163 
 

 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 

with a Set of New Boundary Conditions 

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 

A static spherically symmetric solution to Einstein’s field equations has been found with the standard boundary 

conditions for a spherical distribution of electrically counterpoised dust in Sashinka Wimaladharma and Nalin 

de Silva. However the solution obtained there was not Lorentzian at infinity. To overcome this problem, a set of 

new boundary conditions has been introduced along with different coordinates for different regions of the matter 

distribution. 

Keywords: Einstein’s field equations; electrically counterpoised dust. 

1. Introduction  

Sashinka wimaladharma and Nalin de silva [1] considered a spherically symmetric distribution of electrically 

counterpoised dust and solved Einstein-Maxwell field equations using the standard boundary conditions which 

says that metric coefficients and their partial derivatives are continuous on the boundary of the sphere. Also 

there the metrics had the same form for the two regions inside and outside the sphere with the same coordinates 

(𝑡𝑡, 𝑟𝑟) both inside and outside the sphere.  

Also it has been also found that the metric in the exterior region of the sphere is not tending to Lorentzian form 

at infinity. 

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

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

164 
 

In order to obtain a Lorentzian metric at infinity for a spherically symmetric distribution of matter, it is assumed 

that the time and radial coordinates are different for the two regions, inside and outside the sphere and a new set 

of boundary conditions has been introduced to solve the Einstein-Maxwell equations and equations for a 

distribution of electrically counterpoised dust (ECD). 

2. Material and the Method 

Closely following the form of the metric obtained in   Sashinka wimaladharma and Nalin de Silva [1], a metric 

for exterior region of the sphere has taken to be of the form 

𝑑𝑑𝑑𝑑2 = 1

�𝐴𝐴1+
𝐴𝐴2
𝑅𝑅 �

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅
�
2

(𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω2)     .                                                            

The density of a sphere of electrically counterpoised dust is taken to be constant and equal to 24
1
lπ

with 

14 2 =lπρ  as
 
in Sashinka wimaladharma and Nalin de Silva [1] . 

Now we introduce two sets of coordinates (𝑡𝑡, 𝑟𝑟)  and Ω and (𝑑𝑑,𝑑𝑑) and Ω for the metrics inside and outside of a 

sphere of constant density, respectively. 

Then the metrics for the two regions  inside and outside of a sphere of constant density 24
1
lπ

ρ =  can be 

written in the form 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝑑𝑑𝑡𝑡2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
(𝑑𝑑𝑟𝑟2 + 𝑟𝑟2𝑑𝑑Ω2)                          0 ≤ 𝑟𝑟 ≤ 𝑎𝑎  

          

 

𝑑𝑑𝑑𝑑2 = 1

�𝐴𝐴1+
𝐴𝐴2
𝑅𝑅 �

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅
�
2
�𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω

2
�

  

             𝐴𝐴 < 𝑑𝑑       .                                                        (1) 

where 𝑑𝑑Ω2 = 𝑑𝑑Θ2 + sin2 Θ  𝑑𝑑Φ2   and  𝑑𝑑Ω
2

= 𝑑𝑑Θ
2

+ sin2 Θ  𝑑𝑑Φ
2
, 𝜃𝜃 �𝑅𝑅

𝑙𝑙
�
 
is the Emden function satisfying the 

Emden equation [2] with 3=n . 

The metric for the inside of the sphere is taken to be of the same form 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝑑𝑑𝑡𝑡2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
(𝑑𝑑𝑟𝑟2 + 𝑟𝑟2𝑑𝑑Ω2)  which has been found to be the metric inside the sphere in 

Sashinka wimaladharma and Nalin de Silva [1]., as there are no differences between the coordinates that were 

used in Sashinka wimaladharma and Nalin de Silva [1], and the present coordinates inside the sphere.  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

165 
 

Also the metric of the space– time outside the sphere is taken to be in  the form  

𝑑𝑑𝑑𝑑2 = 1

�𝐴𝐴1+
𝐴𝐴2
𝑅𝑅 �

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅
�
2
�𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω

2
�                                                                                         (2)

 
so that the metric is isotropic in space coordinates. 

Since the metric has to be Lorentzian at infinity, the constant 𝐴𝐴1  should be equal to one ,i.e .𝐴𝐴1 = 1. 

In general the coordinates in and outside of the sphere do not need to be the same and we take this to be our 

dissociating point from the work that has been carried out previously until now. Intuitively Seneviratne and de 

Silva[3]  have also used different coordinates in their work.    

Therefore in this approach 𝑟𝑟 = 𝑎𝑎 in the matter-filled region corresponds to 𝑑𝑑 = 𝐴𝐴 in the region without matter, 

outside the sphere. 

