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March 3, 2023

Special Relativity and the Mercator Projection

Posted by John Baez

When you look at an object zipping past you at nearly the speed of light, it looks not squashed but rotated.

This phenomenon is well known: it’s called Terrell rotation. But this paper puts a new spin on it:

The abstract is pretty funny:

Abstract. It has been almost half a century since the realization [1,3] that an object moving at relativistic speeds (and observed by light reflected from some point source) is seen not as squashed (as a naive interpretation of the Lorentz-Fitzgerald contraction would suggest) but rather as rotated, through an angle dependent on its velocity and direction of motion. I will try to show here that this subject continues to be worth exploring.

The author asserts his moral right to remuneration for any applications of these ideas to computer games or big-budget sci-fi movies.

But here’s the idea.

Suppose a rocket is moving at velocity v in units where c=1c = 1. Suppose it’s in the upper half of the xyx y plane, moving from left to right, parallel to the xx axis. Suppose you are at the origin.

If the rocket’s position is (r,θ)(r,\theta) in polar coordinates, let

ψ=θπ2 \psi = \theta - \frac{\pi}{2}

so ψ=0\psi = 0 when rocket is ‘just passing you’. Let’s call ψ\psi the line of sight angle.

The rocket looks rotated counterclockwise by an angle ϕ\phi, where

cos(ϕ+ψ)=cosψv1vcosψ \cos(\phi + \psi) = \frac{\cos \psi - v}{1 - v cos \psi}

But Morava found a nicer way to write this formula!

As we were all once forced to learn, the integral of secx\sec x is

λ(x)=ln|tanx+secx| \lambda(x) = \ln\vert\tan x + \sec x\vert

Now you’re finally going to get some good out of this knowledge.

The reason why people figured out this integral in the first place is that in a Mercator projection map, a city with latitude xx lies on a line with yy coordinate λ(x)\lambda(x).

Now, define ‘Mercator addition’ by

λ(xy)=λ(x)+λ(y) \lambda (x \oplus y) = \lambda (x) + \lambda (y)

So, do the Mercator sum of angles xyx \oplus y you think of them as latitudes, you convert them into heights up from the equator on a Mercator map, then add those heights, then convert the answer back into a latitude.

Suppose a rocket whizzing by at speed vv has light of sight angle ψ\psi. Then Morava shows it looks rotated by an angle ϕ\phi, where

ψ+ϕ=ψarcsinv \psi + \phi = \psi \oplus \arcsin v

I’ll admit I haven’t checked his computation.

I don’t understand the deep inner meaning of this. So let me just say it in words!

To get the apparent rotation angle ϕ\phi of a moving object, you take the Mercator sum of its line-of-sight angle ψ\psi and the arcsine of its velocity, then subtract off ψ\psi using ordinary subtraction.

What does it really mean? Why should the Mercator projection be related to special relativity? Can you help me out here?

I’m sure Jack Morava was interested in this because Mercator addition is an example of a formal group law, and formal group laws are important in homotopy theory.

One other weird and appealing fact is that the function λ\lambda, which you hated so much in calculus class, is almost its own inverse! To be precise,

λ 1(x)=iλ(ix) \lambda^{-1}(x) = - i\lambda (i x)

Morava also gets some mileage out of this… but read his paper for details.

Posted at March 3, 2023 8:06 PM UTC

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Re: Special Relativity and the Mercator Projection

Dear John,

Srsly this is reminiscent of Wick rotation, but I don’t know how to make that intuition even vaguely precise. As Private Eye says, I think we should be told… I hope someone can make sense of this (assuming it’s not just false). Thanks, hopefully…

Posted by: jack morava on March 4, 2023 1:10 PM | Permalink | Reply to this

Re: Special Relativity and the Mercator Projection

Wick rotation is the ‘rotation’ of time into imaginary time, but I’ve never heard of anyone talking about it as a continuous process, which is what you seem to be hinting. I also don’t see why it should be related to what we’re seeing here.

What I’m curious about is: what made you try to get the function

λ(x)=ln|secx+tanx|=arctanhsinx \lambda(x) = \ln|\sec x + \tan x| = \mathrm{arctanh} \, \sin x

into the game?

Hmm, the last formula may be related to Wick rotations, since first we start out with a trig function and then do an inverse hyperbolic trig function.

Posted by: John Baez on March 4, 2023 6:00 PM | Permalink | Reply to this

Re: Special Relativity and the Mercator Projection

It’s hard to recall how you untied a knot. It may have come out of trying picture things like composing the cx logarithm function, multiplying by things on the unit circle such as sqrt etc, and then exponentiating (instead of counting sheep).

