New Vacuum Solutions for Quadratic Metric affine Gravity · Deﬁnition A generalised pp-wave is a...

New Vacuum Solutions for Quadratic Metric–affineGravity

Vedad Pasic

University of Bath, University of Tuzla

University College London28 August 2008

Vedad Pasic New Vacuum Solutions for Quadratic Metric–affine Gravity

Structure

Mathematical modelPP-waves with torsionNew vacuum solutions for quadratic metric-affine gravityInterpretation

Structure

Mathematical model

PP-waves with torsionNew vacuum solutions for quadratic metric-affine gravityInterpretation

Structure

Mathematical modelPP-waves with torsion

New vacuum solutions for quadratic metric-affine gravityInterpretation

Structure

Mathematical modelPP-waves with torsionNew vacuum solutions for quadratic metric-affine gravity

Interpretation

Structure

Mathematical modelPP-waves with torsionNew vacuum solutions for quadratic metric-affine gravityInterpretation

Metric–affine gravity

Spacetime considered to be a connected real 4–manifold Mequipped with a Lorentzian metric g and an affine connection Γ,i.e.

∇µuλ = ∂µuλ + Γλµνuν .

Characterisation by an independent linear connection Γdistinguishes MAG from GR - g and Γ viewed as two totallyindependent quantities.10 independent components of gµν and the 64 connectioncoefficients Γλ

µν are the unknowns of MAGDefinition. We call a spacetime {M,g, Γ} Riemannian if theconnection is Levi–Civita (i.e. Γλ

µν ={

λµν

}), and non-Riemannian

otherwise.

µν ={

λµν

otherwise.

Characterisation by an independent linear connection Γdistinguishes MAG from GR - g and Γ viewed as two totallyindependent quantities.

10 independent components of gµν and the 64 connectioncoefficients Γλ

µν ={

λµν

otherwise.

µν are the unknowns of MAG

Definition. We call a spacetime {M,g, Γ} Riemannian if theconnection is Levi–Civita (i.e. Γλ

µν ={

λµν

otherwise.

µν ={

λµν

otherwise.

Quadratic metric–affine gravity

Action isS :=

∫q(R),

where q(R) is a Lorentz invariant purely quadratic form oncurvature.The quadratic form q(R) has 16 R2 terms with 16 real couplingconstants.Action conformally invariant, unlike Einstein–Hilbert.

Action isS :=

∫q(R),

where q(R) is a Lorentz invariant purely quadratic form oncurvature.

The quadratic form q(R) has 16 R2 terms with 16 real couplingconstants.Action conformally invariant, unlike Einstein–Hilbert.

Action isS :=

∫q(R),

where q(R) is a Lorentz invariant purely quadratic form oncurvature.The quadratic form q(R) has 16 R2 terms with 16 real couplingconstants.

Action conformally invariant, unlike Einstein–Hilbert.

Action isS :=

∫q(R),

where q(R) is a Lorentz invariant purely quadratic form oncurvature.The quadratic form q(R) has 16 R2 terms with 16 real couplingconstants.Action conformally invariant, unlike Einstein–Hilbert.

Independent variation of g and Γ produces the system ofEuler–Lagrange equations

∂S/∂g = 0, (1)∂S/∂Γ = 0. (2)

Independent variation of g and Γ produces the system ofEuler–Lagrange equations

∂S/∂g = 0, (1)∂S/∂Γ = 0. (2)

Known solutions for QMAG

Einstein spaces (Yang, Mielke);pp-waves with parallel Ricci curvature (Vassiliev);Certain explicitly given torsion waves (Singh and Griffiths);Triplet ansatz (Hehl, Macıas, Obukhov, Esser, ...);Minimal pseudoinstanton generalisation (Obukhov).

Einstein spaces (Yang, Mielke);

pp-waves with parallel Ricci curvature (Vassiliev);Certain explicitly given torsion waves (Singh and Griffiths);Triplet ansatz (Hehl, Macıas, Obukhov, Esser, ...);Minimal pseudoinstanton generalisation (Obukhov).

Einstein spaces (Yang, Mielke);pp-waves with parallel Ricci curvature (Vassiliev);

Certain explicitly given torsion waves (Singh and Griffiths);Triplet ansatz (Hehl, Macıas, Obukhov, Esser, ...);Minimal pseudoinstanton generalisation (Obukhov).

Einstein spaces (Yang, Mielke);pp-waves with parallel Ricci curvature (Vassiliev);Certain explicitly given torsion waves (Singh and Griffiths);

Triplet ansatz (Hehl, Macıas, Obukhov, Esser, ...);Minimal pseudoinstanton generalisation (Obukhov).

