Lorentzian Geometry from Discrete Causal Structure

Lorentzian Geometry from Discrete Causal Structure Sumati Surya Raman Research Institute FTAG Sept 2013 (RRI) March 2013 1 / 25

Outline An Introduction to the Causal Set Approach to Quantum Gravity LBombelli, JLee, D Meyer and R Sorkin, PRL 1987 Causal Sets as Discrete Lorentzian Geometry Curvature from Order Mriganko Roy, Debdeep Sinha, Sumati Surya, Phys Rev D 87, 044046 (2013) Recovering Locality in a Causal Set Lisa Glaser, Sumati Surya, in preparation (RRI) March 2013 2 / 25

The Causal Set Hypothesis The causal set approach is based on two fundamental building blocks: The Causal Structure Poset + Spacetime Discreteness (RRI) March 2013 3 / 25

The Causal Structure Poset (M, ) z y w A causal spacetime (M, g) (M, ) which is a partially ordered set or poset x M is the set of events is: Acyclic: x y and y x x = y Reflexive: x x Transitive: x y, y z x z (RRI) March 2013 4 / 25

Causal Structure Poset Lorentzian Signature Causal Spacetime It is only for (, +,, +) signature that the chronological future and past of a point are disconnected Future x Past They are connected in any multiple time spacetime (,,, +,, +) This violates local causality Thus the causal structure of a spacetime of signature different from (, +, +, +) is not a poset (RRI) March 2013 5 / 25

Causal Structure Poset Lorentzian Signature Causal Spacetime It is only for (, +,, +) signature that the chronological future and past of a point are disconnected They are connected in any multiple time spacetime (,,, +,, +) t t q p x This violates local causality Thus the causal structure of a spacetime of signature different from (, +, +, +) is not a poset (RRI) March 2013 5 / 25

Causal Structure Poset Lorentzian Signature Causal Spacetime It is only for (, +,, +) signature that the chronological future and past of a point are disconnected They are connected in any multiple time spacetime (,,, +,, +) This violates local causality Thus the causal structure of a spacetime of signature different from (, +, +, +) is not a poset The Causal Structure Poset (M, ) is Unique to Lorentzian Geometry (RRI) March 2013 5 / 25

How primitive is (M, )? Zeeman, Penrose, Kronheimer, Hawking, Geroch, Ellis, Malament, etc (RRI) March 2013 6 / 25

How primitive is (M, )? Zeeman, Penrose, Kronheimer, Hawking, Geroch, Ellis, Malament, etc (M, ) determines the conformal class of the metric (RRI) March 2013 6 / 25

(M, ) determines the conformal class of the metric If f : (M 1, 1 ) (M 2, 2 ) is a bijection, then: ds 2 1 = Ω2 ds 2 2, for spacetimes which satisfy certain causality restrictions S W Hawking, AR King, PJ McCarthy, J Math Phys (1976); D Malament, J Math Phys (1977); O Parrikar, S Surya (2011) Causal structure = 9/10 th of the spacetime geometry Volume element = 1/10 th of the spacetime geometry Causal Structure + Volume = Spacetime geometry (RRI) March 2013 6 / 25

Discreteness Planck scale physics: l p = G /c 3 Black Hole Entropy, Resolution of Singularities, Regularisation of QFTs, etc Discreteness can give the spacetime volume element: y 0000 1111 000000 111111 00000000 11111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 00000000 11111111 0000000 1111111 00000 11111 000 111 01 x (RRI) March 2013 7 / 25

The Causal Set Hypothesis The Causal Structure Poset Spacetime Discreteness The underlying structure of spacetime is a causal set or locally finite poset (C, ) Discreteness implemented via local finiteness: Fut(x) Past(y) < y x (RRI) March 2013 8 / 25

The Causal Set Hypothesis Causal Structure Partially Ordered Set Spacetime Volume Number Order + Number Spacetime geometry Fundamental entity is a causal set The continuum is emergent and is an approximation The continuum limit is not of fundamental importance Causal Set Quantum Gravity (CQG) is a quantum theory of causal sets: Z = e is(c) C Ω (RRI) March 2013 9 / 25

