A practical look at Regge calculus

Dimitri Marinelli

Physics Department -Universita degli Studi di Pavia

and I.N.F.N. - Pavia

in collaboration withProf. G. Immirzi

Karl Schwarzschild Meeting 2013,Frankfurt am Main

Many Quantum Gravity Theories need,either a discrete gravity in the classical limit

ora statistical mechanics of discrete space(-times).

this can be provided byRegge calculus

(Regge 1961)

Many Quantum Gravity Theories need,either a discrete gravity in the classical limit

ora statistical mechanics of discrete space(-times).

this can be provided byRegge calculus

(Regge 1961)

In this talk:

discretizedS3 × R - cylindrical model

Friedmann Robertson Walker space-time with closed universe

proposed by Wheeler - Les Houches Lectures 1963

...several attempts ...until 1994

John W. Barrett, Mark Galassi, Warner A. Miller, Rafael D.Sorkin, Philip A. Tuckey, Ruth M. Williams

gr-qc/9411008

What is Regge calculus?

General Relativity4-dimensional differential manifold M

ma metric tensor gµν with signature (−,+,+,+)

S [gµν ] = − c4

16πG

∫M

R [gµν ]√−g d4x +

∫LM [gµν ]

√−g d4x

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

Curvature in a 2-simplicial complex

Deficit angle ε

A deficit angle is introduced ε = 2π − 6θ

Curvature in a 2-simplicial complex

Deficit angle ε

A deficit angle is introduced ε = 2π − 6θ

Regge actionEinstein-Hilbert action for conic singularities Sorkin-1974

In Minkowski spacetime:t = tx = r · cos (k(ε) · φ)y = r · sin (k(ε) · φ)z = z

k (ε) ≡ 1− ε2π

φ ∈ [0, 2π[ ⇒ g′ =

−1 0 0 00 1 0 0

0 0(

1− θ2π

)2r2 0

0 0 0 1

Regularizing the cusp we can calculate the Ricci scalar

e2λ(r) ={

r2 if r → 0

r2(

1− ε2π

)2if r � 0

⇒ R = 2(λ′′ (r) + (λ′ (r))2

)

and the action:

S = − 116π

∫∫∫∫dφdr dz dt R

√−g = 1

8πε

∫∫dz dt = 1

8πεA

Regge actionEinstein-Hilbert action for conic singularities Sorkin-1974

In Minkowski spacetime:t = tx = r · cos (k(ε) · φ)y = r · sin (k(ε) · φ)z = z

k (ε) ≡ 1− ε2π

φ ∈ [0, 2π[ ⇒ g′ =

−1 0 0 00 1 0 0

0 0(

1− θ2π

)2r2 0

0 0 0 1

Regularizing the cusp we can calculate the Ricci scalar

e2λ(r) ={

r2 if r → 0

r2(

1− ε2π

)2if r � 0

⇒ R = 2(λ′′ (r) + (λ′ (r))2

)

and the action:

S = − 116π

∫∫∫∫dφdr dz dt R

√−g = 1

8πε

∫∫dz dt = 1

8πεA

Regge actionEinstein-Hilbert action for conic singularities Sorkin-1974

In Minkowski spacetime:t = tx = r · cos (k(ε) · φ)y = r · sin (k(ε) · φ)z = z

k (ε) ≡ 1− ε2π

φ ∈ [0, 2π[ ⇒ g′ =

−1 0 0 00 1 0 0

0 0(

1− θ2π

)2r2 0

0 0 0 1

Regularizing the cusp we can calculate the Ricci scalar

e2λ(r) ={

r2 if r → 0

r2(

1− ε2π

)2if r � 0

⇒ R = 2(λ′′ (r) + (λ′ (r))2

)

and the action:

S = − 116π

∫∫∫∫dφdr dz dt R

√−g = 1

8πε

∫∫dz dt = 1

8πεA

Regge actionEinstein-Hilbert action for conic singularities Sorkin-1974

In Minkowski spacetime:t = tx = r · cos (k(ε) · φ)y = r · sin (k(ε) · φ)z = z

k (ε) ≡ 1− ε2π

φ ∈ [0, 2π[ ⇒ g′ =

−1 0 0 00 1 0 0

0 0(

1− θ2π

)2r2 0

0 0 0 1

Regularizing the cusp we can calculate the Ricci scalar

e2λ(r) ={

r2 if r → 0

r2(

1− ε2π

)2if r � 0

⇒ R = 2(λ′′ (r) + (λ′ (r))2

)

and the action:

S = − 116π

∫∫∫∫dφdr dz dt R

√−g = 1

8πε

∫∫dz dt = 1

8πεA

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 δSR[le ] = 0 where

SR = 18π

∑t

Atεt

with At the area of the triangle t and εt its associated deficitangle.

Einstein’s equations (non linear partial differential equations) nowbecome implicit equations.

Can be considered a finite difference method for general relativity.

From 3-d simplicial complex to 4-d

We are interested in a triangulation with topology S3 × R.

for S3:• 5-cell or Pentachoron• 16-cell• 600-cell

“Tent-like” evolutionspace-like triangles

Conditions for the simplicial complexDehn-Sommerville equations

For a simplicial complex Π with boundary ∂Π

Nv (Π)− Nv (∂Π) =

4∑i=0

(−1)i+4(

i + 11

)Ni (M) = Nv − 2Ne + 3Nt − 4Nτ + 5Nσ

Ne (Π)− Ne (∂Π) =

4∑i=1

(−1)i+4(

i + 12

)Ni (M) = −Ne + 3Nt − 6Nτ + 10Nσ

Nt (Π)− Nt (∂Π) =

4∑i=2

(−1)i+4(

i + 13

)Ni (M) = Nt − 4Nτ + 10Nσ

Nτ (Π)− Nτ (∂Π) =

4∑i=3

(−1)i+4(

i + 14

)Ni (M) = −Nτ + 5Nσ

Nσ (Π)− Nσ (∂Π) =

4∑i=4

(−1)i+4(

i + 15

)Ni (M) = Nσ

Combinatorial-symmetric schemepotentially all edges space-like

30× σs

Combinatorial-symmetric scheme

l

l

d

d

l l

l′

dd

interesting triangulation for quantum gravity.

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”.

Preliminary numeric analysistent-like model, 5-cell

where a(t) is the scale parameter.

Preliminary numeric analysistent-like model, 16 and 600-cell

where a(t) is the scale parameter.

Conclusions

Regge calculus can be an important tool both to understandclassical gravity and as a map in the labyrinth of the modernmodels of quantum gravity.

“One can finally hope that Regge’s truly geometric way offormulating general relativity will someday make the content of theEinstein field equation ... stand out sharp and clear...”

J. A. Wheeler

Thank you.

Conclusions

Regge calculus can be an important tool both to understandclassical gravity and as a map in the labyrinth of the modernmodels of quantum gravity.

“One can finally hope that Regge’s truly geometric way offormulating general relativity will someday make the content of theEinstein field equation ... stand out sharp and clear...”

J. A. Wheeler

Thank you.