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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)

ssj3892414
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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).

Hypnosifl
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