This document is available on the Seeing Relativity web site, at

http://www.anu.edu.au/Physics/Searle/
 

Understanding the Videos

The two videos show photorealistic representations of reduced c scenes. This means that the speed of light has been slowed down from over one billion kilometres per hour to a speed of only a few meters per second. The consequences of this fiction have been restricted to optical effects, and allows us to see special-relativistic effects not possible in everyday life.

Einstein used this technique in his thought experiments, attempting to imagine moving with a light beam, that ultimately lead to his theory of Special Relativity. Gamow also used a reduced c world in his classic "Mr. Tompkins in Wonderland", though his work is open to incorrect interpretation, and in some editions the illustrations are simply incorrect. This was a common error until the 1960's, with many physicists simply assuming that what was observed in special relativity (notably length contraction) would also be seen. In 1959 R. Penrose¹ observed that a sphere would present a spherical outline to all observers regardless of their relative motion, and J. Terrell² demonstrated that distant objects would appear rotated (now known as a Terrell Rotation), not contracted as many had thought.

A scene-by-scene description of the video follows, pointing out the features and intent of each scene. It has been designed to provide the basis for a qualitative lecture on relativistic optics. The explanation assumes a basic understanding of special relativity.

1) R. Penrose, Proc. Camb. Phil. Soc. 55, 137 (1959).

2) J. Terrell, Physical Review 116, 1041 (1959).
 
  

Seeing Relativity

10 minutes

Title Sequence

Constant proper acceleration through a starfield. This is how space travel at near-light speeds would appear, contrary to the special effects in Star Trek and Star Wars universes.

A sphere (moving at 0.5c) passes over a screen, onto which the title is projected. Note that the sphere leaves a delayed shadow - cast earlier when it was in a different position - and that light reflected from the moving sphere is Doppler-shifted, appearing a different colour.

As we enter our reduced c world - where the speed of light is only five meters per second (18 kilometers per hour) the effects of slow light are apparent even without motion. A flickering streetlamp does not instantly illuminate its surroundings; rather it casts expanding spherical shells of light into space around it, which we can see when they hit a surface.

If we take a flash photograph inside an object, such as this tram, we can see that the flash does not instantly illuminate the interior, but slowly travels through the tram. As the light must travel from our flash and back to the camera, the illuminated region appears to move radially outward at half the speed of light.

If we now turn the streetlamp on and off, we can see that even though the wall is closer to the streetlamp than the ground, parts of the ground are illuminated first. To understand this, we must take into account not only the distance of the surface to the streetlamp, but also the distance to the camera.

To clarify this, we move to a schematic world where we can track the path of individual photons. As light makes the trip between the lamp, the surface and the camera, the light which follows the shortest path will reach the eye first. As the direct route between lamp and camera is the shortest and thus the fastest, we see the lamp itself turn on before anything else is illuminated. The next quickest route is from the lamp to the ground to the camera, so the ground is illuminated first. The route to the wall is longer, so it appears later. It is interesting to note that the point on any surface which is illuminated first is also the point at which we would see a reflection of the light source, in accordance with Fermat's principle of least time.

We now look at moving objects. A tram moving close to the speed of light (0.866c, a Lorentz factor of 2) displays many effects, becoming distorted, and changing colour and brightness. It's shadow also falls at an unusual angle. The effects of distortion (angular aberration), colour (the Doppler effect) and intensity (the Headlight effect) can be separately treated, so for the moment we "enhance" our image, correcting it so only distortion appears. This correction occurs within the large rectangle labelled "computer corrected".

The tram appears to have been shortened, sheared (with the two ends no longer perpendicular to the sides), and slightly bent.

To explain the shear, we again move to the schematic world. As the tram moves close to the speed of light, it is shortened by the Lorentz contraction. This explains why it appears shorter. As the back side of the tram is further from us than the front side, the light takes longer to reach us. Thus the light we see at any one time must have left the front and back surfaces of the tram at different times. As the tram is moving at a speed comparable to that of light, this time delay also means that we see the front surface of the bus later and further along the track than the back surface, thus we see a shear.

This effect is known as the Terrell rotation, as the degree of contraction and shear combine to the exact proportions that a rotation would produce.

However, this effect is dependent on the object appearing small and distant. If we are too close to the object it's different parts will appear rotated by different amounts, sometimes resulting in extreme distortion.

If we move with the tram, the world around us is subject to the same sort of distortions. While we can understand these in terms of the Terrell effect, there is a more powerful concept - relativistic aberration. [Also note that when we move with the tram, it seems to be moving faster - this is time dilation at work.]

To understand relativistic aberration, consider the ordinary "real" world. When a vehicle moves though rain, to the vehicle the rain seems to fall at an angle. In an analogous process, photons "falling" into the camera appear to come from different angles as the camera moves at different speeds. As the camera moves faster and faster, photons enter it at increasingly steeper angles. This means that things that would appear behind us if we were in their rest frame are wrapped forward into our field of view. The same, reversed, applies to outgoing photons.

