Difficulty level: ★ ★ ★
The following is a natural choice of orthonormal tetrad for an observer moving radially in Schwarzschild spacetime with “energy per unit rest mass” e:
The components are given in Schwarzschild coordinates. (The ± signs are not independent — they must be either both +1 or both -1. Note that e does not distinguish between inward and outward motion. There is additional freedom to define any of these vectors as their negative.)
We normally think of e as invariant, where there is a presumption of freely falling / geodesic motion, but even if not we can regard it as an instantaneous value.
is the 4-velocity computed previously. The other vectors can be obtained from substituting and into the tetrad here. is determined from and the equation for e above, then V follows from inverting . This orthonormal frame is useful for determining the object’s perspective, e.g. tidal forces, visual appearances, etc.
Difficulty level: ☆
2015 is the 100-year anniversary general relativity. Einstein completed the final version of the field equations in November 1915. As today is the last day of the year, had to get this post in!
Difficulty level: ★
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.
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,
To derive this, first consider the 4-velocity of stationary observers:
We know the “moving” body has 4-velocity u of form since the motion is radial. The Lorentz factor for the relative speed is
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…
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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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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!