1. Gravity

Question 1

What does Newton’s law of gravity predict about planet orbits?

Answer 1

That they orbit the sun in elliptical trajectories

Question 2

What is Kepler’s first law?

Answer 2

That all planet orbits are ellipses

Question 3

What is Kepler’s second law?

Answer 3

Planets sweep out equal areas in equal times
This is due to the conservation of angular momentum

Question 4

What is Kepler’s third law?

Answer 4

The square of the orbital period, P is equal to the cube of the semi major axis, R_a

Question 5

What is the perihelion of a planet?

Answer 5

The point at which it is closest to the sun

Question 6

What does Newton’s law of gravity fail to predict?

Answer 6

The precession of the perihelion of planets due to gravitational interaction

Question 7

How is the observation of the precession of the perihelion accounted for?

Answer 7

• Most of it due to gravitational pull from different planets e.g. Jupiter
• Einstein’s GR says the planets move along geodesics in space-time geometry

Question 8

What is a “prograde” perihelion?

Answer 8

A precession in which the direction of the perihelion is in the direction of orbital motion

Question 9

State Einstein’s equivalence principle

Answer 9

Intertial and gravitational masses are the same thing

Question 10

Describe how the equivalence principle can be applied to the motion of light in frame K and K’ for horizontal motion

Answer 10

Frame K (at rest): Light moves along a horizontal line
Frame K’ (Accel. upwards): Light has some deflection below the horizontal line

Equiv. principle: K’ frame is the same as a new frame at rest which is in a gravitational field

Question 11

What does an observer notice about light that travels vertically upwards in frame K vs K’?

Answer 11

Frame K (rest): No change in freq.
Frame K’ (Const accel): Observed freq has changed due to gravitational redshift

Question 12

How is Newton’s field equation obtained?

Answer 12

• Take the gradient of the potential -GM/r, and F = M grad V.
• Use Laplace on Green function

Question 13

Explain how you translaste between Newton’s field equation and Einstein’s equations

Answer 13

Laplace = curvature
Potential = space time metric
Mass density = Energy density

Question 14

What are the two principles of special relativity

Answer 14

• Laws of nature are the same in all intertial frames
• Speed of light is the same in all inertial frames

Question 15

Briefly explain the concept of time dilation for a clock in Frame K

Answer 15

Clocks that are moving run slowly
- Clock in K measures cT = cT
- Clock in K’ measures cT’ = γcT
- Time elapsed in K’ is longer

Question 16

What do measurements of the time or distance depend on ?

Answer 16

The observer

Question 17

What are invariants and why are they useful

Answer 17

Things that are the same in all frames of reference

Question 18

What is the metric of special relativity?

Answer 18

Minkowski invariant
(x’-ct’)(x-ct)

Question 19

Explain why the phase of a wave is an invariant quantity

Answer 19

The phase counts the number of waves in any wave train and all observers must agree on this

Question 20

Describe the Minkowski s-t geometry

Answer 20

Inside cone is time like
Along the cone is light like
Outside cone is space like
Future is above the past

Question 21

State the values of the velocity invariant for space, light and time like trajectories

Answer 21

> 0 is space like
=0 is light like
<0 is time like

Question 22

When is a set “space like”?

Answer 22

If at every point, every vector tangent to it is space like

Question 23

When is a set “time like”?

Answer 23

If there is at least 1 vector tangent to it that is time like

Question 24

Give examples of space and time like sets

Answer 24

Space: A plane - ct = ct’ = 0 and sphere -c^2 t^2 + x^2 + y^2 = -R^2
Time: A plane - x= x’ = 0 and sphere -c^2 t^2 + x^2 + y^2 = +R^2

Question 25

What is the Minkowski metric?

Answer 25

The covariant η_(μν) type (0,2) tensor. 00=1, 11=22=33 = 1q

Question 26

What is the name of the sign convention for the covariant tensors

Answer 26

The signature and we use (-, +, +, +)

Question 27

Explain the similarities and differences between a vector and a tensor

Answer 27

• They both have a magnitude
• A tensor hcan have several directions whereas a vector just has one

Question 28

What is the proper time?

Answer 28

The time measured by an observer who is moving along the trajectory

Question 29

How do free particles travel?

Answer 29

In straight lines at constant speeds (N1)

Question 30

Describe how massive and massless particles move in Minkowski space

Answer 30

Massive particles move along time like trajectories
Massless particles move along null (light) like

Question 31

What is the stress-energy-momentum tensor

Answer 31

The source of the gravitational field in general relativity which replaces the mass of Newton’s theory

Question 32

In the continuity equation, what do the values of mu represent

Answer 32

mu = 0 is the conservation of mass
mu = i are the N-S equations

Question 33

State the similarities and differences between the stress energy momentum tensor and the electromagnetic field tensor

Answer 33

Both symmetric, type (0,2) and conserved
- EM field is also traceless