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1 Summary
2 How these images were made
3 Other images
4 Licensing

[edit] Summary

Demonstration of Aberration of light and Relativistic Doppler effect. In this diagram, the blue point represents the observer. The x,y-plane is represented by yellow graph paper. As the observer accelerates, he sees the graph paper change colors. Also he sees the distortion of the x,y-grid due to the aberration of light. The black vertical line is the y-axis. The observer accelerates along the x-axis.

Velocity is 0.44 speed of light.

Velocity is 0.78 speed of light.

Velocity is 0.89 speed of light.

[edit] How these images were made

Diagram 1
Animation: Image:Lcprojection2m.gif

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

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

where

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

is a 3D velocity vector and

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

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

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

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:

Diagram 2. The grey ellipse is moving relativistic sphere, its oblate shape due to Lorentz contraction. Colored ellipse is visual image of the sphere. Background curves are a xy-coordinates grid which is rigidly linked to the sphere, it is shown only at one moment in time.

$x(a,b) = r\cos a\,$

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

$z(a,b) = r\sin a\sin b\,$

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

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

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

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

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

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

it can be written as

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

At any given observer time

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

the surface in 3-d space after substitution

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

and

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

can be written as

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

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

$t' = t - distance/c\,$

or in our notation

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

Solution to this equation is

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

Hence the observer will see the surface

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

where

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

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

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

[edit] Other images

Moving wave source..
Animation: Media:Waves01.gif

Relativistic Doppler effect in comparison with non-relativistic effect.
Animation: Media:Compare03s.gif

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.

On this image observer (blue point) is inside the subway. Velocity is zero. It is the first frame in an animated model of movement.
Animation: Media:RelSubway2.gif

On this image observer (blue point) moves inside the same subway. Velocity is equal 0.7 speed of light
Animation: Media:RelSubway2.gif

The outside view of the subway. However, this is only a model, there does not exist an observer which can see the tunnel like this.

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.

This sphere moves toward to observer.
Animation: Media:SphereAberration01.gif

On this picture an observer is inside the sphere.
Animation: Media:SphereAberration01.gif

This sphere moves away from observer.
Animation: Media:SphereAberration01.gif

Tachyon visualization. Since that object moves faster then speed of light we can not see it approaching. Only after a tachyon has passed nearby, we could see two images of the tachyon, appearing and departing in opposite directions.
Animation: Image:Tachyon03.gif

Animation: Media:Tachyon03.gif

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

[edit] Licensing

I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
Subject to disclaimers.

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