Anisotropic geometrodynamics: observations and cosmological consequences
Sergey SiparovState University of Civil Aviation, St-PetersburgRussian Federation
“Gamov-2009”, Odessa
Motivation (observational)
Flat rotation curves of spiral galaxies – modern challenge: simple, not small, statistically verified – contradicts the theory!
Attempts to modify the gravitation theory in order to explain flat RC
Einstein-Hilbert action
1. f(R)-theories (De Witt) – where to stop? 2. Additional scalar fields (Brans-Dicke) – still not found 3. Weyl tensor (Mannheim) – no GW 4. Scalar-vector-tensor theory (Moffat) – 5-th force
(repulsive) 5. Phenomenological MOND theory (Milgrem) – arbitrary
choice of functions to fit observations Dark matter notion – inconsistent
Unsatisfactory
RgxdG
cSEH
2/143
)(16
Astrophysical (observational) restrictions for any gravitation theory modifications
1. Flat rotation curves
2. Tully-Fisher law for luminosity:
3. Globular clusters behavior (I): no need for any correction to the gravitation law outside the spiral galaxy plane (anisotropy ?)
4. Globular clusters behavior (II): contrary to the Keplerian
expectations, they are found rather in the vicinity of the galaxy center than at the periphery
5. Lensing effect appears to be 4-6 times larger than predicted
None is explained by the classical GRT
4~ orblum vL
Suggestion: try anisotropic metricReasons:
Geometry: the account for anisotropy is the natural generalization leading to the natural change in the “simplest scalar” in EH action
Physics: 1) velocity dependent gravitation is consistent with the equivalence principle: it is impossible to distinguish the inertial forces (e.g. Corolis!) from gravitational forces;
2) gravitational force must enter the metric
Introduce
where γ_ij - Minkowski metric ε_ij(x,y) - small anisotropic perturbation - directional variable (tangent to trajectory of a probe) u(x) – vector field generating the anisotropy – characterizes the velocities of the distributed gravitation sources
),(),()),(,(~ yxyxgyxuxg ijijijij
ds
dxy
ii
Generalized geodesics and assumptions
Generalized geodesics
Assumptions Use two Einstein’s assumptions:1. The components y2 , y3, y4 can be neglected in comparison
with y1 which is equal to unity within the accuracy of the second order;
2. The motion is slow, therefore, the time x1-derivative in the equations for geodesics can be neglected in comparison to the space x2-, x3-, and x4-derivatives;
Add similar one: 3. On the y-subspace of the phase space (x,y) the y1-
derivative can be neglected in comparison to the y2-, y3-, and y4-derivatives.
0)2
1(
2
lkjtj
klitlki
i
yyyyxds
dy
Generalized geodesics
Generalized geodesics
Use assumptions k=l=1 yk=yl=1
Introduce new tensor:
to obtain
0)2
1(
2
lkjtj
klitlki
i
yyyyxds
dy
11 kl ttA
y
11
2
1
011
jjtiti
i
yx
A
ds
dy
t
j
jt
jt x
A
x
AF
011
jt
jitjtj
itii
yx
AyF
ds
dy
Geometrical “Maxwell equations”
Anti-symmetric rank-2 tensor suffices:
Use designation:
to get
Use designation:
to get
Interpretation: charge q - electromagnetism charge m_g - gravitation
0
x
F
x
F
x
F
)(34
)(12
)(24
)(31
)(14
)(23 ;;;;; g
xg
xg
yg
yg
zg
z BFEFBFEFEFBF
0
0
)(
)()(
g
gg
Bdiv
Erott
B
ArotB
AE
xg
xg
)(
1)()(
x
FI
1432 ;;; IjIjIjI zyx
)()(
)()(
)(
mg
mg
g
Ediv
jt
EBrot
uj mm )()(
Force of gravitation
Equation of motion:
Newton force
Velocity dependent force (analogue: Coriolis (or Lorentz) force)
Third force
),(],[2
11)(
11)(11)(
2)( v
vvrotv
mcF
dt
vdm xxx
g
11)(
2)(
2x
gN
mcF
],[2],[2
11)(
2)(
vm
dvrotv
mcF x
gC
v
rotc
x
11)(
2
4
),(2
11)(
2)3( v
v
mcF x
g
sHsmcHcvcHvaC /1][;/][;/;/];,[2],[2
Predecessors – GRT corrections for a rotating body in an isotropic space-time
Gravitomagnetism: correction to the spherically symmetric mass gravity due to its rotation.
Lense-Thirring: orbit precession. Later: clock effect; Sagnac effect; gravitomagnetic Stern-Gerlach effect;
Gravity Probe B – confirmed theory within less than 10% accuracy
Frame-dragging Einstein – geodesics with 3 terms (includes rotational and
linear frame-dragging, and inertial mass increase when other masses are nearby)
AGD difference: it is the 1-st order theory in an anisotropic space-time
AGD applications – simplified model
Attraction center plus circular contour with current Pay attention to the known one-to-one correlation with Maxwell equations
Effective parameters (R_eff, J, V_eff) can be taken from observations
M
IRMRII effeffeffneff22
eff
effeff I
L neffeffeff LLI
AGD applications - I
Rotation curves (initial goal)
Model gives: z = 0; b = r/R_eff = O(1) B_z(r) J/r def: J = C_2
q = m_g = m
- Newton law
- flat curve -
])1(
1[)1(
2)(
2
2
Eb
bK
bRJrB
effz
212 Cqvr
qCmv orborb
)4
11(2 2
2
12
rC
CCvorb
)4
11(2 2
2
12
rC
CCvorb 2~ Cvorb
AGD applications - II
Tully-Fisher law
Model:
Luminosity:
eff
eff
effeffeff R
R
RKeplerLaw
T
MRJRC
2/3
2
2 ~~~)()(
2~ efflum RL
2~ Cvorb
4/1~ lumorb Lv
AGD applications - III
Applicability region and regimes
Illustrative qualitative limit case (giant Black Hole in the center of a galaxy)
For M = 10^11M_Sol a_C/a_N = 1 at r ~ 10^18 m v ~ 10^5 m/s Consistent with observations no reason to expect Newton law
effeff
eff
N
C
c
vr
R
r
c
vV
a
a
22~
eff
eff
N
C
I
L
c
vr
a
a2
~
GM
cvr
R
r
c
vV
a
a
eff
eff
N
C
2~
2
2
2
c
GMrR Seff cVeff
AGD applications - IV
Numerical modeling 1) Quasi-precession, non-Keplerian behavior of globular
clusters, and lensing problem
2) Spiral arms
AGD qualitative results and general cosmological consequences
GRT results remained valid in its applicability region Flat RC explained Astrophysical restrictions sufficed AGD applicability regions determined, limit case checked Qualitative pictures obtained Specific prediction: change in the OMPR effect conditions
No dark matter for galaxies needed is it the same for galaxy clusters?
Gravitation ceased to be only attraction can there be no dark energy of repulsion?
Hubble red shift is it Universe expansion or gravitational red shift as it could be according to the observed amazingly fast tangent motion of quasars at the periphery of the visible Universe?
Thank you!
S.Siparov, arXiv [gr-qc]: 0809.1817 (2008)
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