What do you picture in your head when you think about spacetime?

[jeebs]

Hi,
I'm trying to learn stuff for my relativity module, and it goes on about spacetime and the 4-dimensional continuum and four-vectors and all that business. I am having trouble imagining whatever it is I am supposed to picture when people talk about this.
Take this thing, i believe its called the "space time interval", ds = (ct,x,y,z) (is this called a Minkowski interval or something?).
Obviously its easy to imagine what the x,y,z part looks like, but then you've got this ct part.
It's clearly a velocity multiplied by a time, so it has dimensions of length and matches the x,y,z part in that respect. What is this ct part actually supposed to represent or tell us though - the distance light can travel in some time period t.
Why do we include this extra (zeroth, i think it gets called?) coordinate?
Which direction is this distance supposed to point, if this makes sense to ask?
Why, if we are using a velocity-time product, has the speed of light been chosen?

I don't even think I've asked this question very well... I'm just going through some notes we were given in class, and all I have been doing for the past couple of days is blindly following the maths through them without really understanding its significance.

Can anyone offer anything that might make me less confused?

 

[tiny-tim]

Hi jeebs!

What I picture in my head is all the equations without "c" in them, and time being just a length dimension.

So (ct,x,y,z) becomes just an ordinary vector, (t,x,y,z), with all components measured in the same units, so that for example you can rotate them however you like.

(And cosmologists tend to use coordinates in which c = 1 anyway)