Sciences
How does the Special Theory of Relativity work?
The Lorentz transformation
Recall that in the
previous section, we tried to model space and time in a way that would
be
consistent with the observed constancy of the speed of light in Nature.
The
resulting model for space and time measurements has observers measuring
time and
space differently when they are moving relatively to one another. The
two basic
differences between what relativelymoving observers measure can be
summarized
as:
Relativistic time dilation:
The
process that occurred in the blue driver's rest frame with in time T_{b}
was perceived by the red driver to have occurred in time
T_{r} = T_{b} /
(1  (U/c)^{2})^{1/2}.
This means
that a clock appears to tick more slowly to an
observer who perceives that the clock is moving than it does to an
observer who is in the rest frame of the clock. 
Relativistic length contraction:
The blue
car measured to have length L_{b}
in the blue driver's rest frame was measured by the red driver to have
have the length
L_{r} = L_{b} (1
 (U/c)^{2})^{1/2}.
This means
that the length of some object appears to have a shorter
length to an observer who perceives that the object is moving
than it does to an observer who is in the rest frame of the object. (This
applies to the length parallel to the direction of motion only.) 
We will now state (without
going through the grunge work of proving) that the mathematical model
that best
encompasses the observed constancy of the speed of light in Nature
(pretending
for now that gravity doesn't exist) is the one where the
spacetime
coordinates of two such observers as described above are
related through
what is now called a
Lorentz transformation:
c T = (1  U^{2}/c^{2})^{1/2} c T' + (U/c)(1  U^{2}/c^{2})^{1/2} X'
X = (U/c)(1  U^{2}/c^{2})^{1/2} c T' + (1  U^{2}/c^{2})^{1/2} X'

This transformation tells
us how some observer
O, who sees another observer
O'
moving at
velocity
U, translates measurements made by
O'
into measurements
valid in the
O frame of reference. In our specific
case, we'd say this is
how the red driver sees things that are happening in the frame of
reference of
the blue driver.
For example, in
time
dilation: in the blue car's frame of reference, the laser
pulse did not
travel in the X
_{b}direction at all, so
X'
= X_{b} = 0,
leaving
T = T' /(1  U^{2}/c^{2})^{1/2}.
In
length
contraction:
in the red car's frame of reference, the red driver must measure the
length of
the blue car at a single moment of the
red driver's time.
This means we
have
T_{r} = T = 0 and so
c
T' =  (U/c) X', leading to
X
= X' (1  U^{2}/c^{2})^{1/2}.
The Lorentz transformation
is the foundation of relativistic geometry, which we will examine next.
The geometry of relativity
In order to talk about
relativistic
geometry, we need some
coordinate systems
to compare. The red and
blue cars in the figure to the right each have a (schematic) coordinate
system
attached to their individual
rest frame.
The
Lorentz
transformation
c T = (1  U^{2}/c^{2})^{1/2} c T' + (U/c)(1  U^{2}/c^{2})^{1/2} X'
X = (U/c)(1  U^{2}/c^{2})^{1/2} c T' + (1  U^{2}/c^{2})^{1/2} X'
gives a mathematical prescription for comparing how the red and blue
drivers
measure space and time.
The path of a light ray is
given by
X = c T or
X =  c T.
You should verify for yourself that
if
X = c T then
X' = c T', so
the paths of light rays do not
change under Lorentz transformation. The purple line in the figure
represents
the path of a light ray.
If we draw the
time
axis in units of c T instead of just plain T, then the paths of light
will
always be lines at 45° angles.
To see relativististic
effects in action, it is helpful to draw one coordinate system in the
other
coordinate system.
The
coordinate axes
of the
red driver's system are formed from the line
X=0 (the
T
axis) and the line
T=0 (the
X
axis), as shown by the red set of
axes on the figure below.
The
coordinate axes
of the
blue driver's system are formed from the
line
X'=0 (the
T'
axis) and the line
T'=0 (the
X'
axis).
If we want to draw the
T'
and
X' axes in the
(T,X)
coordinate system, we have to use the
Lorentz transformation. Plugging into the Lorentz transformation, the
line
X'=0
(the
T' axis) reduces to the formula
cT =
(c/U) X. The line
T'=0
(the
X' axis) reduces to
cT = (U/c) X.
These lines form the blue
set of axes shown on the figure to the left.
