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0F
Study of a Hypothetical Metric Based on
Vacuum Einstein Field Equation
Ehsan Bashiri
NIOPDC, Ilam, Iran, email: [email protected]
Abstract— This research supposes existing a hypothetical metric which is arisen from difference between proper times derived from Schwarzschild metric and what is shall be recorded if no gravity affect _flat metric. Primary purpose of this paper is to find out whether this new metric supports our physical rules or not; hence, this imaginary metric would be analyzed and verified according to vacuum Einstein field equation. Result of this analysis and verification is that Einstein tensor vanishes and solution of vacuum field equation satisfies. Although some parts of this paper are based on imaginary equations, it helps researches and students examine 37Tcompatibility capability37T of hypothetical equations with regard to Einstein tensor.
Keywords—General Relativity,Einstein field equation, spacetime, time delay
I. INTRODUCTION
Special Relativity formulates structure of Minkowski spacetime in a flat vacuum space distant from any massive object with zero cosmological constant and energy-momentum tensor [1]. While in General Relativity, Schwarzschild metric describes existence of gravitational field and its effect on the structure of space around any spherical rotating or non-rotating massive object, again in a vacuum space [2].
It is known to all how gravitational field of the Sun affects and deflects path of a beam of light passing in vicinity of a mass according to the Schwarzschild metric. Moreover, curvature induced due to the gravity causes a delay in time which seems to reduce speed of light examined by Shapiro [3].
What human can observe is that of exists in the real world because of effects of gravitational fields which surrounds him. This means that human could be able to observe different things, or at least at different times sooner or later, if gravitational field subside or become denser. This paper tries to formalize a relationship between these two metrics and analyze it based on Einstein vacuum field equation.
II.NEW HYPOTHETICAL METRIC
A.Basic Relationships; Calculating Time Delay
Looking at the figure below, two beams of light are considered to be emitted from an infinitely distant star passing near
the Sun:
i is a real beam of light corresponding to Schwarzschild Metric,
)
)
(
sin
(
)
2
1
(
)
2
1
(
2 1 2 2 2 2 22
θ
θ
φ
τ
dr
r
d
d
r
m
dt
r
m
d
i ii i
i
i
=
−
−
−
−
+
−
(1)
)
)
(
sin
(
θ
θ
φ
τ
dt
dr
r
d
d
d
j=
j−
j−
j+
(2)Fig. 1 Schematic of two beams of light passing near the Sun
Conditions:
r
φ
Plane and2
π
θ
=
From the figure above, following relationships could be obtained
j j i i j
i
j i
j i
dr
r
R
dr
r
d
r
R
r
R
r
r
R
r
r
=
=
⇒
=
=
=
=
+
=
+
=
λ
δ
φ
δ
φ
δ
λ
λ
δ
cos
&
2 2 2
2 2 2
(3)
Inserting these relationships into the flat metric and rewriting equation (2) in terms of quantities of i will give
−
−
=
−
−
−
−
=
−2 2 2 2
2 2
2
2 2 2 1 2
2
)
(
)
(
)
2
1
(
)
2
1
(
φ
δ
δ
τ
φ
τ
d
r
R
dr
R
dt
d
d
r
dr
r
m
dt
r
m
d
i i
j j
i i i i
i i
(4)
there’s no need to prove that [5],
)
2
1
(
)
(
2i
r
m
R
−
=
δ
(5)this changes equation (4) to
−
−
−
−
=
−
−
−
−
=
−2 2 2
2 2
2 2 2 1 2
2
)
2
1
(
)
2
1
(
)
2
1
(
)
2
1
(
φ
τ
φ
τ
d
r
r
m
dr
r
m
dt
d
d
r
dr
r
m
dt
r
m
d
i i i
i j
j
i i i i
i i
(6)
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−
+
−
=
−
+
−
=
− −2 2 2
2
2 2 1 2
2 2
)
2
1
(
)
2
1
(
)
2
1
(
)
2
1
(
φ
φ
d
r
r
m
dr
r
m
dt
d
r
r
m
dr
r
m
dt
i i i
i j
i i i
i i
inserting (3) quantities and rewriting above relationships based on determining result of division of
dt
iondt
jgives rise to2 2 1 2
2
)
(
)
2
1
(
)
(
)
2
1
(
)
2
1
(
)
(
λ
λ
δ
R
r
m
r
m
r
m
dt
dt
i i i
j i
+
−
−
+
−
=
− −
(7)
Based on the overall deviation of light passing near the Sun resulting from the Einstein prediction [2], it is known that
R
M
4
=
φ
(8)Assuming that the under-experiment beam of the light passes tangent to the Sun, i.e.
