Radial motion in the Schwarzschild metric, relative to stationary observers

Difficulty level:   ★ ★ ★

Last time we derived the 4-velocity u of a small test body moving radially in the Schwarzschild geometry, in terms of e, the “energy per unit rest mass”. Another parametrisation is in terms of the 3-speed V relative to stationary observers. This turns out to be, in Schwarzschild coordinate expression,

    \[u^\mu=\left(\gamma\Schw^{-1/2},V\gamma\Schw^{1/2},0,0\right)\]

To derive this, first consider the 4-velocity of stationary observers:

    \[u_{\rm{Schw}}^\mu=\left(\Schw^{-1/2},0,0,0\right)\]

We know the “moving” body has 4-velocity u of form u^\mu=(u^t,u^r,0,0) since the motion is radial. The Lorentz factor \gamma\equiv(1-V^2)^{-1/2} for the relative speed is

    \[\gamma=-\fvec u\cdot\fvec u_{\rm{Schw}}\]

Evaluating and rearranging yields u^t=\gamma\Schw^{-1/2}. Normalisation \fvec u\cdot\fvec u=-1 leads to u^r=\pm V\gamma\Schw^{1/2}, after some algebra including use of the identity \gamma^2-1=V^2\gamma^2. We allow V<0 also, and define this as inward motion. Carefully considering the sign, this results in the top equation. (An alternate derivation is to perform a local Lorentz boost. Later articles will discuss this… The Special Relativity formulae cannot be applied directly to Schwarzschild coordinates.)

Some special cases are noteworthy. For V=0, γ=1, and u reduces to uSchw. This corresponds to e=-\fvec\xi\cdot\fveclabel{u}{Schw}=\Schw^{1/2}. Also we can relate the parametrisation by V (and γ) to the parametrisation by e via

    \[\gamma=e\Schw^{-1/2}\qquad V=-\frac{1}{e}\sqrt{e^2-1+\frac{2M}{r}}\]

where the leftmost equation follows from the definition e\equiv-\fvec\xi\cdot\fvec u, and subsequently the rightmost equation from γ=γ(V). For raindrops with e=1, the relative speed reduces to V=-\Schwroot.

We would expect the construction to fail for r\le 2M, as stationary timelike observers cannot exist there, and so the relative speed to them would become meaningless. But curiously, it can actually work for a faster-than-light V>1 “Lorentz” boost, as even the authorities MTW (§31.2, explicit acknowledgement) and Taylor & Wheeler (§B.4, implicitly vrel>1 for r<2M) attest. Sometime, I will investigate this further…

Radial motion in the Schwarzschild metric, in terms of e

Difficulty level:   ★ ★ ★

A nice way to parametrise the 4-velocity u of a small test body moving radially within Schwarzschild spacetime is by the “energy per unit rest mass” e:

    \[{u^\mu=\left(e\Schw^{-1},\pm\eroot,0,0\right)\]

For the “±” term, choose the sign based on whether the motion is inwards or outwards. All components are given in Schwarzschild coordinates (t,r,\theta,\phi). The result was derived as follows. In geometric units, the metric is:

    \[\Schwmetric\]

By definition e\equiv-\fvec\xi\cdot\fvec u, where \fvec\xi\equiv\partial_t is the Killing vector corresponding to the independence of the metric from t, and has components \xi^\mu=(1,0,0,0) (Hartle §9.3). For geodesic (freefalling) motion e is invariant, however even for accelerated motion e is well-defined instantaneously and makes a useful parametrisation.

We want to find u^\mu=(u^t,u^r,u^\theta,u^\phi) say. Rearranging the defining equation for e gives u^t=e\Schw^{-1}. Radial motion means u^\theta=u^\phi=0, so the normalised condition \fvec u\cdot\fvec u=-1 yields the remaining component \abs{u^r}. The resulting formula is valid for all 0<r\ne 2M, and for e=1 the 4-velocity describes “raindrops” as expected.

Relative speed

Difficulty level:   ★ ★

Suppose two observers at the same place and time (that is, “event”) move with 4-velocities u and v respectively, then they measure their relative speed as follows. The Lorentz factor is simply

    \[\gamma=-\fvec u\cdot\fvec v\]

(The dot is not the Euclidean dot product, but uses the metric: g_{\alpha\beta}u^\alpha v^\beta where the indices \alpha and \beta are summed over by the Einstein summation convention.) The proof is based on the axiom that some local inertial frame exists, although interestingly one does not need to explicitly construct it.

The relative 3-speed V, may then be recovered via:

    \[V=\sqrt{1-\gamma^{-2}}\]

See for instance Carroll (end of §2.5) who terms it “ordinary three-velocity”. Other sources express the first formula more indirectly, in terms of the energy and momentum measured by an observer \fvec u: E=-\fvec u\cdot\fvec p where \fvec p=m\fvec v is the 4-momentum of another observer/object, and combine this with E=m\gamma (MTW Exercise 2.5 in §2.8 term it “ordinary velocity”, or Hartle §5.6, and Example 9.1 in §9.3).

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This is a resource for general relativity, which is Einstein’s theory of space, time, and gravity. It includes the related fields of astrophysics and cosmology, which use physics to study the universe. Later, I will likely stray into quantum mechanics and philosophy of science.

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