Curvature tensors in a 4D Riemann–Cartan space ...
Transcript of Curvature tensors in a 4D Riemann–Cartan space ...
![Page 1: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/1.jpg)
Curvature tensors in a 4D Riemann–Cartan space:Irreducible decompositions and superenergy
Jens Boos and Friedrich W. [email protected] [email protected] 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
![Page 2: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/2.jpg)
Geometric Foundations of Gravity
![Page 3: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/3.jpg)
Geometric Foundations of Gauge Theory
![Page 4: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/4.jpg)
Geometric Foundations of Gauge Theory Gravity↔
![Page 5: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/5.jpg)
The ingredients of gauge theory: the example of electrodynamics
1/10
![Page 6: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/6.jpg)
The ingredients of gauge theory: the example of electrodynamics
1/10
Phenomenological Maxwell:
redundancyconserved external current j
![Page 7: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/7.jpg)
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
![Page 8: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/8.jpg)
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
![Page 9: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/9.jpg)
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
![Page 10: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/10.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
![Page 11: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/11.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
![Page 12: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/12.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
![Page 13: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/13.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
![Page 14: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/14.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
![Page 15: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/15.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
Riemann curvature tensor
Cartan’s torsion tensor
![Page 16: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/16.jpg)
2/10
electrodynamics Yang–Mills theory General Relativity Poincaré gauge theory
Curvature tensors
rotational curvature
translational curvature
![Page 17: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/17.jpg)
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.
![Page 18: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/18.jpg)
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
![Page 19: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/19.jpg)
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
![Page 20: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/20.jpg)
Young decomposition of the Riemann curvature tensor (2/2)
5/10
GL(4, R) SO(1,3)“Young decomposition” “Irreducible decomposition”
![Page 21: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/21.jpg)
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
![Page 22: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/22.jpg)
Explicit form of the decomposition of the Bel tensor:
An algebraic superenergy tensor in Poincaré gauge theory of gravity (2/3)
7/10
![Page 23: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/23.jpg)
The following is our fnal result:
An algebraic superenergy tensor in Poincaré gauge theory of gravity (3/3)
8/10
![Page 24: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/24.jpg)
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
![Page 25: Curvature tensors in a 4D Riemann–Cartan space ...](https://reader034.fdocuments.us/reader034/viewer/2022042320/625d10c277e34d43dd244160/html5/thumbnails/25.jpg)
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