## Relative velocity in general relativity

Suppose we have two 4-velocity vectors and at the same point in curved spacetime. (This avoids complications such as parallel transport. Physically, think of the two objects as not necessarily overlapping, but close enough that we can neglect curvature etc.) We can calculate the relative velocity each determines of the other, which is not .

Consider firstly inertial frames in Minkowski spacetime. Using coordinates corresponding to some observer , the components of a different observer satisfy:

Here is ‘s proper time, is the Lorentz factor as I have discussed previously, and the are the relative speeds in the coordinate directions. This calculation is inspired by Tsamparlis 2019  §6.2.

With a view to generalisation, we re-express the displayed formula above using vectors in place of coordinate components: . This is more elegant, and explicitly tensorial. The reader may find better notation than , but this is the relative velocity of from ‘s frame. Rearranging,

This vector lies in the local 3-space of , since , so in particular is spatial. It has length , which is the overall relative speed, and satisfies . If you want, for there is also a decomposition , where is a unit vector. Conversely, the relative velocity of with respect to is . This also has length , but lies in ‘s 3-space. Why is unlike in Newtonian physics, aside from the trivial case ? They are in different frames (Tsamparlis §6.4). But with the appropriate Lorentz boost map, such an identity is recovered (Jantzen+ 1992  §4).

All the vector formulae above transfer unchanged to curved spacetime, for 4-velocities at the same event, including worldlines with acceleration. This can be justified using local inertial coordinates. While the formulae do appear in the literature (Jantzen+ 1992; Bini 2014  §6, etc), the topic of observer measurements in general is not widely promoted. I recall two separate conversations with senior relativists who were unfamiliar with use of the Lorentz factor in a curved spacetime context.

In contrast, one quantity which should not be naively ported across from special relativity is acceleration. In curved spacetime, the 4-acceleration of a particle requires the covariant derivative, which depends on curvature. Relative acceleration between observers is more complicated, as it depends on one’s choice of affine connection, for which there are various natural options. For instance, Fermi-Walker transport, or co-rotation with an observer’s frame, see e.g. Jantzen, Carini & Bini (Jantzen+ 1992; Jantzen+ 1995 ; Jantzen+ 2013  draft).

[Finally, the expression given near the start bothered me at first, because time-dilation is mutual, so one might equally argue a case for . But the key point is, the derivative occurs along the direction of , not . Another way to check the expression is to write . This is a contraction of the 1-form with the vector , as I explained previously. The “flat” symbol just means is the 1-form dual to . Conversely, along we have , and this is not a contradiction!]

## Derivative as contraction of a 1-form and vector

Suppose you seek the derivative of a quantity along a curve, such as the rate of change of a scalar by proper time along a worldline: or perhaps the rate of change of pressure by proper distance along a given spatial direction: . These derivatives are conveniently expressed as a contraction between a 1-form (the gradient of the scalar) and a tangent vector to the curve. For the first example,

where is some coordinate system, is the 1-form with components , and is the 4-velocity. is the contraction of the vector and 1-form, yielding a scalar. Schutz 2009  §3.3 gives this derivation.

A spacelike path can be parametrised by proper distance. Then , where is the unit tangent vector. An example of a paper which uses this is Gibbons 1972 , for the change in stress along a rigid cable, see the line after his Equation 4.

For a null path there is no natural parameter, at least not without additional context. But for any chosen parameter , we have as before, where is the tangent vector. Of course this applies to the other cases as well. Note all these calculations occur within a single tangent space.