This is why the tram appears small as it approaches us, and large as it recedes from us.

If we accelerate the camera slowly, we can see the distortion increasing, and objects - like the clouds behind us - sliding into our field of view, and the sun moving closer to our direction of motion.

To see what is happening all around us, we use a mercator projection; a map of everything we can see around us. Accelerating, we can see that objects behind us increase in size. Even though we are moving away from the houses at the end of the street [left and right edges] they grow larger. The patch of blue sky overhead shrinks to a small circle ahead of us.

Returning to the tram we now consider the colour shift.

Even at low speeds (0.1c) objects change their colour significantly. Green lamp posts ahead of us look red, behind us they look green. This is the Doppler shift at work.

Just as the direction of motion of photons change when the camera moves, so do the frequencies which we perceive as colour. Objects we move towards appear blueshifted. Objects we move away from appear redshifted. As we go faster, the effect becomes extreme, and we see a rainbow effect as any sharp spectral features pass through the visible band.

If we remove colour correction from our acceleration scenes, we see the world ahead go blue, and the sun darken as we start to see in the ultraviolet and above. This is matched by a reddening behind us. A rainbow ring appears in the sky as the blue colour is shifted through the regions of the spectrum our eye detects as red and green.

Though the distribution of energy in spectra means that the Doppler effect will sometimes make an object look darker, this is not the only effect changing the brightness of the tram.

At high speeds, without the protection of computer correction, extreme intensity effects occur, with most parts of the scene too dark or too bright to discern any detail. Because aberration concentrates photons in front of us and spreads them out behind us, objects ahead look brighter and objects behind look dimmer. Time dilation also means that the shutter of a moving camera is open longer - collecting more light and making the image brighter.

This effect also means that moving light sources concentrate their output along their direction of motion - the headlights on the front and back of this tram are of equal intensity when it is stationary.

We can see that as we accelerate without the benefit of computer correction, we are swiftly overtaken by the effects of relativity, with the scene changing colour and becoming blindingly bright or too dark to make out. At high velocity, the effect is extreme - the camera is subjected to a beam of high energy electromagnetic radiation.

A more subtle effect occurs with shadows. Note that the shadow of the moving tram lies at a steep angle compared with that of other objects. Viewed from above, we can see that because light moves slowly, shadows are not cast instantaneously. Because the effects of the blocking take time to reach the surface, it can remain illuminated (or vice versa) when it would normally not be. Moving light sources also cast unusual shadows - because they cast different parts of the shadows from different positions at different times.

These effects can be seen in the real universe on a much larger scale - light delay in the "light echoes" of supernovae, the Doppler shift of fast-moving astronomical bodies, and the headlight effect in the incredible brightness of gas jets directed towards us.

While humans cannot yet directly experience these effects, it is possible, though not practical, to build a space ship with today's technology that could travel fast enough to see these effects directly.
 
  

Visualising Special Relativity

7 minutes

There are several elements superimposed on the video, constituting the head-up display (HUD). The number at the top left is the current camera speed relative to the scene. The number at bottom left is the Lorentz or gamma factor, a measure of the degree of relativistic effects. The image at bottom right is a map of the scene with the camera view-pyramid (position, and field of view) shown in blue, and the camera velocity shown in green.

The first scene is a trip down a highway without any relativistic effects. Note the position and orientation of the structures in the desert.

For the next trip, we enable relativistic aberration. As we accelerate, note that the angular compression creates an initial impression of backwards motion. As we pass the sign, it seems to rotate around. This can be viewed as a Terrell rotation, or as angular aberration keeping the sign in our field of view as we pass it. The back walls of the building are also visible, and extreme distortion is visible on all the objects. Note particularly the sky, steadily shrinking down to the vanishing point.

We now enable Doppler shifting. Note that the red desert is blueshifted ahead through the green and red, causing a rainbow effect. As the blue of the sky is further blueshifted, it drains of colour. Near the edges of the image, the opposite happens - the sky takes on a reddish hue and the road is drained of colour as the red desert shifts into the infra-red.

With full relativistic effects (now including the headlight effect) the image quickly turns monotone, with objects near the edge of the screen darkened, and the centre brightly illuminated.

The Terrell effect can be illustrated with this flyby of a cube. Note the orientation of the cube change. Also compare it's apparent position with the position indicated on the HUD map. Remember, we are seeing the cube as it was, not as it is.

If we instead fly through the cube, the structures Terrell rotate independently, seeming to turn the cube inside out. Note that even when we have exited the back of the cube, aberration keeps most of it in view.

Another property of aberration is that it preserves circles - that is, a sphere will always present a spherical outline to any observer regardless of their relative motion. We see this demonstrated by flying a camera around the Earth at high speed. Though the camera is very close to the surface, aberration wraps the Earth into our forward field of view. But because we are so close to the earth, we can see only a small portion of its surface - so small regions, about the size of Borneo seem to bulge out and fill the sphere.