Is the Time Coordinate Timelike or Spacelike?

[vertices]

Sorry, I do realize that this is probably a really stupid question, but can someone please help me understand the following statement:

[For a given metric] ...t is timelike and x spacelike for x > 2m, their roles are swapped for 0 < x < 2m.

This means that the region x > 2m is time-reversal invariant – taking t → −t the pattern of
possible trajectories are the same.

However, for 0 < x < 2m, where x plays the role of ‘time’, we must choose between time
running from left to right, i.e. with increasing x, or from right to left, i.e. with decreasing x.

My understanding of a timelike path, is one which always stays inside a light cone, defined by lines with gradients c and -c that cross at the origin.

What does it mean to say that time coordinate itself is timelike or spacelike (and the same for space-coordinates?) And what does it mean to say "x plays the role of time".

Thanks.

 

[George Jones]

What does it mean to say that time coordinate itself is timelike or spacelike (and the same for space-coordinates?) And what does it mean to say "x plays the role of time".

To find the nature of any coordinate, let that coordinate vary while holding all the other coordinates constant (fixed). This results in a curve in spacetime. At any event p on the curve, the tangent vector to the curve at p gives the nature (spacelike, lightlike, or timelike) of the coordinate at p. Note that even if both p and q are on the same coordinate curve, the nature of the coordinate does not have to be the same at p and q.

 

[JesseM]

My understanding of a timelike path, is one which always stays inside a light cone, defined by lines with gradients c and -c that cross at the origin.

You can also define a timelike path in terms of the metric--use the metric to do a line integral of ds^2 along the path, if it's spacelike the result should be positive, if it's timelike the result should be negative. And as George Jones said, a small increment of the time coordinate (while holding other coordinates constant) results in a path through spacetime as well. Likewise a small increment in the x coordinate will result in a different path, and if this path is timelike, that's what it means to say "x plays the role of time".

 

[vertices]

Thanks George and Jesse.

So to work out, for example, whether time is timelike, I just set all the (dx^\mu)^2 terms in the metric to zero, where \mu\neq0

If I get ds^2=adt^2 where a is positive, t is timelike; if a is negative, t is spacelike?

Thanks.

 

[JesseM]

If I get ds^2=adt^2 where a is positive, t is timelike; if a is negative, t is spacelike?

I think ds^2 is normally defined in such a way that if it's positive, the interval is spacelike, and if it's negative, it's timelike.

 

[DrGreg]

If I get ds^2=adt^2 where a is positive, t is timelike; if a is negative, t is spacelike?

I think ds^2 is normally defined in such a way that if it's positive, the interval is spacelike, and if it's negative, it's timelike.

Unfortunately there is no universally agreed convention which way round this should be; different authors do it different ways. If you're not sure of the author's convention, remember there ought to be 3 spacelike coordinates (all with the same sign for ds²) and 1 timelike coordinate (with the opposite sign) (unless there are null coordinates for which ds=0).