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Avoid Infinity!

In Einstein's theory of general relativity, gravity can become so strong that matter can collapse under its own weight to infinite density. Such an event causes what is called a singularity. There is a conjecture, called cosmic censorship, that proposes that all singularities that might develop in the real universe are hidden away inside event horizons. The horizon marks the boundary of a black hole, behind which no information can escape to the outside universe. Therefore cosmic censorship implies that all singularities are hidden inside black holes.

When attempting to evolve a black hole numerically, we somehow must be able to deal with the singularity inside. But a computer program will "crash" if it tries to compute quantities that are infinite, so some strategy must be developed to hide the singularity from the computer too.

SpacetimeDiagram 1
Avoid Infinity -- Diagram 1

In the diagram above, the wavy line represents a singularity and the red line represents the black hole horizon, evolving in time. Left of the horizon is the black hole's interior. This region can never communicate with the universe outside the horizon.

Now imagine trying to compute the value of the gravitational field at different points in space at different times. The "time slices" in diagram 2 below (marked t = 0,10,20) provide a way by which we can divide up time to do calculations. But, if we took such "straight" horizontal slices, we would intersect the singularity before time t=20 and our code would crash because of the infinities there.

SpacetimeDiagram 2
Avoid Infinity -- Diagram 2

If we want to perform some interesting calculations, out where the gravitational waves are, a cleverer choice would be to take time slices that bend upwards. In diagram 3 below we show two slices (marked t=10 and t=30) that cover more of the spacetime of interest, including the gravitational waves leaving the black hole, while at the same time avoiding the infinities found at the singularity. But, although these bent time slices allow us to study more of the spacetime, they also cause lots of numerical trouble. As portrayed in the diagram, the time slices bend more and more as we reach ever further into the future in one place. While avoiding the singularity inside the black hole, the calculations become more difficult, until eventually the slicing "snaps" and again the code crashes.

SpacetimeDiagram 3
Avoid Infinity -- Diagram 3

In one new approach illustrated below (diagram 4), the horizon is used as an inner boundary. Since information inside the horizon can never get out, in principle we could use slices like those marked below to do our calculations on. These slices do not have any "bending pathology," but their length varies, and data on the left edge of the slice must be specified in some way. Called Horizon Boundary Conditions, this novel computational method is being pursued by many research groups at present.

SpacetimeDiagram 4
Avoid Infinity -- Diagram 4

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Copyright 1995, The Board of Trustees of the University of Illinois

NCSA. Last modified 11/9/95.