GR Schutz: Why r Must Decrease in Black Hole Section?

[Harel]

Schutz states in his book in the black hole section that: At r<2GM, r is a timelike coordnate, while t has become spacelike: even more evidence for the funniness of t and r! Since the infalling particle must follow a
timelike world line, it must constantly change r, and of course this means decrease r.
I understand why r must change but why does it means deacreas in r and not increase?

 

[bcrowell]

This is a great question.

There is nothing in GR that distinguishes past from future. Therefore it is not possible that the unique spherically symmetric solution to the vacuum field equations would distinguish past from future. The equations and coordinate charts you're referring to can be described equally well as a white hole, which only emits matter and radiation, but never accepts it.

It is also possible to extend those coordinate charts. The maximally extended version of the Schwarzschild spacetime includes both a black hole and a white hole. However, the white hole can't be formed by gravitational collapse, so we don't think white holes exist in our universe.

I find this kind of thing impossible to analyze without a Penrose diagram. I have a simple, easy intro to Penrose diagrams in my book Relativity for Poets http://www.lightandmatter.com/poets/ , section 11.5. For a fancier discussion, see section 7.3 of General Relativity http://www.lightandmatter.com/genrel/ . For the maximally extended Schwarzschild spacetime, try Carroll http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html .

 

[Orodruin]

However, the white hole can't be formed by gravitational collapse, so we don't think white holes exist in our universe.

This is true in GR. In loop quantum gravity there are some bounce solutions where the collapse can be turned around. This may be true also in other quantisations, but this is the one I am aware of. (Of course, loop quantum gravity remains unverified to this date.)