Re an upthread discussion of work by Kontsevich and Segal, I think they work out a theory of extensions of Lorentz pseudometrics to complexifications, with bounds on the depth in some sense of these complexifications.

I don’t mean to suggest this is related in any way to my note - I was just hoping someone would work out a relativistic SuperMario extension in which the speed of light is about 15 miles per hour.

Posted by: jack morava on March 4, 2023 6:42 PM | Permalink | Reply to this

Re: Special Relativity and the Mercator Projection

The sin\sin in that last formula for λ\lambda and the arcsin\arcsin in the definition of ϕ\phi cancel out, which means that what we’re really adding is rapidity.

This also suggests that we should treat sin(ψ)\sin(\psi) as a velocity, but I don’t know how that would work out for combining rotations.

Posted by: unekdoud on March 7, 2023 12:25 AM | Permalink | Reply to this

Re: Special Relativity and the Mercator Projection

I seem to recall that the Lorentz group has a compact torus /\mathbb{R}/\mathbb{Z} and a noncompact torus R ×\R^\times, and I always assumed that Wick rotation was some kind of Weyl group conjugation (like exponentiating to the power 1=i\sqrt{-1} = i) that somehow transposes them; but I know absolutely nothing about Lie theory, and I hope for moderation!

Posted by: jack morava on March 7, 2023 9:14 PM | Permalink | Reply to this

Re: Special Relativity and the Mercator Projection

Usually in physics ‘Wick rotation’ means this:

We start with Minkowski spacetime 3,1\mathbb{R}^{3,1}, meaning 4\mathbb{R}^4 with the bilinear form called the Minkowski metric. Then we complexify it and extend the bilinear form to a complex-bilinear form on 3,1\mathbb{C} \otimes \mathbb{R}^{3,1}. Then the Wick rotation map

W: 3,1 3,1W : \mathbb{C} \otimes \mathbb{R}^{3,1} \to \mathbb{C} \otimes \mathbb{R}^{3,1}

is the map

(x,y,z,t)(x,y,z,it) (x,y,z,t) \mapsto (x,y,z,i t)

and it sends 3,1 3,1\mathbb{R}^{3,1} \subset \mathbb{C} \otimes \mathbb{R}^{3,1} to a copy of 4d Euclidean space sitting inside complexified Minkowski spacetime.

So, in short, Wick rotation sends time to imaginary time.

Posted by: John Baez on March 8, 2023 7:45 PM | Permalink | Reply to this

Re: Special Relativity and the Mercator Projection

This goes back to Einstein/Hilbert/Weyl &c but I think it has an interpretation in terms of both the Lie group Sl 2(C)\Sl_2(\C) ( = the double cover of the identity component of the Lorentz group) and its Lie algebra sl 2(C)\sl_2(\C), in terms of maximal abelian subgroups/Lie algs… will try to sort out the details and say something more precise soon.

Posted by: jack morava on March 9, 2023 12:38 AM | Permalink | Reply to this

Some history

Some interesting history behind the formula for the integral of the secant. As Eli Maor recounts in his Trigonometic Delights, in the chapter “A Mapmaker’s Paradise”:

…Edmund Gunter … published a table of logarithmic tangents. Around 1645 Henry Bond, a mathematics teacher and authority on navigation, compared this table with Wright’s meriodional table [of ∫ sec t, constructed via numerical integration for the Mercator projection] and noticed to his surprise that the two tables matched… He conjectured that ∫0φ sec t dt is equal to ln tan(45°+φ/2) … but he could not prove it. Soon his conjecture became one of the outstanding mathematical problems of the 1650s. …

Finally in 1668 James Gregory … succeeded in proving Bond’s conjecture; his proof, however, was so difficult that Halley denounced it as being full of “complications”. So it befell Isaac Barrow … to give an “intelligent” proof of Bond’s conjecture (1670), And in doing so he seems to have been the first to use the technique of decomposition into partial fractions…

Posted by: Michael Weiss on April 24, 2023 8:02 PM | Permalink | Reply to this

Re: Some history

That’s very interesting! Imagine popping into a time machine and emerging back in the era where proving

0 ϕsectdt=lntan(ϕ2+π4) \int_0^\phi \sec t \, d t = \ln \, \tan\left(\frac{\phi}{2} + \frac{\pi}{4}\right)

was one of the outstanding problems of the day!

There’s some more information about this here. 200 years later, Cayley also worked on this stuff.

Posted by: John Baez on April 24, 2023 10:30 PM | Permalink | Reply to this

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