Seeing Relativity 10 minutes
We begin with a view of constant proper
acceleration through a starfield. This is how space travel to
near-light speeds would appear, contrary to the special effects
in Star Trek and Star Wars universes. C, the speed of light ,
is not exceeded.
A sphere (moving at 0.5c) passes over
a screen, onto which the title is projected. The sphere leaves
a delayed shadow - cast earlier when it was in a different position
. 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
a stationary 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.
The streetlamp turns 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.
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.
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. Its 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 its 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. The map is made by "unwrapping" a sphere
and stretching the areas at the poles to make the map flat. This
is the same kind of map that shows Greenland the size of South
America.
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.
Doppler shift effects are seen again
in this mercator projection.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.
As the tram accelerates, electromagnetic
radiation enters more and more directly from in front. This effect,
when combined with the doppler shift towards shorter wavelengths
of higher frequencies, would produce a lethal high intensity beam
of x-rays and gamma-rays. This is another consequence of near
light travel usually ignored in science fiction.
Here is the same intensity effect shown in mercator projection.
Note that any information from the outside world enters through
a very restricted kind of porthole . What changes would occur
in any radi o communication between 2 trams, travelling one behind
another ?
Moving light sources also cast unusual
shadows because they cast different parts of the shadows
from different positions at different times. Note the shape of
the shadows cast on the ground. Why does this occur ?
These effects can be seen in the real
universe. But the real speed of light is very fast, three hundred
thousand kilometers per second. Because relativity effects are
most pronounced at very high speeds, the real visual effects of
relativity must occur on a much larger scale than the simulations
you've just seen. For example : 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
The first scene is a trip down a highway
without any relativistic effects. Note the position and orientation
of the structures in the desert. There are several elements superimposed
on the video, constituting the head-up display (HUD). The number
at the top right 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.
For the next trip, we enable only 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. Look at the difference between our actual
postion as viewed from above on the lower right, and our apparent
position. 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, adding
it to the aberration effects from before. 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. Doppler and intensity effects are disabled.
Note the orientation of the cube change. Also compare its 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 orbiting a camera around the
Earth at high speed. The camera's position can be seen from above
in the lower right hand corner of the screen.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.