Curvature Invariants of Spherically Symmetric Schwarzschild Solution without Cosmological Constant

Journal of Applied Sciences Research, 4(1): 16-31, 2008
© 2008, INSInet Publication
Curvature Invariants of Spherically Symmetric Schwarzschild
Solution without Cosmological Constant
Md. Haider Ali Biswas, Uzzal Kumar Mollick, Samsad Pervin and Ariful Islam
Mathematics Discipline, Khulna University, Khulna-9208, Bangladesh.
Abstract: Invariant plays a very essential role not only in mathematics and physics but also in almost
every branches of science. It is one of most important tools of tensor analysis as well as of the theory
of relativity and cosmology. This study is an investigation of curvature invariants of the spherically
symmetric and manifestly static Schwarzschild solution of Einstein field equation without cosmological
constant.
Keywords: Curvature tensor, invariant properties, spherically symmetric, static, Schwarzschild line element
INTRODUCTION
An invariant does not change under any change of coordinates. Thus, if
is a function of co-ordinates,
then it is invariant provided, it retains its value under a transformation from x i to new coordinates
, such that,
.
Note that the form of the function may change but its value does not. N ote that the infinitesimal square of
distance ds 2 is an invariant. The study of invariance problems was pioneered in the early part of this century by
Emmy Noether [6 ] influenced by the work of Klein and of Lie on the transformation properties of differential
equations under continuous groups of transformations. Another important technique, Calculus of variation was used
to generalize Euler-Lagrange equation [3 ] which is also invariant under canonical transformations. Due to symmetry,
another important tool of mathematics which has a close connection with invariance [2 ], space time curvature is
invariant.
Riemann-Christoffel tensor (or more commonly, the Riemann tensor, or the Curvature tensor) plays an
important role in specifying the geometrical properties of space time.
The curvature invariants of generalized Schwarzschild solution with cosmological constant were calculated by
several authors [4 ,1 ]. In this connection, it is of some interest to study the curvature invariants of spherically
symmetric and manifestly static Schwarzschild solution of Einstein equation without cosmological constant and also
a brief discussion on its space-time singularities.
Riemann Christoffel Curvature Tensor: Let A be a covariant vector. By the definition of covariant derivative
of a covariant tensor of rank one and covariant derivative of a covariant tensor of rank two.
(1)
Now,
Corresponding Address:
Md. Haider Ali Biswas, Assistant Professor, Mathematics Discipline, Khulna University, Khulna9208, Bangladesh.
Phone: 88-041-720171-3/343,
Mob: 01711948396
Email: [email protected]
16
J. Appl. Sci. Res., 4(1): 16-31, 2008
(2)
Interchanging j and k in this equation
(3)
Subtracting (3) from (2)
W e define
then
is called covariant curvature tensor.
Now
(4)
W e know that
17
J. Appl. Sci. Res., 4(1): 16-31, 2008
Similarly
Putting this value in (4) ,we get
(5)
From the relation between Christoffel symbols of first kind and second kind
and from the definition of Christoffel symbols of first kind
Adding then we get
Putting these values in (5), we get
[Interchanging j and k of
]
18
J. Appl. Sci. Res., 4(1): 16-31, 2008
(6)
Now, the Schwarzschild solution without cosmological constant is given by,
(7)
Then metric for 4-dimensional space is given by
(8)
Comparing (7) and (8), we get
Now Christoffel symbols of second kind are given by
(9)
By the definition of Reciprocal Tensor
19
J. Appl. Sci. Res., 4(1): 16-31, 2008
Now the non-vanishing Christoffel symbols of second kind are
20
J. Appl. Sci. Res., 4(1): 16-31, 2008
From Riemann Christoffel Curvature Tensor +
Now, the non-vanishing components of
are given below
21
J. Appl. Sci. Res., 4(1): 16-31, 2008
(10)
(11)
22
J. Appl. Sci. Res., 4(1): 16-31, 2008
(12)
23
J. Appl. Sci. Res., 4(1): 16-31, 2008
(13)
24
J. Appl. Sci. Res., 4(1): 16-31, 2008
(14)
25
J. Appl. Sci. Res., 4(1): 16-31, 2008
(15)
-------------------------------------------------------------------------------------------------------------------------------------------------
etc.
