(View Cosmos DVD 6, episode 9, on Flatland and curved space.)For a black hole of mass M and angular momentum L, the horizon is a spherical surface located at a distanceNote that since mass is a scalar (having no directionality), it is a constant in every frame of reference.
r = (G M^{2} + (G^{2} M^{4} - c^{2} L^{2})^{1/2}) / (M c^{2})from the center of the black hole. This is a complicated function, so we will specialize for a moment to the (probably) unphysical case of a static black hole, which does not spin. Its horizon is then at
r = 2 G M / c^{2}The escape velocity necessary to escape the gravitational pull of a black hole gets larger as you get closer to the horizon. Using this expression for r, the escape velocity at the horizon is= 2953 meters * M / M_{solar}Note that this implies that G / c^{2} is a conversion factor for converting mass to length.
ev = (2 G M / r)^{1/2}which is consistent with our notions= c
The surface gravity at the horizon is
g = c^{4} / (4 G M)For M = 2 M_{solar}, this is almost 2 million times the surface gravity of Sirius B, and 4 times that of the neutron star in the center of the Crab Nebula. But there is a caveat for the neutron star: it is a pulsar, so we know it spins, and therefore using our simplistic version of the equation for the horizon is not exactly a valid comparison. However, even though the pulsar is spinning 30 times each second, the angular momentum term is less than 0.03% of the mass term, so this result is very close.= 1.52 * 10^{13} m/s^{2} / (M / M_{solar})
Consider now the tidal acceleration experienced by a 2-meter object at the horizon:
G M / r^{2} - G M / (r+2)^{2}Since M is a multiple of M_{solar}, G M is much greater than c^{2}, and this reduces to approximately= (c^{6} / (4 G M)) (c^{2} + 2 G M) / (c^{2} + G M)^{2}
c^{6} / (2 G^{2} M^{2})For M = 2 M_{solar}, this is about 5 * 10^{9} m/s^{2}. This means that our 2-meter object experiences a tidal acceleration over 500 million times Earth's surface gravity! But for M = 10^{6} M_{solar}, the tidal acceleration is 0.02 m/s^{2}: unnoticeable.= 2 * 10^{10} / (M / M_{solar})^{2}
It is expected that many if not most galaxies harbor a supermassive black hole in their cores, probably surrounded by an accretion disc, whose matter is accelerated to relativistic speeds and radiates tremendous amounts of energy as it falls into the horizon:
Here is a movie of the crab pulsar (source). The disc-like disturbances around the pulsar and the accretion discs around the black holes are related in that the curvature of spacetime is essentially the same (except in magnitude) around all spinning masses. It is probable that most if not all of the violent events we see in the universe, from novae to gamma ray bursts, are powered by either
Portfolio Exercise: Find a white dwarf other than Sirius B and a neutron star other than the one at the heart of the Crab Nebula, and compute the following: its density, the escape velocity at its surface, and its surface gravity. For the white dwarf, you will need to know its radius and mass; you will have to find its mass (which means it will probably have to be a binary companion), and you can compute its radius if you find its absolute magnitude and temperature. For the neutron star, your source will have to provide its mass and radius.Now compute the horizon radius for a black hole of mass equal to the masses of the white dwarf and neutron star you found. Using that radius, compute the escape velocity and surface gravity of the "equivalent" black hole. Express all escape velocities in terms of the speed of light, and all surface gravities in terms of Earth's surface gravity (9.8 m/s^{2}).
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.
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.
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.
Portfolio Exercise: Using the applet, 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 masses of 2, 2000 and 2000000 M_{solar}. 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?
Contrary to what you might have heard, this means that you will indeed expire when you cross the horizon of a black hole. Assuming you cross feet first, the future light cone of your feet is pointing into the center of the black hole, so no nerve impulses can reach your head from there. When your heart crosses, no blood can flow from it to your head. And when your head crosses the horizon, the parts of your brain cannot communicate with each other and your consciousness ceases.
©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.
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