T2: 3. The Schwarzchild Solution

Question 1

What does Einstein’s equation reduce to in a vacuum?

Answer 1

R_μν = 0

(since T^μν=0)

Question 2

State the flat space metric in polar coords

Answer 2

ds^2 = -dt^2 +dr^2 +r^2 dΩ^2

where dΩ^2 = dθ^2 + sin(θ)^2 dϕ^2

Question 3

State the most general spherically symmetric metric w/flat space signature.

Answer 3

Put an exp[2A(r)] in front of each term in the polar metric, with label A,B,C respectively.

Question 4

Define asymptotically flat for metric

Answer 4

A metric which tends to flat space when one coordinate is taken to an extreme.

Question 5

What conditions are the Schwarzschild metric define under?

Answer 5

Spherically symmetric, static mass distribution in a vacuum

Question 6

Recall the g_tt component of the metric which is near flat space

Answer 6

g_tt ≈ -(1+2ϕ)

Question 7

State the Schwarzschild radius

Answer 7

r_s = 2GM

Question 8

State the full Schwarzchild metric

Answer 8

ds^2 = -(1-r_s/r)dt^2 +(1-r_s/r)^-1 dr^2 +r^2 dΩ^2

Question 9

What does the Schwarzschild metric descibe? Examples?

Answer 9

The metric outside a static, spherically symmetric matter distribution. E.g. planet, star, blackhole.

Question 10

Where are the singularities in the Schwarzschild metric?

Answer 10

At r = 0 and r = r_s

Question 11

State the Killing equation

Answer 11

∇_μ ξ_ν+∇_ν ξ_μ=0

Question 12

If a metric is independent of coordinate t (x^0) what does that imply about ∂_t (∂_0)

Answer 12

The vector field ∂_t (∂_0) is a killing vector:
ξ^μ (x) = δ_0^μ

Question 13

Given a killing vector, what quantity is conserved? and where?

Answer 13

Q_ξ = ξ_μ u^μ is constant along the geodesic x^μ (s)

Question 14

What is the product of a symmetric and antisymmetric tensor?

Answer 14

Zero!!

Question 15

Recall how to choose parameter λ? Why?

Answer 15

We choose λ such that the magnitude of the velocity/tangent vector = 1. This gives us ‘straight line’s

I.e. g_μν u^μ u^ν=-ϵ

Question 16

Given a magnitude =1 of tangent vector: g_μν u^μ u^ν=-ϵ. What are the possible values of ϵ and what kind of curve do they indicate?

Answer 16

ϵ=1 indicates a timelike path; ϵ=0 indicates a null path.

Question 17

To find geodesics in the Schwarzschild metric, what two conserved quantities do we take?

Answer 17

Time and angular momentum.

Question 18

What is the time killing vector and conserved quantity?

Answer 18

Time-translation invariance:

H^μ = (∂_t )^μ = (1,0,0,0)

K = -H_μ u^μ = (1-2GM/r) dt/dλ

Question 19

What is the angular momentum killing vector and conserved quantity?

Answer 19

Translation in ϕ invariance:

R^μ = (∂_ϕ)^μ = (0,0,0,1)

J = R_μ u^μ = r^2 sin^2(θ) dϕ/dλ = r^2 dϕ/dλ

Question 20

What restriction do we place on the conservation of angular momentum?

Answer 20

We want conservation about the equatorial plane: θ=π/2.

Question 21

What convenient equation pops out of the killing vector/metric situe?

Answer 21

The conservation of energy for a particle moving in a 1d potential V(r) with effective energy E.

Question 22

How does the potential we find for the Schwarzschild metric differ from Newtonian?

Answer 22

There is an additional 2GMJ^2/r^3 term, which tends to zero at large r (where our metric tends flat).

Question 23

At what radius is the potential for the Schwarzschild metric always zero?

Answer 23

The Schwarzschild radium r_s = 2GM

Question 24

In what case is a particle in a perfect, circular orbit?

Answer 24

When it sits exactly at the bottom of the potential well.

Question 25

What are the two circular orbit radii for large J? Which one is stable

Answer 25

Stable: J^2/GM

Unstable: 3GM

Question 26

As J increases, how do the stable and unstable orbits relate to J and r.

Answer 26

The stable orbit moves out with J^2 while the unstable orbit plateaus to 3GM independent of J.

Question 27

As J decreases, what happens to the roots of r_c? What solution do we get?

Answer 27

The roots merge to a single solution for r_c:
r_c=6GM

Question 28

What is the small value of J which retrieves the stable orbit r_c = 6GM?

Answer 28

J = 2√3 GM

Question 29

What is the range for unstable orbits?

Answer 29

3GM ≤ r_c ≤ 6GM

Question 30

What quantities do we define considering the Schwarzschild equations for light?

Answer 30

u ≡ 1/r
b ≡ J/K

Question 31

Define impact parameter

Answer 31

The closest approach of a light ray travelling past a mass in a straight line

Question 32

How do we describe redshift?

Answer 32

Consider a photon trying to leave gravitational field, expending energy as it does so. Universality of the speed of light means its speed cannot change, so by E=ℏω, its frequency must decrease and colour shift.

Question 33

State the redshift formula giving the ratio of proper time intervals between inner and outer observers

Answer 33

(Δτ_i)/(Δτ_0 )=√((1-2GM/r_i )/(1-2GM/r_o ))

Question 34

Define coordinate singularity and give example

Answer 34

A singularity caused by a poor choice in coordinates, E.g. r = 2GM on the Schwarzschild metric

Question 35

Define curvature singularity and give example

Answer 35

A singularity where the geometry is actually singular e.g. infinitely curved. r=0 on the Schwarzchild metric is an example; no redefinition of coordinates can fix.

Question 36

Define light cone

Answer 36

The cone created by taking light rays from r=0. They follow t=+-r and define a region where timelike objects can exist (between the +-t pole and the curve)

Question 37

How do we find Eddington-Finkelstein coordinates?

Answer 37

Considering the light cone in Schwarzschild metric

Question 38

Given an ss metric, what coords do we define to find Schwarzschild?

Answer 38

r- = e^2C(r) r
Differentiate and rewrite in r-
Relabel r- as r

Question 39

What is the small change in angle for a lightray passing an object?

Answer 39

Δϕ = 4GM/b