The Absolute Derivative
In relativity we typically deal with two types of quantities – fields, which are defined everywhere – and particle properties, which are defined only along a curve or world line. The familiar covariant derivative is appropriate when we need to differentiate a field. A field is a function of all four coordinates, and the covariant derivative of φ(x) consists of the four partial derivatives ∂φ/∂xµ plus correction terms involving the Christoffel symbols, one for each tensor index on φ.
A particle property φ(s), on the other hand, is a function only of a single parameter s running along the curve. In this situation, the partial derivatives of φ with respect to the four coordinates do not exist. (Unfortunately many references miss this point!) Writing partial derivatives would require that φ be defined everywhere in a neighborhood of the curve, which is not the case. The correct derivative to use for a particle property is called the absolute derivative, and is written either δφ/δs or Dφ/Ds. Similar in form to the covariant derivative, it consists of an ordinary derivative dφ/ds plus a correction term for each tensor index. For example, if the world line is given parametrically as xµ(s), the absolute derivative of a contravariant vector φµ is Dφµ/Ds = dφµ/ds + φν Γµνσ dxσ/ds.
For a given timelike curve xµ(s) we define the vector vµ = Dxµ/Ds. This 4-vector is clearly tangent to the curve. If the parameter s is chosen to measure proper time, then vµ will be a unit vector, v·v = -1. Next we define an acceleration 4-vector aµ = Dvµ/Ds. Since D(vµvµ)/Ds = 0 = vµaµ, aµ is orthogonal to vµ and therefore lies in the instantaneous rest frame of the particle.
Given a vector W at a single point on the world line, we can define it at other points along the curve by specifying how it is to be propagated or transported along the curve. The simplest method is parallel transport: DW/Ds = 0. For example if the particle is nonaccelerating, then the velocity vector is parallel-transported: Dv/Ds = 0. But a generally more useful concept is Fermi-Walker transport. We often need to deal with a previously 3-dimensional quantity. In 4 dimensions it will be confined to the instantaneous rest frame of the particle and represented by a vector orthogonal to v. As the particle accelerates, the rest frame changes. Consequently for W to remain orthogonal it must undergo a Lorentz transformation in the v-a plane. The definition of Fermi-Walker transport is
DWµ/Ds = Wν(vµaν – aµvν)
Note that using F-W transport, if W·v = 0 initially, it will remain so. Physically, Fermi-Walker transport defines what we mean by a nonnrotating instantaneous rest frame. To say it again, the only way W changes is a Lorentz boost applied in the direction of the particle’s acceleration.
The spin vector S is such a quantity. The fact that S evolves according to Fermi-Walker transport is the source of precession effects in both special and general relativity.
Precession in Flat Space
Consider a particle moving in a circular orbit of radius r in the equatorial plane. Define a comoving orthonormal basis of 4-vectors:
er = (cos ωt, sin ωt, 0, 0)
eφ = (-sin ωt, cos ωt, 0, 0)
et = (0, 0, 0, 1)
Thus er points radially outward, and eφ points tangentially along the circumference.
Let τ be the particle’s proper time. Then
Der /Dτ = γω eφ
Deφ /Dτ = – γω er
where γ = dt/dτ.
The particle’s position, velocity and acceleration 4-vectors are:
x = r er + t et
v = Dx/Dt = rγω eφ + γ et
a = Dv/Dt = – rγ2ω2 er
From this we can derive that the magnitude of the 3-velocity is v = rω, and that γ2 = 1/(1 – r2ω2).
Now let’s suppose that the particle carries a spin: a unit 4-vector S orthogonal to v. In general S can be written as some time-dependent linear combination of er, eφ and et:
S = a(τ) er + b(τ) eφ + c(τ) et
S·v = 0 ⇒ c(τ) = rω b(τ)
S·a = – rγ2ω2 a
The time derivative of S comes from derivatives of a, b, c and also from the fact that the basis vectors er, eφ are rotating:
DS/Dτ = (da/dτ er + db/dτ eφ + dc/dτ et) + (γωa eφ – γωb er)
Finally, this must be inserted into the Fermi-Walker transport equation:
DS/Dτ = (S·a)v = – r2γ3ω3a eφ – rγ3ω2a et
and the coefficients matched:
coefficients of er: da/dτ – γωb = 0
coefficients of eφ: db/dτ + γωa = – r2γ3ω3a
coefficients of et: dc/dτ = – rγ3ω2a
Which simplify to:
da/dτ = γω b
db/dτ = – γ3ω a
(The dc/dτ equation is identically satisfied by our earlier result, c(τ) = rω b(t).)
