I review how to derive Newton’s law of universal gravitation from the Weyl strut between two Chazy-Curzon particles. I also briefly review Causal Dynamical Triangulations (CDT), a method for evaluating the path integral from canonical quantum gravity using Regge calculus and restrictions of the class of simplicial manifolds evaluated to those with a defined time foliation, thus enforcing a causal structure. I then discuss how to apply this approach to Causal Dynamical Triangulations, in particular modifying the algorithm to keep two simplicial submanifolds with curvature (i.e. mass) a fixed distance from each other, modulo regularized deviations and across all time slices. I then discuss how to determine if CDT produces an equivalent Weyl strut, which can then be used to obtain the Newtonian limit. I wrap up with a brief discussion of computational methods and code development.

1. The Newtonian Limit in CDT
Adam Getchell (UC Davis) acgetchell@ucdavis.edu

2. Path Integral
Credit: NASA/WMAP Science Team

3. CDT Path Integral
3D Delaunay Triangulation: 256 Timeslices, 7473 Vertices, 47021 Simplices2D Icosahedron: 1 timeslice, 30 Vertices, 20 Simplices

4. Does CDT have a Newtonian Limit?
• CDT looks like GR at cosmological scales,
does it have a Newtonian limit?
• 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
F =
Gm1m2
r2

5. A trick from GR
• The static axisymmetric Weyl metric
• With two-body solutions
• Leads to a “strut”
• With a stress
• That can be integrated to get the Newtonian force
ds2
= e2
dt2
e2(⌫ )
dr2
+ dz2
r2
e 2
d 2
(r, z) =
µ1
R1
µ2
R2
, Ri =
q
r2 + (z zi)
2
Tzz =
1
8⇡G
⇣
1 e ⌫(r,z)
⌘
2⇡ (r)
F =
Z
TzzdA =
1
4G
⇣
1 e ⌫(r,z)
⌘
=
Gm1m2
(z1 z2)
2 for µi = Gmi
⌫(0, z) =
4µ1µ2
(z1 z2)
2
⌫(r, z) =
µ2
1r2
R4
1
µ2
2r2
R4
2
+
4µ1µ2
(z1 z2)
2

r2
+ (z z1) (z z2)
R1R2
1

6. Measurements in CDT
Mass ➔ Epp quasilocal energy
• 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)
Einstein Tensor in Regge Calculus (Barrett, 1986)
EE ⌘
1
8⇡G
Z
⌦
d2
x
p
| |
⇣p
k2 l2
p
¯k2 ¯l2
⌘
l ⌘ µ⌫
lµ⌫ k ⌘ µ⌫
kµ⌫

7. Some Computational Methods
Distance
• Calculate single-source shortest path between the two
masses using Bellman-Ford algorithm in O(VE)
• Modify allowed moves in a sweep to not permit
successive moves which increase or decrease distance
Hausdorff Distance
• Calculate Voronoi diagram of Delaunay
triangulation
• Use Voronoi diagram to find minimum Hausdorff
distance for sets (Huttenlocher,︎︎︎︎ Kedem︎︎︎︎︎ and
Kleinberg) in O((m+n)
6
log(mn))︎︎︎

8. My Work
Re-implement CDT
• Rewrite in modern C++
• C++14 standard
• Use well-known libraries
• CGAL
• GeomView
• Eigen
• MPFR, GMP
• Intel TBB

9. My Work
Use current toolchains
• LLVM/Clang
• Hosted on GitHub
• Travis-CI for continuous testing
• GoogleMock
• CMake for cross-platform building
• Doxygen for document generation
• Others
Easy to evaluate, use, and contribute

10. Fast Foliated Delaunay Triangulations
8 timeslices, 68 vertices,
619 faces, 298 simplices
Creation Time: 0.043336s
(MacBook Pro Retina, Mid 2012)

11. Fast Foliated Delaunay Triangulations
256 timeslices, 222,136 vertices,
2,873,253 faces, 1,436,257 simplices
Creation Time: 284.596s
(MacBook Pro Retina, Mid 2012)

12. Interested? Please join!

13. Interested? Please join!