Inside the event horizon of a Schwarzschild black hole the radial coordinate becomes timelike and the time coordinate becomes spacelike (they change signatures).

I read that timelike coordinates are one-directional and spacelike ones are bidirectional because an object with positive mass can only move along a timelike path, and that that means that inside the event horizon the radial coordinate can only decrease as an observer's proper time increases, leading to an inevitable encounter with the singularity. Does that mean that coordinate time can decrease if it's spacelike in the current metric?

Could an observer equipped with a rocket engine manipulate their time coordinate, or otherwise enter a geodesic with a fluctuating time coordinate? What would it look like?

  • $\begingroup$ Only forth, no back.. You can't go back in time because, if you go back to the time and you kill yourself..., then think what's wrong $\endgroup$ – user208739 May 5 '19 at 13:01
  • $\begingroup$ @AbhasKumarSinha You wouldn't be able to kill your past self because it is further away from the back hole and you can only approach it. Distance from the back hole takes the role of time in separating your personal past and future, but what happens to the original time coordinate? $\endgroup$ – Dorijan Lendvaj May 5 '19 at 13:05
  • $\begingroup$ Apologies, I don't know GR, I just watched the Episode of Stephen Hawking in Discovery, where he said, that one cannot Travel back in time using a simple experiment. It is also called Shakespeare's Paradox. Suppose you write a poem and you go back to time and give it to Shakespeare to publish it, then who's the writer of the poem. A varied version is available here - quora.com/What-is-the-Shakespeare-Paradox $\endgroup$ – user208739 May 5 '19 at 13:13

The $t$ and $r$ coordinates you're referring to are in a coordinate system called the Schwarzschild coordinates. General relativity allows us to use any system of coordinates, and there is no preferred system of coordinates. That's different from Newtonian mechanics, in which time is absolute, so everybody has to agree on time. The Schwarzschild coordinates are not even particularly convenient coordinates for describing a Schwarzschild black hole; they misbehave at the horizon.

Yes, you can go back and forth in $t$ if you're inside the horizon. But that doesn't mean anything special. The $t$ coordinate is spacelike there, so you're just going back and forth in space.

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