Equations of Motion in
an Expanding Universe
Sergei Kopeikin
18-19/04/2013
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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Expanding Universe =
Conformally-flat Manifold
𝑑𝑠 2 = −𝑑𝑡 2 + 𝑅2 𝑡 𝛿𝑖𝑗 𝑑𝑦 𝑖 𝑑𝑦 𝑗
𝑑𝑡 = 𝑎 𝜂 𝑑𝜂
𝑑𝑠 2 = 𝑎2 𝜂 𝑓𝛼𝛽 𝑑𝑦 𝛼 𝑑𝑦 𝛽
𝑦 𝛼 = 𝜂, 𝑦 𝑖 − conformal coordinates ;
𝑎 𝜂 ≡ 𝑅[𝑡 𝜂 ]
t - proper time of Hubble observers (physical time)
𝜂 − conformal time (mathematically convenient,
18-19/04/2013
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
unphysical)
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We are interested in studying the laws of local physics on
the conformally-flat manifold.
Focus on the (Einstein) equivalence principle.
It states that there exists a diffeomorphism 𝑦 = 𝑦 𝑥 from
global, 𝑦 𝛼 , to local inertial coordinates, 𝑥 𝛼 = (𝜏, 𝑥 𝑖 ), such
that
𝑑𝑠 2 = 𝑓𝛼𝛽 𝑑𝑥 𝛼 𝑑𝑥 𝛽
and
𝛼
Γ𝛽𝛾
=0
and
local equations of motion for test particles:
2
𝛼
𝑑 𝑥
=0
2
𝑑𝜏
18-19/04/2013
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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We simplify theory by sticking to the linearized Hubble
approximation in all equations.
Local diffeomorphism:
𝜇 𝜕𝑦 𝜈
𝜕𝑦
𝑎2 (𝜂)𝑓𝜇𝜈 𝛼 𝛽
𝜕𝑥 𝜕𝑥
= 𝑓𝛼𝛽
Special conformal transformation:
𝜇 𝜕𝑦 𝜈
𝜕𝑦
Ω2 (𝑥)𝑓𝜇𝜈 𝛼 𝛽
𝜕𝑥 𝜕𝑥
= 𝑓𝛼𝛽
Ω 𝑥 = 1 − 2𝑏𝛼 𝑥 𝛼 + 𝑏 2 𝑥 2
𝛼
𝛼 2
𝑥
−
𝑏
𝑥
𝛼
𝑦 =
Ω 𝑥 𝛼 𝐻 𝛼
Matching: 𝑏 = 𝑢 ;
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18-19/04/2013
Ω 𝑥 = 𝑎 𝜏 = 1 + H𝜏
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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Local inertial metric: 𝑑𝑠 2 = −𝑑𝜏 2 + 𝛿𝑖𝑗 𝑑𝑥 𝑖 𝑑𝑥 𝑗
Local inertial frame: 𝜏 − proper time of static observers
𝑥 i − normal Gaussian coordinates
Static observers with fixed 𝑥 𝑖 move with respect to the
Hubble observers which have fixed 𝑦 i , and vice versa
𝑖
𝑥 =𝑎 𝜏 𝑦
𝑖
𝜏 = 𝑡 + 𝑂 𝐻2
18-19/04/2013
𝑑𝑥 𝑖
𝑑𝜏
= 𝐻𝑦 𝑖
Hubble law
proper time 𝜏 of static observers
coincides with the cosmic time t
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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Focus on motion of photons (light).
