Frame of Ref: dx Observed as dt?

[eljose79]

given a frame of reference s In General Relativity in which you meassure an interval of space dx, could we have another frame S`so the observer in that S`see that dx observed in S as a time interval dt?

 

[robphy]

given a frame of reference s In General Relativity in which you meassure an interval of space dx, could we have another frame S`so the observer in that S`see that dx observed in S as a time interval dt?

If you are asking
"Could [in a small neighborhood] a displacement interval be determined to be spacelike (i.e., have negative square-norm) to one observer but timelike (i.e., have positive square-norm) to another observer?",
the answer is no.
The character of the displacement interval is determined by the metric tensor at the event, not by the choice an observer at that event.

 

[pervect]

given a frame of reference s In General Relativity in which you meassure an interval of space dx, could we have another frame S`so the observer in that S`see that dx observed in S as a time interval dt?

I don't think so. A spacelike interval will remain spacelike for all observers, a timlike interval will remain timelike.

 

[LURCH]

In anither thread (about Black Holes) somone mentioned reading that, at the EH, spacelike worldlines become timelike and vise-versa. Is this incorrect?

 

[selfAdjoint]

In the Schwartzschild metric, the "radial" direction out side the horizon switches to a time direction inside. This again is a feature of that coordinate system.

 

[pervect]

In anither thread (about Black Holes) somone mentioned reading that, at the EH, spacelike worldlines become timelike and vise-versa. Is this incorrect?

No, it is correct that at the event horizon the 'r' coordinate switches roles from being space-like to time-like. However, the original question was whether some specific interval could be considered space-like by one observer, and time-like by another. The fact that the r-coordiante is time-like inside the event horizon and space-like outside does not imply that different observers see the same interval differently. _All_ observers will agree that some vector pointing in the 'r' direction is spacelike when r is outside the event horizon. And they'll agree that a different vector r1 located inside the event horizon is timelike.