Fundamentals of Time and Relativity

Moving Sphere

Consider first the situation of a stationary sphere. Let $d$ be the three-vector from the center of the sphere to an inertial receiver at the origin, and let the sphere subtend an angle $2 θ$ at the receiver. Photons emitted from the outline of the sphere have frequency four-vectors $k = (κ, k)$ , with $κ = | k |$ . If a photon appears to the receiver to come from the outline of the sphere, then $k$ necessarily satisfies the geometric equation

6.6

κ | d | \cos θ - k \cdot d = k \cdot d = 0

where $d : = (| d | \cos θ, d)$ is a spacelike four-vector in the receiver’s frame. For the stationary receiver, outline events are characterized by condition (6.6).

To another inertial receiver $O^{'}$ (e.g., aboard a fast spacecraft), who sees the sphere in the direction $- d^{'}$ , photons arrive from directions $- k^{'}$ . Because the Minkowski product in (6.6) is an invariant, the same condition for outline events holds. Therefore, photons appear to the moving receiver to come from a sphere with outline subtending an angle $2 θ^{'}$ .

Visual Appearance — Fig.6.2.Comparison of the measured length contraction of a globe versus its visual appearance, as viewed from a distance of three diameters of the globe from the eye to the red cross [7].

If the second receiver moves with the speed $v = γ (v) (1, v)$ parallel to $d$ away from the sphere, we have $v = v (d / | d |)$ and $v \cdot d = | d^{'} | \cos θ^{'} = γ (v) | d | (\cos θ - v)$ . Calculating the ratio $| d^{'} | / | d |$ from the definition of the four-vector $d$ (or by a Lorentz transformation), one obtains the following relation between the subtended angles:

6.7

\cot θ^{'} = γ (v) (\cot θ - v \csc θ)

So $θ^{'} \to π$ as $v \to 1$ . As the receiver accelerates away from the sphere, the outline grows until it fills the whole sky, apart from a small hole directly ahead.

To a moving receiver, a sphere still has a circular outline, but of a different size (Penrose, 1959). The sphere is the only geometrical shape that appears the same to all receivers.
The result (6.7) is just one other form of the relativistic aberration formula (6.4).