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I am new to physics and was trying to learn more especially about general relativity. The Schwarzschild metric, changes the sign of the time and radial parts of the metric once we cross the event horizon. Someone on stack exchange had noted that as a result inside the horizon the decrease in radius is the direction of increasing time.

My question is if radius (i.e space) becomes timelike due to change in metric sign, then does time also become spacelike. I mean can then inside the horizon one move freely in time?

NOTE: As far as my understanding of the linked question goes, that question is about cosmology of interiro of ablack hole (it asks about a spacetime originating form blackhole horizon). My question i think is different, its really more on the lines of time-travel.Since the signature of the time corrdinate is spacelike, it should behave like space and we should be able to move in it in the backward direction as we can in space. Some say that it is a coordinate artifact, but I think the other coordinates dont tend to have the physical interpretation of time.(I may have understood it incorrectly)

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marked as duplicate by John Rennie, Brandon Enright, JamalS, Neuneck, Pranav Hosangadi Jan 7 '15 at 11:59

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    $\begingroup$ Yes, time and space are "interchanged." As another feature, energy and momentum are interchanged. However, it is not good to say that we can move "freely," because it can be shown that we have a finite proper time during which we can be inside the horizon before we hit the singularity. $\endgroup$ – Ryan Unger Jan 7 '15 at 3:07
  • $\begingroup$ As the linked question explains, you're taking the space-time flip too literally. Time doesn't become space and we can't move both directions in time. $\endgroup$ – John Rennie Jan 8 '15 at 7:25
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Yes, the "time" coordinate of Schwarzschild coordinates is spacelike inside the horizon (meaning a path of constant radial coordinate and varying time coordinate is a spacelike curve). But this is just an artifact of the coordinate system chosen, with no particular physical significance; in special relativity one could likewise design a non-inertial coordinate system where some coordinate switched from being timelike to being spacelike past some arbitrarily-chosen boundary. And if you use a coordinate system called Kruskal-Szekeres coordinates on the Schwarzschild black hole spacetime, rather than Schwarzschild coordinates, then the radial coordinate is spacelike both inside and outside the horizon, and the time coordinate is timelike both inside and outside (another nice feature of these coordinates is that all light rays moving in a radial direction are represented as straight lines 45 degrees from the vertical, just as in inertial frames in SR, so the light cone structure is obvious).

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    $\begingroup$ So what's the physics of the "inside", though? It has to be independent of the choice of coordinates. $\endgroup$ – CuriousOne Jan 7 '15 at 4:03
  • $\begingroup$ @CuriousOne - If we're just talking about the physics as predicted by classical GR, the coordinate-independent stuff would be things like proper time between passing the event horizon and your worldline terminating at the singularity, tidal forces as a function of proper time, the proper time at which you would receive light signals from various other events inside or outside the horizon (like different ticks of a clock hovering or orbiting at constant radius outside the horizon), etc. $\endgroup$ – Hypnosifl Jan 7 '15 at 4:08
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    $\begingroup$ Yep, that's pretty much what I am talking about. All of that should be the same in all choices of coordinates. So the question is how we are getting this sign flip in the metric between the Schwarzschild and Kruskal-Szekeres. My GR classes were too long ago to remember... $\endgroup$ – CuriousOne Jan 7 '15 at 4:42
  • $\begingroup$ @CuriousOne: the transformation between Schwarzschild and KZ coordinates is different on the two sides of the event horizon. That's how $u$ remains spacelike and $v$ timelike everywhere. $\endgroup$ – John Rennie Jan 8 '15 at 7:24

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