Impacts of a New Spatial Variable on a Black Hole Metric Solution

International Journal of Scientific Research and Engineering Development-– Volume 3 - Issue 5, Sep - Oct 2020
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 133
Impacts of a New Spatial Variable on a Black Hole Metric
Solution
Anish Giri
Physics Initiatives in Nepal, Kathmandu, Nepal
Email : girianish2021@gmail.com
----------------------------------------************************----------------------------------
Abstract:
Mathematical formulations are often correlated with the truth; however, there are a few cases where
mathematics is forcefully used to predict something. As a result, fiction is introduced.“The particle
problem in the General Theory of Relativity”,one of the historical papers by Einstein and Rosen in 1935
used a simple spatial variable in the blackhole solutions to omit the concept of singularity and infinity. As
a result, the concept of spacetime portal—a topological configuration with two asymptotically flat regions
connected through a bridge—was mathematically backed up. In this paper, I have introduced a similar
kind of variable = − 2 . Referencing this variable, I have logically stated that = − 2
introduced in Einstein and Rosen’s paper might not necessarily correlate to reality. Such introduction of
variable creates different fiction possibilities in theoretical physics and those possibilities are discussed in
this paper.
Keywords —Black hole, General Theory of Relativity, Spacetime portal, topological configuration.
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I. INTRODUCTION
The wormhole is a special topological configuration
that connects two distant spatial points of the
universe via a bridge. It is probably one of the most
fascinating ideas in physics since everyone is
exhilarated by the fictional idea of time travel. [1]
The idea of wormhole first appeared on a paper
“Comments on Einstein Theory of Gravity” by
Ludwig Flamm in 1916, a year after General
Theory of Relativity was published. Later in 1935,
[2] Albert Einstein and Nathan Rosen proposed an
atomistic theory of matter and electricity excluding
singularity which provided a mathematical
framework for wormholes. In that paper, Einstein
and Rosen introduced a variable = − 2 in
the Schwarzschild metric that got rid of the concept
of singularity. We will discuss briefly how that
variable was introduced and then we will analyze
the introduction of a similar variable that disobeys
the idea of Einstein and Rosen’s paper.
II. DEVELOPMENT OF = −
VARIABLE
[3] When Einstein formulated his Gravitational
field equation at the end of 1915, he needed to
approximate the solution so that Arthur Eddington
could test the deflection of starlight due to the
gravitational field of the sun to prove General
relativity true. It was Karl Schwarzschild who
formulated a metric which was a quite exact and
non-trivial solution in a spherically symmetric
coordinate system.
The simplest form of Schwarzschild metric
involves familiar variables like space, time, and
mass variables as:
= = 1 −
2
−
1
(1 − )
− (
+ sin ")
RESEARCH ARTICLE OPEN ACCESS

