How can I understand $\mathrm ds^2 = -c\,\mathrm dt^2 + [\mathrm dx-v_s(t)f(r_s)\mathrm dt]^2 +\mathrm dy^2 +\mathrm dz^2 $ in the simplest way? How can I understand this equation $$\mathrm ds^2 = -c\,\mathrm dt^2 + [\mathrm dx-v_s(t)f(r_s)\mathrm dt]^2 +\mathrm dy^2 +\mathrm dz^2 $$ in the simplest way?
I am a 13 year old boy who is totally embarrassed after reading the equation from Harold white's warp field mechanics.
I think you would help me to understand this equation.
 A: It expresses infinitesimal distance in space-time coordinates.
What's interesting is the long term, representing some kind of wave moving on the $x$ axis.
I assume you know the pythagorean theorem $ds^2 = dx^2 + dy^2$.
Well, this just adds the $z$ and $t$ coordinates.
The long term defines a coordinate that replaces the $x$.
It is a combination of $x$ and $t$, much like a velocity, and combines it into the distance.
It's as if your coordinate system (in which you're measuring distance) were moving along the $x$ axis at speed $v_s(t)f(r_s)$.
Answering comment: The axes are whatever you choose them to be, as long as they're locally orthogonal. You choose the $x$ axis as the direction of movement.
A system moving at velocity $V$ can be described at $dx - Vdt$, meaning if $V=5$, for example, time increases by $1$ and $x$ increases by $5$, it's still the same thing. OK, so what is $V$? It's that funny thing $v_s(t)f(r_s)$ that varies with time $t$, distance $s$, and that function $f$ that depends on $r_s$, whatever that is. So maybe it's slowing down, or speeding up, or something.