The Continuum Approximation Regular lattice does not preserve Number-Volume correspondence (RRI) March 2013 10 / 25

t The Continuum Approximation Regular lattice does not preserve Number-Volume correspondence Random lattice generated via a Poisson process: P V (n) 1 n! e ρv (ρv ) n, < N >= ρv 1 embeddingasc 09 08 07 06 05 04 03 02 01 0 0 01 02 03 04 05 06 07 08 09 1 x A causal set that is approximated by a spacetime is obtained via a Poisson sprinkling, with the partial order on elements induced by the causal order (RRI) March 2013 10 / 25

Causal Set Kinematics or How to Recognise a Manifold-like Causal Set C is approximated by (M, g) if it admits a faithful embedding Φ : C (M, g) Order relation in C induced causal order in Φ(C) Φ(C) (M, g) is a high probability Poisson sprinkling in (M, g) A generic causal set looks nothing like spacetime: N/4 N/2 N/4 How does one reconstruct spacetime geometry and topology purely order theoretically? (RRI) March 2013 11 / 25

The Inherent Non-locality of a Causal Set The Poisson process P V (n) = 1 n! e ρv (ρv ) n is Lorentz invariant LBombelli, JHenson, R Sorkin, ModPhysLett 2009 In Minkowski spacetime every element has an infinite valency Unlike Euclidean space, there are an infinite number of nearest neighbours in a sprinkling into Minkowski spacetime This makes it highly non-trivial to extract local information about a manifold (RRI) March 2013 12 / 25

Reconstructing Spacetime From a Causal Set Example: Timelike distance d(x, y) = Length of the longest chain from x to y x y G Brightwell, R Gregory, Phys Rev Lett (1991) (RRI) March 2013 13 / 25

Reconstructing Spacetime From a Causal Set Dimension DA Meyer, PhD thesis (MIT, 1988); J Myrheim, CERN preprint TH-2538 (1978) Homology S Major, DP Rideout, S Surya, JMathPhys (2007); S Surya, Theoretical Computer Science, (2008), S Major, DP Rideout, S Surya, CQG (2009) Timelike and Spacelike Distance G Brightwell, R Gregory, Phys Rev Lett (1991); DP Rideout, P Wallden; arxiv:08101768 The D Alembertian operator RD Sorkin, arxiv:gr-qc/0703099, D Benicasa and F Dowker, PRL (2010), F Dowker and L Glaser, arxiv:13052588 Advanced and Retarded Green s function and Feynman s Green s function S Johnston; Class Quantum Grav (2008), PRL (2009) Action functional DBenincasa and F Dowker, PRL (2010), F Dowker and L Glaser, arxiv:13052588 Scalar Curvature Mriganko Roy, Debdeep Sinha and Sumati Surya, PRD (2013) Locality Lisa Glaser and Sumati Surya, (in preparation) (RRI) March 2013 13 / 25

Curvature from Order Mriganko Roy, Debdeep Sinha and Sumati Surya, PRD (2013) A chain c k = {x 1, x 2, x k } is a totally ordered subset of C of length k x 1 x 2 x k C k : the abundance of c k s in C (RRI) March 2013 14 / 25

Curvature from Order Mriganko Roy, Debdeep Sinha and Sumati Surya, PRD (2013) Myrheim-Meyer Dimension Estimator for C (M, η) f (n) C 2 η C 1 2 = Γ(n + 1)Γ( n 2 ) η 4Γ( 3n 2 ) 1 ( N ) 1 2 R 2 C 2 η: Expectation value for the number of relations in C DA Meyer, PhD thesis (MIT, 1988); J Myrheim, CERN preprint TH-2538 (1978) How do curvature corrections affect this result? (RRI) March 2013 14 / 25