In this equation, the coast signature (+ − − −) has
been used. Since it is a solution in the spherical
coordinate system, θ means the latitude and
φ refers to the co-longitude. The diagonal elements
of the metric tensor are: %% = 1 − , =
%
&''
=
%
%(
)*
+
, ,, = , and -- = sin . Here,
= ./. G is gravitational constant and is a
Mass parameter. [4]The theoretical black hole’s
solution that we deal with has the event horizon
specifically called ‘bifurcated killing horizon’,
which is a smooth space-like manifold with a union
of two null hypersurfaces that intersect
transversally to form a bifurcate null surface.
Noticing the singularity at the event horizon i.e.
= 2/ turned out to be a perplexing discrepancy
to Schwarzschild. It can also be seen that the metric
has intrinsic curvature singularity at = 0. [3] So,
Schwarzschild replaced the r coordinate with a
preferable ′ coordinate such that 3
=
( ,
− (2 ),)
'
4. This was a commendable approach
of Schwarzschild to have used a simplified solution
by introducing ′ where it is just the cube root of
the result gained by subtracting the (2 ),
from the
total space-time volume enclosed by . Now, the
singularity is contained at the origin.
Singularity is the biggest adversary of physics and
mathematics. To eradicate the problem of
singularity, Einstein and Rosen, in the same 1935
paper, suggested an introduction of a plain variable
that would change the future of physics and science
fiction.
With the introduction of the new variable = −
2/, the Schwarzschild metric becomes:
= 5
+ 2/
6 − 4( + 2/) − ( + 2 ) ( + sin ")
It can be seen that new has been obtained and
the coefficient of becomes zero when = 0.
Now, this new metric can take any value for , and
hence, there is no singularity or infinity at any finite
point. This solution gives a field free from a
singularity that contains the electrically neutral
elementary particle. This erudite move did not just
solve the vexing problem of singularity, but also put
forth a new possibility. It can be noted that varies
from +89:898 ; to – 89:898 ; , varies from
+89:898 ; to 2/ and again from 2/ to +89:898 ;.
Hence, two similar sheets are connected to by
hyperplane at = 2/ or = 0.
[5]We can also consider that a two-sided
Schwarzschild solution gives two asymptotically
flat regions, each having an identical black hole. In
that case, those black holes are entangled and those
bifurcated horizons touch at the center of the origin.
Similarly, there is another metric that was
formulated including the electromagnetic field in
the General theory of relativity that incorporated
inhomogeneous and homogeneous Maxwell
equations. [6]Reissner-Nordström metric is the
solution of General Relativity which describes the
space-time around a spherically symmetric charged
body with mass / and charge = as:
d = >1 − +
?)
)
@d −
%
(%(
)A
+
B?)/ ))
d − (dθ + sin θ dφ )
Here, E is an integration constant (E =
F)G
-HIJKL)
gained while finding %% and , denoted as
electric charge. When we assign = = 0 , the
Reissner–Nordströmmetric transforms into the
Schwarzschild metric. In the 1935 paper, Einstein
discussed the consequence of introducing the new
variable in this solution as in the previous
Schwarzschild metric. [6]Since and E are two
integration constants that are independent of each
other, mass is not determined by charge. We try to
assume that there is no mass in the field and assign
= 0, so that, we can conceive a solution without
singularity.
So, we get the following solution:
= (1 − E / ) −
%
(%( ?)/ ))
− ( θ + sin θ φ )
When we plug in a new variable = −
?)
, we
get:
= (
M)
( N)B ?))
) − – ( + Q /2) ( θ + sin θ φ )

This metric represents an elementary electrical
particle without a mass and this solution is free
from the singularity. This solution is valid for every
point in the 2 asymptotical sheets where the charge
signifies the Einstein-Rosen Bridge.
Similarly, there are other famous metrics like [7]
Morris-Throne metric (involving gravitational red-
shift function and shape function), [8]Kerr metric
(dealing with space-time near an uncharged
spherically symmetric rotating black hole) ,
[9]Vacuum Static Brane-Dickes wormholes
solution (dealing with torsion), [10]Kruskal-
Szekeres Metric(a maximally extended
Schwarzschild solution) and many other elegant
solutions. However, all of them hold a congruency
because they all use the -like variable of various
forms to eliminate the singularity and claim that the
revised metric including “ term” gives two
asymptotically flat regions joining each other via a
bridge.
III.INTRODUCTION AND CONSEQUENCE OF NEW
VARIABLE P
= −
One intriguing question can prove that introduction
of the term wouldn’t necessarily uphold
mathematical consonance and claim that it settles
the problem of singularity and actuates a new
concept-wormhole. What if we use ,
or -
or
where can be any constant? If we take = −
2 , then the Schwarzschild metric becomes:
= >
MQ
MQB R
@ − (uT
+ 2M)uT(
− (uT
+ 2M) ( θ + sin θ φ )
In this equation, it is noticeable that the value of
changes. When varies from +89:898 ; to
– 89:898 ; then there are two possibilities for . If
is even, then the condition for is the same as the
previous one. But, when is odd, then there can be
negative as well. This yields a new contradiction
in the concept since the integrity of negative can
be cross-interrogated.
Another insight into it is that if we use the
fundamental theorem of algebra, we can see that
there is a maximum number of possible solutions
for and there may not be only two asymptotical
flat regions, but many regions connecting each
other. Presumably, the most fascinating part of this
idea is that it results in a concept ‘probability of
spatial coordinate’ since we will not be able to
locate the exact point to be reached after the
opening of another side of the wormhole.
This concept might introduce a new problem. Since
every mathematical solution of u will not be real,
there will be an imaginary solution of spatial
coordinate . [11] As we know that imaginary
numbers also have some physical reality
applications like[12] radio electronics, special
relativity, and superluminal speeds, a hypothesis for
the possibility of ‘spatial solution of different
universe’ might be put forward.
The Kruskal metric also predicts the possibility of a
wormhole connecting two universes, strictly called
‘white hole’. [13] The wormhole structure and
topology of this model can be modified in such a
way that this model can produce an infinite number
of universes along with both space and time
direction.
The idea of the spatial solution for a different
universe is pretty interesting although we are not
apprised of the reality. We can still proceed with
our view of why the ‘different universe’ concept is
insensible. Since a black hole may not be eternal as
suggested by hawking due to [14] the Hawking
radiation, the existence of such a universe is not
possible.
[15]Another point to be noted is that three alleged
big bang universes are either spatially finite or
spatially infinite. Spatially infinite black hole
universe cannot fit inside the spatially finite big
bang universe. The two different spatially infinite
big bang universes either have a constant negative
curvature or a constant zero curvature. Nevertheless,
no black hole universe possesses such a constant
curvature and hence can’t fit into either of the
spatially infinite big bang universes. And joining
such arbitrary universes by one bridge is an absurd
idea. Thus, it is better to say that the black hole, if it
does not end to singularity, takes us to a different
point of the same universe.

