Image:XYCoordinates.gif
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[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.
[edit] How these images were made
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
where
is a 3D velocity vector and
The component is the timelike component of while the other three components are the spatial components and
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:
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
or
where is timelike history parameter. In a frame which moves relatively to the object with velocity after Lorentz transformations
it can be written as
At any given observer time
the surface in 3-d space after substitution
and
can be written as
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
or in our notation
Solution to this equation is
Hence the observer will see the surface
where
On Diagram 2 surface is flattened (this effect is called Lorentz contraction) grey ellipse, while the visible sphere image 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
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 |
[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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Date/Time | Dimensions | User | Comment | |
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current | 23:50, 12 August 2006 | 454×189 (1.53 MB) | TxAlien (Talk | contribs) | (Demonstration of light aberration and Relativistic Doppler effect.) |
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