<<

Curvature in a 4D –Cartan space: Irreducible decompositions and superenergy

Jens Boos and Friedrich W. Hehl [email protected] [email protected] University of Alberta University of Cologne & University of Missouri

Tuesday, August 29, 17:00

Geometric Foundations of in Tartu Institute of , University of Tartu, Estonia

Geometric Foundations of Gravity

Geometric Foundations of

Geometric Foundations of Gauge Theory ↔ Gravity

The ingredients of gauge theory: the example of electrodynamics

1/10 The ingredients of gauge theory: the example of electrodynamics

Phenomenological Maxwell:

redundancy conserved external current j

1/10 The ingredients of gauge theory: the example of electrodynamics

Phenomenological Maxwell: Complex feld:

redundancy invariance conserved external current j conserved U(1) current

1/10 The ingredients of gauge theory: the example of electrodynamics

Phenomenological Maxwell: Complex spinor feld:

redundancy invariance conserved external current j conserved U(1) current

Complete, gauge-theoretical description:

• local U(1) invariance

1/10 The ingredients of gauge theory: the example of electrodynamics

Phenomenological Maxwell: Complex spinor feld: rs carrie mi force crosco ry of ent pic d theo l curr mat escrip terna ter; N tion en ex oeth of gredundancyiv invariance er cu rrents conserved external current j conserved U(1) current

Complete, gauge-theoretical description: tter and lete description of ma gauge theory = comp via gauge bosons how it interacts • local U(1) invariance

1/10 tensors

electrodynamics Yang–Mills theory Poincaré gauge theory

2/10 Curvature tensors

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

2/10 Curvature tensors

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

2/10 Curvature tensors

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

2/10 Curvature tensors

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

2/10 Curvature tensors

Riemann curvature

Cartan’s

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

2/10 Curvature tensors

rotational curvature

translational curvature

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

2/10 Curvature tensors

rotational curvature

translational curvature

A Riemann–Cartan geometry U4 is a four-dimensional , whose torsion tensor and curvature tensor satisfy

The frst Bianchi identity links dynamical properties of torsion to algebraic properties of curvature.

→ In the presence of non-vanishing torsion, the curvature tensor has diferent algebraic properties. Analyze this.

2/10 Young decomposition of a general rank-p tensor

Based on Schur–Weyl duality that links representations of Sn and GL(4, R), see literature.

Here, is the J-th allowed Young diagram, and is the k-th Young tableaux of the Young diagram . Lastly, denotes the Young symmetrizer associated with a certain Young tableaux.

This decomposition is block diagonal in the sense that .

→ Let us apply this to the Riemann tensor of a U4 geometry with curvature and torsion!

3/10 Young decomposition of the (1/2)

Symmetries of the Riemann tensor: # • double 2-form: (algebraic curvature tensor) 36 • Bianchi identity (if torsion vanishes) 16 • implications: (if torsion vanishes) 15 + 1

Young decomposition of the Riemann tensor ( ):

Weyl tensor “paircom” tensor pseudoscalar symmetric tracefree Ricci tensor antisymmetric Ricci tensor Ricci scalar 4/10 Young decomposition of the Riemann curvature tensor (2/2)

GL(4, R) SO(1,3)

“Young decomposition” “Irreducible decomposition” 5/10 An algebraic superenergy tensor in Poincaré gauge theory of gravity (1/3)

The Bel tensor can be defned in terms of the duals of the Riemann tensor:

The Young decomposition is

In General Relativity, the Bel–Robinson tensor is constructed analogously from the . It is also related to superenergy: a positive defnite quantity for a timelike observer. How to generalize to Poincaré gauge theory?

→ Introduce Bel tensor and subtract traces to defne an algebraic Bel–Robinson tensor.

6/10 An algebraic superenergy tensor in Poincaré gauge theory of gravity (2/3)

Explicit form of the decomposition of the Bel tensor:

7/10 An algebraic superenergy tensor in Poincaré gauge theory of gravity (3/3)

The following is our fnal result:

8/10 Some remarks on the Bel trace tensor

The Bel trace tensor lists how diferent curvature ingredients contribute to traces: • In General Relativity, implies . • In other theories (diferent Lagrangian, diferent geometry with torsion, ...), the vacuum feld equations may impose other constraints on the curvature. • Only the Weyl tensor does not appear in the Bel trace tensor. This is because it is traceless, , and it also satisfes .

Conclusions

The Bel trace tensor allows us to defne a tensor that has the same algebraic properties as the Bel–Robinson tensor. Further work needs to be done:

• Would a spinorial treatment give rise to a deeper algebraic understanding? • What about diferential properties of the algebraic Bel–Robinson tensor?

Thank you for your attention. 9/10 Outlook: further decompositions in four

SO(1,3) GL(4,R) R T GL(4,R) SO(1,3)

X Z Y Z' V W

V4 geometry with vanishing torsion. U4 geometry with non-vanishing torsion. 10/10