Toward a non-metric theory of gravity

Transcript

1 oward a non-metric theory of gravity playing around with the (affine) connection! Oscar Castillo-Felisola o.castillo.felisola(at)gmail.com UFSM, Valparaiso O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 1/8

2 Facts to consider: General Relativity (GR) has proven to be the most successful theory of gravity. t 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, S RNGS, B RANS -D CKE, ELEPARALLEL GRAVY, M ODFED N EWONAN DYNAMCS (MOND), C AUSAL SES, R EGGE C ALCULUS, DYNAMCAL RANGULAONS, P ENROSE WSOR HEORY, P ENROSE SPN NEWORKS, C ONNES N ON COMMUAVE GEOMERY, C AUSAL F ERMON S YSEMS... O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 2/8

3 Facts to consider: General Relativity (GR) has proven to be the most successful theory of gravity. t 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, S RNGS, B RANS -D CKE, ELEPARALLEL GRAVY, M ODFED N EWONAN DYNAMCS (MOND), C AUSAL SES, R EGGE C ALCULUS, DYNAMCAL RANGULAONS, P ENROSE WSOR HEORY, P ENROSE SPN NEWORKS, C ONNES N ON COMMUAVE GEOMERY, C AUSAL F ERMON S YSEMS... O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 2/8

4 Defining our playground Consider an affine connection Γ ρ µ σ Curvature and Ricci tensor depend on Γ. No R can be constructed (needs a metric). Parallel transport is well-possessed. Decomposition: Γ µ ρσ = Γµ (ρσ) + µ [ρσ] = Γµ ρσ + ρσλκ µ,λκ + A[ρ δ µν] We consider: a power-counting renormalizable, and diffeomorphism invariant model. O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 3/8

5 hree-dimensional model Z S[Γ,, A] = d x A1 Rµν µ ρ νρ + A2 µνρ Rµν σ σ Aρ 3 + A3 µνρ Aµ ν Aρ + A4 µν µ Aν + A5 µν Aµ Aν + A6 det( µν ) + A7 µνλ Γσ µρ ν Γρ λσ 2 + Γτ µρ Γρ νσ Γσ λτ + A8 µνρ Γσ µσ ν Γτ ρτ. 3 Eddington s trick dentify δ S[Γ] gg µν δr(µν) O. Castillo-Felisola (UFSM, Valparaiso) S[g, Γ, A] = d x g A1 R + A4 µ Aµ + A5 Aµ Aµ + A6 + A2 µνρ Rµν σ σ Aρ + A3 LCSv + A7 LCSna + A8 LCSa Z 3 oward a non-metric theory of gravity 4/8

6 And not! Something completely different. " Z S[Γ,, A] = 4 µ B1 Rµν ρ d x µ + B3 Rµν ρ ν,ρσ µ + C1 Rµρ ν σ + D2 α,µν + D4 + D7 + α,βγ + F3 κ,ρσ ρ,γδ σ αβγδ + B2 Rµν ρ σ Aσ + B4 Rµν ρ ν,ρσ λ,βγ + C2 Rµν ρ σ λ δ,ρσ ρ,στ λ,στ ρ,γδ ρ + D1 αβγδ µνρσ + D3 ρ,µν β,µν Aσ + B5 Rµν ρ σ,µν (λ Aκ) µνρσ + D5 γ,µν ρ,µν ρ Aλ [µ Aν] + E1 (ρ δ,ηκ ρ,αβ ν,αβ σ) α,µν µ,αβ [λ σ,λκ Aσ β,ρσ γ λ,νγ κ,ρσ σβγδ σ,µν λ δ,ρσ (λ,κ)γ αβγδ µνρσ Aκ] µνρσ + D6 µνλκ + E2 (λ βµνλ αρσκ Aν (λ Aµ) µ) Aν (F1 βγηκ αρµν δλστ + F2 βληκ γρµν αδστ ) # Aτ αβγλ µνρσ + F4 η,αβ κ,γδ Aη Aκ αβγδ Up to a boundary term! O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 5/8

7 What does this action say? Scalar perturbations around a static, homogeneous and isotropic solution Modifies the geodesic Effective potential is Keplerian!! his is a general result!!! Contact with GR: ake the limit µ λρ 0 (at EOM level). he limit is a consistent truncation. Only an EOM remains C1 [σ Rρ]µ µ ν C2 ν Rρσ µ µ = 0 [σ Rρ]ν = 0. Only valid for vanishing torsion!!! O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 6/8

8 he vanishing torsion case [σ Rρ]ν = 0. t is a generalisation of Einstein s equation Rµν = 0. Rµν gµν, with or without Λ. But Rµν 6= Rµν (g). We would like to study the physics in the new kind of solutions. Currently: Analysing the duality provided by the Eddington trick. Counting the number of degrees of freedom (new method ). Coupling matter to gravity (EOM from ). O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 7/8

9 hank you!!! Any questions? (based on with Aureliano Skirzewski but an improved version on the way) O. Castillo-Felisola (UFSM, Valparaiso) oward a non-metric theory of gravity 8/8