Causal Dynamical Triangulations with CGAL

Adam Getchell

acgetchell@ucdavis.edu

University of California, Davis

CGAL Developers Conference, Nancy, Sept 29-October 2, 2015

Causal Dynamical Triangulations

A candidate theory of quantum gravity

CGAL

You are the experts!

Path Integral

Credit: NASA/WMAP Science team

Path Integral

\langle B|T|A\rangle=\int\mathcal{D}[g]e^{iS_{EH}}

\langle B|T|A\rangle=\int\mathcal{D}[g]e^{iS_{EH}}

S_{EH}=\frac{1}{16\pi G_{N}}\int d^{4}x\sqrt{-g}(R-2\Lambda)

S_{EH}=\frac{1}{16\pi G_{N}}\int d^{4}x\sqrt{-g}(R-2\Lambda)

Equations of Motion

\partial S = 0 \rightarrow R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G_{N}T_{\mu\nu}

\partial S = 0 \rightarrow R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G_{N}T_{\mu\nu}

Ricci scalar

Cosmological constant

Ricci tensor

Ricci scalar

Stress-Energy tensor

Transition probability amplitude

Calculating the Path Integral

\langle B|T|A\rangle=\int\mathcal{D}[g]e^{iS_{EH}}

\langle B|T|A\rangle=\int\mathcal{D}[g]e^{iS_{EH}}

S_{EH}=\frac{1}{16\pi G_{N}}\int d^{4}x\sqrt{-g}(R-2\Lambda)

S_{EH}=\frac{1}{16\pi G_{N}}\int d^{4}x\sqrt{-g}(R-2\Lambda)

Almost impossible to calculate!

Calculating the Path Integral

Quickly gets complicated!

Perturbative sums, renormalization, etc.

CDT Path Integral

3D Delaunay Triangulation: 256 Timeslices, 7473 Vertices, 47021 Simplices

2D Icosahedron: 1 timeslice, 30 vertices, 20 Simplices

CDT Path Integral

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{iS_{R}(T)}

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{iS_{R}(T)}

S_{R}=\frac{1}{8\pi G_{N}}\left(\sum\limits_{hinges}A_{h}\delta_{h}-\Lambda\sum\limits_{simplices}V_{s}\right)

S_{R}=\frac{1}{8\pi G_{N}}\left(\sum\limits_{hinges}A_{h}\delta_{h}-\Lambda\sum\limits_{simplices}V_{s}\right)

Inequivalent Triangulations

Regge Action

Partition Function

Transition probability amplitude

Calculating the CDT Path Integral

S_{R}=\frac{1}{8\pi G_{N}}\left(\sum\limits_{hinges}A_{h}\delta_{h}-\Lambda\sum\limits_{simplices}V_{s}\right)

S_{R}=\frac{1}{8\pi G_{N}}\left(\sum\limits_{hinges}A_{h}\delta_{h}-\Lambda\sum\limits_{simplices}V_{s}\right)

Area of hinge

Volume of Simplex

Deficit Angle

Regge Action

CDT Action

\begin{array}{l} S^{(3)} &=& 2\pi k\sqrt{\alpha}N_1^{TL} \\ &+& N_3^{(3,1)}\left[-3k\text{arcsinh}\left(\frac{1}{\sqrt{3} \sqrt{4\alpha +1}}\right)-3k\sqrt{\alpha}\text{arccos}\left(\frac{2\alpha+1} {4\alpha+1}\right)-\frac{\lambda}{12}\sqrt{3\alpha+1}\right] \\ &+& N_3^{(2,2)}\left[2k\text{arcsinh}\left(\frac{2\sqrt{2}\sqrt{2\alpha+1}} {4\alpha +1}\right)-4k\sqrt{\alpha}\text{arccos}\left(\frac{-1}{4\alpha+1} \right)-\frac{\lambda}{12}\sqrt{4\alpha +2}\right] \end{array}

\begin{array}{l} S^{(3)} &=& 2\pi k\sqrt{\alpha}N_1^{TL} \\ &+& N_3^{(3,1)}\left[-3k\text{arcsinh}\left(\frac{1}{\sqrt{3} \sqrt{4\alpha +1}}\right)-3k\sqrt{\alpha}\text{arccos}\left(\frac{2\alpha+1} {4\alpha+1}\right)-\frac{\lambda}{12}\sqrt{3\alpha+1}\right] \\ &+& N_3^{(2,2)}\left[2k\text{arcsinh}\left(\frac{2\sqrt{2}\sqrt{2\alpha+1}} {4\alpha +1}\right)-4k\sqrt{\alpha}\text{arccos}\left(\frac{-1}{4\alpha+1} \right)-\frac{\lambda}{12}\sqrt{4\alpha +2}\right] \end{array}

