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- Mach and Evolution of Universe MACH'S PRINCIPLE AND EVOLUTION OF THE UNIVERSE Michel Bonnardeau (1) Revised May 1, 2006
ABSTRACT Within the framework of Mach's principle and of D.W. Sciama's physical interpretation of it, a quantitative evaluation of the inertial mass as a function of the density of the universe and of the epoch is obtained. The result shows that the inertial mass varies as the universe expands except for a flat, Euclidean universe. This gives insights on the Principle of Equivalence.
(1) Michel Bonnardeau 116 Jonquille Arzelier 38650 Chateau Bernard FRANCE
[email protected]
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I. INTRODUCTION A physical interpretation of Mach's principle and of the origin of the inertial mass has been proposed by D.W. Sciama [1]. In this theory, derived from an analogy with Electromagnetism, the gravitation gives rise to a force proportional to the acceleration and inversely proportional to the distance. The inertial force is then the resultant of this force from all the gravitational masses in the universe applied to an accelerating mass. Within this framework, the inertial forces depend upon the distribution of masses in the universe. However, during the evolution from the Big Bang to the continuing present expansion, this repartition of masses is modified. Then, the inertial masses are expected to vary as a function of time. In this study, the inertial forces will be calculated along the lines of D.W. Sciama's theory [1], quantitatively, as a function of the density and of epoch [2] and their variation during the evolution of the universe will be made clear. It will be shown that, in the particular case of a flat, Euclidean universe, the inertial forces are constant with time. This leads to insights on the origin of the Principle of Equivalence.
II. CALCULATION OF THE INERTIAL MASS IN AN EXPANDING UNIVERSE Following D.W. Sciama [1], the inertial force on a gravitational mass mg due to a mass M g is written as:
F
K⋅ mg ⋅ M g 2
(1)
⋅γ ⋅Φ
c ⋅D
where γ is the acceleration, Φ the angular dependence (of the order of 1), c the velocity of light, D the distance (actually a distance-luminosity"over a cosmological scale) between mg and M g . K is the coupling constant of the inertial force and it will be identified with the gravitational constant G. Integrating over all the masses in the universe, we have: F
K⋅ γ ⋅ Φ c
2
⌠ ⋅ mg ⋅ ⌡
z =∞
ρ( z) D
(2)
dV
z =0
where ρ(z) is the density at the redshift z and dV is the element of volume.
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The Robertson-Walker metric can be written as [3]:
ds
2
2
R( t)
2
c ⋅ dt −
2 2
1 + k⋅ r 4
2
2
(
⋅ dr + r ⋅ dθ + sin ( θ) ⋅ dφ 2
2
2
2
)
(3)
where R(t) is the radius of the universe and the values for k may be -1 (pseudo-Euclidean open universe), 0 (flat universe), +1 (closed universe). The element of volume is then: 3 2
dV
4⋅ π ⋅ R ⋅ r ⋅ dr 2 1 + k⋅ r 4
(4)
3
Let us calculate the distance-luminosity D. The gravitational forces are assumed to be due to particles (may be gravitons) which propagate along geodesics. Let be a source at coordinate r where the radius of the universe is R(t) and the redshift is z (or Y) with
1 1+ z
Y
R R0
(underscript 0 is for the
( )
present time). Let also be an observer at coordinate r1 where the radius of the universe is R t1
R1 with
( )
redshift z1 Y1 where t 1 is not necessarily the present time but may be some epoch in the past or the future. The source emits N particles/s/st. The observer receives 4⋅ π ⋅ N⋅
R1 R
particles/s. The observer is at 2
the surface of the sphere centered on the source and the surface of which is
r z, z1
(
)
1+
r 2
(
)
with 2 r z, z1
( ( ))
. The distance-luminosity between the source and the observer is then:
k⋅ r 4
−3
1
D z, z1
4⋅ π ⋅ R1
2 2 R0⋅ r z, z1 ⋅ ( 1 + z) ⋅ 1 + z1
(
)
(
)
(5)
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We also have from (3) along a geodesic: −R⋅ dr
c⋅ dt
(6)
2
1+
k⋅ r 4
Then we have a first evaluation of the dV/D term in the integral (2): 5
−3
2 2 −4⋅ π ⋅ r z, z1 ⋅ c⋅ dt ⋅ R0⋅ Y ⋅ Y1
dV
(
D
)
5
−3
c R'0 2 2 −4⋅ π ⋅ r z, z1 ⋅ ⋅ ⋅ dY⋅ R0⋅ Y ⋅ Y1 H0 R'
(
)
(7)
where H=R'/R is the Hubble's parameter. The Friedmann's equations [3] will allow us to express R(t) as a function of the density ρ and of the pressure p. For the metric (3) they are: 3
8⋅ π ⋅ G⋅ ρ
R
8⋅ π ⋅ G⋅ p c
2
(
−1
2
R
2
2
⋅ k⋅ c + R'
2
(
2
)−Λ
(8a)
)
2
(8b)
⋅ k⋅ c + R' + 2⋅ R''⋅ R + Λ
with Λ the cosmological constant. A dust universe (p=0) is considered. The mass within a given volume, or 3
ρ( t ) ⋅ R( t) , is then constant with time.
