PHYS 650: ``General Relativity''

Questions on Chapter 8:
Geodesics

Summarize the variational principle for free test particle motion.
Why do we use an affine parameter s rather than proper time for the more general way to find equations of motions for free particles?
What is the general form of the Lagrangian for free particle motion?
How does the reformulation of the geodesic equation in terms of Christoffel symbols help us to solve the equations of motion?
Verify the expressions for the Christoffel symbols for flat space in polar coordinates.
Verify equation (8.20) for the general formula to find the Christoffel symbols in 2D polar coordinates in flat space.
What is a Killing vector?
What are the Cartesian expressions for the Killing vectors corresponding to rotations around the x and y axes?
Which is the conserved quantity corresponding to a Killing vector?
Explain the statement that (8.33) is a first integral corresponding to u·u = 1.
Explain how symmetries associated with Killing vectors help you to find the equations of motion of a particle.
What is the difference between the geodesic equation for light rays (8.42) and for particles 98.14)?
Explain why the Christoffel symbols vanish (locally) in a local inertial frame.
What are Riemann Normal Coordinates?
Why does equation (8.46) hold?
Why does equation (8.46), together with (8.45) prove that Riemann Normal Coordinates parametrize a local inertial frame?
What is the connection between a local inertial frame and a freely falling frame?