Sciences
Why was the Special Theory of Relativity needed?
The velocity addition problem
The observed constancy of
the speed at which light travels tells us that the
Newtonian
model of space and time is flawed. But the flaws don't become
noticeable until we start trying to describe things moving near the
speed of light.
One reason why Einstein's
Special Theory of Relativity was needed was because of the special
problems that cropped up when scientists noticed that the
speed
of light is a constant everywhere in every direction. This
caused problems with the Newtonian model for measuring time. One of
these problems is called the
velocity addition problem.
The velocity addition
problem is illustrated in the figures above and below. In the top
figure we see the red driver's frame of reference or
rest
frame. To the driver of the red car in her rest frame, the
blue car is travelling forward at
velocity U and
some purple object is flying out of the blue car at
velocity V.
In the figure below the
point of view of the blue driver is illustrated. To the driver of the
blue car, the purple object is flying forwards out of her window at
velocity
V', and the red car is driving backwards.
The velocity addition
problem asks the question:
Given U and V',
what is V?
If we use Newton's model for
time as being experienced exactly the same for all observers, we wind
up with the answer:
V = U + V'. How?
Suppose, as Newton believed,
the red and blue drivers measure time and distance precisely the same.
According to the blue driver, in time T the distance of the purple ball
from the blue car is
Xball = V' T.
In the same time T in the reference frame of the red car, the blue car
has travelled the distance
Xcar = U T.
According to the red driver, the total distance the purple ball
travelled is the distance the blue car travelled from the red car plus
the distance the purple ball travelled from the blue car, or
X
= Xcar + Xball = U T + V'
T = (U + V') T.
But
X = V T,
and that gives us the velocity addition formula:
V = U + V'
So, for example, if the blue
car is going 30 mph and the driver of the blue car measures the purple
ball going 60 mph, then the driver of the red car should measure the
purple ball going 90 mph, because
U = 30 mph,
V'
= 60 mph and hence
V = U + V' = 90 mph.
Sounds reasonable, right?
Okay, now suppose that
U
= half the speed of light, and
V' = the speed of
light (maybe the purple object is a laser pulse). Now what is
V?
The above formula tells us
V = one and a half times the speed
of light.
This is contrary to observed
behavior of Nature. Therefore the Newtonian model of time as being
experienced equally by all observers must not be a good model for
Nature.
Time must be relative
If the
speed of
light is the same even if the light source itself is moving
at some speed relative to the person doing the measuring, then how does
this affect the way different observers measure time and space and
combine them into spacetime?
Let's take the red and blue
cars from the previous example, and let's mount a laser on top of the
blue car. We'll put a mirror on the blue car as shown in the figures,
and aim the laser at the mirror perpendicular to the direction the blue
car is travelling.
 |
Here is the laser pulse viewed in the frame of
reference of the blue car.
|
Within the time interval
T'
measured by the driver of the blue car operating the laser, the laser
light travels a distance
2 L from the laser to the
mirror and back.
Meanwhile, the driver of the
red car sees the blue car go whizzing by. According to the red driver,
the laser took a total time
T to hit the mirror and
return. (She's willing to agree with the driver of the blue car that
the distance between the laser and the mirror is
L,
since neither driver has any velocity in that direction.) She measures
the blue car to have travelled a distance
X = U T
and the laser pulse to have travelled a total distance of
2 D = c T
in time
T.
Here is the laser pulse viewed in the frame of
reference of the red car.
|
 |
Here is where the problem
arises. Since we just deduced above that
c T' = 2 L
and
c T = 2 D, the only way we could have
T'
= T would be if
D = L.
Therefore, the red
and blue drivers do not measure time equally..
Now how can we sort this out
and find out how their measurements of time should differ in order to
account for the observed constancy of the speed of light?
What both drivers agree on
is the distance
L perpendicular to the motion of
the cars. Using the Pythagorean Rule on the laser pulse's path in
space, we get
L2 + (X/2)2
= (c T/2)2 .
From the blue driver's point
of view
L2 = (c T'/2)2
, and putting these together and using
X = U T
we get:
(c T')2 + (X)2
= (c T)2 or
T = T' / (1 - (U/c)2)1/2
This is stupendous!
The
drivers of the blue and red cars don't measure the same time for the
laser to hit the mirror and come back! We are forced to
conclude this if we want to be consistent with the observed constancy
of the speed of light (so far) in Nature.
This has profound
implications for the mathematical modelling of space and time as
observed in Nature. This means we have to expand the idea of Euclidean
analytic geometry to include the observer-dependent
relativity
of measurements of time and space. This opens up a gigantic can of
mathematical worms, eventually bringing us to
black holes,
wormholes and at least the abstract mathematical possibility of time
travel.
| What we've learned here is called Relativistic
time dilation. The process that occurred in the blue driver's
rest frame with in time T' was perceived by the red
driver to have occurred in time T = T' / (1 - (U/c)2)1/2,
which can be much much greater than T' if U
is close to the speed of light c. |
Space must be relative, too
In the previous frame we
learned that if we want to use geometry to model space and time
together, in order to account for the observed constancy of the speed
of light,
observers moving at constant velocity relative to
one another perceive the passage of time differently.
Now we will figure out
whether these observers measure space differently as well.
Let's suppose the driver of
the blue car below has measured the length of her car with a ruler to
be length
L'. She sees the red car driving by her
at velocity
-U. The blue driver sees that it takes
a time
T' for the front of the red car to pass from
the front end to the back end, and she calculates that therefore
L'
= U T'.
 |
L' is the blue car length, and T' is the time
for the red car to pass, as measured by the blue driver. |
The driver of the red car,
on the other hand, sees the blue car rushing past her. She measures the
time
T it takes for the blue car to pass her front
bumper. She then calculates the length
L of the red
car to be
L = U T.
So we have
L/L' =
T/T'. Now we just need to know the relationship between
T
and
T'. But we calculated that already in the
previous frame.
 |
L is the blue car length, and T is the time
for the blue car to pass, as measured by the red driver. |
Recall that in the previous
section, the blue driver was timing a laser pulse and its reflection
from a mirror. These were events that happened at the same place
according to the blue driver, and their time separation was measured to
be
T', with
only a single clock needed
for the measurement.
The red driver saw the laser
pulse and return happening in
different places
(because she saw the laser as moving with the blue car) and so the red
driver's time measurement
T could only be made with
a minimum of two (synchronized) clocks.
Constraining the speed of
light to be constant gave us this relationship between the time
T'
measured by the blue driver and the time
T measured
by the red driver:
T' = T (1 - (U/c)2)1/2.
The blue driver with the
laser is like the red driver in this example, trying to measure a time
interval between two events happening at the same place. The blue
driver in this example is like the red driver from the previous
section, trying to measure a time interval between two events in two
different places.
Therefore, the relationship
T
= T' (1 - (U/c)2)1/2
must hold between
T and
T' in
this
example. (The roles of
T and
T'
are switched compared with the last section because the roles of the
red and blue drivers with respect to time measurement has switched, as
explained above.)
But this means that the
relationsip between the lengths
L and
L'
must be
L = L' (1 - (U/c)2)1/2
Therefore, if we want to
make a mathematical model for spacetime that is consistent with the
observed constancy of the speed of light, we have to conclude in this
model that
measurement of space and time is not the same for
all observers.
| What we've learned here is called Relativistic
length contraction. The blue car measured to have length L'
in the blue driver's rest frame was measured by the red driver to have
have the length L = L' (1 - (U/c)2)1/2,
which can be much much smaller than L' if U
is close to the speed of light c. |
Is there anything that is
the same for all observers? That is what we'll look at next.
