Conformal Tensors in Lovelock Gravity Lovelock gravity shares - PowerPoint PPT Presentation

Conformal Tensors in Lovelock Gravity Lovelock gravity shares important features with Einstein gravity … But exactly which features? Do we know them all? Formulate some basic questions … a) Higher Curvature Bianchi Identities b) Analogues of 3D GR Answers supplied by 1. Intro to Lovelock new higher curvature 2. Riemann-Lovelock Tensor & “Lovelock Flatness” constructs 3. Weyl-Lovelock et. al. What about “conformal 4. Further (interesting?) questions Lovelock flatness”? In abundance … David Kastor 2016 Northeast Gravity Workshop 1

Coupling constants 1) Introduction to Lovelock Higher curvature analogues of scalar curvature k D Z d D x √− g X a k R ( k ) S = k =0 b 1 b 2 . . . R a 2 k − 1 a 2 k R ( k ) = δ a 1 ...a 2 k b 2 k − 1 b 2 k b 1 ...b 2 k R a 1 a 2 δ b 1 ...b n a 1 ...a n = δ b 1 [ a 1 · · · δ b n a n ] R (0) = 1 Cosmological constant term R (1) = R Einstein-Hilbert term R (2) = 1 c + R a cd R cd be − 2 R ad bc R c d − 2 R c b R a b R Gauss-Bonnet term � � R ae 6 Euler density in D=2k dimensions R ( k )  D − 1 � - vanishes for D<2k k D = 2 - variation vanishes in D=2k David Kastor 2016 Northeast Gravity Workshop 2

Equations of motion depend only The nice thing about Lovelock … on Riemann tensor and not its derivatives k D - no 4 th derivatives of metric Z d D x √− g X a k R ( k ) S = - same initial data as GR k =0 - no ghosts k D X a k G ( k ) a b = 0 Higher curvature analogues of k =0 Einstein tensor b = (2 k + 1) α k d 1 d 2 . . . R c 2 k − 1 c 2 k G ( k ) δ bc 1 ...c 2 k d 2 k − 1 d 2 k ad 1 ...d 2 k R c 1 c 2 a 2 r a G ( k ) a Covariantly conserved b = 0 G (1) a Einstein tensor b G ( k ) a Vanishes for D < 2k+1 b David Kastor 2016 Northeast Gravity Workshop 3

Questions … A) r a G ( k ) a Covariantly conserved b = 0 Follows from twice contracted Bianchi identity k=1 de = 0 bc 0 = r [ a R bc ] r [ a R bc ] = 1 3( r a R � 2 r b R b a ) = � 2 3 r b G b a Is there an analogue of the uncontracted Bianchi identity for k>1? Is there a higher curvature Lovelock analogue of the Riemann tensor in this sense? David Kastor 2016 Northeast Gravity Workshop 4

Questions … B) Vacuum GR in D=3 cd = 0 All solutions to Einstein’s G ab = 0 R ab equation are flat Both Riemann and Ricci tensors have 6 3 x 3 symmetric tensors independent components Or simple Lovelock-type construction … Relation is true in gh = 1 ⇣ ⌘ d ] + δ cd cd − 4 δ [ c δ cdef abgh R ef R ab [ a R b ] ab R all dimensions 6 LHS vanishes in D=3, determining Riemann tensor in terms of its contractions David Kastor 2016 Northeast Gravity Workshop 5

Questions … B) Vacuum GR in D=3 cd = 0 All solutions to Einstein’s G ab = 0 R ab equation are flat Is there an analogue of this for k>1? Look at “pure” k th order R ( k ) Euler density in D=2k dimensions Lovelock gravity in D=2k+1 Z Trivial in D=2k, Only k th order Lovelock d D x √− g R ( k ) S pure = like GR in D=2 term in action This is highest order D=2k+1 Lovelock term available Expect all solutions to Lovelock will asymptote to solutions of pure k th order theory in high curvature regime David Kastor 2016 Northeast Gravity Workshop 6

Questions … B) Vacuum GR in D=3 cd = 0 All solutions to Einstein’s G ab = 0 R ab equation are flat Is there an analogue of this for k>1? Look at “pure” k th order R ( k ) Euler density in D=2k dimensions Lovelock gravity in D=2k+1 Z Only k th order Lovelock d D x √− g R ( k ) S pure = term in action Is there a higher curvature Lovelock flatness condition, such that all solutions to pure k th order Lovelock in D=2k+1 are k th order Lovelock flat? Like question 1, this calls for a higher curvature analogue of the Riemann tensor David Kastor 2016 Northeast Gravity Workshop 7

2) Riemann-Lovelock tensors & “Lovelock flatness” Call this Riemann (k) tensor R ( k ) c 2 d 2 · · · R a k b k ] [ c 1 d 1 R a 2 b 2 c k d k ] c 1 d 1 ...c k d k ≡ R [ a 1 b 1 a 1 b 1 ...a k b k Tensor of type (2k,2k), vanishes Like all 1D spaces are k=1 Lovelock flat for D<2k and satisfies … R ( k ) a 1 ...a 2 k b 1 ...b 2 k = R ( k ) [ a 1 ...a 2 k ] b 1 ...b 2 k = R ( k ) a 1 ...a 2 k [ b 1 ...b 2 k ] = R ( k ) b 1 ...b 2 k a 1 ...a 2 k R ( k ) b 2 ...b 2 k = 0 [ a 1 ...a 2 k b 1 ] Bianchi identities Symmetries r [ c R ( k ) b 1 ...b 2 k = 0 a 1 ...a 2 k ] Analogous to familiar properties of Riemann tensor R ( k ) c 1 d 1 ...c k d k = 0 k th order Lovelock flatness a 1 b 1 ...a k b k or Riemann (k) flat David Kastor 2016 Northeast Gravity Workshop 8

