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UNIT 9: RELATIVITY CONCEPTS AND POSTULATES OF SPECIAL RELATIVITY
Yves, a student teacher in year two once was moving in a pick up as shown in
the figure above. She had a small ball that she projected upwards when the
car was moving at a speed of 60km/h.
Basing on the statement above and figure 9.1, answer the following questions.
a) Do you think Yves was able to catch the ball 3 seconds later after projection
assuming the car continued at a steady speed of 60 km/h?
b) What do you think was the shape of the path described by the ball as
observed by Yves while in the car?
c) Yves a stationary observer at the banks of the road observes the projected
ball right at a time when Yves projected it.Do you think the path of the ball as
observed by Yves was similar to that of Diana? If not, can you describe what
you think would be the observed path by him.
d) While still in the moving car, Yves moves at 5 km/h with respect to the car.
Do you think as observed by Diana, Yves was moving at 5 km/h? If not, whatis the estimation of speed of Yves as observed by Diana?
9.1.1. Introduction to special relativity
Physics as it was known at the end of the nineteenth century is referred to as
classical physics:
- Newtonian mechanics beautifully explained the motion of objects on
Earth and in the heavens. Furthermore, it formed the basis for successful
treatments of fluids, wave motion, and sound.
- Kinetictheory explained the behavior of gases and other materials.
- Maxwell’s theory of electromagnetismdeveloped in 1873 by James
Clerk Maxwell, a Scottish physicist embodied all of electric and magnetic
phenomena,
Soon, however, scientists began to look more closely at a few inconvenient
phenomena that could not be explained by the theories available at the time.
This led to birth of the new Physics that grew out of the great revolution at the
turn of the twentieth century and is now called Modern Physics (the Theory
of Relativity and Quantum Theory).
Most of our everyday experiences and observations have to do with objects
that move at speeds much less than the speed of light. Newtonian mechanics
was formulated to describe the motion of such objects, and this formalism is
still very successful in describing a wide range of phenomena that occur at low
speeds. It fails, however, when applied to particles whose speeds approach that
of light.
Experimentally, the predictions of Newtonian theory can be tested at high
speeds by accelerating electrons or other charged particles through a large
electric potential difference. For example, it is possible to accelerate an electron
to a speed of 0.99 c (where c is the speed of light) by using a potential difference
of several million volts.
According to Newtonian mechanics, if the potential difference is increased by
a factor of 4, the electron’s kinetic energy is four times greater and its speed
should double to 1.98 c. However, experiments show that the speed of the
electron—as well as the speed of any other particle in the Universe—always
remains less than the speed of light, regardless of the size of the accelerating
voltage. Because it places no upper limit on speed, Newtonian mechanics is
contrary to modern experimental results and is clearly a limited theory.
In 1905, at the age of only 26, Einstein published three papers of extraordinary
importance:
- One was an analysis of Brownianmotion;
- A second (for which he was awarded the Nobel Prize) was on the
photoelectriceffect.
- In the third, Einstein introduced his special theory of relativity.
Although Einstein made many other important contributions to Science, the
special theory of relativity alone represents one of the greatest intellectualachievements of all time.
With this theory, experimental observations can be correctly predicted over the
range of speeds from to speeds approaching the speed of light. At low speeds,
Einstein’s theory reduces to Newtonian mechanics as a limiting situation
(principle of correspondence).
It is important to recognize that Einstein was working on Electromagnetism
when he developed the special theory of relativity. He was convinced that
Maxwell’s equations were correct, and in order to reconcile them with one of
his postulates, he was forced into the bizarre notion of assuming that space
and timeare not absolute.
In addition to its well-known and essential role in theoretical Physics, the
Special Theory of Relativity has practical applications, including the design of
nuclear power plants and modern global positioning system (GPS) units. Thesedevices do not work if designed in accordance with non-relativistic principles.
9.1.2. Galilean transformation equation
(a) Principle of Galilean relativity
You’ve no doubt observed how a car that is moving slowly forward appears
to be moving backward when you pass it. In general, when two observers
measure the velocity of a moving body, they get different results if one observer
is moving relative to the other. The velocity seen by a particular observer is
called the velocity relativeto that observer, or simply relative velocity.
To describe a physical event, it is necessary to establish a frame of reference.
You should recall from Mechanics that Newton’s laws are valid in all inertial
frames of reference. Because an inertialframe frame is defined as one in which
Newton’s first law is valid, we can say that an inertial frame of reference is one
in which an object is observed to have no acceleration when no forces act on
it. Furthermore, any system moving with constant velocity with respect to an
inertial system must also be an inertial system.
There is no preferred inertial reference frame. This means that the results
of an experiment performed in a vehicle moving with uniform velocity will
be identical to the results of the same experiment performed in a stationary
vehicle. The formal statement of this result is called the principle of Galilean
relativity: “The laws of Physics must be the same in all inertial frames of
reference.”
Let us consider an observation that illustrates the equivalence of the laws of
Mechanics in different inertial frames. A pickup truck moves with a constantvelocity, as shown in Fig. 9.2a.
