What is the relative acceleration composition law in General relativity? In Euclidean geometry we have the following relative acceleration composition law:
$$ \vec a_{DE} + \vec a_{EF} = \vec a_{DF} $$
Where the relative acceleration between $i$ and $j$ for any $i$ and $j$ is given by:
$$ a_{ij} = a_i - a_j$$
with $a_i$ being the acceleration of $i$ and $a_j$ being the acceleration of $j$.
Is there a nice geometric way to calculate the relative acceleration composition law for $3$ intersecting (at a point) geodesics? I know the separation vector $n$ between $2$ neighboring geodesics obey:
$$ \nabla_u^2 n = R (u,v) n $$
Where $ R(u,v) = (\nabla_u \nabla_v - \nabla_v \nabla_u)$ and $\nabla_u v$ is the derivative of $v$ along $u$.
 A: Precisely this question has been asked and answered in the following paper:

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*Bini, D., Carini, P., & Jantzen, R. T. (1995). Relative observer kinematics in general relativity. Classical and Quantum Gravity, 12(10), 2549, doi:10.1088/0264-9381/12/10/013, free pdf at archive.org.


Abstract. The straightforward reformulation of special relativistic concepts about relative observer kinematics in the context of the flat affine geometry of Minkowski spacetime, so that they respect the manifold structure of that spacetime, allows one to derive the general relativistic ‘addition of acceleration law’. This transformation law describes the relationship between the relative accelerations of a single test particle as seen by two different families of test observers.

A: What are you trying to achieve with this? (Four-) Acceleration is essentially the curvature of the world-line of an object. If your question is about four-acceleration, then you are asking how to add two curvatures. When would you want to do this?
If you simply want to construct something it might help to work from basics. Specify one or two world-lines. Specify location on these curves where the two accelerations would be defined. If these locations would correspond to the same point in space, the two vectors would be from the same tangent space, so no issues in adding them. If the points in space are different, then you will need to employ some way of transporting between these two tangent-spaces. Parallel transport along the shortest-path line between the points, may be one way, but you could also choose to do something like Lie transport along one of the world-lines you have defined (create a congruence of these world-lines etc). There are different options. So this is why you need to be more specific.
