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:

If *y* and *z* dimensions are included:

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

The **wave vector (or propagation vector)** has dimensions of meter^{-1}. Its spatial parts use the unit vector in the direction of motion of the photon:

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

The **Doppler neutral angle**, where D=1:

The **aberration** of light:

In this diagram explaining the geometry, D is the detector-direction, P is the photon-direction:

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).

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

- the galaxy recedes from the Earth because of the Hubble expansion of the Universe
- the jet moves with some high speed relative the galaxy

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):