Milne Koordinaten Transformation

Milne Universum: Proper Distance FLRW {r,t} ↔ Minkowski {x,T} Transformation

{dr,dt} → {dx,dT} {dx,dT} → {dr,dt} gᵤᵧ(x,T) → gᵤᵧ(r,t) gᵤᵧ(r,t) → gᵤᵧ(x,T)

{x,T} & {dx,dT} in terms of {r,t} & {dr,dt}

Input 1a:

Output 1a:

dT=dt·cosh[r/c/t]+(dr·t-dt·r)·sinh[r/c/t]/c/t;

Input 2a:

{r,t} & {dr,dt} in terms of {x,T} & {dx,dT}

Input 1b:

r=R·a=c·t·arctanh[vₓ]=√[(c·T)²-x²]·arctanh[x/T/c]; t=T/γ=√[T²-(x/c)²];

Output 1b:

dr=[c·(dx·T-dT·x)+(c²·dT·T-dx·x)]·arccoth[c·T/x]/√[(c·T)²-x²];

dt=(c²·dT·T-dx·x)/c²/√[T²-x²/c²];

Input 2b:

Transformation: Minkowski → FLRW

Input 1c:

ds²=c²·dT²-dx² → ds²=(c²-H²r²)·dt²+H·r·dt·dr-dr²

Output 0c:

Output 1c:

Input 2c:

Transformation: FLRW → Minkowski

Input 1d:

ds²=(c²-H²r²)·dt²+H·r·dt·dr-dr² → ds²=c²·dT²-dx²

Output 0d:

Output 1d:

Input 2d:

Mehr:

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Simon Tyran, Vienna

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