The Relativity Playground

This page contains discussions and a Java applet which are designed to help you understand the geometric nuances of General Relativity. You will need the Java Runtime Environment (JRE); if you do not have it, you can get it free at java.sun.com.

There are a lot of web sites with evocative pictures that illustrate the ideas of Relativity. This is not one of them. Relativity is fundamentally about using mathematics to understand the nature of spacetime: the thing containing the entire universe, in all its past and future moments. So we are not going to shy away from numbers here. Our aim is to provide a layperson's interface to the full mathematical apparatus of Relativity. This page is for those willing to roll up their sleeves and get a little dirt on their hands. That said, Enjoy!

Curvature and Geodesics around a Black Hole

The hardest thing to conceptualize in General Relativity is how spacetime can be curved: how can something without substance have shape? With enough play time around a black hole, perhaps this can become clear, or at least clearer than it is now...

Our black hole is electrically neutral, so it is described by only two parameters: its mass and its angular momentum. Angular momentum is a measure of an object's rate of spin, but also takes into account its bulk. A collapsing star must have a mass "M" greater than 1.86 times the mass of the Sun to form a black hole, and its angular momentum "a" must be less than or equal to "M" for the black hole to have a horizon. The escape velocity necessary to escape the gravitational pull of a black hole gets larger as you get closer to the horizon; at the horizon, it is equal to the speed of light. So once something crosses the horizon, it can never return.

In the following applet, you can choose "M" and "a" for the black hole. You can also choose the initial position (relative to the horizon) and speed of a pair of probes, and their initial orbital direction: either azimuthal (around the "equator") or polar (along a "great circle" through the "north pole"). The "Replot" button starts the computations. Please be patient: each plot requires evaluation at over 3.5 million points! The "Replot" button will be labeled with ellipsis while the evaluation process is taking place.

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

The plot on the left is a 3-D plot showing the curvature of spacetime around the black hole, and the paths of the probes, which follow geodesics. The Geodesic Equations describe the path of any object whose own mass is negligible compared to the mass of the black hole (including light rays). The equations become very "stiff" near the horizon, so the last segment or two of the geodesic plots may not be smooth. The curvature is measured by the Kretschmann Invariant. Spacetime is described by a metric, which literally tells how the measurement of intervals varies from event to event. The Kretschmann Invariant is an algebraic function of the metric and its derivatives, which measures the curvature in a way that does not depend on the coordinate system used to describe the metric.

The 3-D plot is colored magenta where the Kretschmann Invariant is positive, and cyan were it is negative. Where the plot appears blue or white, you are seeing positive regions behind negative ones, or vise versa. The axes on the three dimensional plot are not drawn inside the horizon of the black hole. The region inside the horizon on the 3-D plot does not appear to be completely black because you are seeing it through some nonzero values of the invariant. The plot scale automatically changes so that the width of the plot (at "r scale" = 1) is four times the radius of the horizon assuming "a" is 0. So when "a" is greater than 0, you may want to decrease the "r scale" value and replot, which has the effect of taking a closer look.

The applet also provides a two dimensional plot (on the right) of time vs. position of the geodesics. The axes menus allow you to control the two dimensional plot. The origin of this plot is (0, 0) except when plotting radial distance, when it is (horizon radius, 0). The coordinates of the upper right hand corner are determined by the maximum values begin plotted (given in the "Final conditions" window at the bottom of the applet). For the polar and azimuthal angles, the right hand edge is at p/2 and 2 p, respectively. The polar plot includes a yellow line which marks the value of the polar angle where the curvature is zero at the horizon.

You can rotate the 3-D image by dragging on it with the mouse (note that rotation will take a little while; see above). "Theta" is the polar angle (theta=0 is above the positive z axis, theta=p/2 is on the x-y plane, and theta=p is below the -z axis). "Phi" is the azimuthal angle (rotating around the z axis from 0 to 2p on the x axis).

Things to try...

Lower the initial speed of the probes to 30% of the speed of light. This is less than the escape velocity and so the probes cross the horizon; but when? By default, the 2-D graph shows radial distance versus proper time (which is the time the probes measure). You can see that the probes cross the horizon in a finite amount of time, and the "Final conditions" data indicates that it is a very short time indeed. But if you choose "asymptotic time" for the 2-D vertical axis, a very different graph appears. We see that from the point of view of an observer "at infinity", the probes never cross the horizon but only approach it asymptotically. This is an eloquent indication that the coordinates we use to make measurements far from the black hole do not have the expected meaning in the region near the horizon.

You can see that changing "M" is only interesting if "a" is nonzero; this is because the applet automatically changes scale with "M". You can also see that "M" must be small and "a" must be a significant fraction of "M" before the plot changes substantially. This is a physical consequence of the spacetime metric around the hole.

Start the probes in a polar orbit with "a" > 0; observe that near the horizon, in the region called the ergosphere, the angular momentum of the black hole "drags" the probes in the direction of the black hole's rotation. This is called frame dragging.

Note that as "a" increases, regions of "negative curvature" appear at the poles. What can this mean physically? Try "a" = .9 "M", and speed = 30% of the speed of light with both polar and azimuthal geodesics. The paths of the polar geodesics indicate that the regions of "negative curvature" correspond to centrifugal barriers.

Vary the initial position and speed to see how the escape velocity depends on distance from the horizon. Do this first for a = 0 and both large and small masses. Then set a = .9 M and do the same. Does the escape velocity depend on the mass? Does it matter whether the probes follow an azimuthal or polar orbit?

We have really cheated a bit here for dramatic effect: we really don't need a black hole to see the same effects. The Kerr metric used in this applet describes spacetime around any isolated, compact rotating massive object. So the same effects, albeit much less pronounced, can in principle be measured around our Sun, or even around the Earth or the Moon.

Please send comments or suggestions to the author.