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UNIVERSAL MODEL OF GR
An attempt to systematize the study of models in GR
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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.
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Modeling GR
The generic model of GR consists of the following parts:
- Background Riemannian space (gab)
– Kinematics (ua)
– Dynamics (Gab=kΤab)
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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?
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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 )
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GENERIC MODEL
GEOMETRYgab
Bianchi identities
DYNAMICSμ, p, qa,πab
Conservation Law
• KINEMATICS• ua, • ωab, σab, θ, ua;b ub
• propagation eqns • constraint eqns
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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
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1+3 decomposition
• Every non-null four-vector defines a 1+3 decomposition.
• Projection operator:
• Decomposition of a vector:
• Decomposition of a 2-tensor
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Examples of 1+3 decomposition
• Kinematical variables:
• Physical variables
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Types of matterConservation equations
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Propagation and Constrain eqnsKinematics – Ricci identity on ua
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Propagation and Constrain eqnsThe physics
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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
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Examples of collineations
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Collineation tree
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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
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Expressing collineations in terms of the parameters ψ,Ha
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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.
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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:
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To compute the lhs (use the collineation identity and symmetry parameters):
1+3 decomposition:
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Field equations in 1+3 decomposition (to be supplemented by conservation eqns)
• uaub term:
• uahcb term:
• had
hcb term:
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The string fluid
• Definition:
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Physical variables and conservation equations
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• Example of string fluid: The EM field in RMHD approximation with infinite conductivity and vanishing electric field
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Additional assumptions
• ξa=ξua is a symmetry vector
• Symmetry/kinematics:
• Further assumptions required or stop!– Specify ξa to be RIC defined by requirement:
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Kinematical implications of Collineation
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Dynamical implications of the Collineation i.e. field equations. An equation of state still
needed!
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Second approach – Example Tsamparlis GRG 38 (2006), 311
and cyclically for the Tyy, Tzz.
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Specification of ua 1+3 Kinematics
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Physical variables I
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Physical variables II
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Select model: String fluid
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Specification of physical variables
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The final field equations
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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:
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Geometry implies Physics
• Einstein equations imply for the Tab.
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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
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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
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KINEMATICS
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DYNAMICS
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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
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Inconsistency of assumption
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Assume further πab=0