In this section we will use the Minkowski space 4-dimension vectors and units where speed of light c = 1. In this notation velocity vector is

${\displaystyle \mathbf {U} =(u_{0},u_{1},u_{2},u_{3})\equiv (u_{0},\mathbf {u} )=\left(\gamma ,\gamma {\frac {\mathbf {v} }{c}}\right)}$

where

${\displaystyle \mathbf {v} =(c/u_{0})\mathbf {u} \,}$

is a 3D velocity vector and

${\displaystyle u_{0}=\gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$

The component ${\displaystyle u_{0}\,}$ is the timelike component of ${\displaystyle \,\mathbf {U} \,}$ while the other three components are the spatial components and

${\displaystyle u_{0}^{2}-u_{1}^{2}-u_{2}^{2}-u_{3}^{2}=u_{0}^{2}-\mathbf {u} \cdot \mathbf {u} =1\,}$

In spacetime, any small object is the sequence of spacetime events or world lines. Hence a 3D object in space-time can be described as a set of world lines corresponding to all the points that make up that object. Consider a simple objects in the space such as a sphere:

${\displaystyle x(a,b)=r\cos a\,}$

${\displaystyle y(a,b)=r\sin a\cos b\,}$

${\displaystyle z(a,b)=r\sin a\sin b\,}$

where a and b are parameters of the mapping and r is the radius of the sphere. To make the formulas easy to read we will leave out the parameters (a,b). In the frame where the object is at rest, the set of world lines can be represented as the world surface in Minkowski space

${\displaystyle S(a,b,\tau )=\left(\tau ,\ x(a,b),\ y(a,b),\ z(a,b)\right)\,}$

or

${\displaystyle S\left(\tau \right)=(\tau ,x,y,z)}$

where ${\displaystyle \tau \,}$ is timelike history parameter. In a frame which moves relatively to the object with velocity ${\displaystyle \left(u_{0},u,0,0\right)\,}$ after Lorentz transformations

${\displaystyle \tau '=\tau u_{0}-xu\,}$

${\displaystyle x'=xu_{0}-\tau u\,}$

it can be written as

${\displaystyle S(\tau )=(\tau u_{0}-xu,xu_{0}-\tau u,y,z)\,}$

At any given observer time

${\displaystyle t\equiv \tau '=\tau u_{0}-u\,}$

the surface in 3-d space after substitution

${\displaystyle \tau ={\frac {t+xu}{u_{0}}}}$

and

${\displaystyle xu_{0}-\tau u=x{\frac {u_{0}^{2}-u^{2}}{u_{0}}}-u{\frac {t}{u_{0}}}={\frac {x-tu}{u_{0}}}}$

can be written as

${\displaystyle S_{Lorentz}(a,b,t)=\left({\frac {x-tu}{u_{0}}},y,z\right)}$

However, observer can only see the object on light cone in the past when light is emitted by points on the object. Therefore, if observers coordinates are (t,0,0,0) he sees (Diagram 1) the surface points where they were in time

${\displaystyle t'=t-distance/c\,}$

or in our notation

${\displaystyle t'=t-{\sqrt {\left({\frac {x-t'u}{u_{0}}}\right)^{2}+y^{2}+z^{2}}}}$

Solution to this equation is

${\displaystyle t'=u_{0}2t-xu-u_{0}{\sqrt {\left(x-ut\right)^{2}+y^{2}+z^{2}}}}$

Hence the observer will see the surface

${\displaystyle S_{visual}=S_{Lorentz}(a,b,t')=\left(x',y,z\right)=\left({\frac {x-t'u}{u_{0}}},y,z\right)\,}$

where

${\displaystyle x'=\left(x-ut\right)u_{0}+u{\sqrt {\left(x-ut\right)^{2}+y^{2}+z^{2}}}}$

On Diagram 2 surface ${\displaystyle S_{Lorentz}\,}$ is flattened (this effect is called Lorentz contraction) grey ellipse, while the visible sphere image ${\displaystyle S_{visual}\,}$ is colored. Color change is due to the relativistic Doppler effect.

It's still not finished--TxAlien 05:04, 3 September 2006 (UTC)

Next pictures show a model of movement through a subway tunnel at an increasing speed. Observer would not only see changes in colors but the outer walls will also appear convex. This convex appearance is due to seeing parts of the tunnel at different moments in time because of the finite speed of light. This effect is called aberration of light.

This images shows the view on a sphere moving at 0.7c relatively to a stationary observer, which is represented by blue point. Curved lines represent the xy grid coordinates of a moving system.

All these images are graphical solutions of an accurate mathematical model which is based on the Lorentz transformation

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• 2006-08-12 23:50 TxAlien 454×189× (1600354 bytes) Demonstration of light aberration and [[Relativistic Doppler effect]].