1. The Newtonian Limit in CDT
Adam Getchell (UC Davis) acgetchell@ucdavis.edu
Sunday, April 14, 13

2. Does CDT have a Newtonian Limit?
• CDT looks like GR at cosmological scales,
does it have a Newtonian limit?
Gm1 m2
F =
r2
• At first glance, this is hard:
• CDT is not well suited for approximating
smooth classical space-times
• We don’t have the time or resolution to
watch objects fall
Sunday, April 14, 13

3. A trick from GR
• The static axisymmetric Weyl metric
ds2 = e2 dt2 e2(⌫ )
dr2 + dz 2 r2 e 2
d 2
• With two-body solutions
q
µ1 µ2 2
(r, z) = , Ri = r2 + (z zi )
R1 R2
 2
µ2 r 2
1 µ2 r 2
2 4µ1 µ2 r + (z z1 ) (z z2 )
⌫(r, z) = 4 4 + 2 1
R1 R2 (z1 z2 ) R1 R2
• Leads to a “strut”
4µ1 µ2
⌫(0, z) = 2
(z1 z2 )
• With a stress
1 ⇣ ⌫(r,z)
⌘
Tzz = 1 e 2⇡ (r)
8⇡G
• That can be integrated to get the Newtonian force
Z
1 ⇣ ⌫(r,z)
⌘ Gm1 m2
F = Tzz dA = 1 e = 2 for µi = Gmi
4G (z1 z2 )
Sunday, April 14, 13

4. Measuring Mass in CDT
Mass ➔ Epp quasilocal energy
Z p ⇣p p ⌘
1 ¯ ¯2
EE ⌘ d2 x | | k2 l2 k2 l
8⇡G ⌦
µ⌫ µ⌫
l⌘ lµ⌫ k⌘ kµ⌫
• In 2+1 CDT, extrinsic curvature at an edge is proportional to the number
of connected tetrahedra
• In 3+1 CDT, extrinsic curvature at a face is proportional to the number of
connected pentachorons (4-simplices)
Sunday, April 14, 13

5. Measuring Mass in CDT
Mass ➔ Epp quasilocal energy
Z p ⇣p p ⌘
1 ¯ ¯2
EE ⌘ d2 x | | k2 l2 k2 l
8⇡G ⌦
µ⌫ µ⌫
l⌘ lµ⌫ k⌘ kµ⌫
• In 2+1 CDT, extrinsic curvature at an edge is proportional to the number
of connected tetrahedra
• In 3+1 CDT, extrinsic curvature at a face is proportional to the number of
connected pentachorons (4-simplices)
Sunday, April 14, 13

6. Some Computational Methods
• Distance
• Calculate single-source shortest path
between the two masses using Bellman-
Ford algorithm
• Modify allowed moves in a sweep to not
permit successive moves which increase or
decrease distance
• Stress
• Hmmm ...
Sunday, April 14, 13

7. On-Going work
• Based on Rajesh Kommu’s CDT implementation
• arxiv.org/abs/1110.6875
• Code is on GitHub (warning: messy!)
• https://github.com/ucdavis/CDT
• Work in early stages, results “soon”
• Thanks to Markus Afshar, Professor Steve
Carlip, Joshua Cooperman, Colin Cunliff,
Henrique Gomez, Rajesh Kommu, Jonah Miller,
and Charles Pierce for many helpful discussions
Sunday, April 14, 13