III. CONCLUSION
The scientific consortium has procured its first-ever
picture of a ‘black hole’, but we are still speculative
about what lies behind the event horizon. Our
colossal understanding of mathematics genuinely
confronts us to endless possibilities, but all of them
bear some sort of incongruity or mathematics is
forcedly employed to claim that the idea is
veracious and convincing. We do not yet precisely
know what lies behind it but one can efficaciously
resume deriving more plausible equations and
correlating them with the probable reality. We have
encountered what revolution even a simple change
in a variable can bring. Thus a logically strong
mathematical variable must be introduced to make a
hypothesis of what lies beyond the black hole,
otherwise, we have seen what possibilities we have
to consider before taking the u = r − 2M as a
correct variable for metric solutions to end the
concept of singularity.
ACKNOWLEDGMENT
I would like to thank Dr. Robert Scherrer,
Vanderbilt University; Dr. Kapil Adhikari, Physics
Research Initiatives-Pokhara; Mr. Swaraj Sagar
Pradhan, Mr. Pradip Poudel, Mr. Aadarsh Ojha, and
Mr. Kushal Dhakal for their mentorship and
guidance for completing this research paper.
REFERENCES
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Zeitschrift, vol. 17, pp. 448-454, 1916.
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[4] P. T. Chru ́ciel, J. L. Costa, and M. Heusler, “Stationary Black Holes:
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2012.
[5] H. Maldac, L. Susskind, “Cool Horizons For Entangled Black holes,”
Fortschritte der Physik, vol. 61, issue 9, pp. 781-811, 2013.
[6] J. Nordebo, “The Reissner-Nordström metric,” Bachelors dissertation,
Department of Physics, Umeå University, 2016.
[7] M. S. Morris and K. S. Throne, “Wormholes in spacetime and their use
for interstellar travel: A tool for teaching General Relativity,”
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[8] E. Neuman, “Kerr-Newman metric,” Scholarpedia, vol. 10, issue. 9,
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[10] S. Crothers, “The Kruskal-Szekeres "Extension": Counter-Examples,”
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[11] A. A. Antonov, “Resonance on Real and Complex Frequencies,”
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[12] A. A. Antonov, “Hidden Multiverse: Explanation of Dark Matter and
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[13] A. A. Shatskiy, “A dynamic model of the wormhole and the Multiverse
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[14] D. N. Page, “Hawking radiation and black hole thermodynamics,” New
Journal of Physics, vol. 7, pp. 203-203, 2005.
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10.13140/2.1.3682.8163, 2014.

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