Haranas - Redshift Calculations in the Dynamic Theory of Gravity.pdf

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The Dynamic Theory of Gravity
1
Journal of Theoretics
Redshift Calculations in the
Dynamic Theory of Gravity
Ioannis Iraklis Haranas
Department of Physics and Astronomy
York University
128 Petrie Science Building
York University
Toronto – Ontario
CANADA
E mail: ioannis@yorku.ca
Abstract:
In a new theory called Dynamic Theory of Gravity, the cosmological
distance to an object and also its gravitational potential can be calculated.
We first measure its redshift on the surface of the Earth. The theory can be
applied as well to an object in orbit above the Earth, e.g., a satellite such as
the Hubble telescope. In this paper, we give various expressions for the
redshifts calculated on the surface of the Earth as well as on an object in orbit,
being the Hubble telescope. Our calculations will assume that the emitting
body is a star of mass M = M X-ray(source) = 1.6 × 10 8 M solar masses and a core radius R
= 80 pc, at a cosmological distance away from the Earth. We take the orbital
height h of the Hubble telescope to be 450 Km.
Introduction :
There is a new theory of gravity called Dynamic Theory of Gravity
[DTG]. Based on classical thermodynamics Ref:[1] [2] [3] [9] it has been shown
that the fundamental laws of Classical Thermodynamics also require Einstein’s
177883268.004.png
2
postulate of a constant speed of light. DTG describes physical phenomena in
terms of five dimensions: space, time, and mass. Ref[4] The theory makes its
predictions for redshifts by working in the five dimensional geometry of space,
time, and mass, and determines the unit of action in the atomic states in a
way that can be calculated with the help of quantum Poisson brackets when
covariant differentiation is used:
[ ]
x
µ
,
p
ν
Φ
=
i
=
g
ν
q
{
δ
+
[ ]
µ
,
q
x
s
} Φ
.
(1)
µ
q
In (1) the vector curvature is contained in the Chrisoffel symbols of the
second kind and the gauge function Φ is a multiplicative factor in the metric
tensor g ν q , where the indices take the values ν , q = 0,1,2,3,4. In the
commutator, x µ and p ν are the space and momentum variables respectively,
and finally δ µ q is the Cronecker delta. In DTG the momentum ascribed as a
variable canonically conjugated to the mass is the rate at which mass may be
converted into energy. The canonical momentum is defined as follows below:
p =
4
mv
4
,
(1a)
where the velocity in the fifth dimension is given by:
•
γ
v
=
,
(1b)
4
α
and is a time derivative where gamma itself has units of mass density or
•
γ
kg/m
3 , and α o is a density gradient with units of kg/m 4 . In the absence of
curvature, (1) becomes:
[ ] Φ
x
µ
,
p
ν
Φ
=
i
=
δ
ν
.
(2)
Γ
s
3
From (2) we see that the unit of action is the product of a multiple of
Cronecker’s δ µ q function and the gauge function Φ . It can be also shown that
if we use gauge field equations Ref:[6] then the gauge function Φ is of the
form:
Φ
=
exp
+
k
( )
A
Bt
exp
−
λ
.
(3)
R
R
Assuming conservation of photon energy and expanding the
exponentials and then comparing this expression with (11), we need then to
evaluate the constants A, B, and k. Recalling that the emission time t e = 0 and
the received time t r = L /c, the expression for the redshift reduces to the
following: Ref[5]
−
λ
ob
−
λ
em
M
ob
λ
r
R
R
−
R
∆
λ
G
M
e
ob
M
e
em
HL
r
e
ob
R
z
=
=
exp
−
−
+
e
r
−
1
,
λ
c
2
R
R
c
M
e
ob
em
⊕
R
⊕
(4)
where
M
⊕
is the gravitational potential of the earth,
M
ob
is the
R
R
⊕
ob
reduced gravitational potential at the detection point, and
M
em
is at the
R
em
emission point of the radiation. Since λ << R, expression (4) can be simplified
for the earth’s surface (Es): Ref [5].
177883268.005.png 177883268.006.png
4
[ ]
ln
1
+
z
=
−
G
M
ob
−
M
em
+
HL
,
(5)
Es
c
2
R
R
c
ob
em
and for orbiting Hubble telescope (ht) of a height h the following expression:
[ ]
ln
1
+
z
=
−
G
M
⊕
−
M
em
+
HL
R
⊕
.
(6)
ht
( )
c
2
R
+
h
R
c
R
+
h
⊕
em
⊕
As a result of the analysis in Ref[5], we solve two equations with two
unknowns, the gravitational potential GM/R and the cosmological distance L
of the emitting object. These can be found from:
GM
=
c
2
+
1
R
⊕
ln
[ ]
1
+
z
−
R
⊕
ln
[
1
+
z
]
(7)
ht
Es
R
h
R
+
h
⊕
and
L
=
c
(
ln
[ ]
1
+
z
−
ln[
1
+
z
]
)
+
1
R
⊕
+
GM
⊕
.(8)
Es
ht
H
h
2
c
R
⊕
In this theory, the predicted redshifts are significantly different when
measured on the surface of the Earth, or at a height of 450 km for example
above the surface. In Einstein’s theory of relativity, the redshift of an object
may be written as follows:
z
=
−
G
M
ob
−
M
em
,
(9
c
2
R
R
ob
em
177883268.007.png 177883268.001.png 177883268.002.png
 
5
where the subscripts specify the emitter and observer gravitational
potentials respectively. Since the redshift of an object at cosmological distance
L is given by:
z = ,
H
L
(10)
c
then the total redshift will be given from: Ref[4]
z
=
−
G
M
ob
−
M
em
+
H
L
,
(11)
c
2
R
R
c
ob
em
where H is Hubble’s constant, c is the speed of light, and L the cosmological
distance to the object. Any difference in the redshift will come from the
difference between the gravitational potential at the surface of the earth and
at some height above the surface. However, this difference will be small due
to the small size of the earth compared with cosmological objects. Compared
with the Sun, this effect would be of the order of 10 -5 . In the case z Es ≈ z ht (7)
and (8) simplify as follows:
GM
em
=
c
2
ln
[
+
z
Es
] ,
(11a)
R
em
L
=
c
GM
⊕
2
.
(11b)
H
R
c
⊕
1
177883268.003.png
 

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