**ASSIGNMENT # 4
**

**(2 Parts)**

__PART I __ [ DUE: Tues FEB 16, 2010 ---
Reading Week ]

__1. __Do all of the derivations necessary to substantiate the equations given in
the following webpages:

**(a) The Interior Solution, **

**(b) Newtonian Cosmology, **

**(c) Cosmic Assumptions, **

**(d) Constant Curvature. **

**If an equation was assumed outside the context of the page you do not have
to derive it (example: the constant curvature Riemann expression).**

__2.__ Do problem 23.6 in D'Inverno.

**PART II**

__3.__ Redo all of the equations from the Friedmann Equations webpage
to the Light Propagation webpage filling in all derivation details.

__4.__ Prove that for a physically possible perfect fluid, no solution
of the Einstein equations is homogeneous, everywhere-isotropic, and static.

__5.__ Consider a universe that is isotropic and homogeneous. Assume
that it contains only dust and a constant cosmological stress energy.

** (a) Show that there is a static metric solution.**

**(b) Perturb this solution about the static solution to investigate its
stability. If the solution is unstable then a time dependent perturbation
of both the scale factor and the energy density should imply that the scale
factor perturbation grows with time. If it's stable then this perturbation
should be bounded in time.**