Equations of motion of light in global coordinates:
𝑑2𝑦𝛼
𝑑𝜂 2
=0
We expect in local (Gaussian) coordinates:
𝑑2𝑥𝛼
𝑑𝜏2
=0
Coordinate transformation: 𝜏 = 𝑎 𝜂 𝜂
𝑥𝑖 = 𝑎 𝜂 𝑦𝑖
However, operationally 𝜏 = 𝑡. Hence, we conclude
𝑡 = 𝑎 𝜂 𝜂 = 𝜂 + 𝐻𝜂2 + 𝑂(𝐻2 )
But the cosmic and conformal
times are related by
𝐻 2
𝑡 = ∫ 𝑎 𝜂 𝑑𝜂 = 𝜂 + 𝜂 + 𝑂(𝐻2 )
2
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7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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We cannot postulate that the affine parameter on light geodesic is
the proper time of static observers. It must differ from 𝜏
Conformal transformation: 𝜆 = 𝑎 𝜂 𝜂
𝜆 = 𝑎 𝜂 𝜂 = 𝜂 + 𝐻𝜂2 + 𝑂(𝐻2 )
𝐻 2
𝑡 = ∫ 𝑎 𝜂 𝑑𝜂 = 𝜂 + 𝜂 + 𝑂(𝐻2 )
2
𝑥𝑖 = 𝑎 𝜂 𝑦𝑖
𝑑𝜆 = 𝑎 𝜏 𝑑𝜏
No contradiction! But the price to pay is that the local inertial
metric differs from the optical metric on light cone (Synge, Perlick)
𝑑𝑠 2 = −𝑑𝜆2 + 𝛿𝑖𝑗 𝑑𝑥 𝑖 𝑑𝑥 𝑗
𝑑2 𝑥 𝛼
𝑑𝜏2
=𝐻
𝑑𝑥 𝛼
𝑑𝜏
18-19/04/2013
− 𝑢𝛼
𝑑𝑠 2 = −𝑎2 (𝜏)𝑑𝜏 2 + 𝛿𝑖𝑗 𝑑𝑥 𝑖 𝑑𝑥 𝑗
local equations of motion for photons
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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Radar and laser ranging: solving 𝑑𝑠 = 0
Radial geodesics, observer at the origin of the local coordinates
𝑑𝑟 = ±𝑎 ±𝜏 𝑑𝜏
+ outgoing ray,
− incoming ray
𝜏1
𝐻 2
𝑟 = 𝑟0 ± 𝜏 + 𝜏
2
∗
(𝜏 , 𝑟)
𝜏
𝐻
𝑟 = 𝑟0 ± 𝜏 − 𝜏0 + 𝐻(𝜏 − 𝜏0 )2
2
𝑟1 = 𝑟0 ± 𝜏1 − 𝜏 + (𝜏1 − 𝜏)2
2
1
𝜏0
ℓ = 𝜏1 − 𝜏0
For central observer:
ℓ=𝑟 ;
2
𝜏∗
18-19/04/2013
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
𝐻 2
=𝜏+ 𝑟
2
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Doppler Effect (static observers)
𝜔 = −𝑘𝛼 𝑢𝛼
𝜔2
𝜔1
=
frequency of light
𝑘0 (𝜏2 )
𝑘0 𝜏1
𝑑𝑘𝛼
𝑑𝜆
=
1 𝜕𝑔𝜇𝜈
−
𝑘𝜇 𝑘𝜈
𝛼
2 𝜕𝑥
𝜔2 , 𝑎2
𝑘0 (𝜏)
= const.
𝑎(𝜏)
𝜔2
𝜔1
=
𝑎(𝜏2 )
𝑎(𝜏1 )
18-19/04/2013
𝜔1 , 𝑎1
blue shift
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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Do static observers exist in an expanding universe?
The answer is affirmative. Scrutiny analysis of electromagnetic
and gravitational forces reveals that they are not subject to the
Hubble expansion in the linearized Hubble approximation.
Atomic and planetary orbits are stable and can be used to
materialize the reference frame of static observers. Hence, the
local cosmological expansion (in the theoretical model proposed)
can be detected, at least, in principle.
Pioneer anomaly effect is naturally explained by the property of
the conformal invariance of light geodesics as contrasted with
time-like geodesics which are not conformally-invariant.
18-19/04/2013
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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18-19/04/2013
7-th Gulf Coast Gravity Meeting
University of Missisipi, Oxford
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