Einstein spaces (Yang, Mielke);pp-waves with parallel Ricci curvature (Vassiliev);Certain explicitly given torsion waves (Singh and Griffiths);Triplet ansatz (Hehl, Macıas, Obukhov, Esser, ...);

Minimal pseudoinstanton generalisation (Obukhov).

Einstein spaces (Yang, Mielke);pp-waves with parallel Ricci curvature (Vassiliev);Certain explicitly given torsion waves (Singh and Griffiths);Triplet ansatz (Hehl, Macıas, Obukhov, Esser, ...);Minimal pseudoinstanton generalisation (Obukhov).

Classical pp-waves

Definition. A pp-wave is a Riemannian spacetime which admitsa non-vanishing parellel spinor field (∇χ = 0).Definition. A pp-wave is a Riemannian spacetime whose metriccan be written locally in the form

ds2 = 2 dx0 dx3 − (dx1)2 − (dx2)2 + f (x1, x2, x3) (dx3)2

in some local coordinates.Well known spacetimes in GR, simple formula for curvature - onlytrace free Ricci and Weyl pieces.

Classical pp-waves

Definition. A pp-wave is a Riemannian spacetime which admitsa non-vanishing parellel spinor field (∇χ = 0).

Definition. A pp-wave is a Riemannian spacetime whose metriccan be written locally in the form

Classical pp-waves

in some local coordinates.

Well known spacetimes in GR, simple formula for curvature - onlytrace free Ricci and Weyl pieces.

Classical pp-waves

Generalised pp-waves

Consider the polarized Maxwell equation

∗dA = ±idA.

Plane wave solutions of this equation can be written down as

A = h(ϕ) m + k(ϕ) l ,

ϕ : M → R, ϕ(x) :=

∫l · dx .

Definition A generalised pp-wave is a metric compatiblespacetime with pp-metric and torsion

T :=12

Re(A⊗ dA).

∗dA = ±idA.

A = h(ϕ) m + k(ϕ) l ,

ϕ : M → R, ϕ(x) :=

∫l · dx .

T :=12

Re(A⊗ dA).

∗dA = ±idA.

A = h(ϕ) m + k(ϕ) l ,

ϕ : M → R, ϕ(x) :=

∫l · dx .

T :=12

Re(A⊗ dA).

∗dA = ±idA.

A = h(ϕ) m + k(ϕ) l ,

ϕ : M → R, ϕ(x) :=

∫l · dx .

T :=12

Re(A⊗ dA).

Curvature of a generalised pp-wave is

R = −12

(l ∧ {∇})⊗ (l ∧ {∇})f +14

(h2)′′ (l ∧m)⊗ (l ∧m)).

Torsion of a generalised pp-wave is

T = Re ((a l + b m)⊗ (l ∧m)) ,

wherea :=

h′(ϕ) k(ϕ), b :=12

h′(ϕ) h(ϕ).

R = −12

(l ∧ {∇})⊗ (l ∧ {∇})f +14

(h2)′′ (l ∧m)⊗ (l ∧m)).

T = Re ((a l + b m)⊗ (l ∧m)) ,

wherea :=

h′(ϕ) k(ϕ), b :=12

h′(ϕ) h(ϕ).

R = −12

(l ∧ {∇})⊗ (l ∧ {∇})f +14

(h2)′′ (l ∧m)⊗ (l ∧m)).

T = Re ((a l + b m)⊗ (l ∧m)) ,

wherea :=

h′(ϕ) k(ϕ), b :=12

h′(ϕ) h(ϕ).

Main result of the thesis

Theorem Generalised pp-waves of parallel Ricci curvature aresolutions of the field equations (1) and (2).In special local coordinates, ‘parallel Ricci curvature’ is written asf11 + f22 = const.Generalised pp-waves of parallel Ricci curvature admit a simpleexplicit description.

Theorem Generalised pp-waves of parallel Ricci curvature aresolutions of the field equations (1) and (2).

In special local coordinates, ‘parallel Ricci curvature’ is written asf11 + f22 = const.Generalised pp-waves of parallel Ricci curvature admit a simpleexplicit description.

Theorem Generalised pp-waves of parallel Ricci curvature aresolutions of the field equations (1) and (2).In special local coordinates, ‘parallel Ricci curvature’ is written asf11 + f22 = const.

Generalised pp-waves of parallel Ricci curvature admit a simpleexplicit description.