Now we can see from the
spacetime geometry how
relativistic length contraction
happens. The blue
bar in the figure represents the blue car. The first animation sequence
shows
how the blue car moves in the blue driver's spacetime coordinate
system. The
length
in space for the blue driver is defined at a
specific
moment of the blue
driver's time T'.
How does the red driver
measure the length of the blue car? She defines the
length in
space of
the moving blue car as
distance in her Xcoordinate
that the blue car
occupies at a
specific moment of the red driver's time T.
To the red driver, the
blue car looks shorter than the blue driver says it is.
It is also true that to
the blue driver, the red car looks shorter than what the red driver
says it is.
Neither observer has a privileged or more true version of reality. That
is an
important part of the mathematical model for spacetime called
Einstein's
Special Theory of Relativity.
The velocity addition problem, solved
Recall the
Pythagorean
Rule (which the Mesopotamians knew at least a millenium
before the
Pythagoreans). By taking differentials the Pythagorean Rule can be
written as
what is called the
Euclidean metric
dL^{2} = dX^{2}
+ dY^{2}
(Note: this is for
two space dimensions. In higher
space dimensions you
just add more terms to the formula. In three space dimensions it's
dL^{2}
= dX^{2} + dY^{2} + dZ^{2}.)
Now what to do about
spacetime?
It turns out that finally
about 4000 years after the Mesopotamians experimentally (but not
theoretically)
figured out the Euclidean metric, Albert Einstein hit on the spacetime
analog,
which we now call the
Minkowski metric:
dS^{2} = c^{2}
dT^{2}  dL^{2}
where
dL^{2} is the Euclidean
metric on the space part of the
spacetime. For example, in two space and one time dimensions, the
Minkowski
metric becomes:
dS^{2} = c^{2}
dT^{2}  dX^{2}  dY^{2}
In the
cases we've been
looking at, the relevant Minkowski metric is simply
dS^{2} = c^{2}
dT^{2}  dX^{2}
Recall the
Lorentz
transformation (using
b = (U/c) for shorthand):
c T = c T' / (1  b^{2})^{1/2}
+ X' b/ (1  b^{2})^{1/2}
X = c T' b/ (1  b^{2})^{1/2} +
X' / (1  b^{2})^{1/2}
The
Lorentz transformation
has the amazing property that it leaves the Minkowski metric unchanged
so that
c^{2} dT^{2}
 dX^{2} = c^{2} dT' ^{2}
 dX' ^{2},
which is demonstrated in the above figure. The two purple lines
represent
different values of the Lorentzinvariant interval
dS^{2}.
In
particular, the bright
purple line representing
dS^{2} = 0
is important because it
represents the
path of light (which is described by
both
X = cT and
X' = cT'). This shows that the
Minkowski
metric invariance under
Lorentz transformations really does encode the observed
constancy of the
speed of light in Nature. So we're on the right track here in trying to
model
the natural world using the language of mathematics.
How does
the
Minkowski
metric help us solve the
velocity addition problem
and come up with a
new rule for adding velocities that will
encode the observed
behavior of the
constant speed of light as a maximum observable speed in nature?
dX = dX' / (1  b^{2})^{1/2}
+ c dT' b / (1  b^{2})^{1/2} =
dT' (V' + U)/(1  b^{2})^{1/2}
c dT = c dT' / (1  b^{2})^{1/2}
+ dX' b /(1  b^{2})^{1/2} = c
dT' (1 + U V'/c^{2})/(1  b^{2})^{1/2}
The velocity we're looking for is
V = dX/dT.
Dividng the top line
dX
above by the bottom
c dT, we get for
V:
Relativistic velocity addition
rule
dX/dT
= V = (U + V')/(1 + U V'/c^{2}) 
as the
velocity addition rule consistent with the
observed behavior of
nature.
Let's
check: if the blue
car is rolling at
U = c/2 and a laser on the blue
car is travelling at
V'
= c then the red driver sees the laser travelling at
V
= (c/2 + c)/(1 +
1/2) = c so this Minkowski metric models very well the
observed behavior of
Nature that one can't seem to make light travel faster than
c,
not by
putting a laser on a moving car, or a rocket or any other type of
moving
vehicle. If either
U=c or
V'=c,
the velocity addition formula
reduces to
V=c.
Isn't
that clever?
How
could nature be that smart?