A new set of boundary conditions  has been introduced as  

��𝑔𝑔𝑖𝑖𝑖𝑖𝛿𝛿𝑥𝑥𝑖𝑖�𝑟𝑟=𝑎𝑎 = ��𝐺𝐺𝑖𝑖𝑖𝑖𝛿𝛿𝑋𝑋𝑖𝑖�𝑅𝑅=𝐴𝐴                                                                                                                      (3) 

� 1
�𝑔𝑔𝑖𝑖𝑖𝑖

� 𝑑𝑑
𝑑𝑑𝑥𝑥𝑖𝑖

��𝑔𝑔𝑖𝑖𝑖𝑖�� 𝛿𝛿𝑥𝑥𝑖𝑖�
𝑟𝑟=𝑎𝑎

= � 1
�𝐺𝐺𝑖𝑖𝑖𝑖

� 𝑑𝑑
𝑑𝑑𝑋𝑋𝑖𝑖

��𝐺𝐺𝑖𝑖𝑖𝑖�� 𝛿𝛿𝑋𝑋𝑖𝑖�
𝑅𝑅=𝐴𝐴

                                                                (4) 

instead of  standard boundary conditions which says that metric coefficients and their partial derivatives are 

continuous across the boundary. 

Here 𝑔𝑔𝑖𝑖𝑖𝑖 , 𝑥𝑥𝑖𝑖   𝑖𝑖 = 0,1,2,3 and 𝐺𝐺𝑖𝑖𝑖𝑖 , 𝑋𝑋𝑖𝑖  𝑖𝑖 = 0,1,2,3 are metric coefficients and coordinates for the inside and the 

outside of the sphere, respectively. 

The  boundary condition (3)  is introduced guiding by the notion of what may be called proper distances and 

proper times of two observers on either side of the sphere. 

The  boundary condition (4)  is introduced replacing ordinary partial derivatives 
Φ∂
∂

Θ∂
∂

∂
∂ ,,
r

 by generalized 

partial derivatives in curvilinear coordinates  in the form

 

� 1
�𝑔𝑔𝑖𝑖𝑖𝑖

� 𝑑𝑑
𝑑𝑑𝑥𝑥𝑖𝑖

��𝑔𝑔𝑖𝑖𝑖𝑖�� 𝛿𝛿𝑥𝑥𝑖𝑖�
𝑟𝑟=𝑎𝑎

= � 1
�𝐺𝐺𝑖𝑖𝑖𝑖

� 𝑑𝑑
𝑑𝑑𝑋𝑋𝑖𝑖

��𝐺𝐺𝑖𝑖𝑖𝑖�� 𝛿𝛿𝑋𝑋𝑖𝑖�
𝑅𝑅=𝐴𝐴

  

These boundary conditions have not been used previously by any other authors, but have been used intuitively 

by Seneviratne  and de Silva [3]. 

First let us write the metrics in (1) at the boundary 𝑟𝑟 = 𝑎𝑎 and 𝑑𝑑 = 𝐴𝐴 as  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

166 
 

𝛿𝛿𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝛿𝛿𝑡𝑡2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
�𝛿𝛿𝑟𝑟2 + 𝑟𝑟2(𝛿𝛿Θ2 + sin2 Θ  𝛿𝛿Φ2)�  

𝛿𝛿𝑑𝑑2 = 1

�𝐴𝐴1+
𝐴𝐴2
𝑅𝑅 �

2 𝑐𝑐2𝛿𝛿𝑑𝑑2 − �𝐴𝐴1 + 𝐴𝐴2
𝑅𝑅
�
2
�𝛿𝛿𝑑𝑑2 + 𝑑𝑑2 �𝛿𝛿Θ

2
+ sin2 Θ  𝛿𝛿Φ

2
��                                                          (5) 

From isotropy Θ has to be equal to Θ and Φ has to be equal to Φ which  implies that Ω=Ω .  

Hence the metrics (1) take the form 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝑑𝑑𝑡𝑡2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
(𝑑𝑑𝑟𝑟2 + 𝑟𝑟2𝑑𝑑Ω2)                          0 ≤ 𝑟𝑟 ≤ 𝑎𝑎  

𝑑𝑑𝑑𝑑2 = 1

�1+𝐴𝐴2𝑅𝑅 �
2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �1 + 𝐴𝐴2

𝑅𝑅
�
2

(𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω2)

  

             𝐴𝐴 < 𝑑𝑑                                                                    (6) 

Hence (5) now takes the form  

𝛿𝛿𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝛿𝛿𝑡𝑡2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
�𝛿𝛿𝑟𝑟2 + 𝑟𝑟2(𝛿𝛿Θ2 + sin2 Θ  𝛿𝛿Φ2)�  

𝛿𝛿𝑑𝑑2 = 1

�1+𝐴𝐴2𝑅𝑅 �
2 𝑐𝑐2𝛿𝛿𝑑𝑑2 − �1 + 𝐴𝐴2

𝑅𝑅
�
2
�𝛿𝛿𝑑𝑑2 + 𝑑𝑑2(𝛿𝛿Θ2 + sin2 Θ  𝛿𝛿Φ2)�

                                                                

(7) 

at the boundary of the sphere. 

