝑎𝑎                                                                              (28) 

If the rear of the pulse is observed at 𝑑𝑑0 + Δ𝑑𝑑0 at 𝑅𝑅 = 𝑅𝑅0, then integration of (27) gives 

∫ 𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑0+Δ𝑑𝑑0
𝑑𝑑0′+Δ𝑑𝑑0′

= ∫ �𝜃𝜃 �𝑎𝑎
𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
+ 𝑚𝑚

𝑅𝑅
�
2
𝑑𝑑𝑅𝑅𝑅𝑅0

𝑅𝑅=𝑎𝑎  

                                    

(29)

  From the equations (28) and (29) we obtain 

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

Δ𝑑𝑑0′ = Δ𝑑𝑑0                                                 (30) 

Then using the metrics in (15), the proper times of the two observers corresponding to Δ𝑑𝑑0′ and Δ𝑑𝑑0 can be 

written as  

Δ𝜏𝜏0′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�
� Δ𝑑𝑑0′     and    Δ𝜏𝜏0 = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅0

�  Δ𝑑𝑑0  

                      

(31) 

where Δ𝜏𝜏0′  and Δ𝜏𝜏0  respectively are the proper time intervals of the two observers at 𝑅𝑅 = 𝑎𝑎,outside the sphere, 

and at 𝑅𝑅 = 𝑅𝑅0, a large distance outside of the sphere  observing the pulse . 

Since the number of cycles of the pulse remain the same throughout its way, we obtain 

𝜐𝜐0
𝜐𝜐0′

= Δ𝜏𝜏0′

Δ𝜏𝜏0
                                                                               (32) 

Substitution (31) in (32) gives 

𝜐𝜐0
𝜐𝜐0′

=
�𝜃𝜃�𝑎𝑎𝑙𝑙�−

𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅0
�

�𝜃𝜃�𝑎𝑎𝑙𝑙��
�Δ𝑑𝑑0

′

Δ𝑑𝑑0
�                  (33)

 Substituting Δ𝑑𝑑0′ = Δ𝑑𝑑0 in (33), we obtain 

𝜐𝜐0
𝜐𝜐0′

=
�𝜃𝜃�𝑎𝑎𝑙𝑙�−

𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅0
�

�𝜃𝜃�𝑎𝑎𝑙𝑙��
                              (34) 

Using the fact 𝜈𝜈0
𝜈𝜈𝑒𝑒

= 𝜐𝜐0
𝜐𝜐0′

𝜐𝜐0′
𝜐𝜐𝑒𝑒′

𝜈𝜈𝑒𝑒′
𝜈𝜈𝑒𝑒

  and the equations (23), (26) and (34), we obtain 

𝜈𝜈0
𝜈𝜈𝑒𝑒

=
�𝜃𝜃�𝑎𝑎𝑙𝑙�−

𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅0
�

�𝜃𝜃�𝑎𝑎𝑙𝑙��

�𝜃𝜃�𝑎𝑎𝑙𝑙��

�𝜃𝜃�𝑅𝑅𝑒𝑒𝑙𝑙 ��

               

(35) 

Since the red shift z corresponds to the change of the wave length, we have 

1 + 𝑧𝑧 = 𝜆𝜆0
𝜆𝜆𝑒𝑒

=
𝜃𝜃�𝑅𝑅𝑒𝑒𝑙𝑙 �

�𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎+

𝑚𝑚
𝑅𝑅0
�
                                                                            

        

(36) 

It is clear   that the red shift as observed by an observer at infinity is given by  

1 + 𝑧𝑧 = 𝜆𝜆0
𝜆𝜆𝑒𝑒

=
𝜃𝜃�𝑅𝑅𝑒𝑒𝑙𝑙 �

�𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎�

                     (37)

 
Since 𝑅𝑅𝑒𝑒 < 𝑎𝑎 and 𝜃𝜃 �𝑅𝑅

𝑙𝑙
� is a decreasing function, it can be shown that the pulse of light is red shifted.

                                                         

 

163 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No  1, pp 155-164 

4. Conclusion 

We obtained the metrics for an electrically counterpoised dust sphere using what are known as standard 

boundary conditions. In this formalism we use the same coordinates in the two regions inside the spherical 

distribution of matter as well as outside. 

Also it is shown that a pulse of light which is emitted from the inside of the sphere as observed by an observer at 

infinity is red shifted. So the solution obtained for the Einstein’s field equations is physically acceptable. 

However, the metric we obtained for the external region is not Lorentzian at infinity (large distances) in general 

and it becomes Lorentzian only when  𝑎𝑎, the  radial  coordinate and 𝑚𝑚, the mass of the ECD distribution 

becomes zero. That is when there is no matter distribution and which makes the space time flat and Lorentzian 

throughout. 

In our next article, we will overcome this difficulty by using different coordinates for the two regions, inside of 

the sphere and outside of the sphere. Then we will require more equations to solve Einstein’s field equations. 

Therefore we will apply a new set of boundary conditions other than standard boundary conditions to find exact 

solutions to Einstein’s field equations. Also we are going to compare the results obtained in this article with the 

results which will be obtained using new set of boundary conditions. 

Acknowledgements 

I express my sincere thanks and profound gratitude to my research supervisors, Dr. L.N.K de Silva and late Prof. 

S.B.P Wickramasuriya, for their invaluable advice, guidance and encouragement throughout the course of this 

work. 

References 

[1] S.B.P. Wickramasuriya. “On static distribution of matter in general relativity.”  PhD thesis, Queen 

Elizabeth College, London, 1972. 

[2] P.M.N. Dharmawardana. “Study of the Lane Emden equation and some of the associated equations.”                          

MPhil thesis, University of Kelaniya, Sri Lanka, 2005. 

[3] N.A.S.N Wimaladharma. “A study of  static relativistic electrically charged spherical distributions with 

old and new boundary conditions.” MPhil thesis, University of Kelaniya, Sri Lanka, 2013.       

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