Hence
(A)
Now,
(16)
(17)
26
J. Appl. Sci. Res., 4(1): 16-31, 2008
(18)
(19)
(20)
(21)
27
J. Appl. Sci. Res., 4(1): 16-31, 2008
------------------------------------------------------------------------------------------------------------------------------------------------
---------------------------------------------------------------------------------------------------------------------------------------------------
Hence
(B)
where g levi-Civita tensor.
28
J. Appl. Sci. Res., 4(1): 16-31, 2008
Now
Therefore
(22)
Therefore
(23)
Therefore
(24)
29
J. Appl. Sci. Res., 4(1): 16-31, 2008
Therefore
(25)
Therefore
(26)
Therefore
(27)
Hence
(C)
-------------------------------------------------------------------------------------------------------------------------------------------------
30
J. Appl. Sci. Res., 4(1): 16-31, 2008
----------------------------------------------------------------------------------------------------------------------------------------------
Hence
(D)
Discussion: The Schwarzschild solution (7) was the
first physically significant solution of the field
equations of general relativity. It showed how space
time is curved around spherically symmetric
distribution of matter[5 ]. The problem solved by
Schwarzschild was basically a local problem, in the
sense that the deviations of space time geometry from
the Minkowski geometry of special relativity gradually
diminish to zero as we move further and further away
from gravitating sphere. This result can be easily
verified from the line element (7) by letting the radial
coordinate r go to infinity. In technical jargon a space
time satisfying this property is called asymptotically
flat. Another view point of Schwarzschild solution was
that our universe would be static. So there is no scope
for a dynamical solution such as one involving
gravitational radiation, even if our spherical source is
expanding, contracting, or oscillating. This remarkable
result is known as Birkhoff’s theorem. An observer
viewing the universe from any vantage point will find
that it looks the same in all directions and that it
presents the same aspect from all advantage points.
These two properties are known as isotropy and
homogeneity; they will turn out a play simplifying
roles in relativistic cosmology.
. Also the Riemann tensor
scalar invariant which reduces from (7), when we take
equation (7) to be describing the patch
,
then it is seen that as
, the curvature scalar,
which is finite at r = 2M . All
these are shown in equations, (A), (B), (C), and (D).
Since it is scalar, its value remains the same in all coordinate systems; also this invariant blows up at the
origin r = 0. The singularity at the origin is indeed
irremovable and variously called an intrinsic curvature,
physical, essential, genuine, real or naked singularity.
REFERENCES
1.
2.
Conclusion: From a mathematical point of view,
Schwarzschild solution describes a space-time
singularity namely at
. If we
compute the components of curvature tensor
3.
and
4.
construct invariants out of these, such as,
5.
, , ……….
6.
these invariants diverge and it follows that the
point r = 0 is a real space-time singularity. In
the Schwarzschild space-time there is an essential
curvature singularity at r = 0 in the sense that along
any non-space-time trajectory falling into the
singularity, as
, the Krestschman scalar
31
Azad, M.A.K., 1985. Curvature Invariants of the
G e n e r a li z e d S c h w a r z s c hild S o lutio n w ith
Cosmological Constant. Chitta gong Univ. Stud.,
Part II: Sci., 9(2): 129-133.
Biswas, M.H.A., M.M.K. Alam, M.S. Masud and
M.F. Siddika, 2006. A Review of Symmetries
Conserved Quantities and Invariance Properties.
Khulna University Studies, 7(1): 107-118.
Biswas, M.H.A., 2005. Generalization of EulerLagrange’s Differential Equation for the Functional
of Higher Order Derivatives. Journal of Science
and Technology, 3: 28-35.
Islam, J.N., 1983. The cosmological constant and
classical tests of general relativity. Physics Letters,
97A: 6.
Narlikar, J.V., 2002. An Introduction to
Cosmology. Third Edition, Cambridge University
Press, Cambridge, United Kingdom, pp: 38-95.
Neother, E., 1918. Nachrichten von der
Gesellschaft der W issenschaften zu Göttingen.
pp: 235-257 (in German).