Application of the first boundary condition (3)  for Tt and  on the boundary (𝑟𝑟 = 𝑎𝑎 and  𝑑𝑑 = 𝐴𝐴 ) gives 

1

𝜃𝜃�𝑎𝑎𝑙𝑙�
𝑐𝑐 𝛿𝛿𝑡𝑡 = 1

�1+𝐴𝐴2𝐴𝐴 �
𝑐𝑐 𝛿𝛿𝑑𝑑  

𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

=
𝜃𝜃�𝑎𝑎𝑙𝑙�

�1+𝐴𝐴2𝐴𝐴 �
                                                                                                                                                            (8) 

Application of the second boundary condition (4) for Tt and  on the boundary (𝑟𝑟 = 𝑎𝑎 and 𝑑𝑑 = 𝐴𝐴 ) gives 

1

𝜃𝜃�𝑎𝑎𝑙𝑙�
�− 1

�𝜃𝜃�𝑎𝑎𝑙𝑙��
2
1
𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�� 𝑐𝑐 𝛿𝛿𝑡𝑡 = 1

�1+𝐴𝐴2𝐴𝐴 �
�− 1

�1+𝐴𝐴2𝐴𝐴 �
2 �−

𝐴𝐴2
𝐴𝐴2
�� 𝑐𝑐 𝛿𝛿𝑑𝑑  

𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

=
−𝐴𝐴2𝑙𝑙�𝜃𝜃�

𝑎𝑎
𝑙𝑙��

3

𝐴𝐴2𝜃𝜃′�𝑎𝑎𝑙𝑙���1+
𝐴𝐴2
𝐴𝐴 ��

3                                                                                                                                             (9) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

167 
 

where “ ′ ” denotes the differentiation with respect to 𝑟𝑟 . 

Application of  the first boundary condition (3)  for Rr and  on the boundary (𝑟𝑟 = 𝑎𝑎 and 𝑑𝑑 = 𝐴𝐴 ) gives 

𝜃𝜃 �𝑎𝑎
𝑙𝑙
� 𝛿𝛿𝑟𝑟 = �1 + 𝐴𝐴2

𝐴𝐴
� 𝛿𝛿𝑑𝑑  

𝛿𝛿𝑟𝑟
𝛿𝛿𝑅𝑅

=
�1+𝐴𝐴2𝐴𝐴 �

𝜃𝜃�𝑎𝑎𝑙𝑙�
                                                                                                                                                          (10) 

Application of  the second boundary condition (4)  for Rr and  on the boundary (𝑟𝑟 = 𝑎𝑎 and  𝑑𝑑 = 𝐴𝐴 ) gives 

1

𝜃𝜃�𝑎𝑎𝑙𝑙�
�1
𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�� 𝛿𝛿𝑟𝑟 = 1

�1+𝐴𝐴2𝐴𝐴 �
�− 𝐴𝐴2

𝐴𝐴2
� 𝛿𝛿𝑑𝑑  

𝛿𝛿𝑟𝑟
𝛿𝛿𝑅𝑅

=
𝜃𝜃�𝑎𝑎𝑙𝑙�

�1+𝐴𝐴2𝐴𝐴 �
� −𝐴𝐴2𝑙𝑙

𝐴𝐴2𝜃𝜃′�𝑎𝑎𝑙𝑙�
�                                                                                                                                          (11) 

Similarly application  of the same boundary conditions (3) and (4) for the angular coordinate Ω  in the metric 

(7)  on the boundary (𝑟𝑟 = 𝑎𝑎 and  𝑑𝑑 = 𝐴𝐴 ) gives 

𝜃𝜃 �𝑎𝑎
𝑙𝑙
� 𝑎𝑎 = �1 + 𝐴𝐴2

𝐴𝐴
�𝐴𝐴                                                                                                                                           (12) 

1

𝜃𝜃�𝑎𝑎𝑙𝑙�
�𝑎𝑎
𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� +  𝜃𝜃 �𝑎𝑎

𝑙𝑙
�� = 1

�1+𝐴𝐴2𝐴𝐴 �
��1 + 𝐴𝐴2

𝐴𝐴
� + �− 𝐴𝐴2

𝐴𝐴2
𝐴𝐴��                                                                                   (13) 

Using  the equations (8) and (9) , the constant 𝐴𝐴2 can be obtained as 

𝜃𝜃�𝑎𝑎𝑙𝑙�

�1+𝐴𝐴2𝐴𝐴 �
=

−𝐴𝐴2𝑙𝑙�𝜃𝜃�
𝑎𝑎
𝑙𝑙��

3

𝐴𝐴2𝜃𝜃′�𝑎𝑎𝑙𝑙���1+
𝐴𝐴2
𝐴𝐴 ��

3  

i.e   𝐴𝐴2 =
𝐴𝐴2
𝑙𝑙 �1+

𝐴𝐴2
𝐴𝐴 �

2
𝜃𝜃′�𝑎𝑎𝑙𝑙�

�𝜃𝜃�𝑎𝑎𝑙𝑙��
2                                                                                                                                        (14) 

The equation (12) can be rewritten in the form 

�1+𝐴𝐴2𝐴𝐴 �

𝜃𝜃�𝑎𝑎𝑙𝑙�
= 𝑎𝑎

𝐴𝐴
                                                                                                                                                            (15) 

Then using equation (15), equation (14) can be simplified in to the form  

𝐴𝐴2 = −𝐴𝐴2

𝑙𝑙
�𝑎𝑎

2

𝐴𝐴2
� 𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�                                                                                                                 (16) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

168 
 

Substitution of the value of 2A in equation (12) gives the value of 𝐴𝐴 as  

𝐴𝐴 = 𝑎𝑎 𝜃𝜃 �𝑎𝑎
𝑙𝑙
� + 𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�                                                                                                                                      (17) 

