OBJECTIVES
After you have finished this lesson you should understand and know about
* Einsteins Special Theory of Relativity.
* time dilation, space contraction, and mass dilation.
* Einsteins General of Theory of Relativity.
* the Principle of Equivalence.
* the effect of gravity on light.
* curvature of Space-Time
* the effect of gravity on time.
COMMENTARY
In the nineteenth century, after YoungÕs Double Slit experiment, most scientists
accepted the fact that light was of a wave nature. James Clarke Maxwell and Heinrich
Hertz later in the century showed that light was actually electromagnetic waves. Every
form of wave phenomena up to that time could be explained as a vibrating disturbance
passing through a material medium. The question, ÒWhat medium carries light waves?Ó
was thus posed, and scientists set out on a quest to find that medium, which they called
the Aether.
If two different observers of a wave are travelling through a common medium at
different speeds with respect to each other, they will assign different speeds to the wave;
so will a single observer if the observer is moving at a different speed with respect to the
medium. But how do we measure speed? We must first set up a Frame of Reference in
which to measure position and thus distance. This
entails choosing an origin and three rigid (i.e., non-stretching) perpendicular lines
(called axes) that intersect at the chosen origin. The origin may or may not be moving
with respect to you, and the axes may or may not be rotating. If neither the origin is
moving nor the axes rotating, one says that the frame of reference is at rest with respect
to you or that the frame is a Rest Frame for you. If two frames of reference are moving
with respect to each other, then speeds and direction of motion measured from the two
different frames will be different. Think about a car travelling at 40 mph overtaking your
car travelling at 35 mph; its speed measured from a frame of reference that is stationary
with respect to your car (e.g., you) is 5 mph, while its speed measured from a frame of
reference fixed at the road side is 40 mph. Speed is the rate at which distance is travelled.
Thus one must measure distance and the time taken to travel the distance. Distances
can be measured in a rest frame by using, for instance, a standard meter measure and
marking off the number of times it fits into the length travelled in a time duration
measured by any oscillatory process (i.e., any process that repeats itself in a regular
fashion) that is stationary in the same rest frame as the observer of the motion.
If the Aether exists, it would produce an absolute frame of reference to which all
motions can be related.1 A. A. Michelson and E. W. Morely constructed a very sensitive
experiment designed to demonstrate the relative motion of the earth through the Aether
by observing the change in the speed of light produced by the earth's motion through the
Aether. Alas for Michelson and Moreley, they could observe no such effect; they
constructed even more refined experiments, but always failed to measure any such
effect. This had a profound negative effect on the lives of these two scientists, producing
many personal problems. However, Albert Einstein took a positive view of this negative
result and concluded that the Aether did not exist (It had no measurable effect, therefore
could not be a part of physical reality.) and that the speed of light in a fixed material was
the same to all inertial observers. An inertial observer is one that is not accelerating (i.e.,
going at constant velocity2). Further he concluded that there could be no such concept as
absolute motion for without the Aether there was no way it could be measured. Einstein
then took these empirical facts and analyzed how we measure distance and time
durations and thus speed to produce his Theory of Special Relativity. Formally stated he
built his theory on the following three empirically verified facts:
0. Only relative velocities can be measured.
1. The speed of light in a fixed medium is the same to all inertial observers in
that
medium.
2. The laws of physics do not depend on the inertial frame of reference in which
they are measured.
(I have denoted the first ÒfactÓ or axiom with a Ò0Ó as it is not usually considered one of
the axioms of special relativity, but it is a basic tenet.)
Einstein was interested in relating observations made by observers in inertial
frames of reference that were moving with respect to each other. Inertial frames of
reference are ones that have a constant velocity. But how does one know if one is
travelling at a constant velocity if all speed is relative? I might be travelling at a constant
velocity compared to an object whose velocity is changing, accelerating! Newton,
however, had already solved this problem by his second law,
Force = mass x acceleration
which is to be interpreted as a force acting on an object causes an acceleration of the
object, the magnitude of acceleration depending on the mass of the object. Observers in
inertial frames of reference always measure the same forces acting on and thus the
same acceleration of a given object even if the inertial frames are different! Forces can be
classified into contact forces, due to collisions and friction, and forces due to gravity
fields, electromagnetic fields, or strong or weak nuclear fields. The first two can, of
course, also be described in terms of the latter four. There are no other known forces. It
is known, however, that from the viewpoint of non-inertial (i.e., accelerating) frames of
reference, objects can start to accelerate even when none of the above forces act on them!
