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Geometry of a homogeneous isotropic universe. Friedman metric.

Asume that the universe has always been homogeneous and isotropic (Friedman model). The metric can then be chosen as ds2=dt2−dl2, where the isotropic three−dimensional line element dl2 can be written as
dl2 = B(r)dr2 + r2(dθ2+sin2θ dφ2) .
The Ricci tensor (borrowed from the Schwarzschild metric calculation) is then equal
Rrr= B'/rB  , Rθθ=1− 1/B + rB'/2B2  .
In the isotropic space (= space with constant curvature) the Ricci tensor must have the form Rab=2λgab , where λ is some constant. This gives B= 1/1−λr2 . Lambda can be negative (open universe), positive (closed universe), or equal to zero (flat universe). For the closed universe denoting λ= 1/a2 >0 and making a substitution r = a sinχ, where 0<=χ<=π, gives the metric of a four−dimentional sphere
dl2=a2(dχ2+sin2χ(dθ2+sin2θdφ2)) .
For the closed universe denoting λ=− 1/a2 <0 and making a substitution r = a sinhχ, where 0<=χ<=∞, gives the metric of a four−dimentional hyperboloid
dl2=a2(dχ2+sinh2χ(dθ2+sin2θdφ2)) .
This type of metric is called Friedman metric (or Freidman−Lemaitre−Robertson−Walker metric).

Friedman equation.

Friedman metric describes a homogeneous and isotropic universe. For the closed universe the interval is
ds2 = a2(dη2 − dχ2 −sin2χ (dθ2 +sin2θ dφ2)) ,
where r=asinχ, η is the scaled time coordinate, dt=adη, and a(η) is the scale parameter of the universe (the radius of the 4−sphere). 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− 1/2 Rδab = κTab will then have the ηη component

( 3/a4 )(a2+a'2) = κε ,
also called Friedman's equation, and the three identical spatial equations ( 1/a4 )(a2+2aa''−a'2)=−κp .

Exercises

  1. Calculate the Ricci tensor and the Ricci scalar for the metric
    ds2=a2(η)(dη2−dχ2−sin2χ (dθ2 +sin2θ dφ2))

  2. Calculate the volume of the closed and open universes.

  3. Prove that for a perfect fluid, Tab=(ε+p)uaub−pgab, no solution of the Einstein's equation is homogenius, isotropic and static. Hints: ε>0, p>0, Taa>=0.

  4. Prove, that with a cosmological constant, Rab− 1/2 Rgab=κTab+Λgab , a static solution does exist.

Copyleft © 2005 D.V.Fedorov (fedorov @ phys au dk)