Geodesics around Black Holes

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 where 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 is four times the radius of the horizon assuming "a" is 0. To zoom, turn the scroll wheel; to change the perspective, drag the mouse across the window.

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 π/2 and 2 π, respectively. The polar plot includes a yellow line which marks the value of the polar angle where the curvature is zero at the horizon.

Things to try...

Lower the initial speed of the probes to 50% 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" = 1 (an extremal black hole); 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? 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 the initial position. (You have found the escape velocity when the 2-D radial distance plot for the inner probe is a vertical line, or leans just slightly to the right.) 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 found that the escape velocity at the horizon is the speed of light. Yet you may have noticed that even when starting the probes from a distance, at almost the speed of light, sometimes they enter the horizon. While it is true that the Newtonian escape velocity is the speed of light at the horizon, Einstein's General Relativity includes additional effects which imply that there is a region around the black hole for which there are no circular orbits. Since our probes start in a direction tangent to a circle around the black hole, in that region they must fall in!

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.