Let us introduce the dimensionless density and deceleration parameters defined as:
σ
q
Ω
4⋅ π ⋅ G⋅ ρ⋅ R
2
3⋅ R'
2
−R''⋅ R 2
R'
2
4⋅ π ⋅ G⋅ ρ
(9a)
2
3⋅ H
−R''
(9b)
2
R⋅ H
We have then:
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k⋅ c
- Mach and Evolution of Universe -
2
2
2
R0 ⋅ H0
R' R' 0
2
3⋅ σ0 − 1 − q 0
1 Y
(10a)
(
)
(
)
3
k⋅ c
2
⋅ 2⋅ σ0 + q 0 + 1 − 3⋅ σ0 ⋅ Y − q 0 − σ0 ⋅ Y
−3 3
dV D
Y ⋅ Y1
2
⋅ dY⋅
(10b)
3⋅ σ0 − q 0 − 1 c −4⋅ π ⋅ r z, z1 ⋅ ⋅ 2 3 H0 2⋅ σ0 + q 0 + 1 − 3⋅ σ0 ⋅ Y − q 0 − σ0 ⋅ Y
(
)
(
)
(
If the cosmological constant is 0 then we have σ0
)
Ω0 2
(11)
q 0 and the universe is flat, Euclidean for Ω 0
1,
closed for Ω 0 > 1, open and pseudo-Euclidean for Ω 0 < 1. This gives: −3
dV D
3
2
2 Y ⋅ Y1 ⋅ dY c k⋅ c −4⋅ π ⋅ r z, z1 ⋅ ⋅ ⋅ 2 Ω + 1 − Ω ⋅Y Ω − 1 0 0 0 H0
(
)
(
(12)
)
Let us evaluate the coordinate r z, z1 . In a dust universe with a zero cosmological constant, we
(
)
have [3]: r z, z1 sin ( χ ) if k χ if k 0
(
)
sh ( χ ) if k
(13)
1 −1
where the χ are:
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χ
⌠ ⌡
c R⋅ R'
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2R0
arccos
dR
α ⋅ ( 1 + z)
(
4R0
)
α ⋅ 1 + z1
8⋅ π ⋅ G
with α
3c
2
if k
α ⋅ ( 1 + z)
α ⋅ ( 1 + z1)
2R0
α ⋅ ( 1 + z1)
4R0
−
2R0
argch
− 1 − arccos
− 1 if k
1
(14)
0
2R0
α ⋅ ( 1 + z)
+ 1 − argch
+ 1 if k
−1
3
⋅ ρ0⋅ R0 . This gives:
(Ω 0 − 1) k r z, z1 ⋅ 1 + 2⋅ q 0⋅ z⋅ q 0⋅ 1 − z1 − 1 − 1 + 2⋅ q 0⋅ z1⋅ q 0⋅ ( 1 − z) − 1 2 q 0 ⋅ ( 1 + z) 1 + z1
(
)
(
(
)
)
(15)
and, from (12), for the dV/D term:
dV
−4⋅ π ⋅ c
2
D
2
2
Ω 0 Y ⋅ Ω 0 − 1 ⋅ Y1 − ⋅ ... 2 Y1 Y1 Ω Ω − Ω 0 − 1 ⋅ Y1 + − Ω − 1 ⋅ Y − 0 ⋅ 0 0 2 Ω 0 − Ω 0 − 1 ⋅Y
3
⋅
H0 ⋅ q 0
Y ⋅ dY
(
)
(
( (
)
) )
(16)
For a dust universe, equation (2) can be written as:
F
K⋅ γ ⋅ Φ c
2
2
0
3⋅ H0 ⋅ σ0 ⌠ 1 ⋅ dV ⋅ mg ⋅ 3 4⋅ π ⋅ G ⋅ D Y ⌡Y
(17)
1
Using (16) this can readily be integrated and the result is:
F
K⋅ γ ⋅ Φ ⋅ mg ⋅
G⋅ Y1 Ω 0 − 1 2
⋅
Ω0
⋅ 1 − 1 −
Ω0 − 1 Ω0
⋅ Y1
(18)
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The inertial force is also F
γ ⋅ mi with mi the inertial mass. Identifying K with G and with Φ of the
order of 1, the inertial mass can be related to the gravitational mass through the relation [4] : mi
(
)
mg ⋅ f Ω 0 , Y1
(
)
with f Ω 0 , Y1
Y1 Ω 0 − 1 2
⋅
Ω0
⋅ 1 − 1 −
Ω0 − 1 Ω0
⋅ Y1
(19)
III. DISCUSSION AND CONCLUSIONS The result (19) implies that, at the present epoch Y1
mi
mg ⋅
1 , we have:
2 Ω0
(20)
Ω0 + 1
The Universe is observed with a density of matter of Ω m0
0.24 (both visible and dark) and with