The Geometry of a Small Causal Diamond Alexandrov Interval I[p, q]: q T/2 y 1 T1 x 1 p Riemann Normal Coordinates: g ab (x) = η ab (0) 1 3 x c x d R acbd (0) + O(x 3 ), RT 2 << 1 C 1 = ρv = ρ I[p,q] g1 dx 1 = ρv 0 ( 1 + α 1 R(0)T 2 + β 1 R 00 (0)T 2 ) J Myrheim, CERN preprint TH-2538 (1978), G W Gibbons and S N Solodukhin, Phys Lett B 649, 317 (2007) (RRI) March 2013 15 / 25

A General Expression Recent results on boundary term contributions help us to find: S Khetrapal and S Surya, Class Quant Grav 30, 065005 (2013) Lemma To the lowest order correction in the flat spacetime expression, the average number of k-element chains in a small causal diamond is ] C k = C k η [1 + T 2 α k R(0) + T 2 β k R 00 (0) + O(T kn+3 ), where nk α k = 12(kn + 2)((k + 1)n + 2), β nk k = 12((k + 1)n + 2) (RRI) March 2013 16 / 25

Discrete Curvature and a New Dimension Estimator: 2(n + 2)(2n + 2)(3n + 2) R(0) = n 3 T 3n+2 (K 1 2K 2 + K 3 ) ( ) Ck 3/k Q k ρ k, K k ((k + 1)n + 2)Q k ζ k A new dimension estimator Continuum Scalar Curvature is Encoded in C 1, C 2, C 3 ( f 2 (n) 1 3 (n + 2) (3n + 2) (4n + 2) (2n + 2) ( ) 4 C3 3 1 χ 3 C 1 4 + 1 (4n + 2)(5n + 2) 3 (2n + 2)(3n + 2) = C 2 2 C 1 4 ) C 4 1 χ 4 C 1 4 Continuum Dimension is Encoded in C 1, C 2, C 3, C 4 (RRI) March 2013 17 / 25

Causal Structure and Locality In the continuum a local region is one that is approximately flat A topology based on Alexandrov Intervals: I[x, y]{z x z y} is more appropriate for Lorentzian spacetime than that based on open balls B n In the continuum, Vol(I[x, y]) and τ(x, y) are the only covariant measures of I[x, y] Insufficient to capture locality or smallness of a region: (RRI) March 2013 18 / 25

Locality from Discreteness The discrete geometry C of an Alexandrov Interval contains more covariant information: Number of Relations in C: How many related pairs in C from ( N 2) possible pairs? Number of Links in C: x is linked to y if z x, y such that x z y Numbers of Chains of size k: C k Numbers of Order-intervals of size m: N m Order-intervals are intrinsically local in the Lorentzian sense: (RRI) March 2013 19 / 25

The Discrete Einstein-Hilbert Action in terms of Order-Intervals Benincasa-Dowker-Glaser Action for Causal Sets: D Benincasa and FDowker PRL, (2010), F Dowker and L Glaser, arxiv:13052588 S (d) (C) N i : # of i-element order intervals = ζ d ( N + β d α d ( Example of d = 2 : S(ɛ)/ = 4ɛ N 2ɛ ) N 2 n=0 Nn f (n, ɛ) ( ) 2 lp Mesoscale l k >> l p : ɛ = l [0, 1] k n i=1 C (d) i N i ) f (n, ɛ) = (1 ɛ) n 2ɛn(1 ɛ) n 1 + 1 2 ɛ2 n(n 1)(1 ɛ) n 2 For ɛ = 1 1 S(2) (C) = N 2N 0 + 4N 1 2N 2 This strongly suggests a definition of locality based on the N i S Surya, ClassQuantGrav 29 (2012) (RRI) March 2013 20 / 25

Locality from Discreteness: Abundance of m-element Order Intervals Let C be obtained from I[p, q] ( d M, η) via a Poisson sprinkling for some ρ The probability that there is a link from x to y, for x, y I[p, q] is given by P xy = e ρvxy, Expectation value of the number of links in I[p, q]: N0 d (ρ, V ) = ρ2 dv x dv y e ρvxy x (RRI) March 2013 21 / 25

Locality From Discreteness: Characteristic Curves Nm (ρ, d (ρv )m+2 Γ (d) 2 V ) = ( ) (m + 2)! d 2 (m + 1) + 1 1 ( d 2 m + 1 ) d 1 d 1 1 + m, 2 d F d d + m, 4 2(d 1) + m,, + m) d d 3 + m, 2 d + m + 2, 4 2(d 1) + m + 2,, + m + 2 ρv, d d N m d 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 0 10 20 30 40 50 m 2d 3d 4d 5d (RRI) March 2013 22 / 25

Locality From Discreteness: Characteristic Curves Nm d (ρv )m+2 Γ (d) 2 (ρ, V ) = ( ) (m + 2)! d 2 (m + 1) + 1 1 ( d 2 m + 1 ) d 1 d 1 1 + m, 2 d F d d + m, 4 2(d 1) + m,, + m) d d 3 + m, 2 d + m + 2, 4 2(d 1) + m + 2,, + m + 2 ρv, d d Continuum Limit: lim ρ Nd m (ρ, V ) N 2 2/d, N lim m (ρ, d V ) Γ ρ N0 d (ρ, V ) = Γ ( 2 d ( 2 d + m ) ) Γ (m + 1) (RRI) March 2013 22 / 25

A Proposal For Locality Let C be an N-element causal set If N m(c) N d m (N ± N) for some d then C is local or approximated by flat spacetime If C C with C = N such that N m(c ) N d m (N ± N ) Then C is a local subset of C (RRI) March 2013 23 / 25

Results from Simulations Flat Spacetime: N m d 600 500 400 300 200 100 simulated analytic 2d simulated analytic 3d simulated analytic 4d 0 0 5 10 15 20 m (RRI) March 2013 24 / 25

Results from Simulations Cut-Trousers Spacetime: N m d 400 analytic 350 300 250 200 150 100 50 0 0 10 20 30 40 50 m (RRI) March 2013 24 / 25

Results from Simulations Spacetimes with Toroidal Slices: 400 350 torus analytic interval 16000 14000 torus analytic interval 300 12000 250 10000 N d m 200 N d m 8000 150 6000 100 4000 50 2000 0 0 10 20 30 40 50 0 0 10 20 30 40 50 m m (a) 2-d 100 elements (b) 3-d 1000 elements 600 local non-local 500 analytic 400 N d m 300 200 100 0 0 2 4 6 8 10 m (c) 3-d local and non-local small intervals (RRI) March 2013 24 / 25

Results from Simulations FRW Spacetimes 10000 desitter FRW w = 0 FRW w = 1/3 flat analytic N d m 1000 100 0 10 20 30 40 50 m (RRI) March 2013 24 / 25

Results from Simulations FRW Spacetimes 450 400 350 analytic flat curved 300 250 N d m 200 150 100 50 0 0 2 4 6 8 10 m (d) FRW with w = 1 3 (RRI) March 2013 24 / 25

Results from Simulations Non-Manifoldlike Causal Sets N d m 9 8 7 6 5 4 3 2 1 N d m 10000 1000 100 10 1 01 0 0 2 4 6 8 10 12 14 m (e) chain 001 0 5 10 15 20 25 30 m (f) KR-order 100000 10000 percolated 2d 3d 4d N d m 1000 100 10 5 10 15 20 25 30 35 40 45 50 m (g) Transitive Percolation: p=001 N=2000 (RRI) March 2013 24 / 25

Concluding Remarks: Characteristic curves for N d m provide: A definition of locality A test for manifoldlikeness A new dimension estimator Suggests a rigidity of discrete Minkowski spacetime: N m(c) as a necessary and sufficient condition for manifoldlikeness (and in particular flatness) Conjecture: An N-element causal set C is approximated by an Alexandrov interval of volume V in d dimensional Minkowski spacetime iff N m(c) N d m (N ± N) (RRI) March 2013 25 / 25