Then the metrics (6) become 

𝑑𝑑𝑑𝑑2 = 1

�𝜃𝜃�𝑅𝑅𝑙𝑙 ��
2 𝑐𝑐2𝑑𝑑𝑡𝑡2 − �𝜃𝜃 �𝑅𝑅

𝑙𝑙
��

2
(𝑑𝑑𝑟𝑟2 + 𝑟𝑟2𝑑𝑑Ω2)                                                  0 ≤ 𝑟𝑟 ≤ 𝑎𝑎  

𝑑𝑑𝑑𝑑2 = 1

�1−1𝑅𝑅�
𝑎𝑎2
𝑙𝑙 𝜃𝜃′�

𝑎𝑎
𝑙𝑙���

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �1 − 1
𝑅𝑅
�𝑎𝑎

2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
���

2

(𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω2)

  

             𝐴𝐴 < 𝑑𝑑                                   (18) 

where we have replaced 𝐴𝐴 by (𝑎𝑎 𝜃𝜃 �𝑎𝑎
𝑙𝑙
� + 𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�)    . 

The values  of  𝐴𝐴  which is the radial coordinate just outside the sphere can be found using the values of  𝑎𝑎
𝑙𝑙
. 

The  relationship between 𝑎𝑎
𝑙𝑙
 and 𝐴𝐴 is tabulated in the Table 1 given below. 

Table 1: the relationship between 𝑎𝑎
𝑙𝑙
 and 𝐴𝐴 

 

 

 

.                                                                  

   

 

                                                                                                                                                            

Substitution of the values for 𝐴𝐴2 and 𝐴𝐴 in  (8) gives 

𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

= 𝜃𝜃 �𝑎𝑎
𝑙𝑙
� + 𝑎𝑎

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�                                                                                                                                         (19) 

The values of 𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

 for few different values of 
l
a

 has been tabulated in Table 2 as given below. 

𝑎𝑎
𝑙𝑙
 

  

𝐴𝐴 = (𝑎𝑎 𝜃𝜃 �𝑎𝑎
𝑙𝑙
� + 𝑎𝑎

2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�)   

 

1 0.602929 l 

2 0.119738 l 

3 -0.57877 l 

4 -1.08588 l 

5 -1.44905 l 

6 -1.75514 l 

7 -2.04842 l 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

169 
 

Table 2: Few values of  𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

 

 

 

 

 

 

 

 

 

Since 𝛿𝛿𝑡𝑡  increases as  , 𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

 should be  positive  ,i.e. 𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

> 0.  

Hence according to Table 2, there is a maximum value that 𝑎𝑎
𝑙𝑙
 can take, which is nearly 16.2 . 

Figure 1 which have the graphs of 𝜃𝜃 �𝑅𝑅
𝑙𝑙
�  and 𝜃𝜃 �𝑅𝑅

𝑙𝑙
� + 𝑅𝑅

𝑙𝑙
𝜃𝜃′ �𝑅𝑅

𝑙𝑙
�  versus 𝑅𝑅

𝑙𝑙
 in  Sashinka wimaladharma and Nalin 

de Silva[1] also confirms this result and a more accurate value for the maximum of  𝑎𝑎
𝑙𝑙  

 is 2.157 as obtained using 

the graph in Figure 1 in Sashinka wimaladharma and Nalin de Silva [1]. 

Considering the values of 𝐴𝐴 in Table 1, it can be found that they are positive only when 𝑎𝑎
𝑙𝑙
≤ 2.16

 
as 𝐴𝐴   differs 

from 𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

 by a factor of 𝑎𝑎 which is positive being the radial coordinate as measured inside the sphere at its surface.   

Therefore spheres whose radial coordinate is greater than 2.16 𝑙𝑙 with suitable units, as measured just inside the 

spheres can not be exist.  

Now the metric (18) outside the sphere can be written as 

𝑑𝑑𝑑𝑑2 = 1

�1−1𝑅𝑅�
𝑎𝑎2
𝑙𝑙 𝜃𝜃′�

𝑎𝑎
𝑙𝑙���

2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �1 − 1
𝑅𝑅
�𝑎𝑎

2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
���

2

�𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω
2
�  

But in Wickramasuriya and Bonnar [5] , it has been shown that 𝑚𝑚 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� where 𝑚𝑚 is the mass of the 

sphere. 

𝑎𝑎
𝑙𝑙
 

 
𝛿𝛿𝑡𝑡
𝛿𝛿𝑑𝑑

= 𝜃𝜃 �
𝑎𝑎
𝑙𝑙
� +

𝑎𝑎
𝑙𝑙
𝜃𝜃′ �

𝑎𝑎
𝑙𝑙
� 

 

 

1 0.602929 

 
2 0.059869 

 
3 -0.19292 

 
4 -0.27139 

 
5 -0.28981 

 
6 -0.29252 

 
7 -0.29252 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

170 
 

Hence the metric (18) outside the sphere takes the form 

𝑑𝑑𝑑𝑑2 = 1

�1+𝑚𝑚𝑅𝑅�
2 𝑐𝑐2𝑑𝑑𝑑𝑑2 − �1 + 𝑚𝑚

𝑅𝑅
�
2

(𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω2)  

Thus introducing different coordinates in the regions inside and outside the sphere gives us  a metric which is 

Lorentzian at infinity, unlike in Sashinka wimaladharma and Nalin de Silva [1] where was a metric that was not 

Lorenzian at infinity.  