Think what happens when you turn a corner in carÑobjects on the back window start to
slide; or what happens on a train when it suddenly startsÑobjects can start to fly
around, even though no force is acting on them! Thus in non-inertial frames of reference
Newton's second law is not obeyed. We can turn this around and define an inertial frame
of reference to be one in which Newton's second law is obeyed. Thus we have an
experimental test to check whether we are in an inertial frame of reference or not and
determine whether Newton's second is obeyed or not.
In order to compare measurements in two inertial frames of reference travelling
with respect to each other, we can construct the following ÒthoughtÓ experiment.
Consider two people, A and B. Person A is in a transparent spaceship travelling at a
speed v as measured by B, who is standing on the surface of some planet. Naturally A
could also measure B's velocity and find that B is moving with a speed v in the opposite
direction; one should even note the trivial facts that with respect to A, A is not moving
and with respect to B, B is not moving. Let A and B both have Òlight clocksÓÑa
combination of perfectly reflecting mirrors separated by a fixed distanceÑthat are
stationary with respect to their owners.
A pulse of light continually reflects between the mirrors at one second intervals. The
following diagram shows what observer B sees as A flies past him at a uniform speed v
In a time tB, as measured by B's clock, B sees that A has travelled a distance vtB and that
the light pulse has travelled a distance of ctB. As measured by A, however, the distance
the light pulse has travelled in that time is ctA, and according to A's clock, tA is actually
\F(1,2) of a second. We should note two important points: both A and B are observing the
same event and both observe that the speed of light is c3. It is also important to
emphasize that even though A and B are moving with respect to each other, they both see
light travelling at the same speed. (This would not be true for sound.)
In the diagram the points K, L, and M form a right angled triangle. We can relate
the lengths of the sides of the triangle by PythagorasÕ theorem and arrive at a
relationship between tA (the time measured by A) and tB (the time measured by B) by the
following algebraic maneuvers:
Surprisingly tA does not equal tB unless v=0! This result is a direct consequence of the
invariance of the speed of light between inertial observers. From the relationship between
tA and tB one can observe that tA £ tB; thus processes that are at rest in AÕs frame of
reference take longer when observed from BÕs frame of reference when v¹0. Hence when
viewed by an observer in motion, the passage of time in AÕs frame of reference slows
down. In general one concludes that processes in motion take a longer time than the
same processes do when they are stationary. The duration of processes thus depends on
the motion of the observer with respect to the process. If the observer is stationary with
respect to the process, the time measured for the duration is called proper time. And if
frames of reference are stationary with respect to the objects being measured, the
quantities measured are called proper quantities. Thus the distance travelled by A in the
preceding diagram, which is measured by B, is the proper distance or proper length, L0,
travelled by A. This distance is stationary with respect to B, but not to A. By using the
definition of speed and time duration measured by B, one obtains
L0 = vtB
while using the same definition of speed and time duration measured by A leads to the
distance travelled as
L = vtA
Thus substituting the preceding expression for tA in terms of tB obtained previously, one
obtains the following expression for L which shows that L£L0:
In other words, the magnitude of lengths in the direction of motion decreases as the
speed of the object increases! This is called space contraction. Both time dilation and
space contraction are real; they are not just effects of perception. Time dilation has been
demonstrated, for instance, in cesium clocks orbiting the earth and in the decay of
cosmic muons entering the earth's atmosphere, which shows that special relativistic
effects are as real as life and death. The Special Theory of Relativity demonstrates that
the duration of processes and sizes of objects depend on their velocities. Velocity depends
on both distance and time, thus distance and time must be related and are aspects of a
more general entity called Space-Time.
Previously we noted that an object can have internal, potential, and kinetic energy.
EinsteinÕs theory of special relativity shows that the total energy of a stationary object is
related to its mass by the formula
\X(E = m0c2)
The rest mass m0 is the proper mass as it is measured in the rest frame of the object.