CDT Action

\begin{array}{l} S^{(4)}\!\! &=&\!\! \frac{k \pi}{2} \sqrt{4\alpha +1}\ N_{2}^{TL} + N_{4}^{(4,1)}\cdot \\ && \!\!\Biggl( -\sqrt{3} k\ {arcsinh} \frac{1}{2\sqrt{2}\sqrt{3\alpha +1}} -\frac{3 k}{2} \sqrt{4\alpha +1} \arccos\frac{2\alpha +1}{2 (3\alpha +1)} -\lambda \frac{\sqrt{8\alpha +3}}{96} \Biggr) \\ &+&\!\! N_{4}^{(3,2)} \Biggl( \frac{\sqrt{3} k}{4}\ {arcsinh}\frac{\sqrt{3}\sqrt{12 \alpha +7}} {2 (3 \alpha+1)} - \frac{3 k}{4} \sqrt{4\alpha +1}\\ &&\!\!\biggl( 2\arccos\frac{-1}{2\sqrt{2}\sqrt{2\alpha +1}\sqrt{3\alpha +1}} + \arccos\frac{4\alpha +3}{4 (2\alpha +1)}\biggr) -\lambda \frac{\sqrt{12\alpha +7}}{96}\Biggr) \end{array}

\begin{array}{l} S^{(4)}\!\! &=&\!\! \frac{k \pi}{2} \sqrt{4\alpha +1}\ N_{2}^{TL} + N_{4}^{(4,1)}\cdot \\ && \!\!\Biggl( -\sqrt{3} k\ {arcsinh} \frac{1}{2\sqrt{2}\sqrt{3\alpha +1}} -\frac{3 k}{2} \sqrt{4\alpha +1} \arccos\frac{2\alpha +1}{2 (3\alpha +1)} -\lambda \frac{\sqrt{8\alpha +3}}{96} \Biggr) \\ &+&\!\! N_{4}^{(3,2)} \Biggl( \frac{\sqrt{3} k}{4}\ {arcsinh}\frac{\sqrt{3}\sqrt{12 \alpha +7}} {2 (3 \alpha+1)} - \frac{3 k}{4} \sqrt{4\alpha +1}\\ &&\!\!\biggl( 2\arccos\frac{-1}{2\sqrt{2}\sqrt{2\alpha +1}\sqrt{3\alpha +1}} + \arccos\frac{4\alpha +3}{4 (2\alpha +1)}\biggr) -\lambda \frac{\sqrt{12\alpha +7}}{96}\Biggr) \end{array}

CDT Path Integral

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{iS_{R}(T)}

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{iS_{R}(T)}

CDT Path Integral

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{-S_{R}(T)}

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{-S_{R}(T)}

Wick rotation

Metropolis-Hastings

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{-S_{R}(T)}

\langle B|T|A\rangle=\sum\limits_{triangulations}\frac{1}{C(T)}e^{-S_{R}(T)}

Inequivalent Triangulations

Partition Function

Wick rotation

Pick an ergodic (Pachner) move
Make that move with a probability of a1a2, where:

a_{1}=\frac{move[i]}{\sum\limits_{i}move[i]}

a_{1}=\frac{move[i]}{\sum\limits_{i}move[i]}

a_{2}=e^{\Delta S}

a_{2}=e^{\Delta S}

3D Ergodic Moves

(2,3) & (3,2)

(4,4)

(2,6) & (6,2)

Simplices involved

Move name

(3,1) & (2,2)

2 (1,3) & 2 (3,1)

(1,3) & (3,1)

4D Ergodic Moves

(2,4) & (4,2)

(3,3)

(4,6) & (6,4)

(2,8) & (8,2)

Does CDT have a Newtonian Limit?

CDT looks like GR at cosmological scales, does it have a Newtonian limit?

F=G_{N}\frac{m_{1}m_{2}}{r^2}

F=G_{N}\frac{m_{1}m_{2}}{r^2}

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

A Trick from GR

ds^{2} = e^{2\lambda}dt^{2} - e^{2\left(\nu-\lambda\right)}\left(dr^{2}+dz^{2}\right) - r^{2}e^{-2\lambda}d\phi^{2}

ds^{2} = e^{2\lambda}dt^{2} - e^{2\left(\nu-\lambda\right)}\left(dr^{2}+dz^{2}\right) - r^{2}e^{-2\lambda}d\phi^{2}

The static axisymmetric Weyl metric:

With two-body Chazy-Curzon (circa 1924) solutions of the Einstein field equations:

