Contents

Collected Formulas

For reference, this page collects all of the formulas mentioned elsewhere on this site. It's worth noting that β, Γ, and D are all dimensionless quantitities: they are pure numbers, with no units attached to them.

The speed is usually expressed as a fraction of the speed limit:

When used as a velocity along some axis, then β will also have a sign, either positive or negative.

The square of the space-time interval between any pair of events, with Δd for the distance:

The warp factor (or Lorentz factor) for time dilation and other effects:

The boost transformation (or Lorentz transformation), relating event coordinates in one grid (unprimed), to a second boosted grid (primed), moving along the x axis at speed β (-1<β<1) with respect to the first:

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If y and z dimensions are included:

Boost for velocity, with each component of the speed expressed (as usual) as a fraction of the speed limit:

The Doppler factor, with θ as the angle between the line-of-sight and the line-of-motion.

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The Doppler neutral angle, where D=1:

The aberration of light:

In this diagram explaining the geometry, S is the source emitter, P is the photon:
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The apparent velocity of an object, as seen in a detector. The angle θ is between 0° and 180°. It's the angle between the line of sight and the object's direction of motion. There are two components, transverse (t) and radial (r).

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For galaxies, astronomers usually prefer to separate the motion into two parts:

In that case, they use this variation on the formula for apparent transverse motion:

Here, D is the Doppler factor for the Hubble recession, and β is the speed of the jet with respect to the galaxy.

The change in apparent magnitude of a star, approximated with a blackbody spectrum of temperature T expressed in kelvins (McKinley, Doherty 1979):