Index header
Intermediate Membranes

Quantum Gravity Simulation

This is a visualization of a simulation of a two-dimensional simplicial quantum gravity model. The surface is a dynamical triangulation, that is, as the simulation progresses, the way that the vertices of the triangular lattice are connected is constantly changing.

When the simulation began, we started with a lattice made of just four triangles, or simplices. Immediately, the computer began building the lattice up by, at random, choosing a triangle, putting a new point in the center and connecting the new point to the corners of the old triangle. Then there were three triangles where there was previously only one. This process was repeated until the lattice contained 30 points and 56 triangles.

Once the lattice is built up, the computer varies its geometry randomly. It does this in two ways:

  • Moving Nodes

    The computer can grab one of the nodes, or points on the lattice at random and try to move it a little bit in any direction.

  • Flipping Links

    The computer may pick two triangles that are next to each other and try to take out the side or link that they share. Then it replaces it so that it now connects the node on each \ triangle that was opposite the old link that they used to share.
With the "Slow" button depressed, watch the simulation and notice examples of moving nodes and link flips.

There are two general rules to the way this model behaves:

  1. The model we are simulating is one where all of the links are like little springs. The surface wants to be in a state where all of the links are as short as possible.
  2. Where two triangles meet along a link, the link behaves like a restaurant door hinge with a spring in it that tries to keep the door shut and flat with the wall. In other words, these springy hinges try to flatten out the surface.
By moving the slide bar labeled "kappa", at the bottom of the applet, you can adjust the strength of the springy hinges in rule (2). When kappa is big the hinges are very strong. When kappa is zero, the springy hinges have no effect, and rule (2) goes away. Kappa is called the bending rigidity. See how the geometry of the surface changes when you change the bending rigidity.

Pressing the "Thermalize" button causes the computer to attempt to make 4800 link flips and 4800 node moves before redrawing the surface. See how much different the surface looks after pressing thermalize once. Try to figure out roughly how many link flips and node moves it takes for the surface to look completely different.

Pressing either of the two "Measurement" buttons will allow you to see a plot of different quantities measured on the lattice as the simulation progresses. The top button shows the average length the links on the lattice. The bottom button shows the average cosine of the angles of the bends across the hinges. Reload the page and try pressing the measurement buttons as soon as the link flips start. How long does it take to reach an equilibrium? What happens to the quantities when you change the bending rigidity, kappa?