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Anisotropic Charged Spheres in Higher Dimensional Rosen's Bimetric 

Relativity Theory 

 

R. T. Katre1*, R. K. Agrawal2, S. D. Tade3 

1Department of Mathematics, Bajaj College of Science, Wardha, India 

2Department of Mathematics, Hislop College, Nagpur, India 

3Ex-Department of Mathematics, Jawaharlal Nehru College, Wadi, Nagpur (India) 

*E-mail: katrert9@gmail.com 

 

Article History: 

Received: 1-11-2024 

Revised: 17-12-2024 

Accepted: 25-12-2024 

Abstract:  

In the present study, we presented the field equations for anisotropic charged spheres in 

higher dimensional spherically symmetric space-time in the framework of Rosen’s 

bimetric relativity theory (Rosen 1979, 1980). To obtain the determinate solutions of 

the higher dimensional field equations, the strategy we adopted is to assume that the 

space-time admits one parameter group of conformal motions generated by the vector 

field 𝝃𝝁 if L𝜉𝑔𝜇𝜈 = 𝜓 𝑔𝜇𝜈 .  

Keywords: Bimetric theory, spherically symmetric model 

 

1.  Introduction 

Rosen [1, 2, 3] proposed a new theory of gravitation called the bimetric general relativity theory. In 

this theory, in addition to the metric tensor 𝑔𝑖𝑗 (the gravitational or physical metric), a second metric 

tensor 𝛾𝑖𝑗 (the background metric) has been introduced describing the geometry of space-time, which 

one would have if there were no matter in the universe. In this respect the theory is similar to an 

earlier version of bimetric theory formulated by Rosen [4, 5, 6]. The field equations of the present 

theory of gravitation are taken to be the same as in general relativity, except for the fact that ordinary 

derivatives of the physical metric are replaced by covariant derivatives with respect to the 

background metric. The background metric 𝛾𝑖𝑗 is taken to describe a space-time of constant 

curvature. It was found by Rosen [2] that these field equations could be written in the form of 

Einstein field equations, but with an additional term on the right-hand side,  

                                                 𝐺𝑖𝑗 = 𝑆𝑖𝑗 −  8𝜋 𝑇𝑖𝑗,     (1)             

where the Einstein tensor 𝐺𝑖𝑗 = 𝑅𝑖𝑗 −
1

2
𝑔𝑖𝑗𝑅 and 𝑇𝑖𝑗 is the energy-momentum tensor of the matter 

distribution and  

                                 𝑆𝑖𝑗 =
3

𝑎2 (𝛾𝑖𝑗 −
1

2
𝑔𝑖𝑗𝑔𝑙𝑘𝛾𝑙𝑘) ,     (2)                                 

where, 𝑎 is a constant scale parameter, of the order of the size of the universe.  

mailto:katrert9@gmail.com


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From this one sees that, in general, if one is dealing with a physical system that is small compared to 

the size of the universe, e.g., the solar system, then the term 𝑆𝑖𝑗 is negligible in the field equations (1) 

and the present theory gives agreement with Einstein general relativity theory.  

In recent years, the study of higher dimensional space-time has gained momentum because of the 

fact that the cosmos at eats early stage of evolution might have had a higher dimensional era. The 

extra space reduced to volume with passage of time, which is beyond the ability of experimental 

observations at the moment. Also, the possibility that the world may have more than 4-dimensions is 

due to Kaluza [7] and Klein [8], who used one extra dimension to unify gravity and 

electromagnetism in a theory that was essentially 5-dimensional in general relativity. Sabbatta [9], 

Lee [10], Appelquist and Chodos [11], and Collins et al. [12] accepted this idea and constructed 

cosmological models in higher dimensions by using various phenomena of particle physics and 

cosmology. Overduin and Wesson [13] presented an excellent review of Kaluza Klein theory and 

higher-dimensional unified theories, in which the cosmological and astrophysical implications of 

extra dimensions were studied. Mohanty and Mahanta [14, 15], Mohanty and Samanta [16] are some 

of the authors who have studied higher-dimensional cosmology in various theories of gravitation. 

There is now extensive literature dealing with different aspects of higher-dimensional cosmology. 

