Toward a non-metric theory of gravity playing around with the (affine) connection!
Oscar Castillo-Felisola o.castillo.felisola(at)gmail.com
UTFSM, Valparaiso
4 de febrero de 2015
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 1 / 8 Facts to consider:
General Relativity (GR) has proven to be the most successful theory of gravity. It is not renormalizable, but there are problems with the choice of variables to be quantized and the choice of the Hilbert space to be used. Sum over all possible field configurations of the metric seems to be wrong (Euclidean + Minkowski).
How do people bypass (some) problems Developing new alternatives theories: LQG,STRINGS,BRANS-DICKE,TELEPARALLEL GRAVITY,MODIFIED NEWTONIANDYNAMICS (MOND),CAUSALSETS,REGGE CALCULUS, DYNAMICAL TRIANGULATIONS,PENROSE TWISTOR THEORY,PENROSE SPINNETWORKS,CONNES NON–COMMUTATIVE GEOMETRY,CAUSAL FERMION SYSTEMS...
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 2 / 8 Facts to consider:
General Relativity (GR) has proven to be the most successful theory of gravity. It is not renormalizable, but there are problems with the choice of variables to be quantized and the choice of the Hilbert space to be used. Sum over all possible field configurations of the metric seems to be wrong (Euclidean + Minkowski).
How do people bypass (some) problems Developing new alternatives theories: LQG,STRINGS,BRANS-DICKE,TELEPARALLEL GRAVITY,MODIFIED NEWTONIANDYNAMICS (MOND),CAUSALSETS,REGGE CALCULUS, DYNAMICAL TRIANGULATIONS,PENROSE TWISTOR THEORY,PENROSE SPINNETWORKS,CONNES NON–COMMUTATIVE GEOMETRY,CAUSAL FERMION SYSTEMS...
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 2 / 8 Defining our playground
µ Consider an affine connection Γˆρ σ Curvature and Ricci tensor depend on Γˆ. No R can be constructed (needs a metric). Parallel transport is well-possessed. Decomposition:
ˆµ µ µ µ µ,λκ µ Γ ρσ = Γ (ρσ) + T [ρσ] = Γ ρσ + ρσλκT + A[ρδν]
We consider:
I a power-counting renormalizable, and I diffeomorphism invariant model.
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 3 / 8 Three-dimensional model
Z 3 µ νρ µνρ σ S[Γ,T,A] = d x A1Rµν ρT + A2 Rµν σAρ µνρ µν µν + A3 Aµ∂νAρ + A4T ∇µAν + A5T AµAν µν µνλ σ ρ + A6 det(T ) + A7 Γ µρ∂νΓ λσ 2 + Γτ Γρ Γσ + A µνρΓσ ∂ Γτ . 3 µρ νσ λτ 8 µσ ν ρτ
Z Eddington’s trick 3 √ µ S[g, Γ,A] = d x g A1R + A4∇ Aµ Identify µ µνρ σ + A5AµA + A6 + A2 Rµν σAρ δ √ µν S[Γ] ≡ gg δR(µν) + A3LCSv + A7LCSna + A8LCSa
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 4 / 8 And not! Something completely different.
Z " 4 µ ν,αβ ρ,γδ σ β,µν ρ,γδ S[Γ,T,A] = d x B1Rµν ρT T αβγδ + B2Rµν ρT T σβγδ
µ ν,ρσ σ ρ,µν ρ σ,µν + B3Rµν ρT Aσ + B4Rµν ρT Aσ + B5Rµν ρT Aσ
µ ν,ρσ ρ σ,µν α,µν β,ρσ (λ,κ)γ + C1Rµρ ν ∇σ T + C2Rµν ρ∇σ T + D1T T ∇γ T βµνλαρσκ
α,µν λ,βγ δ,ρσ µ,αβ λ,νγ δ,ρσ + D2T T ∇λT αβγδ µνρσ + D3T T ∇λT αβγδ µνρσ
λ,µν κ,ρσ λ,µν κ,ρσ λ,µν + D4T T ∇(λAκ)µνρσ + D5T ∇[λT Aκ]µνρσ + D6T Aν ∇(λAµ)
λ,µν ρ,µν σ,λκ λ,µν + D7T Aλ∇[µAν] + E1∇(ρT ∇σ)T µνλκ + E2∇(λT ∇µ)Aν
α,βγ δ,ηκ λ,µν ρ,στ + T T T T (F1βγηκαρµν δλστ + F2βληκγρµν αδστ ) # ρ,αβ γ,µν λ,στ η,αβ κ,γδ + F3T T T Aτ αβγλµνρσ + F4T T Aη Aκαβγδ
Up to a boundary term!
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 5 / 8 What does this action say?
Scalar perturbations around a static, homogeneous and isotropic solution
I Modifies the geodesic I Effective potential is Keplerian!! This is a general result!!! Contact with GR: µ I Take the limit T λρ → 0 (at EOM level). I The limit is a consistent truncation. I Only an EOM remains
µ µ C1∇[σRρ]µ ν − C2∇ν Rρσ µ = 0
∇[σRρ]ν = 0.
Only valid for vanishing torsion!!!
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 6 / 8 The vanishing torsion case
∇[σRρ]ν = 0.
It is a generalisation of Einstein’s equation
I Rµν = 0. I Rµν ∝ gµν , with or without Λ. I But Rµν 6= Rµν (g). I We would like to study the physics in the new kind of solutions. Currently:
I Analysing the “duality” provided by the Eddington trick. I Counting the number of degrees of freedom (new method 1406.1156). I Coupling matter to gravity (EOM from 1411.2424).
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 7 / 8 Thank you!!! Any questions? (based on 1410.6183 with Aureliano Skirzewski but an improved version on the way)
O. Castillo-Felisola (UTFSM, Valparaiso) Toward a non-metric theory of gravity 4 de febrero de 2015 8 / 8