ABSTRACT
We will take a specification of a space-time as a set of coordinates xµ with a non-singular metric gµν(x) with Lorentzian signature, given as an infinitesimal invariant interval, also known as line element, and study some of its properties1. Specifically, we consider,
Minkowski (No gravity) ∆s2 = −∆t2 + ∆x2 + ∆y2 + ∆z2
Rindler (Uniform) ∆s2 = −g20z2∆t2 + ∆x2 + ∆y2 + ∆z2, z > 0
Rotating Disk (Centrifugal) ∆s2 = −f(ρ)∆t2 + 2h(ρ)∆t∆φ+ g(ρ)∆φ2 + ∆ρ2 + ∆z2
f(ρ) := e−ω 2ρ2 − ρ2ω2e+ρ2ω2 ,
h(ρ) := −ωg(ρ) , g(ρ) := ρ2e+ρ2ω2
Schwarzschild
(Spherical) ∆s2 = − (1− 2GMr )∆t2 + (1− 2GMr )−1 ∆r2 + r2∆Ω2 FRW
(Cosmological) ∆s2 = −∆t2 + a2(t) {
1−κr2 + r 2∆Ω2
} where, ∆Ω2 :=
( ∆θ2 + sin2θ∆φ2
) Plane wave ∆s2 = (ηµν + hµν)∆x
µ∆xν , where, (Undulating) hµν(x) = µν(k)e
ik·x + ¯(k)µνe−ik·x and ηµν = diag (-1, 1, 1, 1) .
