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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 $u$.

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2 Answers 2

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Precisely this question has been asked and answered in the following paper:

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.

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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.

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    $\begingroup$ What are you trying to achieve with this? Ummm ... I have this argument (consistent with MB distribution) that gives me the relative momentum distribution of a gas - that is if I randomly look a collision what is the probability of a particular relative momentum . I'm trying to extend it. $\endgroup$ Aug 8 at 8:51
  • $\begingroup$ Specify one or two world-lines - I'm thinking of $3$ intersecting geodesics? You know the relative acceleration between say $D$ and $E$ and $E$ and $F$. Surely there should be some way to find the acceleration between $D$ and $F$ $\endgroup$ Aug 8 at 8:52
  • $\begingroup$ In one comment you are talking about relative momentum, in another, about relative acceleration. I think these are different, and neither is useful. Four-momentum is a vector, so is four-acceleration. You can represent these vectors in different basis. One way to define basis is to specify which bodies are at rest in this frame. If the bodies are accelerating, you can try to define instantaneous rest-frame. You may need to provide more details (MB distribution, your argument etc) in your main question to proceed, right now it still is too ill-defined $\endgroup$
    – Cryo
    Aug 8 at 9:27
  • $\begingroup$ With two or three world-lines intersecting, what you can seek, IMHO, is a joint probability distribution for exit four-velocities, or four-momenta. You can choose to work in the frame where the sum of the entry four-momenta is zero (i.e. Center of momenum) $\endgroup$
    – Cryo
    Aug 8 at 9:35
  • $\begingroup$ I am aware of the difference. Which is why I wrote "I'm trying to extend it." (to relative acceleration). It's a long argument (4-5 Pages Latex pages long). $\endgroup$ Aug 8 at 9:44

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