This applet illustrates the gravitational-lensing wavefront around a Schwarzschild black hole.
You should see a button to the left saying Show wavefronts window. (If not, see here.) To download, see below. |
Once the main window is up, press the run button to start a wavefront. Or rather, an animation of a bunch of photon trajectories, which together represent a wavefront. The large circle has radius R, and at its centre is a Schwarzschild black hole of unit gravitational radius (that is, horizon radius of 2). Photons start on the large circle, move closer to the black hole, and out again. Photons that would fall in are not shown at all.
As each photon leaves the large circle, the exit angle is recorded on the right panel. On the right panel, the horizontal coordinate is the exit angle and the vertical coordinate is the unlensed exit angle. Actually, the exit angles of only the upper half of the wavefront are considered by the the right panel, for easier reading. Thus, the unlensed angle is always between 0° and 180°, whereas the lensed exit angle has a maximum of 180° but can go below 0°, that is, a lensed photon can move into the lower half of the circle. The slope of the right plot is a component of magnification, hence the name.
On the right panel, you can zoom in or out, and re-centre with a mouse click.
Now try changing the right panel from magnification to flux. This changes the axes from angle to solid angle (in units of π). Imagine revolving the large circle on the left to generate a sphere sliced in half by the screen, then measure solid angle with respect to the same origin as before. The slope on this plot is the scalar magnification. Notice how it becomes singular at lensed exit angle of zero. At this point, the unlensed exit angle is slightly positive. (Zero unlensed exit angle would fall into the black hole.) Photons are conserved, obviously. But the lensed solid angle has a larger range than the unlensed. This means that the mean slope of the curve must be less than unity. Even asymptotically at the upper right, the slope is slightly less than unity.
Next, try increasing the number of photons to 1000 or more. This sends more photons closer to the r=3 circle. So you get photons doing multiple orbits of the black hole before emerging, and multiple folds to the wavefront. The horizonal (lensed) angles in magnification and flux can also go deeply negative.
Whenever the wavefront crosses the horizontal axis, the magnification becomes singular. This is indicated by inflexion points in the solid-angle curve at -2, and -4, and so on. These are easier to see if you set R=20 or so.
Finally, we see a generalization of the arrival-time surface. Try going setting tref=0.1 with R=20. The wavefront will run for a tenth as long as before and pause, and a second wavefront will emerge from a point at angle beta. When the two wavefronts cross, it is registered as a point on the arrival-time plot. The two main minima (the upper one being spatially a saddle point) are easy to see. By zooming and re-centring you can see one or two additional minima as well.
Thanks to Olaf Wucknitz for the remark that inspired this work.
With a Schwarzschild lens you just need one lensed supernova to have fun forever. (Provided you have a very big telescope.)