# Time coordinate = timelike?

1. Apr 25, 2010

### vertices

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

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.

2. Apr 25, 2010

### George Jones

Staff Emeritus
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.

3. Apr 25, 2010

### JesseM

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".

4. Apr 26, 2010

### 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.

5. Apr 26, 2010

### JesseM

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.

6. Apr 26, 2010

### DrGreg

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).