Sciences
Space Time Theory
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 relatively-moving observers measure can be
summarized
as:
| Relativistic time dilation: The
process that occurred in the blue driver's rest frame with in time Tb
was perceived by the red driver to have occurred in time Tr = Tb / (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 Lb
in the blue driver's rest frame was measured by the red driver to have
have the length Lr = Lb (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 - U2/c2)-1/2 c T' + (U/c)(1 - U2/c2)-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 Xb-direction at all, so X'
= Xb = 0,
leaving T = T' /(1 - U2/c2)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 Tr = T = 0 and so c
T' = - (U/c) X', leading to X
= X' (1 - U2/c2)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 - U2/c2)-1/2 c T' + (U/c)(1 - U2/c2)-1/2 X'gives a mathematical prescription for comparing how the red and blue drivers measure space and time.
X = (U/c)(1 - U2/c2)-1/2 c T' + (1 - U2/c2)-1/2 X'
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 X-coordinate
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
dL2 = dX2
+ dY2
(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 dL2
= dX2 + dY2 + dZ2.) 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: 
dS2 = c2
dT2 - dL2
where dL2 is the Euclidean
metric on the space part of the
spacetime. For example, in two space and one time dimensions, the
Minkowski
metric becomes:
dS2 = c2
dT2 - dX2 - dY2
In the
cases we've been
looking at, the relevant Minkowski metric is simply
dS2 = c2
dT2 - dX2
Recall the
Lorentz
transformation (using b = (U/c) for shorthand):
c T = c T' / (1 - b2)1/2
+ X' b/ (1 - b2)1/2X = c T' b/ (1 - b2)1/2 + X' / (1 - b2)1/2
The
Lorentz transformation
has the amazing property that it leaves the Minkowski metric unchanged
so that
c2 dT2
- dX2 = c2 dT' 2
- dX' 2,
which is demonstrated in the above figure. The two purple lines
represent
different values of the Lorentz-invariant interval dS2. In
particular, the bright
purple line representing dS2 = 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 - b2)1/2
+ c dT' b / (1 - b2)1/2 =
dT' (V' + U)/(1 - b2)1/2c dT = c dT' / (1 - b2)1/2 + dX' b /(1 - b2)1/2 = c dT' (1 + U V'/c2)/(1 - b2)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'/c2) |
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?