\lambda(r,z)=-\frac{\mu_1}{r_1}-\frac{\mu_2}{r_2}

\lambda(r,z)=-\frac{\mu_1}{r_1}-\frac{\mu_2}{r_2}

\nu(r,z)=\frac{1}{2}\frac{\mu_{1}^{2}r^2}{r_{1}^{4}}-\frac{1}{2}\frac{\mu_{2}^{2}r^2}{r_{2}^{4}}+\frac{2\mu_1\mu_2}{(z-z_2)^2}\left[\frac{r^2+(z-z_1)(z-z_2)}{r_{1}r_{2}}-1\right]

\nu(r,z)=\frac{1}{2}\frac{\mu_{1}^{2}r^2}{r_{1}^{4}}-\frac{1}{2}\frac{\mu_{2}^{2}r^2}{r_{2}^{4}}+\frac{2\mu_1\mu_2}{(z-z_2)^2}\left[\frac{r^2+(z-z_1)(z-z_2)}{r_{1}r_{2}}-1\right]

r_1=\sqrt{r^2+(z-z_1)^2}

r_1=\sqrt{r^2+(z-z_1)^2}

Leads to a strut:

With a stress:

That can be integrated to get the Newtonian force!

\nu(0,z)=\frac{4\mu_{1}\mu_{2}}{z_{1}^2-z_{2}^2}

\nu(0,z)=\frac{4\mu_{1}\mu_{2}}{z_{1}^2-z_{2}^2}

T_{zz}=\frac{1}{8\pi G_{N}}\left(1-e^{-\nu(r,z)}\right)2\pi\delta

T_{zz}=\frac{1}{8\pi G_{N}}\left(1-e^{-\nu(r,z)}\right)2\pi\delta

F=\int T_{zz}dA=\frac{1}{4G_{N}}\left(1-e^{-\nu(r,z)}\right)=G_{N}\frac{m_1m_2}{z_1^2-z_2^2}\:\text{for}\:\mu_i=G_Nm_i

F=\int T_{zz}dA=\frac{1}{4G_{N}}\left(1-e^{-\nu(r,z)}\right)=G_{N}\frac{m_1m_2}{z_1^2-z_2^2}\:\text{for}\:\mu_i=G_Nm_i

A Trick from GR

ds^{2} = e^{2\lambda}dt^{2} - e^{2\left(\nu-\lambda\right)}\left(dr^{2}+dz^{2}\right) - r^{2}e^{-2\lambda}d\phi^{2}

ds^{2} = e^{2\lambda}dt^{2} - e^{2\left(\nu-\lambda\right)}\left(dr^{2}+dz^{2}\right) - r^{2}e^{-2\lambda}d\phi^{2}

g_{\mu\nu}=\left(\begin{array}{cccc} e^{2\lambda} & 0 & 0 & 0\\ 0 & -e^{2\left(\nu-\lambda\right)} & 0 & 0\\ 0 & 0 & -e^{2\left(\nu-\lambda\right)} & 0\\ 0 & 0 & 0 & -\frac{r^{2}}{e^{2\lambda}} \end{array}\right)

g_{\mu\nu}=\left(\begin{array}{cccc} e^{2\lambda} & 0 & 0 & 0\\ 0 & -e^{2\left(\nu-\lambda\right)} & 0 & 0\\ 0 & 0 & -e^{2\left(\nu-\lambda\right)} & 0\\ 0 & 0 & 0 & -\frac{r^{2}}{e^{2\lambda}} \end{array}\right)

\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)

\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)

R_{\sigma\mu\nu}^{\rho}=\partial_{\mu}\Gamma_{\nu\sigma}^{\rho}-\partial_{\nu}\Gamma_{\mu\sigma}^{\rho}+\Gamma_{\mu\lambda}^{\rho}\Gamma_{\nu\sigma}^{\lambda}-\Gamma_{\nu\lambda}^{\rho}\Gamma_{\mu\sigma}^{\lambda}

R_{\sigma\mu\nu}^{\rho}=\partial_{\mu}\Gamma_{\nu\sigma}^{\rho}-\partial_{\nu}\Gamma_{\mu\sigma}^{\rho}+\Gamma_{\mu\lambda}^{\rho}\Gamma_{\nu\sigma}^{\lambda}-\Gamma_{\nu\lambda}^{\rho}\Gamma_{\mu\sigma}^{\lambda}

R_{\mu\nu}=R^{\rho}_{\mu\rho\nu}

R_{\mu\nu}=R^{\rho}_{\mu\rho\nu}

R=R^{\mu}_{\mu}

R=R^{\mu}_{\mu}

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G_{N}T_{\mu\nu}

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G_{N}T_{\mu\nu}

Metric

Affine connection

Riemann tensor

Ricci tensor & Ricci scalar

Einstein field equations

My Work

Find the Newtonian Limit, if it exists

Test case, no limit in 3D

Re-implement CDT

Rewrite in modern C++
Use well-known libraries

My Work

Use current tools

LLVM/Clang
Hosted on GitHub
Continuous integration with Travis-CI
Google Style Guide
GoogleTest/GoogleMock
CMake for cross-platform building
Literate programming with Doxygen
Others (Gitter, Ninja, etc.)