Particularly, Karade and khadekar [17] have shown non-existence of higher dimensional axially 

symmetric field in Rosen's theory of gravitation. Reddy and Venkateswara [18] have shown non-

existence of higher dimensional anisotropic cosmological model in bimetric theory of 

gravitation. Katore et al. [19] have shown the non-existence of n-dimensional static plane 

symmetric solutions in bimetric relativity theory. Khadekar and Tade [20] studied string 

cosmological models in five dimensional bimetric theory of gravitation.  Herrera et al. [21] studied 

the consequences of inclusion of one parameter group of conformal motion of anisotropic matter in 

the Einstein’s general relativity and obtained analytical solutions of field equations for static and 

spherically symmetric distribution of isotropic and anisotropic matter. The study of static anisotropic 

spheres is important in relativistic astrophysics [22].   

Sah and Prakash Chandra [23] have studied class of charged fluid balls in general relativity. Hasmani 

and Pandya [24] have investigated cosmological models for the static spherically symmetric 

spacetime with charged anisotropic fluid distribution in (n+2)-dimensions in context of Rosen's 

Bimetric General Relativity. Mukesh Kumar Madhukar [25] presented the study of some charged 

fluid spheres in General Relativity solutions and conditions for charged fluid spheres in General 

Relativity. Francesco Bajardi and et. al. [26] studied the higher dimensional static and spherically 

symmetric solutions in Extended Gauss–Bonnet Gravity. Recently, Zahraa and Mardanb [27] have 

presented five-dimensional analysis of electromagnetism with heat flow in the post-quasi-static 

approximation.  

With the above motivation, in this paper, we have presented the field equations in higher dimensional 

static spherically symmetric space-time in the framework of bimetric theory of relativity proposed by 

Rosen [2, 3] for anisotropic distribution of matter and also obtain the solutions for the physical 

metric with the assumption that the space-time admits a one-parameter group of conformal motions. 

This work is a generalization of the results of Hererra et al. [21] obtained in general relativity to the 



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higher dimensional bimetric theory of relativity proposed by Rosen. The solution gives agreement 

with Einstein’s general relativity for physical system known in the universe, such as the solar system.  

The work done, in this paper, is organized as follows: In section-2, we obtain the field equations in 

the presence of an anisotropic charged spherically symmetric matter in the frame work of higher 

dimensional bimetric theory of relativity proposed by Rosen [2, 3]. Section-3 deals with the solutions 

of the field equations under the assumption that the space time admits a one-parameter group of 

conformal motions [Herrera et al. (1985)] generated by the vector field 𝜉𝜇 if 

  L𝜉𝑔𝜇𝜈 = 𝜓 𝑔𝜇𝜈  , (1) 

where L.H.S. of (1) is the Lie derivative operator and 𝜓 is conformal factor.  Section-4 contains 

some conclusions. 

2.  Metric and Field Equations 

We consider the higher dimensional static distribution of matter represented by charged spherically 

symmetric fluid which may be anisotropic. In terms of Schwarzschild coordinates the physical metric 

takes the form 

  𝑑𝑠2 = 𝑒𝜈𝑑𝑡2 − 𝑒𝜆𝑑𝑟2 − 𝑟2𝑑𝛺2, (2) 

where 

𝑑𝛺2 = 𝑑𝜃1
2 + 𝑠𝑖𝑛2 𝜃1 𝑑𝜃2

2 + 𝑠𝑖𝑛2 𝜃1  𝑠𝑖𝑛2 𝜃3 𝑑𝜃3
2 + ⋯ + [ ∏

𝑖=1
𝑛−1

𝑠𝑖𝑛2 𝜃𝑖]  𝑑𝜃𝑛
2 

and 𝜆, 𝜈 are the functions of 𝑟 only. 

