The Little Book of Black Holes by Steven S. Gubser & Frans Pretorius

Author:Steven S. Gubser & Frans Pretorius [Gubser, Steven S. & Pretorius, Frans]
Language: eng
Format: epub
Tags: Science, Physics, Astrophysics, Cosmology, Space Science, Relativity, Gravity, General, Astronomy
ISBN: 9780691163727
Google: mmuYDwAAQBAJ
Publisher: PrincetonUP
Published: 2017-10-10T19:35:22+00:00

FIGURE 4.2. Illustration of the Penrose process, looking down the spin axis of the black hole onto the equatorial plane, where the mining vessel and projectile orbit.

Actually, so far nothing about our discussion of the Penrose process is unusual or remarkable. In fact, if we’d run this thought experiment with the black hole replaced by the Sun, the same conservation arguments would apply. The Sun, on consuming the projectile, would have its angular momentum lessened, while the vessel gains an equivalent amount and so gains kinetic energy. However, in this case, the vessel can never gain enough kinetic energy to compensate for the energy-equivalent of the mass carried away by the projectile. For the rotating black hole something unusual is happening: If the orbit is tuned carefully and the projective aimed well, the vessel can gain so much kinetic energy that it compensates for the loss of the projectile with extra to spare. It’s not easy to come up with an intuitive explanation of all that happens in the black hole case. Let us instead describe one key piece of the calculation that illustrates another bizarre property of the extreme warping of space and time around black holes and explains why it’s crucial for the Penrose process that the projectile is fired from inside the ergosphere.

First, we need to make a brief digression to talk about the energy of an object in an orbit. Energy can take different forms. Rest energy is the energy of mass itself, and that is what the equation E = mc2 refers to. There’s also kinetic energy, which is the energy of motion. And, at least in Newtonian gravity, there’s potential energy, which describes how deep in a gravitational well an object is located. Potential energy is negative because it’s the energy you would have to add to an initially stationary object to lift it out of the gravitational well in which you find it. In Newtonian gravity, the total mechanical energy of an orbiting object (that is, its kinetic plus gravitational potential energy) never changes, provided the only force acting upon the object is the gravitational pull from a large, stationary mass like the sun. Any change in the kinetic energy is balanced by an equal and opposite change in the potential energy. In general relativity, it’s trickier to give a definition of potential energy that makes sense in all spacetimes, but at least for an object moving in the Kerr geometry it is possible to do so. The result is consistent with the Newtonian definition far from the black hole. So the upshot is that around a Kerr black hole the total mechanical energy of an object in orbit (which now includes its rest mass energy) can be defined, and because the orbiting object follows a geodesic, this total energy never changes.

Now here’s where the bizarre property of frame dragging comes into play. There are geodesic orbits in the Kerr geometry, entirely confined within the ergosphere, with the property that particles following them

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