ASSIGNMENT 1, due date Fri. January 31st.

Note: this assignment will have an extra low weight in the final mark, unless you do it perfectly !

Problem I:

While in class we have used a metric for homogeneous and isotropic Universe in the form

$$ds^2 = c^2 dt^2 - a^2(t) \left( dr^2 + S_k^2(r) d\Omega^2 \right)$$ (1)

where time variable $t$ measures the proper time of a stationary observer ( $dr=d\Omega=0 \to ds = c dt$), this metric is often useful to write be using using so-called conformal time $\eta$, related to $t$ by differential relation $d \eta = c dt/a(t)$. In conformal time, interval is given by

$$ds^2 = a^2(\eta) \left( d\eta^2 - dr^2 - S_k^2(r) d\Omega^2 \right)$$ (2)

Rederive the relation between the scale factors of emission $a(\eta_e)$ and absorption $a(\eta_o)$ of the light and the redshift $z$ it experiences while propagating radially, by working exclusively in conformal time coordinates. Start with the presentation of an accurate space-time diagram for propagation of light between emitter and observer in this coordinate frame.

Problem II:

GR refresher

For the Universe with flat spatial sections, with metric specified by the interval

$$ds^2 = a^2(\eta) \left( d\eta^2 - dx^2 - dy^2 - dz^2 \right)$$ (3)

(I have denoted the spatial coordinates as $x^1=x, x^2=y, x^3=z$ in Cartesian tradition) calculate

1. Inverse contravariant metric tensor $g^{\alpha\beta}$
2. Ricci tensor component $R_0^0$

Note:

we use Latin letters for spatial indexes. On my website there is a summary of useful formulae. Be careful with Einstein summation convention