r
i≈
r
j=
R
:
)
(
&
4
sin
R
R
M
R
δ
λ
φ
φ
≈
⇒
=
=
considering
R
M
k
=
, and using an important principle in mathematics [3]ε
ε
p=
+
p
+
)
1
1
(
(9)(7) simplifies to
2 3 2
2
16
1
96
20
4
1
)
(
k
k
k
k
dt
dt
j i
+
+
+
+
=
ignoring 2P nd
P
and 3P rd
P
powers of k that are puny, 2
2
)
2
1
(
4
1
)
(
≈
+
≈
−
−R
M
k
dt
dt
j i
in other words;
k
R
M
dt
dt
j
i
=
(
1
−
2
)
−1=
1
+
2
or
1
)
2
1
(
−
−=
i j
i
r
m
dt
dt
(10)
The only parameter of j metric that remained unchanged with respect to the i terms in (6) is
dt
j. However it could be rewritten by inserting (10) relationship as follows
−
−
−
−
−
=
−
−
−
−
=
− 2 2 2 2 2 2 2 2 2 1 2 2)
2
1
(
)
2
1
(
)
2
1
(
)
2
1
(
)
2
1
(
φ
τ
φ
τ
d
r
r
m
dr
r
m
dt
r
m
d
d
r
dr
r
m
dt
r
m
d
i i i i i i j i i i i i i (11)As mentioned in previous section, relationship (7) was derived by ignoring the General Relativistic effects of light.
Nevertheless, in reality proper time and distance of light in Schwarzschild metric is not zero (
d
τ
light≠
0
). This fact permitsto subtract both equations of scalar form of metrics as follows
2 2 2 1 2 2 2 2 2
)
2
(
)
2
1
(
)
2
1
(
)
2
1
(
)
2
1
(
φ
τ
τ
τ
d
r
r
m
dr
r
m
r
m
dt
r
m
r
m
d
d
d
i i i i i i i i j i−
−
−
−
−
−
−
−
=
−
=
− 2 2 2 2 2)
2
(
)
4
(
)
2
(
φ
τ
r
d
r
m
dr
r
m
dt
r
m
d
i i i i i i−
−
=
(12)This is an assumed equation for subtraction of real Schwarzschild and imaginary flat metrics that is used for discussion of
the next section.
On the other hand _there is no need to prove that_ in reality, proper time of a beam of light in the flat metric is zero.
Although this fact contradicts the condition above (
d
τ
light≠
0
), based on the mathematics it does not interfere in process ofobtaining (12). This hint results to
2 2 2 2 2 2 2 2 2 2
)
2
1
(
0
)
2
1
(
)
2
1
(
)
2
1
(
φ
φ
τ
d
r
dr
dt
r
m
d
r
r
m
dr
r
m
dt
r
m
d
i i i i i i i i i i j+
=
−
⇒
=
−
−
−
−
−
=
(13)Inserting this result in
d
τ
iequation:2 2 2 1 2 2 2 2 2 2 1 2 2
)
2
1
(
)
2
1
(
)
2
1
(
φ
φ
φ
τ
d
r
dr
r
m
d
r
dr
d
r
dr
r
m
dt
r
m
d
i i i i i i i i i i i−
−
−
+
=
−
−
−
−
=
− − 2 22
i i idr
r
m
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2 2
2 2
)
2
(
)
4
(
)
2
(
2
φ
d
r
r
m
dr
r
m
dt
r
m
dr
r
m
i i i i i i i i
−
−
=
−
simplifying will give
0
0
)
2
(
)
2
(
)
2
(
2 2 2 2
2 2 2
2
=
−
−
⇒
=
−
−
φ
φ
d
r
dr
dt
d
r
r
m
dr
r
m
dt
r
m
i i i
i i i i i i
(15)
that is the quantity of the proper time of a beam of light in a flat surface. Result (15) confirms validity of the equation (12)
that is deductive frame of the next section.
