Transforming the Celestial Sphere

A spherical spaceship swooshes by at 99% of the speed of light. What will it look like? Squashed because of Lorentz contraction – like an ellipsoid?

No. The outline of a moving sphere will remain spherical. Roger Penrose explained this first in 1958 – 50 years after Einstein’s formulation of the theory of special relativity. Light rays come from the front and the rear end of the sphere, traveling for different times before they hit the eye. This compensates for Lorentz contraction. But Lorentz contraction is not completely invisible: a shape painted on the sphere will appear distorted.

But we learned that the observer sees shortened rulers – how can these views be reconciled? The concept of an observer is often a vague one. In introductory books on special relativity observers are explorers with clocks and rulers. They are traveling on trains and sending pulses of light. But the true observer in special relativity – the one actually used in calculations – is not a guy with a clock and a ruler. As David Tong says, an observer is more of a big brother / God-like entity – a sea of clocks and rulers available at every point in space and time. Penrose talks about a human or camera-like observer: A film or an eye that takes a picture of the light rays coming from the moving object. The so-called paradoxes in Special Relativity are NOT due to the time it takes light to reach the observer (and these ‘paradoxes’ are not what I am covering in this article). What a human-like observer actually sees is something to be covered on top / in addition to time dilation and length contraction.

The mathematics best suited to describe this effect is related to spinors and stereographic projection.

This article is also the ‘making of’ for my last post and these images:

… and it is an attempt to present mathematical formulas in a way matching these images.

A particle traces out a worldline in four-dimensional spacetime. Whatever happens, it happens at a certain time, in a certain place. An event is encoded in 4 numbers – three spatial dimensions x, y, z and a point of time t (or ct, c being the speed of light).

( ct , x , y , z )

(You can call this 4-tuple a 4-vector, but you should be careful. Only velocities are true vectors in differential geometry: they remain vectors also if space(time) is curved.)

Lorentz Transformations describe how to change co-ordinate charts when switching to another reference frame: a frame that is rotated in three dimensions (not rotating right now) or that is moving at constant velocity – boosted. The math of these transformations makes all the paradoxes go away. They are like generalized rotations that may mix also space and time.

Rotations in 3D space keep lengths and angles invariant: x2 + y2 + z2 remains the same in the rotated frame. The generalized rotations in this 4D Minkowski spacetime also keep length intact, but here the definition of ‘length’ has an interesting change of signs:

c^2 t^2 - x^2 - y^2- z^2

If this length is zero in one frame, it remains so in every frame. This is just stating that the speed of light hast the same value of c in every frame!

You can only speak of a length (and of angles) in a vector space if there is an inner product. The definition of the inner product is encapsulated in a matrix – the spacetime metric. Two 4-vectors are ‘multiplied’ by squeezing the matrix between them. Multiply a vector with itself: the result is its length squared. The minus sign of the spatial components stems from the minus signs in the metric:

\left( {\begin{array}{rrrr}  1 & 0 & 0 & 0 \\  0 & -1 & 0 & 0 \\  0 & 0 & -1 & 0 \\  0 & 0 & 0 & -1  \end{array} } \right)

Rotations are isomorphic to matrices. Rotating about the z-axis means to multiply (ct, x, y, t) with:

\left( {\begin{array}{cccc}  1 & 0 & 0 & 0 \\  0 &  \cos\theta & \sin\theta & 0 \\  0 & -\sin\theta & \cos\theta & 0 \\  0 & 0 & 0 & 1 \\  \end{array} } \right)  \left( {\begin{array}{c}  ct \\  x \\  y \\  z  \end{array} } \right)

Time t and co-ordinate z remain the same. x and y are mixed up.

With boosts, time and space are rotated into each other, in a sense, but cosine and sine functions are replaced by their hyperbolic counterparts. This is a boost to a system moving along the z-direction. The speed v is expressed in terms of rapidity φ.

\left( {\begin{array}{cccc}  \cosh\varphi & 0 & 0 & -\sinh\varphi \\  0 & 1 & 0 & 0 \\  0 & 0 & 1 & 0 \\  -\sinh\varphi & 0 & 0 & \cosh\varphi \\  \end{array} } \right)  \left( {\begin{array}{c}  ct \\  x \\  y \\  z  \end{array} } \right)
\cosh\varphi = \gamma = \sqrt{\frac{1}{1-\frac{v^2}{c^2}}}

The four numbers that form a spacetime event can be arranged in a different way – in a 2×2 complex-valued matrix:

\hat{X} = \left( {\begin{array}{cc}  ct + z & x - iy \\  x + iy & ct - z  \end{array} } \right)

This matrix is, by construction, Hermitian: It is equal to its conjugate transpose. The determinant of this event matrix …

c^2t^2 - x^2 - y^2 - z^2

… is equal to the length squared of the corresponding 4-vector!

