It is shown that, by defining a suitable energy momentum tensor, the field
equations of general relativity admit a line element of Yukawa potential as an
exact solution. It is also shown that matter that produces strong force may be
negative, in which case there would be no Schwarzschild-like singularityComment: Latex 4 pages (some changes in notation, and a discussion on a
  repulsive term that may be related to the charge of a particle

Ho, Vu B

English

arXiv

It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case there would be no Schwarzschild-like singularit

Ho, V B

CERN Document Server

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A metric of Yukawa potential as an exact solution
to the eld equations of general relativity
Vu B Ho
Department of Physics
Monash University
Clayton Victoria 3168
Australia
Abstract
It is shown that, by dening a suitable energy momentum tensor, the eld equa-
tions of general relativity admit a line element of Yukawa potential as an exact
solution. It is also shown that matter that produces strong force may be negative,
in which case there would be no Schwarzschild-like singularity.
1
1. Approximate solution
It has been shown that the line element of Yukawa potential of the form
e
 
= 1   
e
 r
r
(1)
approximately satises, at short range r  1, the vacuum eld equations of general
relativity, T

= 0,
R

 
1
2
g

R = T

; (2)
assuming a centrally symmetric spacetime metric
ds
2
= e

dt
2
  e

dr
2
  r
2
(d
2
+ sin
2
d
2
): (3)
In general, the quantity R = 1= species a range so that the line element of Yukawa
potential can be approximated as a solution to the eld equations of general relativity for
the region r  R. It is seen that for the maximal possible range of R ! 1, the metric
of Yukawa form reduces to the familiar Schwarzschild metric, which is used to describe
the gravitational eld. In the case of short range of nuclear physics, the quantity R can
be assigned a value in terms of the fundamental constants h and c, and the rest mass of
Yukawa force carrier. The result has shown that within the short range of strong force,
the eld equations of general relativity admit a line element that takes the form of Yukawa
potential for strong interaction. This leads to the conclusion that if there is no other form
of matter, besides the mass and the charge, that characterises strong interaction, then it
would be possible to consider strong interaction also a manifestation of general relativity
at short range.
Assume the motion of a particle in a strong force eld is governed by the geodesic
equations
d
2
x

ds
2
+  


dx

ds
dx

ds
= 0; (4)
then with the metric of Yukawa potential of the form
ds
2
=
 
1   
e
 r
r
!
c
2
dt
2
 
 
1  
e
 r
r
!
 1
dr
2
  r
2

d
2
+ sin
2
d
2

; (5)
the equations for the geodesics can be written explicitly as [1, 2]
 
1  
e
 r
r
!
 1
 
dr
d
!
2
+ r
2
 
d
d
!
2
+ r
2
sin
2

 
d
d
!
2
  c
2
 
1 
e
 r
r
! 
dt
d
!
2
=  c
2
; (6)
d
d
 
r
2
d
d
!
  r
2
sin  cos 
 
d
d
!
2
= 0; (7)
d
d
 
r
2
sin
2

d
d
!
= 0; (8)
d
d
" 
1  
e
 r
r
!
dt
d
#
= 0: (9)
2
By choosing spherical polar coordinates and considering the motion in the plane  = =2,
the equations (8) and (9) are reduced to
d
d
=
l
r
2
;
dt
d
=
kr
r   e
 r
; (10)
where l and k are constants of integration. With these relations, the equation for the orbit
can be obtained from the equation (6) as
 
l
r
2
dr
d
!
2
+
l
2
r
2
= c
2
(k
2
  1) +
c
2
r
e
 r
+
l
2
r
3
e
 r
: (11)
In the case when the condition r 1 is satised, with the approximation e
 r
= 1  r,
the equation for the orbit becomes
 
l
r
2
dr
d
!
2
+
l
2
(1 + )
r
2
= c
2
(k
2
  (1 + )) +
c
2
r
+
l
2
r
3
: (12)
By letting u = 1=r and dierentiating the resulting equation with respect to the variable
, it is found
d
2
u
d
2
+ (1 + )u =
c
2
2l
2
+
3l
2
2
u
2
: (13)
Hence, the dynamics of a particle under the inuence of strong force of Yukawa potential
is similar to that of a particle in the Schwarzschild gravitational eld.
2. Exact solution
Unlike the assumed massless force carriers of the gravitational eld, hence the vacuum
solution to the eld equations of general relativity could be justied, the force carriers of
the strong eld are assumed to be massive. Therefore, any solutions to the eld equations
of general relativity that may be used to describe the strong eld should be a nonvacuum
solution. This means that in order to nd a more appropriate solution to describe strong
force, a strong energy momemtum tensor must be specied. In the following it will be
discussed a particular form of strong energy momentum tensor so that the eld equations
of general relativity will admit the line element of Yukawa potential as an exact solution.
Consider a strong energy momentum tensor of the form
T


=
0
B
B
B
B
@
 


e
 r
r
2
0 0 0
0  


e
 r
r
2
0 0
0 0

2
2
e
 r
r
0
0 0 0

2
2
e
 r
r
1
C
C
C
C
A
(14)
3
With this energy momentum tensor, the eld equations of general relativity reduce to the
system of equations [3]
e
 
 
d
dr
 
1
r
!
+
1
r
=  
e
 r
r
; (15)
 e
 
 
d
dr
+
1
r
!
+
1
r
=  
e
 r
r
; (16)
d
dt
= 0; (17)
 e
 
0
@
2
d
2

dr
2
+
 
d
dr
!
2
+
2
r
 
d
dr
 
d
dr
!
 
d
dr
d
dr
1
A
+e
 
0
@
2
d
2

dt
2
+
 
d
dt
!
2
 
d
dt
d
dt
1
A
= 
2
e
 r
r
:
(18)
These equations are not independent since it can be veried that the last equation
follows from the rst three equations. Furthermore, the rst two equations give d=dr +
d=dr = 0 that also leads to  +  = 0. This system of equations when integrated gives
the metric of Yukawa potential.
The form of the energy momentum tensor considered above is mathematically accept-
able since it can be veried to satisfy the conservation laws
T

;
=
1
p
 g
@T


p
 g
@x

 
1
2
g

@x

T

= 0; (19)
using the metric g

of Yukawa form.
However, there emerges an important feature that relates to the nature of the quantity
 in the line element and that of the energy momentum tensor. That is, since the quantities
 and  are positive, the energy component T
0
0
and the quantity  always have opposite
signs. Therefore, since g
00
is positive, if the energy component T
00
is considered to be
positive, then the energy component T
0
0
= g
00
T
00
must be positive, and, in this case, the
quantity  must be negative and there would be no Schwarzschild-like singularity.
References
[1] Bergmann, P. G. Introduction to the Theory of Relativity (Dover, 1976).
[2] Lawden, D. F. An Introduction to Tensor Calculus, Relativity and Cosmology (John
Wiley & Sons, 1982).
[3] Landau, L. D. & Lifshitz, E. M. The Classical Theory of Fields (Pergamon Press,
1975).
4


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