3. General Relativity

Question 1

How is Newton’s equation for particle motion related to the geodesic equation?

Answer 1

Due to Einstein’s equivalence principle
- Gravity and intertia are equivalent
- Motion of a test particle in a gravitational field is equivalent to motion along a geodesic

Question 2

How is the mass density from Newton’s field equation replaced in GR?

Answer 2

Replaced with the stress energy momentum tensor

Question 3

How are the stress energy momentum tensor and mass density related?

Answer 3

The 00 component of the stress energy momentum tensor is equal to the energy density (mass density x c^2)

Question 4

What are the 4 considerations Einstein took into account when translating from Newton’s field equation to his own?

Answer 4

• Second order derivatives of the metric (g) = Some constant (Kappa) x stress-energy-momentum tensor (T)
• Equation must have the geometrical meaning of curvature
• Linear in second derivatives
• Conserve all physics, namely that the divergence of T vanishes (conserves energy and momentum)

Question 5

How is Kappa determined?

Answer 5

Act on Field equation with the inverse metric (g^mu nu) to obtain
-R = Kappa x g ^(mu nu) T_(mu nu)
Take the 00 component and get R_00 = 1/2 Kappa rho c^2. Ignore higher order terms and the time derivatives
- Take the spatial (i) components and Christofel symbol and solve

Question 6

What is the action principle?

Answer 6

That the action should be a scalar quantity constructed out of the metric and its derivatives