Re: Timelike and spacelike?

Paul W says...

Hello,

Reading about black holes got me confused by the words "Timelike" and
"Spacelike"; What do they actually mean?

Let e1 and e2 be two events (by "event" I mean a location in space *and*
time---so an event in Special Relativity is characterized by 4 numbers,
x,y,z and t). For definiteness, assume that in some frame, e1 occurs before
e2. In SR, there are three possible relationships between e1
and e2: (1) It is possible for a slower-than-light object to travel
from e1 to e2. (2) It is not possible for a slower-than-light object
to travel from e1 to e2, but it is possible for a light signal travel
from e1 to e2. (3) It is not possible for even light to travel from
e1 to e2. These three cases are called (1) a timelike separation,
(2) a lightlike separation, and (3) a spacelike separation.

In SR, if e1 and e2 have a timelike separation, then that means that
there is a rest frame in which e1 and e2 take place at the same location,
but at different times. If e1 and e2 have a spacelike separation, that
means that there is a rest frame in which e1 and e2 take place at the
same time, but at different locations.

In terms of components, if e1 has coordinates x1, y1, z1, t1, and
e2 has coordinates x2, y2, z2, t2, then you look at the quantity

s^2 = (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2 - c^2 (t2 - t1)^2

If this quantity is > 0, then the events are spacelike separated.
If it is < 0, then the events are timelike separated. If the quantity
is 0, then the events are lightlike separated.

--
Daryl McCullough
Ithaca, NY

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