Taking traces … Tracing over all pairs of indices gives back scalar R ( k ) = R ( k ) a 1 ...a 2 k Lovelock interaction terms a 1 ...a 2 k Ricci (k) tensor is an R ( k ) b = R ( k ) bc 1 ...c 2 k − 1 analogue of Ricci tensor a ac 1 ...c 2 k − 1 b = k R ( k ) Einstein (k) tensor appears in G ( k ) b R ( k ) b − (1 / 2) δ a Lovelock equation of motion a a Fully contracted Bianchi identity 0 = r [ a R ( k ) b 1 ...b 2 k yields vanishing divergrance for b 1 ...b 2 k ] Einstein (k) tensors 1 ⇣ ⌘ r a R ( k ) � 2 k r b R ( k ) b = a 2 k + 1 Answers 1 st question 2 2 k + 1 r b G ( k ) b = � Demonstrates some a relevance for Riemann- Lovelock tensors David Kastor 2016 Northeast Gravity Workshop 9

Analogue of Pure k th order Lovelock in D=2k+1 vacuum GR in D=3 Z d 2 k +1 x √− g R ( k ) S pure = b = k R ( k ) b R ( k ) = 0 G ( k ) b − (1 / 2) δ a Yes a a 2 nd Question Are all solutions k th order Lovelock flat? R ( k ) c 2 d 2 · · · R a k b k ] [ c 1 d 1 R a 2 b 2 c k d k ] c 1 d 1 ...c k d k ≡ R [ a 1 b 1 a 1 b 1 ...a k b k Same number of independent components D=2k+1 as symmetric (2k+1)x(2k+1) tensor R ( k ) b = 0 c 1 d 1 ...c k d k = 0 Can show R ( k ) a 1 b 1 ...a k b k a David Kastor 2016 Northeast Gravity Workshop 10

Are there interesting spacetimes that are higher order Lovelock flat, but not Riemann flat? Large set of examples … Riemann (k) tensor vanishes for any spacetime of dimension D < 2k Can build higher dimensional Riemann (k) flat spacetimes by adding flat directions Interesting example in D=2k+1 … Static, spherically symmetric solutions of pure k th order Lovelock are missing solid angle spacetimes 2 k +1 = − dt 2 + dr 2 + α 2 r 2 d Ω 2 ds 2 2 k − 1 David Kastor 2016 Northeast Gravity Workshop 11

Static, spherically symmetric solutions of pure k th order Lovelock are missing solid angle spacetimes 2 k +1 = − dt 2 + dr 2 + α 2 r 2 d Ω 2 ds 2 2 k − 1 Angular 2 ρσ = coordinates α 2 r 2 (1 − α 2 ) δ ρσ µ, ν = 1 , . . . , 2 k − 1 R µ ν on sphere µ ν Only nonzero curvature components Curved for α 6 = 1 Riemann (k) tensor c k d k ] = 0 R ( k ) c 2 d 2 · · · R a k b k ] [ c 1 d 1 R a 2 b 2 c 1 d 1 ...c k d k ≡ R [ a 1 b 1 a 1 b 1 ...a k b k Involves anti-symmetrization over 2k indices, but only 2k-1 are available … David Kastor 2016 Northeast Gravity Workshop 12

Static, spherically symmetric solutions of pure k th order Lovelock are missing solid angle spacetimes 2 k +1 = − dt 2 + dr 2 + α 2 r 2 d Ω 2 ds 2 2 k − 1 Angular 2 ρσ = coordinates α 2 r 2 (1 − α 2 ) δ ρσ µ, ν = 1 , . . . , 2 k − 1 R µ ν on sphere µ ν Only nonzero curvature components Curved for α 6 = 1 Missing angle Flat 3 = − dt 2 + dr 2 + α 2 d φ 2 ds 2 GR in D=3 k=1 Global flat space with identifications Further question … Missing solid angle General case Riemann (k) Flat Can we classify all k>1 Global flat space Riemann (k) flat with identifications spacetimes? David Kastor 2016 Northeast Gravity Workshop 13

Conformal tensors in Lovelock … Next step Riemann (k) flatness Conformal (k) flatness A spacetime is Conformal (k) flat if it is related to to a Riemann (k) flat spacetime via a conformal transformation Weyl tensor vanishes D ≥ 4 Conformal flatness Trace free part of Riemann tensor Consider trace free part of Weyl (k) tensors Riemann (k) tensors Do Weyl (k) tensors determine Conformal (k) flatness? David Kastor 2016 Northeast Gravity Workshop 14

First recall some other constructs … cd = R ab cd − 4 δ [ c d ] Weyl tensor W ab [ a S b ] 1 ✓ 1 ◆ b = b − 2( D − 1) δ b Schouten tensor S a R a a R D − 2 c = 2 r [ a S b ] c C ab Cotton tensor b = 0 C ab g ab = e 2 f g ab ˜ Conformal transformations cd = e − 2 f W ab ˜ cd W ab c = e − 2 f � c � W ab ˜ cd r d f � C ab C ab David Kastor 2016 Northeast Gravity Workshop 15

g ab = e 2 f g ab ˜ Conformal transformations cd = e − 2 f W ab ˜ cd W ab c � W ab c = e − 2 f � ˜ cd r d f � C ab C ab Vanishes in D=3 cd = 0 gh = (1 / 6) W ab δ cdef cd D=3 W ab abgh W ef Cotton tensor is conformally invariant Conformal flatness c = 0 C ab condition in D=3 Weyl tensor not defined D=2 All metrics are locally conformally flat David Kastor 2016 Northeast Gravity Workshop 16