A simple but useful identity is:
This follows from the definition of the Lorentz factor γ in terms of a relative speed V between two frames:
Alternatively raise both sides of the γ defining formula to the power of -2, obtaining , then multiply both sides by γ2 and rearrange.
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,
To derive this, first consider the 4-velocity of stationary observers:
Evaluating and rearranging yields . Normalisation leads to , after some algebra including use of the identity . We allow 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 . Also we can relate the parametrisation by V (and γ) to the parametrisation by e via
where the leftmost equation follows from the definition , and subsequently the rightmost equation from γ=γ(V). For raindrops with e=1, the relative speed reduces to .
We would expect the construction to fail for , 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…
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:
For the “±” term, choose the sign based on whether the motion is inwards or outwards. All components are given in Schwarzschild coordinates . The result was derived as follows. In geometric units, the metric is:
By definition , where is the Killing vector corresponding to the independence of the metric from t, and has components (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 say. Rearranging the defining equation for e gives . Radial motion means , so the normalised condition yields the remaining component . The resulting formula is valid for all , and for e=1 the 4-velocity describes “raindrops” as expected.
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
(The dot is not the Euclidean dot product, but uses the metric: where the indices and 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:
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 : where is the 4-momentum of another observer/object, and combine this with (MTW Exercise 2.5 in §2.8 term it “ordinary velocity”, or Hartle §5.6, and Example 9.1 in §9.3).
What would you like to learn about?
Leave a question in the comments.
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.
The content is aimed at all ability levels. For beginners, introductory articles and videos introduce the field and even my research. For experts, I provide technical help in bite-sized nuggets. Please leave your feedback in the comments, and request future topics!
I am a university student studying physics, currently in a Master of Science degree. I intend to start a PhD in relativity soon. Join me on the journey, and we’ll see where it leads!