In this complex-valued picture, a Lorentz Transformation is a 2×2 matrix with determinant 1; it is called A in the following. To get the same transformation laws as for the 4-vector, the event matrix has to be multiplied from left and right by the transformation matrix and its conjugate transpose. As the determinant of a product of matrices is the product of the determinants, the determinant of the resulting matrix is again 1.

\hat{X} \to A \hat{X} A^{\dagger}

The transformation matrices feature half the angles and rapidities. Again, we see θ/2 come up! Rotating about the z-axis, by an angle θ:

\left( {\begin{array}{cc}  e^{+i\theta/2} & 0 \\  0 & e^{-i\theta/2} \\  \end{array} }  \right)  \left( {\begin{array}{cc}  ct+z & x-iy \\  x+iy & ct-z \\  \end{array} }  \right)  \left( {\begin{array}{cc}  e^{-i\theta/2} & 0 \\  0 & e^{+i\theta/2} \\  \end{array} }  \right)

Boosting along the z-axis, with a velocity corresponding to rapidity φ:

\left( {\begin{array}{cc}  e^{-\varphi/2} & 0 \\  0 & e^{+\varphi/2} \\  \end{array} }  \right)  \left( {\begin{array}{cc}  ct+z & x-iy \\  x+iy & ct-z \\  \end{array} }  \right)  \left( {\begin{array}{cc}  e^{-\varphi/2} & 0 \\  0 & e^{+\varphi/2} \\  \end{array} }  \right)

The determinant (of the event matrix) must stay the same. As the determinant of a product of matrices is the product of the determinant, the determinant of the Lorentz Transformation matrix has to be 1.

Any 2×2 complex matrix with determinant 1 …

\left( {\begin{array}{cc}  a & b \\  c & d \\  \end{array} }  \right)
ad - bc = 1

… is a valid Lorentz Transformation, containing combinations of boosts and rotations for any direction.

Counting dimensions and constraints: The group of Lorentz Transformation is built from rotations about 3 different axes and from boost along these axes. Any transformation can be parameterized by 3 angles and 3 rapidities. The complex matrix has 4 spots, thus 8 real numbers. The determinant being equal to real number 1 is a complex equation – two more conditions, one for real and one for imaginary part. 8 minus 2 is 6! \o/

Finally, we can speak about spheres. A sphere with radius R in three-dimensional space is defined by all points having the same distance from the center. Placing the center of the sphere at the zero point (we do not look at translations, only rotations and boots):

x^2 + y^2 + z^2 = R^2

Light travels a distance ct in time t. Starting out at time 0, light rays reach the surface of a sphere with radius ct after time t:

x^2 + y^2 + z^2 = c^2t^2

The definition of this spherical surface is the same as saying that the length of 4-vector (ct, x, y, z) is zero:

c^2t^2 - x^2 - y^2 - z^2 = 0

This equation talks about one event, a single point. Follow a light ray in 4D Minkowski space, the worldline of light traveling at speed c: If you know one (ct,x,y,z) event point on the light ray, you get other points on the same light ray by increasing ct,x,y, and z by the same factor. This is a general property of any ray, any straight line through zero: The ratios of co-ordinates with respect to each other remain constant. In a four-dimensional space, these are three equations relating four co-ordinates.

A light ray is a null ray, it also has to meet the ‘zero length’ constraint for all points: c2t2 – x2 – y2 – z2 = 0. In total, there are two constraints for four co-ordinates: A light ray is defined by two numbers.

The light ray in four dimensions is the ‘trace’ light leaves in three-dimensional space plus the information at which time each (3D) point was hit. Think of all 3D light rays emerging from the zero point, at any direction. At time t, each light ray hits the surface of a sphere defined by x2 + y2 + z2 = c2t2. Each of the spheres is punctured by the same light rays. You partition three-dimensional space into spheres of radius ct. Each of the concentric spheres represents the set of all light rays. It is called a celestial sphere. It is like a photo of the night sky on a spherical film – one sphere, belonging to a certain time t, whose value is not important). As David Tong says: we lose the information about where we are along the light ray. The celestial spheres are equivalence classes of light rays.

John Synge says it this way (p. 95, Relativity: The Special Theory, 1956): The tetrad of components in any four-vector can be expressed via three by null rays.

Any ray (not necessarily null) is determined by the three ratios ct : x : y : z of the coordinates of any event on it. On a null ray these three ratios are connected by virtue of the relation c2t2 – x2 – y2 – z2 = 0, and so it follows that a null is determined by two numbers. Hence a triad of null rays are determined by (and determine) six numbers, and so such triad has the same number of degrees of freedom as a unit orthogonal tetrad.

He refers to the degrees of freedom in a Lorentz Transformation – the six numbers that can be seen as 3 angles and 3 rapidities … or as a set of 3 arbitrary complex numbers. Matrix components a,b,c,d are reduced to 6 real numbers because of the det=1 constraint. Any arbitrary ray is determined by three numbers as the ratios between co-ordinates are fixed. Six numbers are sufficient to transform any (3-parameter) ray into any other, by mapping 3 numbers onto 3 arbitrary, different numbers.

