Can somebody explain the Warp Theory Equation?

[xXIHAYDOIXx]

Could somebody please explain the following equation for me?
ds^2 = −c^2 dt^2 + [dx − v_s (t)f(r_s)dt]^2 + dy^2 + dz^2 where f(r_s)=(tanh(σ(r_s + R)) − tanh(σ(r_s − R)))/(2tanh(σR).

I've wrestled with it for a while, but quite frankly do not get it. I'm only up through Calc II, but I'm more than happy to learn a few new concepts, should the need arise. This uses the familiar coordinates, (t, x, y, z) and curve x = x_s(t), y = 0, z=0 where x is analogous to what is commonly referred to as a spacecraft ’s trajectory.

 

[pervect]

That equation is a line-element, or metric. It provides a map of a space-time geometry. (t,x,y,z) tell you where and when you are on the space-time map. The metric equation gives the "distances" (actually the Lorentz interval, because it's a space-time map and not just a spatial map) between nearby points. I.e. the "distance" ds between two points at (t,x,y,z) and (t+dt, x+dx, y+dy, z+dz) is given by the formula and is the value of ds.

Distances are a convenient way of describing a space-time geometry, just as the table of distances between nails in a rowboat can be used to describe it's shape as in http://www.eftaylor.com/pub/chapter2.pdf.

Or, as Taylor says, "distances determine geometry". So the map, or metric, is a specification of a space-time geometry.

If you're not familiar with the Lorentz interval, it's a very basic concept of special relativity, which is very much recommended to tackle before you try to tackle general relativity.