Schwarzschild equation physical meaning

Question

When you pass an event horizon of a black hole according to the Schwarzschild equation time and space swap the physical meaning. So you can no longer move away from a black hole, in similar way as you can't go back in time. Does this mean that when you pass a horizon you can go back in time, in similar way as you can go back in space?

Answer 1

Not exactly a swap, time becomes an imaginary number a square root of a negative number.

"Formula

The Schwarzschild radius is proportional to the mass with a proportionality constant involving the gravitational constant and the speed of light:

$r_\mathrm{s} = \frac{2 G M}{c^2}$

where:

rs is the Schwarzschild radius; G is the gravitational constant; M is the mass of the object; c is the speed of light in vacuum.$

Gravitational slowing of time

"In general relativity, clocks at rest run slower inside a gravitational potential than outside.

In the case of the Schwarzschild metric, the proper time, the actual time measured by an observer at rest at radius r, during an interval dt of universal time is (1 - rs/r)1/2 dt, which is less than the universal time interval dt. Thus a distant observer at rest will observe the clock of an observer at rest at radius r to run more slowly than the >distant observer's own clock, by a factor

$( 1 - rs / r )1/2$

This time dilation factor tends to zero as r approaches the Schwarzschild radius rs, which means that someone at the Schwarzschild radius will appear to freeze to a stop, as seen by anyone outside the Schwarzschild radius."

"In gravitational time dilation Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated using the Schwarzschild radius as follows:

$\frac{t_r}{t} = \sqrt{1 - \frac{r_\mathrm{s}}{r}}$ where:

tr is the elapsed time for an observer at radial coordinate "r" within the gravitational field; t is the elapsed time for an observer distant from the massive object (and therefore outside of the gravitational field); r is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object); rs is the Schwarzschild radius."

Source :

http://casa.colorado.edu/~ajsh/schwp.html#kruskal

"No stationary frames inside the Schwarzschild radius

According to the Schwarzschild metric, at the Schwarzschild radius $rs$, proper radial distance intervals become infinite, and proper time passes infinitely slowly. Inside the Schwarzschild radius, proper radial distances and proper times appear to become imaginary (that is, the square root of a negative number).

Historically, it took decades before this strange behaviour was understood properly (see again Kip Thorne's book ``Black Holes and Time Warps'' for an account). The problem with the Schwarzschild metric is that it describes the geometry as measured by observers at rest. It is now realized that once inside the Schwarzschild radius, there can be no observers at rest: everything plunges inevitably to the central singularity. In effect, the very fabric of spacetime falls to the singularity, carrying everything with it. No pressure can withstand the inexorable collapse."

What time is at the singularity is a guess at best. Perhaps singular or stopped, non-existent, or infinite on just the otherside of the singularity, if there is another side.