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

• The project is being undertaken at the Australian National University, in the Physics Department, by undergraduate student Antony Searle as part of the Distinguished Scholars in Science Program
• The project involved the creation of a computer program on DOS/UNIX platforms to simulate the effect of relativistic motion through a scene
• All images featured here were rendered by this program on a home PC

INTRODUCING RELATIVITY

• Special relativity covers the relation between events as they appear to moving observers
• The effects of special relativity include the apparent contraction of length and dilation of time for moving observers, and the direction and wavelength of photons may appear different to different observers
• The theoretical basis for special relativity is that the speed of light must appear the same to all observers regardless of their motion
• The coordinate transformations resulting from this requirement are known as the Lorentz transformations, and a critical function appearing in the equations is the Lorentz factor or Gamma factor,

• The Lorentz factor can be thought of as the degree of relativistic effects. It ranges from one for low velocities, at which relativistic effects are negligible and Newtonian mechanics essentially correct, and approaches infinity as the relative velocity of the observers goes to the speed of light. It features in the animations in the lower left corner. The velocity as a fraction of the speed of light appears in the upper right corner

OBSERVATION AND SIGHT

• An observation is the simultaneous measurement of a phenomena in a reference frame
• What is seen, however, is not simultaneous, as light will have left a different points on an object at different times
• At relativistic speeds, light simultaneously reaching the eye may have started at different times in which the object has moved considerably.
• Thus we may see a greatly distorted image of the object
• View an animation illustrating the difference between sight & observation
• This distortion may be treated in two ways
• By the projection of points on the object, taking into account the motion of the object and the time delay in our observation of that point, by means of the Lorentz transformations
• By transforming the light received back into the rest frame of the object, and tracking its path. As the object is stationary, the time of origin is made irrelevant by these methods
• We use the latter approach.
• This is a relatively new area. Pioneering work was done by P. K. Hsiung in 1989-90 as a Ph.D. thesis using the method of raytracing adopted here, and the alternative method of point transformation was implemented in 1996 by M. C. Chang, F. Lai and W. C. Chen (Image Shading Taking into Account Relativistic Effects, ACM Transactions on Graphics, Vol. 15, No. 4, October 1996, Pages 265-300)
• Some educational programs using simple wireframe graphics are also available
• Algorithmically, the program operates by treating every pixel in an image as a possible photon path. It then transforms this possible photon from the camera frame into the frame of the scene, extrapolates back along the line of motion and determines the photon origin. The colour of the surface the photon originated from is converted into an approximate spectrum, and then redshifted or blueshifted as required. The resultant spectrum is the resampled and used to determine the colour of the image.
• The precise transformations used are explained below and their effects illustrated

RELATIVISTIC ABERRATION
 

	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.
	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.
	Relativistic aberration preserves the speed of light, as at bottom left. Note that light concentrates in the direction of motion, according to the formula 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: Aberration Animations:  

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

• Like sound, light undergoes frequency shifting when and observer moves at comparable speeds.
• For sound, or light in an 'ether', the relative frequency shift is given by

where v is the velocity of the observer, a and n are the angle subtended and frequency of the photon in the rest frame, and n' is the frequency recorded by the moving observer
• The relativistic formula is similar, but corrected for time dilation

It can be seen from this formula that, as we would expect, light in the direction of motion is blueshifted, light following the observer is redshifted, and time dilation is solely responsible for the blueshifting of light perpendicular to the direction of motion in the frame of origin.

Doppler Shift Animations:  
 

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.  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.

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

   Headlight Effect Animations:  
 

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:  
THE IMPLEMENTED ALGORITHM

• Assumptions:
• The camera occupies a vanishingly small size
• The shutter is open for a vanishingly small time
• The scene is not accelerating
• All elements of the scene are at rest with respect to each other
• Space-time is flat

 

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

• Raytracing is an analytically elegant method of computing photo realistic images of scenes using simple geometric optics.
• A camera is established in the scene at c.
• A number of objects, each described by a surface equation of the form f(x) = 0 are also present in the scene.
• A ray is projected in direction r from the camera, of the form x = rt + c.
• For each object, the solutions of f(rt + c) = 0 are sought.
• Depending on the optical properties of the object, such as colour, reflectivity, index of refraction, transparency and degree of illumination, the ray is either stopped, transmitted or redirected, or splits into some combination of the three.
• Using the information of the surface intersections, a colour and intensity are allocated to the ray.
• This colour forms the pixel corresponding to the direction of the ray produced by the camera.
• The Relativistic Raytracer is a simple raytracer, where all objects are unreflective and block light, and surfaces are limited to planes, blocks and quadrics. Shadows are also neglected.
• An excellent freeware raytracer may be obtained from www.povray.org.

 

Raytracer Page

Download Page

e-mail Antony Searle