T2: 3. The Schwarzchild Solution Flashcards
What does Einstein’s equation reduce to in a vacuum?
R_μν = 0
(since T^μν=0)
State the flat space metric in polar coords
ds^2 = -dt^2 +dr^2 +r^2 dΩ^2
where dΩ^2 = dθ^2 + sin(θ)^2 dϕ^2
State the most general spherically symmetric metric w/flat space signature.
Put an exp[2A(r)] in front of each term in the polar metric, with label A,B,C respectively.
Define asymptotically flat for metric
A metric which tends to flat space when one coordinate is taken to an extreme.
What conditions are the Schwarzschild metric define under?
Spherically symmetric, static mass distribution in a vacuum
Recall the g_tt component of the metric which is near flat space
g_tt ≈ -(1+2ϕ)
State the Schwarzschild radius
r_s = 2GM
State the full Schwarzchild metric
ds^2 = -(1-r_s/r)dt^2 +(1-r_s/r)^-1 dr^2 +r^2 dΩ^2
What does the Schwarzschild metric descibe? Examples?
The metric outside a static, spherically symmetric matter distribution. E.g. planet, star, blackhole.
Where are the singularities in the Schwarzschild metric?
At r = 0 and r = r_s
State the Killing equation
∇_μ ξ_ν+∇_ν ξ_μ=0
If a metric is independent of coordinate t (x^0) what does that imply about ∂_t (∂_0)
The vector field ∂_t (∂_0) is a killing vector:
ξ^μ (x) = δ_0^μ
Given a killing vector, what quantity is conserved? and where?
Q_ξ = ξ_μ u^μ is constant along the geodesic x^μ (s)
What is the product of a symmetric and antisymmetric tensor?
Zero!!
Recall how to choose parameter λ? Why?
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
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?
ϵ=1 indicates a timelike path; ϵ=0 indicates a null path.
To find geodesics in the Schwarzschild metric, what two conserved quantities do we take?
Time and angular momentum.
What is the time killing vector and conserved quantity?
Time-translation invariance:
H^μ = (∂_t )^μ = (1,0,0,0)
K = -H_μ u^μ = (1-2GM/r) dt/dλ
What is the angular momentum killing vector and conserved quantity?
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
What restriction do we place on the conservation of angular momentum?
We want conservation about the equatorial plane: θ=π/2.
What convenient equation pops out of the killing vector/metric situe?
The conservation of energy for a particle moving in a 1d potential V(r) with effective energy E.
How does the potential we find for the Schwarzschild metric differ from Newtonian?
There is an additional 2GMJ^2/r^3 term, which tends to zero at large r (where our metric tends flat).
At what radius is the potential for the Schwarzschild metric always zero?
The Schwarzschild radium r_s = 2GM
In what case is a particle in a perfect, circular orbit?
When it sits exactly at the bottom of the potential well.
What are the two circular orbit radii for large J? Which one is stable
Stable: J^2/GM
Unstable: 3GM
As J increases, how do the stable and unstable orbits relate to J and r.
The stable orbit moves out with J^2 while the unstable orbit plateaus to 3GM independent of J.
As J decreases, what happens to the roots of r_c? What solution do we get?
The roots merge to a single solution for r_c:
r_c=6GM
What is the small value of J which retrieves the stable orbit r_c = 6GM?
J = 2√3 GM
What is the range for unstable orbits?
3GM ≤ r_c ≤ 6GM
What quantities do we define considering the Schwarzschild equations for light?
u ≡ 1/r
b ≡ J/K
Define impact parameter
The closest approach of a light ray travelling past a mass in a straight line
How do we describe redshift?
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.
State the redshift formula giving the ratio of proper time intervals between inner and outer observers
(Δτ_i)/(Δτ_0 )=√((1-2GM/r_i )/(1-2GM/r_o ))
Define coordinate singularity and give example
A singularity caused by a poor choice in coordinates, E.g. r = 2GM on the Schwarzschild metric
Define curvature singularity and give example
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.
Define light cone
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)
How do we find Eddington-Finkelstein coordinates?
Considering the light cone in Schwarzschild metric
Given an ss metric, what coords do we define to find Schwarzschild?
r- = e^2C(r) r
Differentiate and rewrite in r-
Relabel r- as r
What is the small change in angle for a lightray passing an object?
Δϕ = 4GM/b