Now considering the metric (1) without initially assuming that it becomes Lorentzian at infinity, and applying 

the same boundary conditions as above, it can be shown that the value of the constant 𝐴𝐴1 and 𝐴𝐴2  are such that 

𝐴𝐴1 = 𝑎𝑎
𝐴𝐴
�𝜃𝜃 �𝑎𝑎

𝑙𝑙
� + 𝑎𝑎

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
��                                                                                                                                 (20) 

𝐴𝐴2  = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� .                                                                                                                                              (21) 

Thus it is found that the value for 2A  is the same for the metrics obtained in  Sashinka wimaladharma and Nalin 

de Silva [1]  , (16) and (21) irrespective of whether the metric is Lorentzian at infinity.  

This could have been expected as the expression for 2A  gives the value of m, the mass of the distribution as 

calculated inside the sphere by integration as in Wickramasuriya and Bonnar [4] . 

If the metric for the outside of the sphere is Lorentzian , then then the constant 𝐴𝐴1 should be equal to unity, i.e 

𝐴𝐴1 = 𝑎𝑎
𝐴𝐴
�𝜃𝜃 �𝑎𝑎

𝑙𝑙
� + 𝑎𝑎

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�� = 1  which implies that 𝐴𝐴 = 𝑎𝑎 �𝜃𝜃 �𝑎𝑎

𝑙𝑙
� + 𝑎𝑎

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
��  which is the value of the radial 

coordinate at the surface of the sphere as measured just outside the sphere. 

As before 𝐴𝐴  becomes zero when  𝑎𝑎
𝑙𝑙
 is nearly equal to 2.16. Thus 𝑎𝑎

𝑙𝑙
  has a maximum value nearly equal to 2.16. 

The graph of 𝐴𝐴𝐴𝐴1 against 𝑎𝑎
𝑙𝑙
 is plotted in Figure 1.    𝐴𝐴𝐴𝐴1 

It can be found from the Figure 1  that 𝐴𝐴𝐴𝐴1  has a maximum value which is equal to 0.6 𝑙𝑙. 

Therefore when the metric is Lorentzian (𝐴𝐴1 = 1)  the maximum value of 𝐴𝐴 equal to 0.6 𝑙𝑙. 

This corresponds to a value 𝑎𝑎 which is equal to 𝑙𝑙. Thus the outer radial coordinate 𝐴𝐴 increases with inner radial 

coordinate 𝑎𝑎   until 0.6 𝑙𝑙  . After that  𝐴𝐴 decreases with 𝑎𝑎 and becomes zero when 𝑎𝑎 = 2.16 𝑙𝑙 . 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

171 
 

𝑎𝑎
𝑙𝑙
 

Figure 1: the graph of 𝐴𝐴𝐴𝐴1 against 𝑎𝑎
𝑙𝑙
 

It can be postulated that 𝐴𝐴  increases and attains a maximum at  𝐴𝐴 = 0.6 𝑙𝑙  when  𝑎𝑎 = 𝑙𝑙 , and  that there is no 

decrease of 𝐴𝐴 as 𝑎𝑎 increases though mathematically it is suggested so. If 𝐴𝐴 decreases with 𝑎𝑎  after that it would 

end up with zero and then with negative values that have no meaning physically.  

The mass of the sphere which is equal to 𝑚𝑚 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�  is plotted against  𝐴𝐴   in the Figure 2. 

   
 

                                             𝑚𝑚 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� 

𝐴𝐴
 

Figure 2: the graph of m against A 

Figure 2 suggests that 𝑚𝑚 increases with 𝐴𝐴
 
until reaches the value 𝐴𝐴 = 0.6 𝑙𝑙  .After that  according to the Figure 

2, 𝐴𝐴 decreases though 𝑚𝑚 increases  , and it can be postulated that 𝑚𝑚 has a maximum  1.12 𝑙𝑙 when 𝐴𝐴 = 0.6 𝑙𝑙  . 

The maximum value of 𝐴𝐴  is the same for both cases that obtained from Figure 1 and Figure 2.  

0 0.080.160.240.32 0.4 0.480.560.640.72 0.8 0.880.961.041.12 1.2 1.281.361.441.52 1.6 1.681.761.841.92 2 2.082.162.242.32 2.4 2.48
-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a

A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0

0.2

0.4

0.6

0.8

1

1.2

1.4

A

m



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

172 
 

The graph of 𝑚𝑚 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� versus  𝑎𝑎

𝑙𝑙
 is as shown in Figure 3. 

                               𝑚𝑚 = −𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
� 

𝑎𝑎
𝑙𝑙
 

Figure 3: the graph of 𝑚𝑚 against 𝑎𝑎
𝑙𝑙
 

In Figure 3 , 𝑚𝑚 increases with 𝑎𝑎
𝑙𝑙  without a maximum. However 𝑎𝑎

 
has a maximum 2.16 𝑙𝑙 and the corresponding 

value of 𝑚𝑚 is 1.12 𝑙𝑙. 

Hence it can be postulated that 1.12 𝑙𝑙 is the maximum value that 𝑚𝑚 can take. 

