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Friedmann's equations

In the isotropic and uniform universe the interval ds2 can be conveniently written using the 4-dimentional spherical coordinates (r = asinχ)
ds2 = a2(dη2 - dχ2 - sin2χ (dθ2 + sin2θ dφ2)) ,
where η is the scaled time coordinate (dt = adη) and a(η) is the scale parameter of the universe (the radius of the 4-sphere). In these coordinates the components of the (diagonal) Ricci tensor are
Rηη = (3/a4)(a'2-aa''),   Rχχ = Rθθ = Rφφ = -(1/a4)(2a2+a'2+aa''),   R = -(6/a3)(a+a'')
where prime denotes the derivative with respect to η.

Assuming that the universe is filled with a perfect fluid the energy-momentum tensor of the matter is
Tab = (ε + p)uaub - pgab
where ε is the energy density and p is the pressure. In our frame, where the matter is at rest, the 4-velosity ua={1/a,0,0,0}.

The Einstein's equations (Rab-½Rδab = κTab) will then have the ηη component
(3/a4)(a2 + a'2) = κε,
also called Friedmann´s equation, and the three identical spatial components
(1/a4)(a2 + 2aa'' - a'2) = -κp
Assuming that the universe expands adiabatically, the energy conservation dE+pdV=0 gives another equation
dε = -(ε + p)3da/a,
which can substitute the second Einstein's equation.

Exercises

  1. Prove that for a perfect fluid (Tab=(ε+p)uaub-pgab) no solution of the Einstein's equation is homogenius, isotropic and static. Before Hubble's discovery Einstein considered this a failing of the theory and introduced the cosmological constant Λ as a remedy (Gab=κTab → Gab=κTab+Λgab, where Gab≡Rab-½Rgab).
    Hint: ε>0, p>0, Taa >= 0
  2. Show that with a nonzero cosmological constant a static isotropic solution does exist.

Copyleft © 2003 D.V.Fedorov (fedorov@ifa.au.dk)