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)