Understanding Spacetime Geometry: Timelike and Spacelike Vectors Explained

[jainabhs]

Hi
Can anyone tell me what is timelike vector and what is spacelike vecor?
I read the in a doc for spacelike vector :
For events with spacelike separation, |(x,t)| is called the proper distance between them; an observer who judges them to have happened simultaneously measures t = 0, so |(x,t)| = x.

In the same doc for timelike vector it is written that
For events with timelike separation, |(x,t)| is called the proper time between them; an observer who judges them to have happened at the same place measures x = 0, so |(x,t)| = t.

Can anyone tell me what do these two descriptions mean?
I really didnt understand it, explaining this will help me moving further on this

Thanks in anticipation

Abhishek Jain

 

[pervect]

I talked about this in another theread, but the basic idea is simple. Suppose you have two events in an inertial frame in special relativity (SR). They have coordinates event1:(x1,t1) and event2:(x2,t2). Let dt = t2-t1, and dx=x2-x1. We will chose the ordering and position of events so that dx > 0 and dt>0. Let c be the speed of light.

Then if c dt>dx, the interval is time-like. There is enough time for a light beam to get from event 1 to event 2.

If c dt < dx, the interval is space-like. There is not enough time for a light beam to get from event 1 to event 2. Different observers will not agree about whether event 1 happened first or whether event 2 happened first, but they will all agree that the inverval between event 1 and event 2 is space-like.

if c dt = dx, the interval is a null interval. Light will just have enough time to propagate from event 1 to event 2.

 

[robphy]

To add to pervect's comment...
and to emphasize why there are three types of vectors...

Special Relativity has a "geometry" associated with it, called "Minkowski Geometry", which has analogues with ordinary Euclidean geometry. (Minkowski came up with the names "timelike" and "spacelike").

In three-dimensional Euclidean geometry, the square-norm (which you can think of as the "squared-length") of a vector with components (x,y,z) is given by S^2=x^2+y^2+z^2 (the Pythagorean theorem). In Euclidean geometry, all nonzero vectors have positive square-norm and only the zero-vector has zero square-norm. In four-dimensional Minkowskian geometry, a vector with components (t,x,y,z) has square-norm given by S^2=t^2-x^2-y^2-z^2 [using the so-called +--- signature convention]. In this geometry, there are three classes of nonzero vectors: timelike (where S^2>0), spacelike (where S^2<0) and lightlike [or null] (where S^2=0).

The distinction is best described using the "light cone" at an event. Vectors that point along the interior of the light-cone are timelike, and can represent the path of a [necessarily, massive] inertial observer. Vectors that point into the exterior are called spacelike, which can represent a spatial displacement to some inertial observer. (IMHO, a spacelike vector is best defined as a vector that is [Minkowski-]perpendicular to a timelike vector.) Vectors that point along [i.e. tangent to] the light cone are called lightlike or null, which can represent the path of a free photon.]

In Galilean geometry (the spacetime geometry underlying Galileo's kinematics), there are only two types of vectors... timelike and spacelike-and-null, from spacelike and null collapsing into a single case. (Lightlike, being decoupled from null, is now no longer a useful notion in this geometry.)

 

[SF]

I recommend: Feynman Lectures - V1 - 17 - Space-Time.

 

[jainabhs]

Thank you very much for your replies. It really has helped.
Please send the link for downloading feyman's lectures.

 

[dicerandom]

I don't believe you can download them legally, it's something you're supposed to buy.

 

[actionintegral]

Yes, they are not something you can download legally from a file-sharing neighborhood.