Introduction to General Relativity. Note 12

Introduction to General Relativity. Note « 12 »

Space-time metric of the isotropic universe

The Friedmann's equation for the isotropic universe

(³/_a⁴)(a² + a'²) = κT^η_η

and the energy conservation equation

dε = -(ε + p)3^da/_a

can be integrated for the matter dominated universe, where the pressure is zero, p = 0, and the energy density ε is simply equal to the mass density μ, ε = μ. The energy conservation provides μa³=Const and the subsequent integration of the Friedmann's equation gives for a closed universe (a²>0) a "Big Bang" → "Big Crunch" scenario

a = a₀(1 - cos(η)) , t = a₀(η - sin(η))

For the open isotropic universe the Friedmann's equation provides a "Big Bang" → "Expansion Forever" scenario

a = a₀(cosh(η) - 1) , t = a₀(sinh(η) - η)

For a flat isotropic universe ds² = dt² - b²(t)(dx² + dy² + dz²) the scenario is also "Big Bang" → "Expansion Forever" (see the Exercise)

μb³ = const, b = const t^2/3.

At early stages with high densities the universe was (probably) rather radiation dominated, p = ^ε/₃. This, however, doesn't save us from the singular point at η = 0. Indeed, we have (for η << 1)

εa⁴ = const, a = const t^1/2

Exercises

Consider a flat (Euclidean) isotropic universe with the metric ds² = dt² - b²(t)(dx² + dy² + dz²):

Calculate the Christoffel symbols. [Γ^x_tx = Γ^y_ty = Γ^y_ty = ^b'/_b, Γ^t_xx = Γ^t_yy = Γ^t_zz = bb']
Calculate the Ricci tensor and the Ricci scalar. [R^t_t = -3^b''/_b, R^x_x = R^y_y = R^z_z = - ^b''/_b - 2^b'²/_b²]
Write down the ^t_t component of the Einstein's equations (with perfect fluid). [3^b'²/_b² = κε]
Write down the energy conservation equation ^dV/_V = - ^dε/_(ε+p). [3 ln(b) = - ∫^dε/_(ε+p)]
Integrate the equations for a matter dominated universe (p = 0, ε = μ). [μb³ = const, b = const t^2/3]
Integrate the equations for a radiation dominated universe (p = ^ε/₃). [εb⁴ = const, b = const t^1/2]