If I rotate a triangle lying on the two-dimensional plane both the lengths of the sides and the angles formed are invariant (ie, they are the same before and after the rotation). In a 2D Lorentz transformation on the (t,x) plane, we know that the spacetime interval between points is invariant. What is in this case the invariant quantity corresponding to angles in the Euclidean case?

  • $\begingroup$ I don't think there is anything analogous to the angle, and the reason is that in Minkowski space, a vector can have a zero dot product with itself. $\endgroup$
    – user4552
    Apr 2, 2018 at 1:08

1 Answer 1


The Special relativity analogue of the angles in a triangle is the “rapidity” $\theta$ between timelike segments. The relative velocity $v=c\tanh\theta$ and the time dilation factor is $\gamma=\cosh\theta$ and the Doppler factor is $k=\exp\theta$.

Analogous to an "angle" as the intercepted arc length of the "unit circle", the "rapidity" can be interpreted as the intercepted arc length [using the Minkowski metric] of the "unit hyperbola".

In the Clock Effect, we have a triangle with timelike legs.
We can define the "relative rapidity between future timelike legs" as the Minkowski-arclength of the intercepted arc on the future unit-hyperbola.

In this diagram, we have rapidities $\theta$, $\phi$, and $\eta$.
Given the rapidity $\theta$,
we have relative-velocity $v_{OP,OP'}=\tanh\theta$
and [relative] time-dilation factor $\gamma_{OP,OP'}=\cosh\theta$
... and similarly for the other rapidities: $\phi$ and $\eta$.

[Note: There is no hyperbolic arc intercepted by $P'O$ and $P'Z$. So, it's not clear how to define the "rapidity at P' inside the triangle".]

With the chosen values,
$v_{OP,OP'}=\tanh\theta=6/10$ (so $\gamma_{OP,OP'}=\cosh\theta=10/8)$
and $v_{PZ,P'Z}=\tanh\phi=-6/10$ (so $\gamma_{PZ,P'Z}=\cosh\phi=10/8$).
This is in analogy to the Euclidean case.

(I've drawn the spacetime diagram using rotated graph paper so that the tickmarks along the various segments can be counted.)

RRGP rotated graph paper Clock Effect triangle rapidity

  • $\begingroup$ thanks for the suggestion but rapidity cannot be the quantity analogous to angles I was asking for... rapidity is analogous to the overall angle of rotation, but I was referring to the three internal angles of the triangle and the quantity I'm looking for must necessarily be dependent on the space-time coordinates of the three vertexes. $\endgroup$
    – Lorents
    Apr 1, 2018 at 23:47
  • $\begingroup$ I updated my answer with a diagram illustrating "rapidities", the [Minkowski-]angles between future-timelike vectors. $\endgroup$
    – robphy
    Apr 2, 2018 at 1:57
  • $\begingroup$ I now see what you mean, that's indeed what I was looking for, thanks! $\endgroup$
    – Lorents
    Apr 2, 2018 at 14:34

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