something (cosmological constant, quintessence field, ...) that makes it accelerate. The Universe is also observed as being flat, with a total density Ω m0 + Ω Λ0 acceleration [5]. We have then mi
1 where Ω Λ0 is the density of what makes the
mg on the condition that what makes the Universe accelerates
contributes to the Mach's principle the same way as the matter does. As expected, the inertial force and, therefore, the inertial mass, depends upon Y1, that is of the evolution of the Universe [4]. However, a variation of the inertial mass is not compatible with the Principle of Equivalence, from which the Friedmann's equations used here are derived. So, this calculation of the inertial mass should be considered as a direction in which a perturbation from the Principle of Equivalence
(
is going on. However, for a flat universe, the inertial mass is constant with time: f Ω 0
)
1 , Y1
1 whatever
the value of Y1 is. Then the Principle of Equivalence is respected as the universe evolves and the calculation applies rigorously. The flatness of the Universe is usually explained by the Inflation. I would like to bring out a qualitative argument, derived from the result that the Principle of Equivalence holds only for Ω=1, that may also lead to a flat universe:
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If Ω1 then mi > mg : now the expansion of the universe is faster than if mi
mg and Ω diminishes.
This suggests a feed-back mechanism which makes the universe having naturally a stable position at Ω=1. This gives an insight on the origin of the Principle of Equivalence: not only Ω=1 is needed for the Principle of Equivalence to hold but Mach's principle implies that Ω=1.
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NOTES AND REFERENCES [1] D.W. Sciama (1964) Rev of Modern Physics 36 463; D.W. Sciama (1969) The Physical Foundations of General Relativity, Heinemann. [2] The calculation is somewhat similar to the one for the sky background (and of Olbers's paradox) except that here there is a
1 D
1
term instead of a
2
term. See P.S. Wesson, K. Valle, R. Stabell (1987) ApJ 317
D
601. [3] S. Weinberg (1972) Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons; S. Mavridès (1973) L'Univers Relativiste, Masson. [4] The function f is independent of the epoch 0: let us consider another epoch 0'. We have:
Y1'
R0 R0'
⋅ Y1
H0 Ω 0' Ω 0⋅ H0'
(
2
R0 ⋅ R0'
3
Ω 0⋅
R0 R0'
1
⋅ Ω 0⋅
R0 R0'
+ 1 − Ω0
) f(Ω 0, Y1) .
and then f Ω 0', Y1'
[5] D.N. Spergel et al (2006) Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology, preprint.
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