Curvature tensors in a 4D Riemann–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 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
Phenomenological Maxwell:
redundancy conserved external current j
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
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 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
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
2/10 Curvature tensors
Riemann curvature tensor
Cartan’s torsion tensor
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 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.
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 Riemann curvature tensor (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 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.
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 dimensions
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