How to get from the outline of a sphere one observers sees to the outline of the sphere another observer sees? Connect these celestial spheres, their ‘images on spherical films’ by Lorentz Transformations. It’s equivalent to say: think about the transformations of light rays.

The matrix equivalent of a light ray has determinant zero. The most general, complex-valued 2×2 matrix with determinant zero can be built from two complex numbers. Take any two complex numbers, ξ1 and ξ2. The tuple ξ = (ξ1, ξ2) that can be considered a complex vector. Form this matrix, where † denotes the complex conjugate:

\left( {\begin{array}{cc}  \xi_1 \xi_1^{\dagger} & \xi_1 \xi_2^{\dagger} \\  \xi_1^{\dagger}\xi_2 & \xi_1 \xi_1^{\dagger} \\  \end{array} }  \right)

Its determinant is zero:

(\xi_1 \xi_1^{\dagger})(\xi_1 \xi_1^{\dagger}) - (\xi_1 \xi_2^{\dagger})(\xi_1^{\dagger}\xi_2) \\\\  = |\xi_1|^2 |\xi_2|^2 - |\xi_1|^2 |\xi_2|^2 = 0

The matrix can also be written as the outer product of the vector ξ with itself

\xi \xi^{\dagger} = \left( {\begin{array}{c}  \xi_1 \\  \xi_2  \end{array} }  \right) \left(  {\begin{array}{cc}  \xi_1^{\dagger} & \xi_1^{\dagger} \\  \end{array} }  \right)

This is reminiscent of the spin-like quantum state described recently! A vector-like set of two numbers can be visualized by looking at the ratio of the numbers – also a complex number. You can change ξ1 and ξ2 by the same (complex) factor without changing the physics. In quantum physics, state vectors are normalized. Only probabilities calculated as the absolute squares ‘do count’. The same applies here, as a light ray is described by one complex number – the ratio of ξ1 and ξ2.

What do ξ1 and ξ2 actually mean? A Lorentz Transformation, turning ξ into a rotated/boosted ξ’ acts on the complex 2×2 matrix via a multiplication with the matrix A for the Lorentz Transformation ‘from both sides’:

A = \left( {\begin{array}{cc}  a & b \\  c & d  \end{array} }  \right)
\xi \xi^{\dagger} \to A \xi \xi^{\dagger} A^{\dagger}

This can be interpreted as transforming the complex vector, the spinor ξ itself:

\xi \to A \xi
\xi^{\dagger} \to (A \xi)^{\dagger} = \xi^{\dagger} A^{\dagger}

Matrix A is a generalized rotation, representing actual rotations and boosts. If a four-vector is rotated by some angle, the corresponding 2×2 matrix ξ ξ is multiplied ‘twice’, from the left and the right. In order to yield the correct rotation or boost in the end, half angles show up in matrix X. This means that spinor ξ is only ‘rotated half-way’.

If ξ1 and ξ2 are multiplied by the same complex factor, the determinant remains zero: The components of the ξξ matrix would all change by the same real number, the real magnitude of this complex factor, and the exponentials containing its phase would cancel. But multiplying each component of the ‘event matrix’ with the same factor means you stay on the same ray! You don’t care where you are on the light ray.

The interesting information, the single complex number ξ1 / ξ2 can be visualized as a point in the Riemann sphere, by projecting the point from the complex plane up to the Riemann sphere. As Penrose says, the celestial sphere is also a Riemann sphere. (Penrose explains this briefly in his book The Road to Reality. This forum discussion has some images of the involved sphere(s) from his textbook on spinors and relativity.)

We can again use the geometric interpretation of a point on the sphere discussed for a spin state. The ‘spin arrow’ is a direction in space. The line of sight of the light ray is, too. Both can be interpreted as a complex number, by using stereographic projection and the Riemann sphere.

Multiplying ξ with a matrix A is equivalent to transforming the ratio ξ1 / ξ2 = ω by an operation called a Möbius Transformation:

\frac{\xi_1}{\xi_2} = \omega
\omega \to \frac{a \omega + b}{c \omega + d}

Möbius transformations map circles on the complex plane to other circles in the complex plane – this is closely tied to stereographic projection mapping circles onto circles. Instead of looking at this function of ω, we can look at what happens to the shapes painted by light rays on the Riemann sphere:

The circle seen by one observer is traced out by light rays, a series of ω values describing a circle. It’s a circle on the observer’s spherical film – all the points hit at the same time t. It is a circle on the Riemann sphere. Think of every point on the Riemann sphere as a spin-like arrow represented by complex coefficients (whose ratio is ω). Apply the 2×2 matrix transformations OR project the circle up down to the complex plane and run the Möbius transformation. The circle on the sphere is tilted and/or its size is changed. The circle in the plane is mapped to another circle.


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