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PHYS 5770 Gravitational Theory Spring 2008. Problem Set 3. Due Tue 26 Feb
- Equations of motion in weak gravity
Consider the Newtonian metric
ds^2 = (1 + 2Φ)dt^2 − (1 − 2Φ)(dx^2 + dy^2 + dz^2 ) (1.1)
where Φ(x, y, z) is the familiar Newtonian gravitational potential, a function only of the spatial coordinates x, y, z, not of time t.
(a) Connection coefficients
Confirm that the non-zero connection coefficients are (coefficients as below but with the last two indices swapped are the same by the no-torsion condition Γκμν = Γκνμ)
Γtti = Γitt = Γijj = −Γjji = −Γiii =
∂xi^
(i 6 = j = x, y, z). (1.2)
[Hint: Work to linear order in Φ. You are welcome to use the mathematica notebook metric.nb posted on the website, but if you do, please tell me.]
(b) Energy of a massive particle
Consider a massive, non-relativistic particle moving with 4-velocity uμ^ ≡ dxμ/dτ = {ut, ux, uy, uz^ }. Show that uμuμ^ = 1 implies that
ut^ = 1 +
u^2 − Φ (1.3)
whereas
ut = 1 +
u^2 + Φ (1.4)
where u ≡ [(ux)^2 + (uy)^2 + (uz^ )^2 ]^1 /^2. One of ut^ or ut is constant. Which one? [Hint: Work to linear order in Φ. Note that u^2 is of linear order in Φ. As regards which of ut^ or ut is constant, notice that the metric is independent of time because Φ(x, y, z) is being assumed to be a function only of the spatial coordinates x, y, z, not of time t. You are welcome to quote the results of Problem Set 2.]
(c) Equation of motion of a massive particle
From the geodesic equation duκ dτ
show that dui dt
∂xi^
i = x, y, z. (1.6)
Why is it legitimate to replace dτ by dt? Show further that
dut dt
= − 2 ui^
∂xi^
with implicit summation over i = x, y, z. Does the result agree with what you’d expect from equation (1.3)? [Hint: A consistent perturbative approach is to keep only the lowest order non-vanishing parts of an equation, discarding the higher order parts as negligible.]
(d) Energy of a massless particle
For a massless particle, the proper time along a geodesic is zero, and the affine parameter λ must be used instead of the proper time. The 4-velocity of a massless particle can be defined to be (and really this is just the 4-momentum up to an arbitrary overall factor) vμ^ ≡ dxμ/dλ = {vt, vx, vy, vz^ }. Show that vμvμ^ = 0 implies that
vt^ = (1 − 2Φ)v (1.8)
whereas vt = v (1.9)
where v ≡ [(vx)^2 + (vy)^2 + (vz^ )^2 ]^1 /^2. One of vt^ or vt is constant. Which one?
(e) Equation of motion of a massless particle
From the geodesic equation dvκ dλ
show that the spatial components v ≡ {vx, vy, vz^ } satisfy
dv dλ
= 2 v ×
v ×
∂x
where boldface symbols represent 3D vectors, and in particular ∂Φ/∂x is the spatial 3D gradient ∂Φ/∂x ≡ ∂Φ/∂xi^ = {∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z}. [Hint: Recall that the 3D vector triple product satisfies a × (b × c) = (a · c)b − (a · b)c.]
(f ) Interpret
Interpret your answer, equation (1.11). In what ways does this equation for the accel- eration of photons differ from the equation governing the acceleration of massive particles? [Hint: Without loss of generality, the affine parameter can be normalized so that the photon speed is one, v = 1, so that v is a unit vector representing the direction of the photon.]
(g) Redshift
Consider an observer who happens to be at rest in the Newtonian metric, so that ux^ = uy^ = uz^ = 0. Argue that the energy of a photon observed by this observer, relative to an observer at rest at zero potential, is
uμvμ = 1 − Φ. (1.12)
Does the observed photon have higher or lower energy in a deeper potential well?
(e) Angular momentum and energy in circular orbit
Show that the angular momentum per unit mass for a circular orbit at radius r satisfies
|L| =
r (r/M − 3)^1 /^2
and hence show also that the energy per unit mass in the circular orbit is
E =
r − 2 M [r(r − 3 M )]^1 /^2
(f ) Drop in orbit
There is a certain circular orbit that has the same energy as a massive particle at rest at infinity. This is useful for starship captains to know, because it is possible to drop into this orbit using only a small amount of energy. What is the radius of the orbit? Is it stable or unstable? [Hint: What is the energy E of a particle at rest at infinity?]
(g) Photon sphere
There is a radius where photons can orbit in circular orbits. What is the radius of this orbit? [Hint: Photons can be taken as the limit of a massive particle whose energy per unit mass E vastly exceeds its rest mass energy per unit mass, which is 1.]
(h) Orbital period
Show that the orbital period t, as measured by an observer at rest at infinity, of a particle in circular orbit at radius r is given by Kepler’s 3rd law (remarkably, Kepler’s 3rd law remains true even in the fully general relativistic case, as long as t is taken to be the time measured at infinity) GM t^2 (2π)^2
= r^3. (2.8)
[Hint: Argue that the azimuthal angle φ evolves according to dφ/dt = uφ/ut^ = LB/(Er^2 ).]
