Video: Mathologer on visual logarithms (and hyperbolic trig)

[robphy]

Mathologer (https://en.wikipedia.org/wiki/Burkard_Polster) has a nice video using known (but not well-known)
geometric motivations of the natural logarithm and the hyperbolic functions... and he makes brief mentions of special relativity

I've been using similar motivations to support geometric reasoning on a spacetime diagram (using "rotated graph paper" and "spacetime trigonometry").
In the beginning, he's describing a Lorentz boost (on unrotated graph paper, in light-cone coordinates).
Near the end, one may recognize a triangle involved in the Bondi k-calculus (although that connection isn't mentioned).

Dictionary:
"angle" (as arc length and as sector-area) is related to the rapidity
"exp(x)" is related to the Bondi k-factor (Doppler factor) .... so rapidity=ln(Doppler)
"cosh(x)" is related to the time-dilation factor $\gamma$ (and the timelilke-component of a 4-vector)
"sinh(x)" is related to the dimensionless-velocity*time-dilation factor $(v/c) \gamma$ (and the spacelike-component of a 4-vector)
"tanh(x)" is related to the dimensionless-velocity (v/c)

 

[vanhees71]

It's only that what you call "angle" (in fact it's "rapitity" in Minkowski space) has a meaning of an area and not as an arclength. The arc length of a unit hyperbola has no simple closed form. It's rather an elliptic function.

 

[robphy]

It's only that what you call "angle" (in fact it's "rapitity" in Minkowski space) has a meaning of an area and not as an arclength. The arc length of a unit hyperbola has no simple closed form. It's rather an elliptic function.

The rapidity is also the Minkowski-arc length on the unit hyperbola.

 

[vanhees71]

Yes, but it's very confusing to call this an arc length. A pseudo-metric is not a metric!

 

[Sagittarius A-Star]

I think "Minkowski-arc length" makes clear, what is meant.

 

[jbergman]

I think "Minkowski-arc length" makes clear, what is meant.

I disagree and think that what @vanhees71 is pointing out is important. I used think of this as an actual arc-length until someone else disabused me of the incorrect notion.

 

[Sagittarius A-Star]

I disagree and think that what @vanhees71 is pointing out is important. I used think of this as an actual arc-length until someone else disabused me of the incorrect notion.

I don't know a better name as "Minkowski-arc length" for it.

 

[vanhees71]

What's meant is "proper time" for hyperbolic motion. It's not an "arc length". Of course the fundamental form of Minkowski spacetime is pretty similar to the scalar product in Euclidean affine space, but it's also different, and the difference is of great physical importance: the indefinite fundamental form of Minkowski spacetime allows for a "causal structure", which Euclidean affine space doesn't.

 

[Sagittarius A-Star]

What's meant is "proper time" for hyperbolic motion. It's not an "arc length".

But if you discuss a segment of the unit hyperbola that is space-like, then it's Minkowski-arc length cannot be called "proper time".

 

[vanhees71]

Indeed, which once more underlines that calling the fundamental form of Minkowski space a metric and derive then "arc-lengths" of curves from it is highly misleading. What's with "null lines". They'd have an arc-length of 0. Proper time for time-like world lines is the only "arc-length-like" covariant quantity related to world lines at least have some physical meaning, being the time a proper clock moving along this worldline indicates.

 

[Sagittarius A-Star]

Proper time for time-like world lines is the only "arc-length-like" covariant quantity related to world lines at least have some physical meaning, being the time a proper clock moving along this worldline indicates.

Yes, but the discussion started about the definition(s) of the hyperbolic angle and it's relation to the Minkowski line element.

 

[Sagittarius A-Star]

A pseudo-metric is not a metric!

To my understanding, the Minkowski metric is not a pseudo-metric, because a pseudo-metric is defined as a non-negative real-valued function:
https://en.wikipedia.org/wiki/Pseudometric_space#Definition

See also:

Properties of pseudo-Riemannian manifolds
Just as Euclidean space $\mathbb {R} ^{n}$ can be thought of as the model Riemannian manifold, Minkowski space $\mathbb{R}^{n-1,1}$ with the flat Minkowski metric is the model Lorentzian manifold.

Source:
https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold#Properties_of_pseudo-Riemannian_manifolds

 

[vanhees71]

I don't know, how else to call it. In the math literature it's sometimes called a "fundamental form". Also sometimes for the signature (1,3) (or (3,1)) one finds "Lorentzian manifold" for the pseudo-Riemannian manifold used in GR. I only want to avoid the misleading terminology of calling it a "metric", which leads to confusion at least when students are introduced to Minkowski spacetime.

 

[Sagittarius A-Star]

I don't know, how else to call it. In the math literature it's sometimes called a "fundamental form". Also sometimes for the signature (1,3) (or (3,1)) one finds "Lorentzian manifold" for the pseudo-Riemannian manifold used in GR. I only want to avoid the misleading terminology of calling it a "metric", which leads to confusion at least when students are introduced to Minkowski spacetime.

I think that "Minkowski metric" (instead of calling it only "metric", as done in many SR books) makes clear, what is meant.