This is an interesting result as in the case of electrically counterpoised dust distribution , there are maximum 

values that are written in terms of 𝑙𝑙 which can in turn be expressed in terms of density as 4𝜋𝜋𝜋𝜋𝑙𝑙2 = 1. 

This implies that given the constant density of the distribution there are maximum values that  

 𝑚𝑚,𝑎𝑎 (and hence 𝐴𝐴) can take. 

3. Results 

3.1. The Red Shift of a Pulse of Light 

An expression for the red shift of a pulse of light emitted at a point  inside of the sphere along a radial direction 

as an observer who is at a large distance away from the sphere has been calculated. 

However in this case, unlike in the case in Sashinka Wimaladharma and Nalin de silva [1] where we used 

standard Lichernowicz boundary conditions[5], the differences in time when a light ray passes through the 

boundary of the sphere have taken into consideration.  

First a pulse of light with front emitted at 𝑟𝑟 = 𝑟𝑟𝑒𝑒 at 𝑡𝑡 = 𝑡𝑡𝑒𝑒 with frequency 𝜈𝜈𝑒𝑒 inside the sphere, and an observer 

at  𝑟𝑟 = 𝑎𝑎 , just inside the boundary of the sphere receiving it at 𝑡𝑡 = 𝑡𝑡𝑒𝑒′  with frequency 𝜈𝜈𝑒𝑒′   has been considered. 

0 0.5 1 1.5 2
0

0.2

0.4

0.6

0.8

1



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

173 
 

The radial null geodesics within the sphere are given by the metric (18) as 

0 = 1

�𝜃𝜃�𝑟𝑟𝑙𝑙��
2 𝑐𝑐2𝑑𝑑𝑡𝑡2 − �𝜃𝜃 �𝑟𝑟

𝑙𝑙
��

2
𝑑𝑑𝑟𝑟2 , 

Taking the plus sign since 𝑟𝑟 increases with time 𝑡𝑡 as the photons are going away from the centre of the sphere 

gives  

𝑑𝑑𝑟𝑟
𝑑𝑑𝛿𝛿

= 𝑐𝑐

�𝜃𝜃�𝑟𝑟𝑙𝑙��
2.                                                                                                                                                         (28) 

Equation (28)   can be integrated considering the front of the pulse as 

∫ 𝑐𝑐 𝑑𝑑𝑡𝑡𝛿𝛿𝑒𝑒′

𝛿𝛿𝑒𝑒
= ∫ �𝜃𝜃 �𝑟𝑟

𝑙𝑙
��

2𝑎𝑎
𝑟𝑟𝑒𝑒

𝑑𝑑𝑟𝑟                                                                                                                                (29) 

Assuming that the rear of the pulse which is emitted at 𝑟𝑟 = 𝑟𝑟𝑒𝑒 at 𝑡𝑡 = 𝑡𝑡𝑒𝑒 + Δ𝑡𝑡𝑒𝑒 with frequency 𝜈𝜈𝑒𝑒 will observe it 

at = 𝑎𝑎 , just inside the boundary of the sphere at 𝑡𝑡 = 𝑡𝑡𝑒𝑒′ + Δ𝑡𝑡𝑒𝑒′  with frequency 𝜈𝜈𝑒𝑒′ , 

Equation (28)   can also be integrated considering the rear of the pulse as 

∫ 𝑐𝑐 𝑑𝑑𝑡𝑡𝛿𝛿𝑒𝑒′+Δ𝛿𝛿𝑒𝑒′

𝛿𝛿𝑒𝑒+Δ𝛿𝛿𝑒𝑒
= ∫ �𝜃𝜃 �𝑟𝑟

𝑙𝑙
��

2𝑎𝑎
𝑟𝑟𝑒𝑒

𝑑𝑑𝑟𝑟                                                                                                                           (30) 

Since the right hand sides of equations (29) and (30) are the same , it can be obtained that 

∫ 𝑐𝑐 𝑑𝑑𝑡𝑡𝛿𝛿𝑒𝑒′

𝛿𝛿𝑒𝑒
= ∫ 𝑐𝑐 𝑑𝑑𝑡𝑡𝛿𝛿𝑒𝑒′+Δ𝛿𝛿𝑒𝑒′

𝛿𝛿𝑒𝑒+Δ𝛿𝛿𝑒𝑒
 .                                                                                                                                      (31) 

Rearrangement and simplification of the equation (31) with the assumption of  Δ𝑡𝑡𝑒𝑒 and Δ𝑡𝑡𝑒𝑒′  are very small gives  

Δ𝑡𝑡𝑒𝑒 = Δ𝑡𝑡𝑒𝑒′                                                                                                                                                              (32) 

Now the proper time intervals from the equation (18) of the two observers corresponding to Δ𝑡𝑡𝑒𝑒 and Δ𝑡𝑡𝑒𝑒′  are 

given by 

Δ𝜏𝜏𝑒𝑒 = � 1

𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �
�  Δ𝑡𝑡𝑒𝑒  and Δ𝜏𝜏𝑒𝑒′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�
� Δ𝑡𝑡𝑒𝑒′                                                                                                           (33) 

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

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

The number of cycles of the pulse remain the same at emission and  observation yields  