C.Vacuum Einstein Field Equation
First purpose of this section is to find Einstein tensor of equation (12) as a hypothetical space metric. As it was derived,
2 2 2
2 2
)
2
(
)
4
(
)
2
(
φ
τ
r
d
r
m
dr
r
m
dt
r
m
d
=
−
−
Here only 1P st
P
and 2P nd
P
dimensions of space metric (g) will be verified and other degrees are assumed negligible for simplification:
φ
θ
d
dx
g
d
dx
g
dr
dx
r
m
g
dt
dx
r
m
g
=
=
=
=
=
−
=
=
=
3 33
2 22
1 11
0 00
,
0
,
0
,
4
,
2
It is known that
1
)
(
−=
ijij
g
g
(16)and the final relationship to be examined is [7]
ab ab
ab
R
Rg
G
2
1
−
=
(17)where
G
ab is the Einstein tensor,R
abRicci tensor and R scalar curvature.Here Christoffel symbols are used to derive Ricci tensor and scalar curvature [7]. Since the purpose is to find
R
00&
R
11,and
aeb e
ab
R
R
=
(18)where
R
eaeb is Riemann (curvature) tensor. Hence
=
=
010 1 00
101 0 11
R
R
R
R
l hi k lj l hj k li
ei hj ej hi hj ei hi ej ke hij
k
hi lj hj li j hi i hj hij
g
g
g
g
g
R
R
Γ
Γ
−
Γ
Γ
+
−
+
−
=
Γ
Γ
−
Γ
Γ
+
Γ
−
Γ
=
)
(
2
1
, ,
, ,
, ,
(19)
and
)
(
2
1
, , ,i hi j ijh jh
kh k
ij
=
g
g
+
g
−
g
Γ
(20)where
k h
ij hk
ij
h ij h ij
x
x
g
g
x
g
g
∂
∂
∂
=
∂
∂
=
2
, ,
(21)
Pondering on equation (19), as the purpose was to obtain
R
00&
R
11it could be understood that)
(
2
1
)
(
2
1
)
(
2
1
1 , 00 00 0
10 0 01
1 , 11 11 1
11
1 , 00 11 1
00
g
g
g
g
g
g
=
Γ
=
Γ
=
Γ
−
=
Γ
also
m
r
g
m
r
g
4
2
11 00
−
=
=
and
2 11 1 , 11
2 00 1 , 00
4
2
r
m
r
g
g
r
m
r
g
g
=
∂
∂
=
−
=
∂
∂
=
so
r
r
r
2
1
2
1
4
1
0 10 0 01 1 11
1 00
−
=
Γ
=
Γ
−
=
Γ
−
=
Γ
(22)
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1 11 1 00 11 , 00 11 010
1
0 01 0 10 11 , 00 00 110
0
)
(
2
1
)
(
2
1
Γ
Γ
+
−
=
Γ
Γ
+
=
g
g
R
g
g
R
where
3 1 , 00 11
, 00
4
)
(
r
m
g
r
g
=
∂
∂
=
then
2 010 1
2 110 0
8
5
4
5
r
R
r
R
=
=
(23)
It is known that
−
=
−
=
001 1 010 1
101 0 110 0
R
R
R
R
(24)
hence
=
=
−
=
=
2 010 1 00
2 101 0 11
8
5
4
5
r
R
R
r
R
R
(25)
Now dealing with Einstein tensor, equation (19) is rewritten
ab ab
ab
R
Rg
G
2
1
−
=
where
mr
R
R
g
R
ab ab8
5
=
=
(26)
So
0
=
ab
G
(27)Therefore Einstein tensor vanishes. This result satisfies solution of the vacuum Einstein equation [8]
ab ab
ab
ab
Rg
g
T
R
−
+
Λ
+
Κ
=
)
2
1
(
0
(28)when
Λ
is cosmological constant,2
8
c
G
π
=
Κ
, andT
abenergy-momentum tensor.III. DISCUSSIONS
Equation (11) that is difference of two times, one which is real time calculated by Schwarzschild metric and another
of light which comes towards the Earth reaches
2
k
it
jseconds later than what would be without existence of the Sun. Thisdemonstrates that amount of delay shall be more because a lot of massive objects exert gravitational field rather than Sun.