The first two equations may be combined into a harmonic oscillator equation:
d2a/dτ2 + γ4ω2 a = 0,
with solution a ~ cos(Ωτ) where Ω = γ2ω.
The important thing to note is the extra factor of γ. Although the orbital position varies like (sin, cos)(ωt) = (sin, cos)(γωτ), the spin vector S varies like (sin, cos)(γ2ωτ), that is, more slowly. Thus, relative to a stationary frame, S exhibits a net precession in a retrograde sense, an effect known as Thomas precession.
Precession in the Schwarzschild Metric
Now consider the same particle moving in a circular orbit in a Schwarzschild field, again with coordinate angular velocity ω = dφ/dt. The same procedure is to be followed as we did in flat space, and we’ll mostly just compare results. The velocity 4-vector is
vμ = (vr, vφ, vt) = (0, γω, γ)
with γ determined by the normalization condition
v·v = 1 = γ2(gtt + ω2 gφφ) ⇒ γ2 = (1 – 2M/r – r2ω2)-1. Before it was just (1 – r2ω2)-1.
The acceleration 4-vector is
aμ = Dvμ/Dτ = dvμ/dt + Γμνσ vνvσ
The only nonzero component of aμ is the radial component:
ar = -½ grr(gtt,r vtvt + gφφ,r vφvφ) = -γ2(r – 2M)(ω2 – M/r3), where before we had ar = – γ2rω2.
(Note that for a free particle following a geodesic, the acceleration is zero, ar = 0, and the orbital velocity is given by ω2 = M/r3. We recognize this as Kepler’s Law, “period squared goes as distance cubed.” It’s remarkable that in terms of the coordinate angular velocity, the circular orbits in the Schwarzschild field obey Kepler’s Law exactly!)
Now consider the spacelike vector Sμ orthogonal to vμ:
Sμ = (Sr, Sφ, St)
S·v = 0 = gttStvt + gφφSφvφ ⇒ St = r3ω/(r – 2M) Sφ, whereas before c(t) = rω b(t)
S·a = grrSrar = rγ2(ω2 – M/r3)Sr, whereas before S·a = rγ2ω2 a
Supposing that Sμ is Fermi-Walker transported, we get evolution equations for each component:
dSr/dτ = γω(r – 3M)Sφ (before it was da/dτ = γω b)
dSφ/dτ = -γ3ω(r – 3M)/r2 (before it was db/dτ = -γ3ω a)
d2Sr/dτ2 + γ4ω2(r – 3M)/r2 Sr = 0,
results in a harmonic oscillator equation with solutions Sr ~ A cos(Ωτ) where Ω = γ2ω(r – 3M)/r. The precession rate Ω for Schwarzschild may be thought of as the combination of two effects:
– a retrograde Thomas precession at a rate γ2ω
– a prograde de Sitter precession at a rate 3γ2ωM/r
Precession in the Kerr Metric
For Kerr, results only.
γ2 = (1 – (r2 + a2)ω2 – 2M(1 – aω)2/r)-1
ar = – Δγ2[ω2 – M(1 – aω)2/r3], where Δ = r2 – 2Mr + a2
St = Sφ[rω(r2 + a2) – 2Ma(1 – aω)]/[r – 2M(1 – aω)]
S·a = rγ2[ω2 – M(1 – aω)2/r3]
d2Sr/dτ2 + Ω Sr = 0 where Ω = γ2ω[r – 3M(1 – aω)]/r + γ2Ma(1 – aω)2/r3
— This article was originally part of Physics Forums member Bill_K‘s PF blog. He may not respond to comments.