Special Relativity: A Simple Matter of Perspective?

Popular belief has it that Eintein's Theory of Special Relativity is
something difficult. It is not. Special Relativity is nothing more than
what we usually call "Perspective", but applied to space and time,
and not only to space.
On this page, you will see how easy Relativity paradoxs appear once
you have accepted this idea. 
The Special Relativity "Paradoxs"
Special relativity introduced a number of well known "paradoxs", such as:

The two rockets paradox: If two
rockets cross, each emitting a pulse of light every second, then every
rocket will see the pulses of light of the other rocket as separated by
more than one second, even after taking into account the time it takes
for light to travel from one rocket to the other. This effect is experimentally
verified for instance when receiving data emitted from satellites.

The red shift: In the above experiment,
the light received by the other rocket will also have a lower frequency
than the known frequency of emitted light. This effect is experimentally
verified on distant stars, where the radiation spectrum of known elements
is shifted down.

The twins paradox: If a twin remains
on earth while another twin travels at near light speed and returns to
earth, the travelling twin will be younger when he returns. This effect
is experimentally verified on particles with known average life in particle
accelerators: particles last longer when they are moving. It is also the
basis of the "Planet of the Apes" novel.

The train paradox: If two lightning
bolts strike the track simultaneously at both ends of a fast travelling
train, the marks on the track will be shorter than the known length of
the train at rest.

The barn and the pole paradox: With
sufficient speed, an object can go through a hole which is too short for
the object.
Explanations for these paradoxs usually involve lengthy calculations. See
for instance the chapter
on the twins paradox in the Relativity
FAQ. The goal of this article is to show how these effects can be explained
in a much simpler way, without any computation, as a simple effect of perspective.
The "magic recipe"
However, all special relativity paradoxs can be explained without any computation
with the help of the following "magic recipe":

Plot all events in the experiment in a plane, where time is an axis, and
space is the other axis, as measured by any given observer.

For any mobile in the diagram, the trajectory on the diagram is called
the time line.

Any time measurement made by this mobile will be made along its own trajectory.

Any space measurement made by the mobile will be made perpendicular to
this trajectory.

Any effect of perspective needs to be inverted. Where contractions happen
on the diagram, special relativity will predict a dilatation. Where dilatations
happen on the diagram, special relativity will predict a contraction.

Time and space don't mix. A rule of the thumb is to look suspiciously at
any angle above 45° on your diagram.
Examples of how to apply the magic recipe are shown below. You can also
read the mathematical explanation of why this
magic recipe works.
The twins paradox
This paradox is the most well known and explained paradox, at least as
far as the web is concerned. Several other very good sites on the Web also
explain this same paradox. See for example the Relativity
FAQ, the explanation of the University
of Virginia, or the nicely presented one of Howard
C. McAllister, as well as some more
puzzling ones...
Our magic recipe, however, makes this paradox look quite simple. If
we apply step 1 of the magic recipe, the twins paradox can be plotted as
follows:
The red curve indicates the trajectory of the travelling twin. The blue
line indicates the trajectory of the twin remaining on earth.
Step 2 indicates that the age of the travelling twin is represented
by the length of the red curve, and the age of the twin remaining on earth
by the length of the blue line.
It is a well known fact that the shortest path between two points is
the straight line. So any trajectory the twin follows on our diagram is
always longer. Applying step 3 of our recipe, the travelling twin will
always age less than the twin remaining on earth.
The two rockets paradox
The two rockets paradox is solved equaly well. Applying step 1, the two
rockets paradox can be plotted as follows (from the point of view of the
red rocket):
The big dots represent the events in spacetime corresponding to each
pulse of light, for each of the two rockets. Since the blue rocket is moving
with respect to the red rocket, its trajectory is slanted (there is a displacement
along space).
Applying step 2, and considering that the blue rocket emits a dot every
second of its own time, we see that the plots are separated by the same
distance along its trajectory, not along our time. However, if the red
rockets tries to measure the interval between two pulses emitted by the
blue rocket, it does so using the red dotted lines, which are perpendicular
to its own red time. Conversely, the blue dotted lines show how the blue
rockets measures the interval between two red pulses.
On the diagram, a contraction appears in both cases. Applying step 3,
we discover that rocket red will see pulses emitted by rocket blue separated
by a larger time interval, but that rocket blue will also see rocket red
pulses separated by a larger time interval !
And that's it...
Red shift
The red shift is exactly the same phenomenon as the rockets paradox, applied
to the frequency of the light emitted by moving distant stars. For a given
element, the possible frequencies of emitted light are known and supposedly
constant accross the universe. The set of possible frequencies is known
as the spectrum of the element.
Compared to the spectrum of the same element measured in a lab on earth,
the spectrum of elements in distant stars appears shifted towards low frequencies.
Since red is the lowest frequency in the visible spectrum, this is called
the red shift.
See also the General
Relativity FAQ on this topic.
The train paradox
The train paradox involves space measurements. To plot it correctly, we
need to plot both the front and the rear of the train. In the figure below,
we plot it from the observer on the track. The lightning bolts strike simultraneously
for the observer on the track (at the same time), so they are on the red
dotted line. We then make the space measurement on the track, so following
the green dotted lines on the diagram.
Applying step 2, we can understand why the lightning bolts are not simultaneous
for an observer in the train. The lightning at the front of the train will
occur after the one at the rear of the train.
The distance d between the two bolts, measured in by an observer
on the track (along the green dotted lines), is longer on our diagram than
the length d’ of the train measured by an observer in the train.
Applying step 3, we can deduce that the distance between the bolts in special
relativity is shorter than the length of the train.
The barn and the pole paradox
The barn and the pole paradox is similar to the paradox of the train. It
can be thought of as follows: in "normal" space, it is more difficult to
enter a garage in diagonal than straight. In special relativity, it becomes
easier.
See also the detailed explanation in the Relativity
FAQ.
© 1999, Christophe de Dinechin