With this choice of coordinates the total energy-momentum tensor 𝑇𝑖𝑗 is assumed to be the sum of 

two parts, 𝑇𝑖𝑗

𝐴

 and  𝑇𝑖𝑗

𝐸

,  

i.e.  𝑇𝑖𝑗 =  𝑇𝑖𝑗

𝐴

+ 𝑇𝑖𝑗

𝐸

.      (3) 

Here 𝑻𝒊𝒋

𝑨

 is an anisotropic spherically symmetric matter distribution and is given by 

  𝑇𝑖𝑗

𝐴

= (𝜌𝑚 + 𝑃⊥) 𝑈𝑖 𝑈𝑗 − 𝑃⊥𝑔𝑖𝑗 + (𝑃𝑟 − 𝑃⊥) 𝜒𝑖 𝜒𝑗, (4) 

where 𝑈𝑖 is the time like (𝑛 + 2) -velocity 𝑈𝑖 = 𝛿(𝑛+2)
𝑖 𝑒−𝜈/2, 𝜒𝑖 a unit space-like vector in the radial 

direction 𝜒𝑖 = 𝛿1
𝑖𝑒−𝜆/2, 𝜌𝑚 the energy density, 𝑃𝑟 the radial pressure in the direction of 𝜒𝑖 and 𝑃⊥ the 

pressure orthogonal to 𝜒𝑖. 

Similarly, the electromagnetic contribution can be written as  

  𝑻𝒊𝒋

𝑬

= −
𝟏

𝟒𝝅
(𝑭𝒊

𝜶𝑭𝒋𝜶 −
𝟏

𝟒
𝒈𝒊𝒋 𝑭𝒍𝒌𝑭𝒍𝒌), (5) 

where 𝑭𝒊𝒋 is the electromagnetic field tensor defined in terms of the (𝒏 + 𝟐)-potential 𝑨𝒊 as 

  𝑭𝒊𝒋 = 𝑨𝒊, 𝒋 − 𝑨𝒋,𝒊 . (6) 



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As the frame is considered to be the rest frame, we adopt the gauge  

  𝑨𝒊 = 𝑨𝒊(𝟎,  𝟎,  𝟎, ⋯ (𝒏 + 𝟏) 𝒕𝒊𝒎𝒆𝒔 , 𝝋(𝒓)). 

Also, 𝑭𝒊𝒋 satisfy the Einstein-Maxwell field equations 

  𝐹𝑖𝑗;𝑘 + 𝐹𝑗𝑘;𝑖 + 𝐹𝑘𝑖;𝑗 = 0, (7) 

  𝐹;𝑗
𝑖𝑗

= −4𝜋 𝐽𝑖, (8) 

where𝐽𝑖 being the charge-current density that becomes ie
i uJ

−
=  ( e

−
 is the proper charge density).  

In the commoving system, we choose 

  𝑈𝑖 = ( 0,  0,  0, ⋯ , (𝑛 + 1)0, 𝑈𝑛+2) 

  𝜒𝑖 = (𝜒1,  0,  0, ⋯ ,0(𝑛 + 1)) 

From 𝑈𝑖 𝑈𝑖 = −𝜒𝑖 𝜒𝑖 = 1, we obtain 

  𝑈(𝑛+2) = 𝑒−𝜈/2,  𝜒1 = 𝑒−𝜆/2. (10) 

The non-vanishing components of energy momentum tensors 𝑇𝑗
𝑖 are 

=−====−=
+

+⊥
+
+

2n

2n

AA
1n
1n

A
3
3

A
2
2r

A
1
1 T,PTTT,PT    

and 
21n

1n

E
4

4

E
3

3

E
2

2

E
22n

2n

E
1

1

E

ETTTT,ETT −=======
+

+

+

+
 . 

Now let us tern to the gravitational field equations. In bimetric theory of reletivity they have the form 

[Rosen, (1980)] 

 







+−=−−

E

ij

A

ijijijijij TT8SRg
2

1
RG , (12) 

where the additional term 𝑆𝑖𝑗 is given by  

 𝑆𝑖𝑗 =
3

𝑎2
(𝛾𝑖𝑗 −

1

2
𝑔𝑖𝑗𝑔𝛼𝛽𝛾𝛼𝛽).  (13) 

If we take the background metric 𝛾𝑖𝑗 in a higher dimensional static de-Sitter form, the line element in 

terms of polar coordinates  

(𝑥1, 𝑥2 ⋯ , 𝑥𝑛+1,  𝑥𝑛+2) = (𝑟, 𝜃1,  𝜃2,  ⋯ , 𝜃𝑛,  𝑡) 

is given by  

 𝑑𝜎2 = (1 −
𝑟𝑛

𝑎2)  𝑑𝑡2 − (1 −
𝑟𝑛

𝑎2)
−1

𝑑𝑟2 −  𝑟2𝑑𝛺2. (14) 

