The following text is that of a seminar presented to the ANU Physics Department on 25 September 1997. Note that the links to animations will not function. Animations may be downloaded elsewhere.

VISUALIZING SPECIAL RELATIVITY
Antony Searle
Australian National University Distinguished Scholars Program
Animations
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INTRODUCING RELATIVITY OBSERVATION AND SIGHT RELATIVISTIC ABERRATION
 
Driving through rain - Road Frame 
Driving through rain - Car Frame
  • Aberration arises because of the transformation between inertial frames.
  • Consider the analogy of driving through rain
  • An observer (top left) standing by the side of the road sees rain falling vertically, and the car moving horizontally.
  • The driver (bottom left) sees the rain falling at an angle, towards the car.
  • The vectors described by the falling rain have angularly contracted in the direction of motion.
Light converging on a moving observer - Ether Frame 
Newtonian aberration of light - Observer Frame
  • Now consider photons converging on the point momentarily occupied by a moving observer (top left)
  • Using the Newtonian velocity addition formula, we obtain a redistribution of the light (bottom left).
  • This model corresponds to the concept of motion through the ether.
  • Observe that the speeds of the light in this model are different in different directions - this model is not compatible with special relativity, in which light always has the same speed.
Light in Frame of Origin - angles ALPHA 
Light in Observer Frame - angles ALPHA PRIME
  • Relativistic aberration preserves the speed of light, as at bottom left.
  • Note that light concentrates in the direction of motion, according to the formula
tan(a'/2) = [(c - v)/(c + v)]^(1/2) * tan(a/2)
    where a is the angle of the light with the velocity v with which the observer is moving and a' is corresponding angle in the rest frame of the observer. See Animation
  • Note that as v < c, it follows that a' < a, and thus all rays are rotated forward to some extent.
 
An explanation of the Head-up Display:
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Aberration Animations:
Non-relativistic Animation Desert Aberration Animation Fly-past Cube Animation Fly-through Cube Animation Near-Earth Orbit Animation Pinwheel Animation
 
  • Aberration causes many visual effects. See Images.
  • All can be understood by noting this is a pure 'lens' transformation: given the view-sphere of any observer at a given instant, we can transform it to the view-sphere of any other observer at that point in space-time regardless of their motion.
  • Note also that the angular contraction will lead to objects already passed by the camera appearing to be in front of the camera.
  • Considering the above, the cause of the famous relativistic rotation of objects becomes clear:
  • We see the front of the object when in front of it
  • We see the side of the object when beside it. However, the right angle it would make with an observer at rest to it becomes an acute angle in the frame of the moving observer, and so it appears still to be in front of us.
  • We see the back of an object when past it. However, the obtuse angle it would make with an observer at rest to it becomes a smaller angle, going from acute, to right, to obtuse as we get further from the object. We associate this angular progression with going past an object; and thus we appear to pass the object only after we have already gone past it.
 
 
 
THE DOPPLER EFFECT
f' / f = 1 + (v / c)cos(a)
f' / f = [1 + (v / c)cos(a)] / (1 - (v/c)^2)^(1/2)
Doppler Shift Animations:
Desert Doppler Animation Fly-by Cube Animation Pinwheel Animation
 
 
Blue Spectrum 
Green Spectrum 
Red Spectrum 
 
  • To model the redshifting of light in such an image, an approximate spectrum must be used.
  • The above wavelength spectrums for blue, green and red light are used to construct a composite spectrum for a general RGB colour by linear combination. 
  • White Spectrum
  • While the spectrums are not accurate, they roughly mimic the behavior of a coloured object reflecting light from a 6000K blackbody, and have the advantage of being swiftly calculated.
 
Redshifted spectrum
Blueshifted spectrum
    Predominantly green spectrum
  • The resulting spectrum is then divided by the Doppler factor calculated above. 
  • The modified spectrum is then resampled at the blue, green and red wavelengths to calculate the colour observed
 
  • Note that the model concentrates all colour detail in a narrow band of the spectrum. This is the cause of the rainbow ring visible in the images. In reality, thermal or ultraviolet characteristics, the equivalent of colour beyond the visible, would be shifted into the visible and observed as additional colour in the areas currently dull red or blue. See Images.
  • Relativistic doppler shifting of light is the principle means of detection of accretion discs about massive objects. In an observation of the disk, any sharp spectral features are redshifted on the receding side of the disk and blueshifted on the approaching side into a characteristic double peak, from which many properties of the disk may be determined.
 
THE HEADLIGHT EFFECT
 
  • The intensity of the image captured by the camera is also affected by relativity.
  • This arises principally from the redistribution of light by the camera. Note that light is concentrated in the direction of motion, which thus seems brighter. Also, time dilation means that the shutter captures more light than it would if stationary. The two effects, calculated by the ratio of transformed infinitesimal solid angles, combine to produce the expression
I / I' = (1 + (v/c)cos(a))^2 / (1 - (v/c)^2)^(1/2)
 
 Headlight Effect Animations:
Desert Doppler Animation
 
 
  • The angular contraction concentrates light in the direction of motion.
  • This leads to brightening in the direction of motion, appearing like the headlights of a car at night, though with completely different cause. See Images.
  • This phenomena is important in the emissions of relativistic sources. Noting that in this case, the light is uniformly distributed in the object frame and moving outwards, opposite to our model, in our frame the object appears to be beaming its light almost entirely in the direction of motion. 
  • It has been theorized that this effect may explain a small number of enormously bright objects in the universe - these may be Active Galactic Nuclei ejecting relativistic jets of plasma which are parallel with our line of sight, and consequently appear much brighter than normal.
 
 
 
 Animations:
Non-relativistic Animation Desert Aberration Animation Fly-past Cube Animation Fly-through Cube Animation
 
THE IMPLEMENTED ALGORITHM  
  • Consider the path of a single, postulated photon reaching the camera.
  • We know the direction the photon came from.
  • Thus we can transform this direction from the camera frame to the scene frame
  • As each photograph is linked to a single point is space-time by the assumptions, we know the position and direction of the photon at some time, and can hence calculate it's path.
  • Using standard raytracing techniques, we can find the colour and intensity of the light sharing the photon's path, which is equivalent to examining a photograph taken in that direction by a stationary camera, as the scene is unchanging and thus the time of origin is not significant.
  • We then use the Doppler shift and intensity transformations to determine the colour and intensity observed by the camera in each direction.
  • By applying this process to the direction corresponding to each pixel, an image can be formed.

  • By rendering multiple images with slightly different camera positions and orientations, animations can be formed.
 
RAYTRACING  
Raytracer Page
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