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
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
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
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”.
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.