The picture of spacing falling into a black hole has a sound mathematical basis, first discovered in 1921 by the Nobel prize-winner Alvar Gullstrand2, and independently by the French mathematician Paul Painlevé3.
It is not necessary to understand the mathematics, but I do want to emphasize that, because the concept of space falling into a black hole is mathematically correct4, inferences drawn from that concept are correct.
The Gullstrand-Painlevé metric is
ds2 = − dtff2 + (dr − v dtff)2 + r2(dθ2 + sin2θ dφ2)
which is just the Schwarzschild metric expressed in a different coordinate system. The free-fall time tff is the proper time experienced by observers who free-fall radially from zero velocity at infinity. The velocity v in the Gullstrand-Painlevé metric equals the Newtonian escape velocity from a spherical mass M
v = − ( 2 G M / r )1/2
with a minus sign because space is falling inward, to smaller radius.
Physically, the Gullstrand-Painlevé metric describes space falling into the Schwarzschild black hole at the Newtonian escape velocity. Outside the horizon, the infall velocity is less than the speed of light. At the horizon, the velocity equals the speed of light. And inside the horizon, the velocity exceeds the speed of light. Technically, the Gullstrand-Painlevé metric encodes not only a metric, but also a complete orthonormal tetrad, a set of four locally inertial axes at each point of the spacetime. The Gullstrand-Painlevé tetrad free-falls through the coordinates at the Newtonian escape velocity.
It is an interesting historical fact that the mathematics of black holes was understood long before the physics. Einstein himself misunderstood how black holes work. He thought that the Schwarzschild geometry had a singularity at its horizon, and that the regions inside and outside the horizon constituted two separate spacetimes. I think that even today research into general relativity is too often dominated by abstract mathematical thinking at the expense of conceptual understanding.
Добавлено: Пт апр 22, 2011 11:05 pm