Einstein tensor of derived hypothetical metric was calculated to be zero. When there exists no cosmological constant,
ab
ab
T
R
=
Κ
However, while Ricci tensor is negligible Einstein tensor vanishes, and subsequently energy momentum tensor vanishes too.
This corresponds to the vacuum Einstein field equation, saying that the metric under study could be either flat corresponding
to the Minkoswki spacetime, or Schwarzschild. It even might possess an intermediate form. It must be denoted that, the
metric derived is simplified form of complete metric. Also as mentioned before, tensors were solved by considering only 1P st
P
and 2P nd
P
dimensions. Solving by entering all dimensions will give similar results certainly.
IV. CONCLUSION
Entire this paper is based upon a hypothesis namely difference of two Schwarzschild and flat metrics as a new metric. It is
clear that this metric does not exist in reality; however, it was created in order to answer a question: How much time is
delayed due to existence of gravitational field of massive objects of the universe? Or: how many seconds, minutes or hours
human are back from collecting information from other galaxies? This is a really controversial question and its answer is not
exact; but effect of the Sun could be assessed.
Time required for a beam of light emitted from an outer source to reach Earth could be easily calculated by knowing the
distance; however, what is more important is to find out how many seconds it will take to reach an earthy observer if gravity
does not exist. Time delay which is calculated in this paper is due to existence of Sun alone.
It was understood that subtracting real Schwarzschild and flat equations of metrics of proper time of light, the result would
be that of exists for Schwarzschild metric (
d
τ
i), of course with different format since proper time of flat metric is zero. Thisdifference is treated with as a hypothetical new metric and subsequently, as a space metric Einstein tensor and vacuum field
equation derived and examined. Solving tensors in terms of hypothetical space metric, Einstein tensor was equal to zero.
Therefore in case of zero cosmological constant, energy-momentum tensor vanishes too, that satisfies solution of vacuum
field equation.
REFERENCES [1] A. Einstein, Relativity; The meaning of Relativity, Khwarizmi Press, Iran, 1984. [2] A. Einstein, General Relativity, Khwarizmi Press, Iran, 1984.
[3] C. M. Will, Was Einstein Right?, Putting General Relativity to the Test, Ghoghnus Publications, Iran, 1995.
[4] E. Bashiri, A Simple Method to Calculate Time Delay Due to the Sun Gravitational Field, Journal of Astrophysics & Aerospace Technology. 06. 10.4172/2329-6542-C1-018.
[5] L. D. Landau, E. M. Lifshitz, A Shorter Course of Theoretical Physics, Volume 1, Mechanics and Electrodynamics, First edition, Razi Press, Iran, 1972.
[6] Christian, Magnan, Deviation of Light Near the Sun in General Relativity, (France, 2007), online at:
23TU
http://www.lacosmo.com/DeflectionOfLight/index.htmlU23T.
[7] P. Dunsby, Tensors and Relativity, University of Leeds, UK, 1996 online at: 23TUhttp://www.mth.uct.ac.za/omei/gr/index.htmlU23T.
[8] S. Waner, Introduction to Differential Geometry and General Relativity, Hofstra University, USA, 2002, online at:
