A practical look at Regge calculus

Transcription

1 A practical look at Regge calculus Dimitri Marinelli Physics Department - Università degli Studi di Pavia and I.N.F.N. - Pavia in collaboration with Prof. G. Immirzi Karl Schwarzschild Meeting 2013, Frankfurt am Main

2 Many Quantum Gravity Theories need, either a discrete gravity in the classical limit or a statistical mechanics of discrete space(-times). this can be provided by Regge calculus (Regge 1961)

3 Many Quantum Gravity Theories need, either a discrete gravity in the classical limit or a statistical mechanics of discrete space(-times). this can be provided by Regge calculus (Regge 1961)

4 In this talk: discretized S 3 R - cylindrical model Friedmann Robertson Walker space-time with closed universe proposed by Wheeler - Les Houches Lectures several attempts... until 1994 John W. Barrett, Mark Galassi, Warner A. Miller, Rafael D. Sorkin, Philip A. Tuckey, Ruth M. Williams gr-qc/

5 What is Regge calculus? General Relativity 4-dimensional differential manifold M a metric tensor g µν with signature (, +, +, +) S [g µν ] = c4 16πG M R [g µν ] g d 4 x + L M [g µν ] g d 4 x

6 What is Regge calculus? Spacetime is replaced by a 4-dimensional simplicial complex: Each block is a 4-simplex (4d generalization of a tetrahedron). Each 4-simplex shares its boundary tetrahedra. The space bounded by tetrahedra is a flat Minkowski spacetime (each block encloses a piece of flat spacetime) Metric structure is replaced by Edge lengths of 4-simplices dynamically fixed

7 Curvature in a 2-simplicial complex Deficit angle ɛ A deficit angle is introduced ɛ = 6θ

8 Curvature in a 2-simplicial complex Deficit angle ɛ A deficit angle is introduced ɛ = 6θ

9 Regge action Einstein-Hilbert action for conic singularities Sorkin-1974 In Minkowski spacetime: t = t x = r cos (k(ɛ) φ) y = r sin (k(ɛ) φ) z = z k (ɛ) 1 ɛ φ [0, [ g = ( ) θ r Regularizing { the cusp we can calculate the Ricci scalar r 2 e 2λ(r) if r 0 ( = r ( ) 2 1 ɛ 2 if r 0 R = 2 λ (r) + (λ (r)) 2) and the action: S = 1 16π dφ dr dz dt R g = 1 8π ɛ dz dt = 1 8π ɛ A

10 Regge action Einstein-Hilbert action for conic singularities Sorkin-1974 In Minkowski spacetime: t = t x = r cos (k(ɛ) φ) y = r sin (k(ɛ) φ) z = z k (ɛ) 1 ɛ φ [0, [ g = ( ) θ r Regularizing { the cusp we can calculate the Ricci scalar r 2 e 2λ(r) if r 0 ( = r ( ) 2 1 ɛ 2 if r 0 R = 2 λ (r) + (λ (r)) 2) and the action: S = 1 16π dφ dr dz dt R g = 1 8π ɛ dz dt = 1 8π ɛ A

11 Regge action Einstein-Hilbert action for conic singularities Sorkin-1974 In Minkowski spacetime: t = t x = r cos (k(ɛ) φ) y = r sin (k(ɛ) φ) z = z k (ɛ) 1 ɛ φ [0, [ g = ( ) θ r Regularizing { the cusp we can calculate the Ricci scalar r 2 e 2λ(r) if r 0 ( = r ( ) 2 1 ɛ 2 if r 0 R = 2 λ (r) + (λ (r)) 2) and the action: S = 1 16π dφ dr dz dt R g = 1 8π ɛ dz dt = 1 8π ɛ A

12 Regge action Einstein-Hilbert action for conic singularities Sorkin-1974 In Minkowski spacetime: t = t x = r cos (k(ɛ) φ) y = r sin (k(ɛ) φ) z = z k (ɛ) 1 ɛ φ [0, [ g = ( ) θ r Regularizing { the cusp we can calculate the Ricci scalar r 2 e 2λ(r) if r 0 ( = r ( ) 2 1 ɛ 2 if r 0 R = 2 λ (r) + (λ (r)) 2) and the action: S = 1 16π dφ dr dz dt R g = 1 8π ɛ dz dt = 1 8π ɛ A

13 Regge calculus To study a gravitational system with Regge calculus one has to: build a 4-dimensional triangulation (fix the topology), find a solution of δs R [l e ] = 0 where S R = 1 8π A t ε t t with A t the area of the triangle t and ɛ t its associated deficit angle. Einstein s equations (non linear partial differential equations) now become implicit equations. Can be considered a finite difference method for general relativity.

14 From 3-d simplicial complex to 4-d We are interested in a triangulation with topology S 3 R. for S 3 : 5-cell or Pentachoron 16-cell 600-cell

15 Tent-like evolution space-like triangles

16 Conditions for the simplicial complex Dehn-Sommerville equations For a simplicial complex Π with boundary Π N v (Π) N v ( Π) = 4 ( 1) i+4 ( i ) N i (M) = N v 2N e + 3N t 4N τ + 5N σ N e (Π) N e ( Π) = i=0 4 ( 1) i+4 ( i ) N i (M) = N e + 3N t 6N τ + 10N σ N t (Π) N t ( Π) = i=1 4 ( 1) i+4 ( i ) N i (M) = N t 4N τ + 10N σ N τ (Π) N τ ( Π) = i=2 4 ( 1) i+4 ( i ) N i (M) = N τ + 5N σ N σ (Π) N σ ( Π) = i=3 4 ( 1) i+4 ( i ) N i (M) = N σ i=4

17 30 σs Combinatorial-symmetric scheme potentially all edges space-like

18 Combinatorial-symmetric scheme l d d l d l l l d interesting triangulation for quantum gravity.

19 Metric structure Topology is fixed (a foliated triangulation of dimension 3 + 1). Initial value approach: We choose the time symmetric condition. In this case choosing initial data means choose edge lengths for the initial 3-sphere. We can choose lapse and shift.

20 Preliminary numeric analysis tent-like model, 5-cell where a(t) is the scale parameter.

21 Preliminary numeric analysis tent-like model, 16 and 600-cell where a(t) is the scale parameter.

22 Conclusions Regge calculus can be an important tool both to understand classical gravity and as a map in the labyrinth of the modern models of quantum gravity. One can finally hope that Regge s truly geometric way of formulating general relativity will someday make the content of the Einstein field equation... stand out sharp and clear... Thank you. J. A. Wheeler

23 Conclusions Regge calculus can be an important tool both to understand classical gravity and as a map in the labyrinth of the modern models of quantum gravity. One can finally hope that Regge s truly geometric way of formulating general relativity will someday make the content of the Einstein field equation... stand out sharp and clear... Thank you. J. A. Wheeler