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d(tau)^2 = ρ^2 dω^2 - dρ^2.

He then goes on to say that this metric is the hyperbolic version of the same formula for Cartesian space, i. e.,

ρ^2 dθ^2 + dρ^2

My question is what does the Cartesian version equal? It cannot equal d(tau)^2. Does it equal dS^2? That is,

dS^2 = ρ^2 dθ^2 + dρ^2

And what is this telling us? Is it the formula for the distance between 2 points on a Cartesian plane in polar coordinates? I looked for a derivation of this formula online and on you-tube and couldn't find it. Can someone direct me to a "B-level" online link I can look at?