Exercise 1:

1. The metric on the sphere is given by

    

   

   Compute the non-zero components of the Christoffel symbol.

2. Compute the non-zero components of the Riemann tensor and the Ricci tensor. Compute the Ricci scalar.
3. Recall that the metric in polar coordinates on R³ is given by

    

   

   The components of this metric are independent of ϕ. Determine the Killing vector associated with rotation around the z axis with angle ϕ.

4. Determine the Killing vectors associated with rotations on the sphere. Hint: use ∂_x, ∂_y, and ∂_z as basis elements.

Solution 1:

Exercise 2:

The metric on the hyperboloid H² (Poincaré half-plane) is given by

 

Compute the length of the line segment between (x₀, y₁) and (x₀, y₂).

Solution 2:

Exercise 3:

1. We consider the metric given by

    

   

   Compute the non-zero components of the Christoffel symbol.

2. Write down the zero component of the geodesic equation.
3. Write down the condition for null geodesics and use it to solve the zero component of the null geodesic equation.
4. What is the energy of a photon as measured by a comoving observer in this expanding observer.
5. Write down the relation between the values of the photon energy at two different scales a₁ and a₂.
6. The matter distribution in the Universe is assumed to be a perfect fluid. Write down the non-zero components of the energy–momentum tensor in this Universe as seen by a comoving observer.
7. By using the conservation law of the energy–momentum tensor ∇_μT^μν = 0 and the equation of state P = wρ derive the form of the mass density ρ.

Solution 3:

Exercise 4:

1. The Schwarzschild metric is given by

    

   

   Compute the time translation and rotational Killing vectors in this spacetime.

2. Compute the energy and the angular momentum of a particle moving in this spacetime.
3. Show that the following quantity

    

   

   is conserved along a geodesic.

4. Write down explicitly the above conserved quantity in Schwarzschild spacetime. Derive the effective potential.
5. Determine the light cones of the Schwarzschild metric. What happens at r = 2GM?

Solution 4:

Exercise 5:

1. Write down the electromagnetic field strength tensor F_μν and the inhomogeneous Maxwell’s equation in curved spacetime.
2. Let g = det g_μν. Show that

    

   

3. Show that the inhomogeneous Maxwell’s equation can be put in the form

    

   

4. Write down the law of conservation of charge in curved spacetime.