Re: Event horizon questions?

Posted by James on May 13, 2002 at 14:37:20:

In Reply to: Event horizon questions? posted by universe on May 13, 2002 at 06:21:17:

On your question of time between the horizon and singularity. Mind that this is speculative (because we are causually disconnected from events inside the horizon) but based on Einstein's theory of gravity we can do something like this.

We can derive

dr/dTau = - (2M/r)^1/2

This is a proper radial velocity because we are evaluating our freefall velocity over the interval dr with respect to our wristwatch time dTau. If you were falling from the horizon to r = 0 this would be the time interval you measured using your wristwatch.

dTau = -(r^1/2 dr)/(2M)^1/2

When you integrate this over the distance r = 2M you get a simple solution which looks like this

Tau_meters = (4/3)M_meters

The time (Tau) and mass (M) are in geometric units of length so a Solar M = 1477 meters.

So the time you would measure on your wristwatch as you fall from the horizon to the singularity of a Solar mass back hole would be

Tau_meters = (4/3)1477_meters = 1969 1/3 meters.

To get this in seconds you divide this by the speed of light and get the answer

Tau_seconds = 1969 1/3_meters /300,000,000 meters/second

= .000006564 seconds

You can imagine (though they probably don't exist) supermassive black holes where it could take years to fall from the horizon to the singularity.

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