𝜈𝜈𝑒𝑒Δ𝜏𝜏𝑒𝑒 = 𝜈𝜈𝑒𝑒′Δ𝜏𝜏𝑒𝑒′   



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

174 
 

𝜈𝜈𝑒𝑒′

𝜈𝜈𝑒𝑒
= Δ𝜏𝜏𝑒𝑒

Δ𝜏𝜏𝑒𝑒′
                                                                                                                                                                 

(34) 

Now equation (33) and (34) give  𝜈𝜈𝑒𝑒
′

𝜈𝜈𝑒𝑒
=

𝜃𝜃�𝑎𝑎𝑙𝑙�

𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �

Δ𝛿𝛿𝑒𝑒
Δ𝛿𝛿𝑒𝑒′

                                                                                                  (35) 

Using equation (32), equation (35) can be simplified to 

𝜈𝜈𝑒𝑒′

𝜈𝜈𝑒𝑒
=

𝜃𝜃�𝑎𝑎𝑙𝑙�

𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �
                                                                                                                                                              (36) 

Now consider the moment which the pulse of ray pass through the boundary of the sphere. 

It is assumed that an observer at 𝑑𝑑 = 𝐴𝐴  but outside of the sphere observed the pulse at 𝑑𝑑 = 𝑑𝑑0′  with 

frequency 𝜈𝜈0′ . 

Then the relationship of the time intervals Δ𝑡𝑡𝑒𝑒′   and Δ𝑑𝑑0′ is given by the equation (19), which is  

Δ𝛿𝛿𝑒𝑒′   
Δ𝛿𝛿0′

= 𝜃𝜃 �𝑎𝑎
𝑙𝑙
� + 𝑎𝑎

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�                                                                                                                                       (37) 

Then using the metrics in (18), the proper times of the two relevant observers corresponding to Δ𝑡𝑡𝑒𝑒′   and Δ𝑑𝑑0′  are 

given by 

Δ𝜏𝜏𝑒𝑒′ = � 1

𝜃𝜃�𝑎𝑎𝑙𝑙�
� Δ𝑡𝑡𝑒𝑒′   and  Δ𝜏𝜏0′ = � 1

1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �

𝑙𝑙𝐴𝐴

�Δ𝑑𝑑0′                                                                                                 (38) 

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

𝐴𝐴,outside of the sphere  respectively receiving the pulse and 𝐴𝐴 is given by 

 𝐴𝐴 = 𝑎𝑎𝜃𝜃 �𝑎𝑎
𝑙𝑙
� + 𝑎𝑎2

𝑙𝑙
𝜃𝜃′ �𝑎𝑎

𝑙𝑙
�. 

Using the similar steps in equation (34),(35) and (36), the ratio between the frequencies 𝜈𝜈𝑒𝑒′  and 𝜈𝜈0′  can be 

obtained in the form 

𝜈𝜈𝑒𝑒′  
𝜈𝜈0′

= 1                                                                                                                                                                  (39) 

Now assuming that an observer at a large distance 𝑑𝑑 = 𝑑𝑑0, outside of the sphere observed the front of the pulse 

which is passed through the boundary at 𝑑𝑑 = 𝑑𝑑0′ outside the sphere, the radial null geodesic for the exterior 

vacuum region can be integrated into the form 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

175 
 

∫ 𝑐𝑐𝑑𝑑𝑑𝑑𝛿𝛿0
𝛿𝛿0′

= ∫ �1 −
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙�

𝑙𝑙𝐴𝐴
�
2

𝑑𝑑𝑑𝑑𝑅𝑅0
𝐴𝐴                                                                                                                      (40) 

If the rear of the pulse is observed at 𝑑𝑑 = 𝑑𝑑0′ + Δ𝑑𝑑0′ at 𝑑𝑑 = 𝑑𝑑0, then the radial null geodesic for the exterior 

vacuum region can be integrated into the form 

∫ 𝑐𝑐𝑑𝑑𝑑𝑑𝛿𝛿0+∆𝛿𝛿0
𝛿𝛿0′+∆𝛿𝛿0′

= ∫ �1 −
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙�

𝑙𝑙𝐴𝐴
�
2

𝑑𝑑𝑑𝑑𝑅𝑅0
𝐴𝐴                                                                                                               (41) 

Since the right hand sides of (40) and (41) are equal and hence it can be obtained  

∫ 𝑐𝑐𝑑𝑑𝑑𝑑𝛿𝛿0
𝛿𝛿0′

= ∫ 𝑐𝑐𝑑𝑑𝑑𝑑𝛿𝛿0+∆𝛿𝛿0
𝛿𝛿0′+∆𝛿𝛿0′

  . 

Rearrangement and simplification with the assumption that ∆𝑑𝑑0  and ∆𝑑𝑑0′ are very small gives 

 
∆𝑑𝑑0 = ∆𝑑𝑑0′                                                                                                                                                            (42) 

Now using the metrics in (18), the proper times of the two observers corresponding to ∆𝑑𝑑0′ and ∆𝑑𝑑0 can be 

written as  

Δ𝜏𝜏0′ = � 1

1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙�

𝑙𝑙𝐴𝐴

�Δ𝑑𝑑0′            and                           ∆𝜏𝜏0 = � 1

1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �
𝑙𝑙𝑅𝑅0

�Δ𝑑𝑑0                                                       (43) 

where Δ𝜏𝜏0′  and ∆𝜏𝜏0 are the proper time intervals of the two observers at 𝐴𝐴,just outside the boundary of the sphere 

and at 𝑑𝑑 = 𝑑𝑑0, a large distance outside of the sphere  respectively receiving the pulse.  

