I have been playing with the geometry of special relativity again! The light cone signifies the invariance of the speed of light. There is a notion of length in four-dimensional spacetime, defined as c2t2 – x2 – y2 – z2. Surfaces of constant length are 4-dimensional hyperboloids. Light rays are null rays, as light travels a distance √(x2 + y2 + z2) in time t at speed c. Null rays trace out a double cone, the limiting case of a hyperboloid.
Lorentz Transformations are what you can to do your co-ordinate chart so that c2t2 – x2 – y2 – z2 remains the same. You can rotate only the spatial co-ordinates. Or you can rotate space and time into each other. The latter is called a boost. You can think of a boost as hopping onto another inertial co-ordinate frame that moves at constant speed with respect to the original system.
There is an alternative way to think about Lorentz Transformations: The four spacetime co-ordinates (t,x,y,z) can be arranged in a complex Hermitian 2×2 matrix. To Lorentz-transform this matrix, it is multiplied by another 2×2 matrix from the left and the right. In these transformation matrices, rotation angles show up as half their actual values, intuitively because of these ‘double’ multiplication (from left and right). Four real numbers went into the creation of the Hermitian matrix. They could also go into two complex numbers. The ratio of these is another complex number – and this can be visualized as a point on the Riemann sphere. This is very similar to the tangible interpretation of the ‘inclination angles’ of spins (and half their values) in quantum mechanics.
The Riemann sphere of special relativity is the Celestial Sphere, which comprises the directions of all the light rays – coming from all angles to paint images of distant stars on a spherical film around the origin. A ray is a line through the origin, described by three parameters (in four-dimensional space). A light ray is a null ray, as it also meets c2t2 – x2 – y2 – z2. Thus, the light ray is described by two parameters. These two parameters form one complex number; this complex number corresponds to a point on the sphere, via stereographic projection.
Rotations and boosts transform one complex number into another. Transformations are conformal: They keep angles intact and map circles on circles. Paint an image in the complex plane, e.g. a circle. Boost it. The boosted shape is again a circle. You can also project the circle up to the Riemann sphere first, then Lorentz Transform, then project the result down again (stereographically). When spatial co-ordinates are rotated into each other, circles on the Riemann sphere are either rotated in the expected way. Boosts correspond to circles that change size while they are drawn ‘down’ to the South Pole (when projecting from the North Pole).
In the following image I am boosting two different circles, a golden and a silver one. The ‘original’ circle is in the Northern hemisphere. Velocity is increased linearly up to 0.99c as the circles move down along the surface of the Riemann sphere. For each circle on the sphere, the stereographical projection in the equatorial plane is also depicted.
Images created with SageMath and custom Python code, then inverted.
Details of the figures (number of circles, view point, colors, size and position of circles…) selected for aesthetics, not for ‘educational merit’.
Circles are parameterized curves. Transforms are done in the complex plane (Möbius Transforms of complex numbers are isomorphous to Lorentz Transform of the corresponding 4-vectors). Resulting curves are projected stereographically from the North Pole, by extending the set of equations that describe the circles implicitly (I hit performance limits of my hardware surprisingly soon.) The original circles in the plane are the largest ones, they correspond to the ‘top circles’ on the sphere.
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I have been creating the images the day I was boosted myself. I played with colors. Only after I had taken screenshots, I noticed they embrace the notion of a celestial sphere – maybe in ways going beyond the fact that shapes are spherical. Colors are festive, hinting at holidays and sparkles. But they are not overly colorful, Hi, Omicron!
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References:
Roger Penrose’s book The Road to Reality, section 18.5 The celestial sphere as a Riemann sphere.
David Tong’s lecture note Dynamics and Relativity, section 7.6 Spinors.
From the Lorentz Group to the Celestial Sphere (Lecture Notes)
I’ve a long article / math write-up in the making including more details.
elkemental Force 