1

UNIVERSAL MODEL OF GR

An attempt to systematize the study of models in GR

2

SCOPE

1. Generic model

Determine the minimum number:

- of variables

- of equations relating these variables

which are sufficient to describe all models in GR.

2. Classify/quantify the constraints which are used to select a specific model in GR.

3. Determine general classes of GR models.

3

Modeling GR

The generic model of GR consists of the following parts:

- Background Riemannian space (gab)

– Kinematics (ua)

– Dynamics (Gab=kΤab)

4

VARIABLES• Each part of the generic model is specified by means of variables

and identities / conditions among these variables.• - Space time: gab,Γα

bc i.e. metric, connection.Identities: Bianchi - Symmetries of curvature tensor, Gab

;b=0• Kinematics: ua,ua;b → ωab,σab,θ, ua;bub (1+3)Ricci identity to ua. Integrating conditions of ua;b.

Propagation and Constraint equations (1+3)• Dynamics: Physical variables observed by ua for Tab (1+3)

– Matter density μ– Isotropic pressure p– Momentum transfer / heat conducting qa

– Anisotropic stress tensor πab

Bianchi Identity: Gab;b=0 → Tab

;b=0. Conservation equations via field equations. These are constraints. No field equations. (1+3)

What it means 1+3?

5

Conclusion: The generic/universal model

• Variables– gab

– ua, ua;b → ωab, σab, θ, ua;b ub

– μ, p, qa,πab

• Identities / constrains– Bianchi (symmetries of curvature tensor)– Ricci (propagation and constraint eqns)

– Bianchi (conservation equation, Gab;b=0 )

6

GENERIC MODEL

GEOMETRYgab

Bianchi identities

DYNAMICSμ, p, qa,πab

Conservation Law

• KINEMATICS• ua, • ωab, σab, θ, ua;b ub

• propagation eqns • constraint eqns

7

Specifying a model

• Generic model has free parameters. More constraints required.

• Types of additional assumptions– Specify four-velocity – observers (1+3)– Symmetries

• Geometric symmetries – Collineations• Dynamical symmetries

– Catastatic equations (matter constraints) – Other; Kinematical or dynamical or geometric

8

1+3 decomposition

• Every non-null four-vector defines a 1+3 decomposition.

• Projection operator:

• Decomposition of a vector:

• Decomposition of a 2-tensor

9

Examples of 1+3 decomposition

• Kinematical variables:

• Physical variables

10

Types of matterConservation equations

11

Propagation and Constrain eqnsKinematics – Ricci identity on ua

12

Propagation and Constrain eqnsThe physics

13

Symmetries - Collineations

General symmetry structure:– Lie Derivative [Geometric object] = Tensor

Collineations:

They are symmetries in which the geometric object is defined in terms of the metric e.g. Γa

bc,Rab,Rabcd,etc.

Form of a Collineation:

Lie Derivative [gab, Γabc,Rab,Ra

bcd,etc] = Tensor

14

Examples of collineations

15

Collineation tree

16

Generic collineation

• All relations of the form:

LX[gab, Γabc,Rab,Ra

bcd,etc]

can be expressed in terms of LXgab. E.g.

This leads us to consider LXgab as the generic symmetry. Introduce symmetry parameters ψ,Hab via the identity:

LXgab=2ψgab+2Hab, Haa=0

17

Expressing collineations in terms of the parameters ψ,Ha

18

Symmetries and the field equations

• Write field equations as: Rab=k[Tab-(1/2)Tgab]

• Then: LXRab=kLX [Tab-(1/2)Tgab]• The LXRab → f(ψ,Hab, and derivatives)• The rhs as LX(μ, p, qa,πab)Equating the two results we find the field

equations in the universal form, that is:

LXμ= fμ (ψ,Hab, derivatives)

LXp= fp (ψ,Hab, derivatives) etc.

19

The 1+3 decomposition of symmetries of field equations

• Example: Collineation ξa=ξua

• To compute the rhs use field eqns and 1+3 decomposition (Note: ). Result:

20

To compute the lhs (use the collineation identity and symmetry parameters):

1+3 decomposition:

21

Field equations in 1+3 decomposition (to be supplemented by conservation eqns)

• uaub term:

• uahcb term:

• had

hcb term:

22

The string fluid

• Definition:

23

Physical variables and conservation equations

24

• Example of string fluid: The EM field in RMHD approximation with infinite conductivity and vanishing electric field

25

Additional assumptions

• ξa=ξua is a symmetry vector

• Symmetry/kinematics:

• Further assumptions required or stop!– Specify ξa to be RIC defined by requirement:

26

Kinematical implications of Collineation

27

Dynamical implications of the Collineation i.e. field equations. An equation of state still

needed!

28

Second approach – Example Tsamparlis GRG 38 (2006), 311

and cyclically for the Tyy, Tzz.

29

Specification of ua 1+3 Kinematics

30

Physical variables I

31

Physical variables II

32

Select model: String fluid

33

Specification of physical variables

34

The final field equations

35

ΤHE RW MODEL (K≠0)

• Symmetry assumptions– Spacetime admits a gradient CKV ξα which is

hypersurface orthogonal. Define cosmic time t with ξα =δa

0.

– The 3-spaces are spaces of constant curvature K≠0

These imply the metric: ds2=-dt2+S(t)dσK2

KINEMATICS:

Choice of observers: u_a=S-1/2 (1,0,0,0)

Kinematic variables:

36

Geometry implies Physics

• Einstein equations imply for the Tab.

37

In order to determine the unknown function S(t) we need one more equation. This is an equation of state. Without it the problem is not deterministic.Note: The RW model is based only on geometrical assumptions and the choice of observers! The equation of state can be both a math or a physical assumption.

Physical variables: 1+3 of Tab

38

The model of a rotating star in equilibrium

• Symmetry assumptions– Two commuting KVs ∂t (timelike), ∂φ

(spacelike) which are surface forming etc

Metric: Stationary axisymmetric metric

39

KINEMATICS

40

DYNAMICS

41

All physical variables are ≠0. Therefore the fluid for these observers is a general heat conducting anisotropic fluid. We have three unknown functions (the α,β,ν) hence we need three independent dependent constraints / equations.In the literature (See ``Rotating stars in Relativity’’ N. Stergioulas www.livingreviews.org for review and references) they assume:

Perfect Fluid ↔ qa=0, πab=0

42

Inconsistency of assumption

43

44

Assume further πab=0