Using the fact that  the number of cycles of the pulse remain the same throughout its way and equation (43) , the 

ratio between the frequencies 𝜈𝜈0 and 𝜈𝜈0′   can be obtained as 

𝜈𝜈0
𝜈𝜈0′  

=
�1−

𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �
𝑙𝑙𝑅𝑅0

�

�1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �

𝑙𝑙𝐴𝐴 �

Δ𝛿𝛿0
Δ𝛿𝛿0′

                                                                                                                                              (44) 

Equation (44) can be simplified using equation (42) into the form 

𝜈𝜈0
𝜈𝜈0′  

=
�1−

𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �
𝑙𝑙𝑅𝑅0

�

�1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �

𝑙𝑙𝐴𝐴 �
                                                                                                                                                    (45) 

Using the fact that 𝜈𝜈0
𝜈𝜈𝑒𝑒

= 𝜈𝜈0
𝜈𝜈0′  

𝜈𝜈0′

𝜈𝜈𝑒𝑒′
𝜈𝜈𝑒𝑒′

𝜈𝜈𝑒𝑒
 and equations (36),(39) and (45) , it can be obtained that 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

176 
 

𝜈𝜈0
𝜈𝜈𝑒𝑒

= �
1−

𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �
𝑙𝑙𝑅𝑅0

1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �

𝑙𝑙𝐴𝐴

� (1)�
𝜃𝜃�𝑎𝑎𝑙𝑙�

𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �
�                                                                                                                              (46) 

Since the red shift 𝑧𝑧 corresponds to the change of the wave length  can be calculated as 

1 + 𝑧𝑧 = 𝜆𝜆0
𝜆𝜆𝑒𝑒

=
�1−

𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙�
𝑙𝑙𝐴𝐴 �𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �

�1−
𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �
𝑙𝑙𝑅𝑅0

�𝜃𝜃�𝑎𝑎𝑙𝑙�
  

When 𝑑𝑑0  tends to infinity, the red shift is equal to 1 + 𝑧𝑧 =
�1−

𝑎𝑎2𝜃𝜃′�𝑎𝑎𝑙𝑙 �
𝑙𝑙𝐴𝐴 �𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �

𝜃𝜃�𝑎𝑎𝑙𝑙�
. 

When using standard boundary conditions in Sashinka Wimaladharma and Nalin de Silva[1],  

the red shift was calculated to be  1 + 𝑧𝑧 =
𝜃𝜃�𝑟𝑟𝑒𝑒𝑙𝑙 �

�𝜃𝜃�𝑎𝑎𝑙𝑙�−
𝑚𝑚
𝑎𝑎�

. 

The red shifts as observed at infinity are all equal provided that 𝐴𝐴 = −
𝑎𝑎3𝜃𝜃′�𝑎𝑎𝑙𝑙�

𝑚𝑚𝑙𝑙
�𝜃𝜃 �𝑎𝑎

𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
�. 

Going by the previous results we may impose this condition on 𝐴𝐴 so that the red shift is an invariant with respect 

to change of coordinates.  

4. Conclusion 

A static spherically symmetric solution to Einstein-Maxwell field equations for a spherical distribution of 

electrically counterpoised dust distribution has been found using a new set of boundary conditions with the 

assumption that the coordinates in inside and outside of the sphere are different each other. Also the new metric 

is Lorentzian everywhere. 

It is also found that the mass 𝑚𝑚, radial coordinate just inside the boundary 𝑎𝑎 and radial coordinate just outside of 

the boundary 𝐴𝐴 have maximum values. Therefore Einstein’s equations have been solved for finite distribution of 

matter. 

Imposing the condition  𝐴𝐴 = −
𝑎𝑎3𝜃𝜃′�𝑎𝑎𝑙𝑙�

𝑚𝑚𝑙𝑙
�𝜃𝜃 �𝑎𝑎

𝑙𝑙
� − 𝑚𝑚

𝑎𝑎
� on 𝐴𝐴, it can be shown that the red shift is an invariant with 

respect to change of coordinates. 

Acknowledgement 

I express my sincere thanks and profound gratitude to my research supervisor, Dr. Nalin de Silva for his 

invaluable advice, guidance and encouragement throughout the course of this work. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 21, No  1, pp 163-177 

 

177 
 

References 

[1] Sashinka Wimaladharma and Nalin de Silva. “A Static Solution to Einstein’s Field Equations for a 

spherical Distribution of Electrically Counterpoised Dust”American Scientific Research Journal for 

Engineering,Technology (ASRJETS), and Sciences ,Volume 17,No1 ,pp 155-164,March,2016. 

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

equations.”MPhil theis, university of Kelaniya,2005. 

 [3] K.W.P.B Seneviratne, and Nalin de Silva.  “The Schwarzschild space-time in the background of the 

flat Robertson-Walker space-time” Annual Research Symposium, University of Kelaniya, Sri 

Lanka,2007. 

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

Elizabeth college,London,1972.