2. 1 UNIVERSAL MODEL OF GR An attempt to systematize the study of models in GR

3. 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.

4. 3 Modeling GR The generic model of GR consists of the following parts: - Background Riemannian space (g ab ) –Kinematics (u a ) –Dynamics (G ab =kΤ ab )

5. 4 VARIABLES Each part of the generic model is specified by means of variables and identities / conditions among these variables. - Space time: g ab,Γ α bc i.e. metric, connection. Identities: Bianchi - Symmetries of curvature tensor, G ab ;b =0 Kinematics: u a,u a;b → ω ab,σ ab,θ, u a;b u b (1+3) Ricci identity to u a. Integrating conditions of u a;b. Propagation and Constraint equations (1+3) Dynamics: Physical variables observed by u a for T ab (1+3) – Matter density μ – Isotropic pressure p – Momentum transfer / heat conducting q a – Anisotropic stress tensor π ab Bianchi Identity: G ab ;b =0 → T ab ;b =0. Conservation equations via field equations. These are constraints. No field equations. (1+3) What it means 1+3?

6. 5 Conclusion: The generic/universal model Variables –g ab –u a, u a;b → ω ab, σ ab, θ, u a;b u b –μ, p, q a,π ab Identities / constrains –Bianchi (symmetries of curvature tensor) –Ricci (propagation and constraint eqns) –Bianchi (conservation equation, G ab ;b =0 )

7. 6 GENERIC MODEL GEOMETRY g ab Bianchi identities DYNAMICS μ, p, q a,π ab Conservation Law KINEMATICS u a, ω ab, σ ab, θ, u a;b u b propagation eqns constraint eqns

8. 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

9. 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

10. 9 Examples of 1+3 decomposition Kinematical variables: Physical variables

11. 10 Types of matter Conservation equations

12. 11 Propagation and Constrain eqns Kinematics – Ricci identity on u a

13. 12 Propagation and Constrain eqns The physics

14. 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. Γ a bc,R ab,R a bcd,etc. Form of a Collineation: Lie Derivative [g ab, Γ a bc,R ab,R a bcd,etc] = Tensor

15. 14 Examples of collineations

16. 15 Collineation tree

17. 16 Generic collineation All relations of the form: L X [g ab, Γ a bc,R ab,R a bcd,etc] can be expressed in terms of L X g ab. E.g. This leads us to consider L X g ab as the generic symmetry. Introduce symmetry parameters ψ,H ab via the identity: L X g ab =2ψg ab +2H ab, H a a =0

18. 17 Expressing collineations in terms of the parameters ψ,H a

19. 18 Symmetries and the field equations Write field equations as : R ab =k[T ab -(1/2)Tg ab ] Then: L X R ab =kL X [T ab -(1/2)Tg ab ] The L X R ab → f(ψ,H ab, and derivatives) The rhs as L X (μ, p, q a,π ab ) Equating the two results we find the field equations in the universal form, that is: L X μ= f μ (ψ,H ab, derivatives) L X p= f p (ψ,H ab, derivatives) etc.

20. 19 The 1+3 decomposition of symmetries of field equations Example: Collineation ξ a =ξu a To compute the rhs use field eqns and 1+3 decomposition (Note: ). Result:

21. 20 To compute the lhs (use the collineation identity and symmetry parameters): 1+3 decomposition:

22. 21 Field equations in 1+3 decomposition (to be supplemented by conservation eqns) u a u b term : u a h c b term : h a d h c b term :

23. 22 The string fluid Definition:

24. 23 Physical variables and conservation equations

25. 24 Example of string fluid: The EM field in RMHD approximation with infinite conductivity and vanishing electric field

26. 25 Additional assumptions ξ a =ξu a is a symmetry vector Symmetry/kinematics: Further assumptions required or stop! –Specify ξ a to be RIC defined by requirement :

27. 26 Kinematical implications of Collineation

28. 27 Dynamical implications of the Collineation i.e. field equations. An equation of state still needed!

29. 28 Second approach – Example Tsamparlis GRG 38 (2006), 311 and cyclically for the T yy, T zz.

30. 29 Specification of u a 1+3 Kinematics

31. 30 Physical variables I

32. 31 Physical variables II

33. 32 Select model: String fluid

34. 33 Specification of physical variables

35. 34 The final field equations

36. 35 ΤHE RW MODEL (K≠0) Symmetry assumptions –Spacetime admits a gradient CKV ξ α which is hypersurface orthogonal. Define cosmic time t with ξ α =δ a 0. –The 3-spaces are spaces of constant curvature K≠0 These imply the metric: ds 2 =-dt 2 +S(t)dσ K 2 KINEMATICS: Choice of observers: u_a=S -1/2 (1,0,0,0) Kinematic variables:

37. 36 Geometry implies Physics Einstein equations imply for the T ab.

38. 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

39. 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

40. 39 KINEMATICS

41. 40 DYNAMICS

42. 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 ↔ q a =0, π ab =0

43. 42 Inconsistency of assumption

44. 43

45. 44 Assume further π ab =0