# Hyperbolic geometry of Minkowski space

• I
Gold Member
Summary:
Is the treatment of the geometry of Minkowski space as in the cited article (angle ϑ between inertial frames ϑ=tanh(v/c) with relative velocity v/c, inner product for timelike vectors as |A|*|B|*cosh(ϑ), etc.) one of the standard treatments?
In "The Geometry of Minkowski Space in Terms of Hyperbolic Angles" by Chung, L'yi, & Chung in the Journal of the Korean Physical Society, Vol. 55, No. 6, December 2009, pp. 2323-2327 , the authors define an angle ϑ between the respective inertial planes of two observers in Minkowski space with a relative velocity of v (with c=1) as arctanh(v), and inner products of vectors denoting events by using this angle : for timelike vectors A and B, denoting |A| and |B| as the corresponding spacetime intervals (+ - - -) , the inner product is |A|*|B|*cosh(ϑ), for spacelike ones the same but negative, and between a spacelike and a timelike vector with a similar expression. (This vaguely reminds me of the discussion of rapidity in https://en.wikipedia.org/wiki/Rapidity, but I'm not sure if that is relevant.) My question is, assuming I have presented it correctly, whether this treatment is one of (albeit not the only) the standard treatments of a geometry in Minkowski space, or whether it is just a curiosity, or whether it is flawed.

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vanhees71
Gold Member
I think the paper, which I would not link here due to Copyright, is overcomplicating standard things. If you have a time-like vector in the Minkowski plane you can parametrize it with a parameter (I'd not call it angle for pedagogical reasons, because it's hard enough for students to forget about Euclidean geometry when looking at a Minkowski diagram) as
$$a=A (\cosh \eta,\sinh \eta) \; \Rightarrow \; a \cdot a=A^2 (\cosh^2 \eta - \sinh^2 \eta),$$
where ##A \in \mathbb{R}## and ##\eta \in \mathbb{R}##. Then products of two time-like vectors are given as
$$a_1 \cdot a_2 =A_1 A_2 (\cosh \eta_1 \cosh \eta_2 - \sinh \eta_1 \sinh \eta_2)=A_1 A_2 \cosh(\eta_1-\eta_2).$$
For a space-like vector the corresponding parametrization is
$$a=A(\sinh \eta,\cosh \eta) \; \Rightarrow \; a \cdot a=-A^2,$$
and
$$a_1 \cdot a_2 = A_1 A_2 (\sinh \eta_1 \sinh \eta_2-\cosh \eta_1 \cosh \eta_2)=-A_1 A_2\cosh(\eta_1-\eta_2).$$
I think there's no more behind this than this simple math with hyperbolic functions.

For time-like vectors the ##\eta## is called rapidity.

• Gold Member
Thank you very much, vanhees71, for that excellent explanation.
Oops, I did not realize I was violating copyright by attaching the file (as it is freely available on the Internet). I have removed it.

vanhees71
Gold Member
I'm not sure, but it looks like a paper from a journal, and then usually you are not allowed to simply share it online. Better remove it, because it can get pretty expensive if the publisher takes legal action.

• robphy
Homework Helper
Gold Member
A comment on the term "hyperbolic geometry" (as in the title and in the tags)...
since searching for "hyperbolic geometry" is not likely to yield useful results for understanding Special Relativity.

"Hyperbolic geometry" is a non-euclidean geometry that doesn't satisfy the parallel postulate, and is thus a curved space. (For example, some works by Escher.)

The spacetime of Special Relativity is flat---it satisfies the parallel postulate. We can talk about parallel lines.
So, it doesn't have a hyperbolic geometry.

However, the spacetime of special relativity is non-Euclidean
because the "angles" between timelike vectors is based on the unit hyperbola (playing the role of a circle in Minkowski spacetime geometry).
Special Relativity uses "hyperbolic trigonometry".
("Hyperbolic geometry" uses "circular trigonometry" since the angle between lines in hyperbolic geometry is based on a circle.)

[In special relativity, "hyperbolic geometry" might be helpful to understand effects like the Thomas procession or velocity-composition in different spatial directions. The 3-dimensional "space of rapidities (which are related to velocities)" and the "mass shell" are hyperboloids and have an intrinsic hyperbolic geometry.]

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• troglodyte, PeterDonis and vanhees71
vanhees71
Gold Member
Well, and this tells you that you don't deal with angles but with areas, which is why the inverse function of the hyperbolic functions are called area functions (like ##\text{arcosh} x##, ##\text{arsinh x}##, ##\text{artanh} x##).

What these areas have to do with Minkowski spacetime and why it is precisely the spacetime we need to fulfill Einstein's "two postulates", you can read in @robphy 's great Insights Blogs (as well as his papers cited therein):

https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/
https://www.physicsforums.com/insights/relativity-rotated-graph-paper/
https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/

• troglodyte
robphy
Homework Helper
Gold Member
Well, and this tells you that you don't deal with angles but with areas, which is why the inverse function of the hyperbolic functions are called area functions (like arcoshxarcoshx\text{arcosh} x, arsinh xarsinh x\text{arsinh x}, artanhxartanhx\text{artanh} x).

In the past year or so,
this "area" (as opposed to "arc[length]") viewpoint of angles (I'll still call it that)
at the level of the inverse-function has been growing on me.

• troglodyte and vanhees71