If a passenger in the truck throws a ball straight up, and if air effects are
neglected, the passenger observes that the ball moves in a vertical path. The
motion of the ball appears to be precisely the same as if the ball were thrown
by a person at rest on the Earth. The law of gravity and the equations of motion
under constant acceleration are obeyed whether the truck is at rest or in
uniform motion.
Now consider the same situation viewed by an observer at rest on the Earth.
This stationary observer sees the path of the ball as a parabola, as illustrated
in Fig. 9.2b. Furthermore, according to this observer, the ball has a horizontal
component of velocity equal to the velocity of the truck.
Although the two observers disagree on certain aspects of the situation, they
agree on the validity of Newton’s laws and on such classical principles as
conservation of energy and conservation of linear momentum. This agreement
implies that no mechanical experiment can detect any difference between the
two inertial frames.
The only thing that can be detected is the relative motion of one frame with
respect to the other. That is, the notion of absolute motion through space ismeaningless, as is the notion of a preferred reference frame.
(b) Galilean space–time transformation equations
Suppose that some physical phenomenon, which we call an event, occurs in an
inertial system. The event’s location and time of occurrence can be specified
by the four coordinates (x, y, z, t). We would like to be able to transform these
coordinates from one inertial system to another one moving with uniform
relative velocity.
Consider two inertial systems S and S’ (Fig. 9.4). The system S’ moves with a
constant velocity v along the xx’ axes, where v is measured relative to S.
We assume that an event occurs at the point P and that the origins of S and S’coincide at t = 0
An observer in S describes the event with space–time coordinates (x, y, z, t),
whereas an observer in S’ uses the coordinates (x’, y’, z’, t’) to describe the same
event.
As we see from Fig. 9.4, the relationships between these various coordinatescan be written
These equations are the Galilean space–time transformation equations.
Note that time is assumed to be the same in both inertial systems. That is, within
the framework of classical mechanics, all clocks run at the same rate, regardless
of their velocity, so that the time at which an event occurs for an observer in S
is the same as the time for the same event in S’. Consequently, the time interval
between two successive events should be the same for both observers.
Although this assumption may seem obvious, it turns out to be incorrect in
situations where v is comparable to the speed of light.
Length and time intervals are absolute
Galilean–Newtonian relativity assumed that the lengths of objects are the same
in one reference frame as in another, and that time passes at the same rate indifferent reference frames.
In classical mechanics, then, space and time intervals are considered to be
absolute: their measurement does not change from one reference frame to
another. The mass of an object, as well as all forces, are assumed to be unchanged
by a change in inertial reference frame.
(c) Galilean-Newton Relative velocity
Now suppose that a particle moves a distance dx in a time interval dt as
measured by an observer in S. It follows that the corresponding distance dx’measured by an observer in S’ is
9.1.3. Einstein’s principle of relativity
The special theory of relativity has made wide-ranging changes in our
understanding of nature, but Einstein based his special theory of relativity ontwo postulates:
1. The principle of relativity: The laws of Physics must be the same in all
inertial reference frames. The first postulate can also be stated as: there is
no experiment you can do in an inertial reference frame to determine if you
are at rest or moving uniformly at constant velocity.
2. The constancy of the speed of light: The speed of light in vacuum has the
same value 8 c ms = × 3.0 10 / , in all inertial frames, regardless of the velocity
of the observer or the velocity of the source emitting the light.
3. Uniform motion is invariant: A particle at rest or with constant velocity in
one inertial reference frame will be at rest or have constant velocity in all
inertial reference frames.
The first postulate asserts that all the laws of Physics dealing with Mechanics,
Electromagnetism, Optics, Thermodynamics, and are the same in all reference
frames moving with constant velocity relative to one another. This postulate is
a sweeping generalization of the principle of Galilean relativity, which refers
only to the laws of mechanics.
Einstein’s second postulate immediately implies the following result: It is
impossible for an inertial observer to travel at c, the speed of light invacuum.
Note that postulate 2 is required by postulate 1: If the speed of light were not
the same in all inertial frames, measurements of different speeds would make it
possible to distinguish between inertial frames; as a result, a preferred, absolute
frame could be identified, in contradiction to postulate 1.
These innocent-sounding propositions have far-reaching implications. Here
are three:
1. Events that are simultaneous for one observer may not be simultaneous
for another.
Two events occurring at different points in space which are simultaneous
to one observer are not necessarily simultaneous to a second observer. The
central point of relativity is this: Any inertial frame of reference can be used
to describe events and do Physics. There is no preferred inertial frame of
reference. However, observers in different inertial frames always measure
different time intervals with their clocks and different distances with theirmeter sticks. A light flash goes off in the center of a moving train (Fig.9.5).
9.2.1. Time dilation: Moving clocks run slowly
We can illustrate the fact that observers in different inertial frames always
measure different time intervals between a pair of events by considering avehicle moving to the right with a speed v, as shown in Fig.9.6.