For a region very small as compared to 𝑎 (i.e. for 𝑟 << 𝑎), this line element has the flat space form  

  𝑑𝜎2 = 𝑑𝑡2 − 𝑑𝑟2 − 𝑟2𝑑𝛺2.  (15) 

(9) 

  (11) 



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Let us consider a region with 𝑟 << 𝑎 and neglect the terms, which are small throughout this region, 

we can write the non-vanishing components of 𝑆𝑖
𝑗
 in higher dimensions (Falik and Rosen, 1980) 

 −𝑆1
1 = −𝑆2

2   =   −𝑆3
3 = ⋯ = −𝑆𝑛+1

𝑛+1 = 𝑆𝑛+2
𝑛+2 =

3

2𝑎
𝑒−𝜈.        (16) 

Rosen’s (1980) bimetric field equations given by equation (12) for the for the line-element (2) (for 

𝑟 << 𝑎) can be written as  

  −𝑒−𝜆 [
𝑛(𝑛−1)

2𝑟2 −
𝑛𝜆′

2𝑟
] +

𝑛(𝑛−1)

2𝑟2 = −
3

2𝑎2 𝑒−𝜈 + .8𝜋𝜌 + 𝐸2  (17) 

  −𝑒−𝜆 [
𝑛(𝑛−1)

2𝑟2
+

𝑛𝜈′

2𝑟
] +

𝑛(𝑛−1)

2𝑟2
=

3

2𝑎2
𝑒−𝜈 − 8𝜋𝑃𝑟 + 𝐸2 (18) 

𝑒−𝜆 [
𝜈″

2
+

𝜈′2

4
−

(𝑛 − 1)(𝜆′ − 𝜈′)

2𝑟
−

𝜆′𝜈′

4
+

(𝑛 − 1)(𝑛 − 2)

2𝑟2
] −

(𝑛 − 1)(𝑛 − 2)

2𝑟2
 

        = −
3

2𝑎2 𝑒−𝜈 + 8𝜋𝑃⊥ + 𝐸2,                                                                           (19)  
 

 

where a prime denotes a derivative with respect to 𝑟. 

We now define the effective density and pressures 𝜌𝑚𝑒
 ,  𝑃𝑟𝑒

 and 𝑃⊥𝑒
 (Harpaz and Rosen, 1985) as  

  𝜌𝑚𝑒
= 𝜌𝑚 −

3

16𝜋𝑎2 𝑒−𝜈 ,    𝑃𝑟𝑒
= 𝑃𝑟 −

3

16𝜋𝑎2 𝑒−𝜈 

and  𝑃⊥𝑒
= 𝑃⊥ −

3

16𝜋𝑎2 𝑒−𝜈,  

so that the field equations (17) – (19) take the form 

  −𝑒−𝜆 [
𝑛(𝑛−1)

2𝑟2 −
𝑛𝜆′

2𝑟
] +

𝑛(𝑛−1)

2𝑟2 = 8𝜋𝜌𝑚𝑒
+ 𝐸2 (21) 

  −𝑒−𝜆 [
𝑛(𝑛−1)

2𝑟2
+

𝑛𝜈′

2𝑟
] +

𝑛(𝑛−1)

2𝑟2
= −8𝜋𝑃𝑟𝑒

+ 𝐸2  (22) 

𝑒−𝜆 [
𝜈″

2
+

𝜈′2

4
−

(𝑛 − 1)(𝜆′ − 𝜈′)

2𝑟
−

𝜆′𝜈′

4
+

(𝑛 − 1)(𝑛 − 2)

2𝑟2
] −

(𝑛 − 1)(𝑛 − 2)

2𝑟2
 

        = 8𝜋𝑃⊥𝑒
+ 𝐸2.                                                                                             (23) 

These equations look like the usual Einstein equations with 𝜌𝑚, 𝑝𝑟 and 𝑝⊥ replaced by 𝜌𝑚𝑒
,   𝑝𝑟𝑒

 and 

𝑝⊥𝑒
 respectively. 

It is very difficult to obtain the solution of equations (21)-(23) due to the non-linearity of the field 

equations and therefore one has to make certain simplifying assumption to derive the useful results.  

