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

Jens Boos and Friedrich W. Hehlboos@ualberta.ca hehl@thp.uni-koeln.deUniversity of Alberta University of Cologne & University of Missouri

Tuesday, August 29, 17:00

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

Geometric Foundations of Gravity

Geometric Foundations of Gauge Theory

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

1/10

Phenomenological Maxwell:

redundancyconserved external current j

The ingredients of gauge theory: the example of electrodynamics

1/10

Phenomenological Maxwell: Complex spinor feld:

redundancy invarianceconserved external current j conserved U(1) current

The ingredients of gauge theory: the example of electrodynamics

1/10

Phenomenological Maxwell: Complex spinor feld:

redundancy invarianceconserved external current j conserved U(1) current

Complete, gauge-theoretical description:

• local U(1) invariance

The ingredients of gauge theory: the example of electrodynamics

1/10

Phenomenological Maxwell: Complex spinor feld:

redundancy invarianceconserved external current j conserved U(1) current

Complete, gauge-theoretical description:

• local U(1) invariance

theory of force c

arriers

given external c

urrentmicroscopic description of matter; Noether currents

gauge theory = complete description of matter and

how it interacts via gauge bosons

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

Riemann curvature tensor

Cartan’s torsion tensor

2/10

electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory

Curvature tensors

rotational curvature

translational curvature

2/10

Curvature tensors

rotational curvature

translational curvature

A Riemann–Cartan geometry U4 is a four-dimensional manifold, 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.

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!

Young decomposition of a general rank-p tensor

3/10

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 ( ):

Young decomposition of the Riemann curvature tensor (1/2)

4/10

Weyl tensor “paircom” tensor pseudoscalarsymmetric tracefree Ricci tensor antisymmetric Ricci tensorRicci scalar

Young decomposition of the Riemann curvature tensor (2/2)

5/10

GL(4, R) SO(1,3)“Young decomposition” “Irreducible decomposition”

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 Weyl tensor. It is also related to superenergy: a positive defnite quantity for a timelike observer. How to generalize to Poincaré gauge theory?

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

An algebraic superenergy tensor in Poincaré gauge theory of gravity (1/3)

6/10

Explicit form of the decomposition of the Bel tensor:

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

7/10

The following is our fnal result:

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

8/10

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 .

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.

Some remarks on the Bel trace tensor

9/10

Conclusions

R T

X ZY Z' V W

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

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

Outlook: further decompositions in four dimensions

10/10