Theorem Generalised pp-waves of parallel Ricci curvature aresolutions of the field equations (1) and (2).In special local coordinates, ‘parallel Ricci curvature’ is written asf11 + f22 = const.Generalised pp-waves of parallel Ricci curvature admit a simpleexplicit description.

Outline of the proof

Proof by ‘brute force’.We write down the field equations (1) and (2) for general metriccompatible spacetimes and substitute the formulae for torsion,Ricci and Weyl into these.Together with ∇Ric = 0, we get the result.This result was first presented in :“PP-waves with torsion and metric affine gravity”, 2005 V. Pasic, D.Vassiliev, Class. Quantum Grav. 22 3961-3975.

Proof by ‘brute force’.

We write down the field equations (1) and (2) for general metriccompatible spacetimes and substitute the formulae for torsion,Ricci and Weyl into these.Together with ∇Ric = 0, we get the result.This result was first presented in :“PP-waves with torsion and metric affine gravity”, 2005 V. Pasic, D.Vassiliev, Class. Quantum Grav. 22 3961-3975.

Proof by ‘brute force’.We write down the field equations (1) and (2) for general metriccompatible spacetimes and substitute the formulae for torsion,Ricci and Weyl into these.

Together with ∇Ric = 0, we get the result.This result was first presented in :“PP-waves with torsion and metric affine gravity”, 2005 V. Pasic, D.Vassiliev, Class. Quantum Grav. 22 3961-3975.

Proof by ‘brute force’.We write down the field equations (1) and (2) for general metriccompatible spacetimes and substitute the formulae for torsion,Ricci and Weyl into these.Together with ∇Ric = 0, we get the result.

This result was first presented in :“PP-waves with torsion and metric affine gravity”, 2005 V. Pasic, D.Vassiliev, Class. Quantum Grav. 22 3961-3975.

Proof by ‘brute force’.We write down the field equations (1) and (2) for general metriccompatible spacetimes and substitute the formulae for torsion,Ricci and Weyl into these.Together with ∇Ric = 0, we get the result.This result was first presented in :“PP-waves with torsion and metric affine gravity”, 2005 V. Pasic, D.Vassiliev, Class. Quantum Grav. 22 3961-3975.

Interpretation

Curvature of generalised pp-waves is split.Torsion and torsion generated curvature are waves traveling at thespeed of light.Underlying pp-space can be viewed as the ‘gravitational imprint’created by wave of some massless field.Mathematical model for some massless particle?

Interpretation

Curvature of generalised pp-waves is split.

Torsion and torsion generated curvature are waves traveling at thespeed of light.Underlying pp-space can be viewed as the ‘gravitational imprint’created by wave of some massless field.Mathematical model for some massless particle?

Interpretation

Curvature of generalised pp-waves is split.Torsion and torsion generated curvature are waves traveling at thespeed of light.

Underlying pp-space can be viewed as the ‘gravitational imprint’created by wave of some massless field.Mathematical model for some massless particle?

Interpretation

Curvature of generalised pp-waves is split.Torsion and torsion generated curvature are waves traveling at thespeed of light.Underlying pp-space can be viewed as the ‘gravitational imprint’created by wave of some massless field.

Mathematical model for some massless particle?

Interpretation

Curvature of generalised pp-waves is split.Torsion and torsion generated curvature are waves traveling at thespeed of light.Underlying pp-space can be viewed as the ‘gravitational imprint’created by wave of some massless field.Mathematical model for some massless particle?

Metric-affine vs Einstein-Weyl

Look at Weyl action

SW := 2i∫ (

ξa σµab (∇µξb) − (∇µξ

a)σµab ξb),

In a generalised pp-space Weyl’s equation takes form

σµab{∇}µ ξa = 0.

There exist pp-wave type solutions of Einstein-Weyl model

SEW := k∫R+ Sneutrino,

∂SEW/∂g = 0,

∂SEW/∂ξ = 0.

Metric-affine vs Einstein-WeylLook at Weyl action

SW := 2i∫ (

a)σµab ξb),

∂SEW/∂g = 0,

∂SEW/∂ξ = 0.

SW := 2i∫ (

a)σµab ξb),

∂SEW/∂g = 0,

∂SEW/∂ξ = 0.

SW := 2i∫ (

a)σµab ξb),

∂SEW/∂g = 0,

∂SEW/∂ξ = 0.

New Vacuum Solutions for Quadratic Metric affine Gravity · Deﬁnition A generalised pp-wave is a...

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