3. Conformal motions and solutions of field equations 

In this section, we assume that the physical metric (2) admits a one-parameter group of conformal 

motions and obtained solution of the field equations for static and spherically symmetric fluid 

distributions of anisotropic matter. 

The field equations (21)-(23) resulting in the new form  

  (20) 



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𝑛(𝑛−1)

2𝑟2 [1 −
𝜓2

𝐶3
2] −

𝑛𝜓𝜓′

𝐶3
2𝑟

= 8𝜋𝜌𝑒 + 𝐸2, (24) 

  −
𝑛𝜓2

𝐶3
2𝑟2 [1 −

(𝑛−1)

2
] +

𝑛(𝑛−1)

2𝑟2 = 8𝜋𝑃𝑟𝑒
+ 𝐸2, (25) 

  
𝑛𝜓𝜓′

𝑟𝐶3
2 +

𝑛(𝑛−1)𝜓2

2𝑟2𝐶3
2 −

(𝑛−1)(𝑛−2)

2𝑟2
= 8𝜋𝑃⊥𝑒

+ 𝐸2, (26) 

We now define a function 𝛥, as a measure of anisotropy, such that   

  𝛥 = 4𝜋(𝑃⊥𝑒
− 𝑃𝑟𝑒

). (27) 

Also, if we take 𝑋 =
𝜓2

𝐶3
2, then in terms of the function defined in (27), the equations (24) –(26) 

reduces to :  

  
𝑛(𝑛−1)

4𝑟2
−

𝑛𝑋

2𝑟2
+

𝑛𝑋′

4𝑟
−

(𝑛−1)(𝑛−2)

4𝑟2
− 𝛥 = 𝐸2 (28) 

            
𝑛(𝑛−1)

4𝑟2 −
3𝑛𝑋′

4𝑟
−

𝑛(𝑛−2)𝑋

4𝑟2 −
(𝑛−1)(𝑛−2)

4𝑟2 + 𝛥 = 8𝜋𝜌𝑚𝑒
 (29) 

  
𝑛𝑋

2𝑟2 −
𝑛(𝑛−1)

4𝑟2 +
𝑛𝑋′

4𝑟
−

(𝑛−1)(𝑛−2)

4𝑟2 − 𝛥 = 8𝜋𝑃𝑟𝑒
. (30) 

From which we obtain 

  
𝑛(𝑛−1)

4𝑟2 −
𝑛𝑋

2𝑟2 +
𝑛𝑋′

4𝑟
−

(𝑛−1)(𝑛−2)

4𝑟2 − 𝛥 = 𝐸2 (31) 

 
𝑛(𝑛−1)

4𝑟2 −
3𝑛𝑋′

4𝑟
−

𝑛(𝑛−2)𝑋

4𝑟2 −
(𝑛−1)(𝑛−2)

4𝑟2 + 𝛥 =
−3

2𝑎2 𝑒𝜈 + 8𝜋𝜌𝑚𝑒
 (32) 

 
𝑛𝑋

2𝑟2 −
𝑛(𝑛−1)

4𝑟2 +
𝑛𝑋′

4𝑟
−

(𝑛−1)(𝑛−2)

4𝑟2 − 𝛥 =
3

2𝑎2 𝑒𝜈 − 8𝜋𝑃𝑟𝑒
. (33) 

Therefore, the line element (2), in terms of   𝑒𝜈 = 𝐶2
2𝑟2, 𝜓 = 𝐶3

2𝑒−𝜆/2 and 𝑋 =
𝜓2

𝐶3
2 now becomes

  

  𝑑𝑠2 = 𝐶2
2𝑟2𝑑𝑡2 − (

1

𝑋
) 𝑑𝑟2 − 𝑟2𝑑𝛺2.    (34) 

4. Conclusions 

In this paper, we have presented the field equations for anisotropic charged spheres in higher 

dimensional Rosen’s bimetric theory of relativity. To get a determinate solution of the field 

equations, we have assumed that the space-time admits one parameter group of conformal motion. It 

is observed that the solutions obtained in bimetric relativity in higher dimensions will reduce to the 

corresponding solutions of Einstein’s field equations obtained by Herrera et al. [21] of a physical 

system, which is very small compared to the size of the universe, when the term 
3

2𝑎2
𝑒−𝜈 that appears 

in the field equations of bimetric relativity is negligible.  

 

 



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