Easy to evaluate, use, and contribute

My Work

Mass = Epp quasilocal energy

E_E\equiv\frac{1}{8\pi G_N}\int_{\Omega}d^2x\sqrt{|\sigma|}\left(\sqrt{k^2-l^2}-\sqrt{\bar{k}^2-\bar{l}^2}\right)

E_E\equiv\frac{1}{8\pi G_N}\int_{\Omega}d^2x\sqrt{|\sigma|}\left(\sqrt{k^2-l^2}-\sqrt{\bar{k}^2-\bar{l}^2}\right)

l\equiv\sigma^{\mu\nu}l_{\mu\nu}

l\equiv\sigma^{\mu\nu}l_{\mu\nu}

k\equiv\sigma^{\mu\nu}k_{\mu\nu}

k\equiv\sigma^{\mu\nu}k_{\mu\nu}

In 1+1 simplicial geometry, extrinsic curvature at a vertex is proportional to the number of connected triangles

In 2+1 simplicial geometry, extrinsic curvature at an edge is proportional to the number of connected tetrahedra

In 3+1 simplicial geometry, extrinsic curvature at a face is proportional to the number of connected pentachorons (4-simplices)

My Work

Non-Euclidean Distance

Calculate single-source shortest path between the two masses using Bellman-Ford algorithm in O(VE)
Modify allowed moves to not permit too many successive moves which increase or decrease distance between masses (hold relatively fixed)

Hausdorff Distance(?)

Calculate Voronoi diagram of Delaunay triangulation
Find minimal Hausdorff distance in diagram for sets in O((m+n)^6 log(mn)) (Huttenlocher, Kedem, and Kleinberg)

Fast foliated Delaunay Triangulations in CGAL

8 timeslices, 68 vertices, 619 faces, 298 simplices

Creation time: 0.043336s

(MacBook Pro Retina, Mid 2012)

Fast foliated Delaunay Triangulations in CGAL

/// @param[in] simplices  The number of desired simplices in the triangulation
/// @param[in] timeslices The number of timeslices in the triangulation
/// @returns A std::unique_ptr to the foliated Delaunay triangulation
auto inline make_triangulation(const unsigned simplices,
                               const unsigned timeslices) {
  std::cout << "Generating universe ... " << std::endl;

#ifdef CGAL_LINKED_WITH_TBB
// Construct the locking data-structure, using the bounding-box of the points
  auto bounding_box_size = static_cast<double>(timeslices+1);
  Delaunay::Lock_data_structure locking_ds(
  CGAL::Bbox_3(-bounding_box_size, -bounding_box_size, -bounding_box_size,
                bounding_box_size, bounding_box_size, bounding_box_size), 50);
  Delaunay universe(K(), &locking_ds);
#else
  Delaunay universe;
#endif

  auto universe_ptr = std::make_unique<decltype(universe)>(universe);

  auto causal_vertices = make_foliated_sphere(simplices, timeslices);

  insert_into_triangulation(universe_ptr, causal_vertices);

  fix_triangulation(universe_ptr);

  // This isn't as expensive as it looks thanks to return value optimization
  return universe_ptr;
}  // make_triangulation()

Fast foliated Delaunay Triangulations in CGAL

256 timeslices, 222,132 vertices, 2,873,253 faces, 1,436,257 simplices

Creation time: 284.596s

(MacBook Pro Retina, Mid 2012)

Remaining Work

Implement remaining 3D Ergodic moves
4D Ergodic moves
Restore Delaunay triangulation after moves(?)
HDF5 format data output
Improve Visualization pipeline
Build data analysis pipeline
Make faster
Parallelize (BOINC, Docker, etc)

Interested? Please Join!

Thank You!

[1] The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 4.6.3 edition, 2015.

[2] Steve Carlip. Why Quantum Gravity is Hard. Conceptual and Technical Challenges for Quantum Gravity, Rome, September 2014.

[3] J. Ambjorn, J. Jurkiewicz, and R. Loll. “Dynamically Triangulating Lorentzian Quantum Gravity.” Nuclear Physics B 610, no. 2001 (May 27, 2001): 347–82.

[4] Rajesh Kommu. “A Validation of Causal Dynamical Triangulations.” arXiv:1110.6875, October 31, 2011. http://arxiv.org/abs/1110.6875.