This total energy does not contain any kinetic energy, as the object is not moving, and
only contains potential energy if a force field is acting on the object. If the object is moving
the total energy is given by
\X(E = mc2)
where
which can be thought of as mass dilation. All the preceding formulae involve square
roots, but must produce real numbers for length, mass, and time; hence
\X(1 - \F(v2,c2) ³ 0)
otherwise we would obtain a square root of a negative number, which would lead to
nonphysical expressions for these quantities. The above inequality implies that
\X(c2 - v2 ³ 0)
\X(Þ c2 ³ v2)
\X(Þ c ³ v)
that is, the maximum value the speed of an object can be is the speed of light, c. This
conclusion is consistent with the formula for mass, which leads to the realization that it
would take infinite energy to accelerate an object with non-zero mass to the speed of light.
This can easily be seen by considering the following graph of mass versus speed
and the relation E = mc2. Can anything, then, travel at the speed of light? The answer is
clearly yes, since light travels at this speed! How do we make sense of mass and time
dilation and space contraction for photons? When v=c, the term \X(1 - \F(v2,c2)) becomes
zero, and when it occurs in the denominator of the expression for mass dilation, we
obtain the absurd answer that the mass, and therefore energy, of the photon is infinite!
The only way we can get a sensible answer, non infinite, is if the rest mass of the photon
is actually zero and the photon is therefore not matter. However, the photon still has
energy and it can be derived that the energy according to the theory of special relativity is
\X(E = pc)
where p is the magnitude of the momentum4 of the photon. From the time dilation
formula one can deduce that processes in a photonÕs frame of reference take infinitely
long from any other inertial frame of reference, thus no processes can be taking place
inside photons; as for everything outside the photonÕs rest frame, time has stopped in/on
the photon and in fact it never started! Further, in the direction of motion of the photon it
is infinitely thin. However these conclusions do not take quantum theory into account,
which insists there is no precise position and no precise time, but only fluctuations about
average values.
At the beginning of this unit we stated that the theory of special relativity had two
main axioms. We have only really made use of the first axiom about the invariance of the
speed of light. The second axiom, that the laws of physics do not depend on the inertial
frame of reference in which they are measured, was essentially accepted before Einstein
for mechanical systems. Einstein's contribution was to extend it to cover electromagnetic
phenomena described by MaxwellÕs equations. Two of the basic mechanical laws of
nature are NewtonÕs Second Law of Motion
\X(Force = mass x acceleration)
which can be written in symbolic form as
\X(F = ma)
and NewtonÕs Law of Gravitation: the magnitude of the gravitational force between two
masses equals the product of the masses and NewtonÕs gravitational constant, G, divided
by the square of the distance between the masses, which written in symbolic form is
\X(F = G \F(M1M2,r2))
One of the implications of the special theory of relativity is that photons do not have mass;
therefore, according to NewtonÕs law of gravitation, they are not affected by gravity fields.
However Einstein decided to ponder another thought experiment: Consider being in a
space vehicle far from any source of gravity, that is, far from any mass, and with no
means to observe outside the spaceship. If the ship were travelling at a constant speed
(i.e., with zero acceleration) and you were standing on the floor of the ship and decided to
let go of a ball you were holding in your hand, the ball would stay floating in exactly the
position you left it. This is due to the fact that the ball, you, and the shipÕs floor are all
travelling at the same constant speed and in the same direction and letting go of the ball
did nothing to change this, so the ball stays at the same distance above the shipÕs floor as
it was.
acceleration a=0
velocity is constant
If, however, the ship starts to accelerate the ball will still be going at the original speed,
but the floor will be going faster and faster and will catch the ball up until the ball hits
the floor. Inside the ship it will appear that the ball just drops to the floor.
acceleration a ¹0
velocity is not constant
On the other hand, if the speed of the spaceship remained constant but a large planet
suddenly appeared on the scene producing a gravity field, again the ball would fall to the
floor.
How can one decide which of the two events occurred? Einstein pondered this and came
to the conclusion there was no way to measure the difference! Therefore, by the principle
of reality, if a difference cannot be measured, then there is no difference, and one must
conclude that the effects of a Newtonian gravity field are identical to the effects of
acceleration. This is called the Principle of Equivalence and is the basis of EinsteinÕs
General Theory of Relativity, which is a generalization of NewtonÕs theory of gravity. The
principle of equivalence has some interesting consequences for photons: If in the
accelerating space ship a pulse of photons were emitted in a direction parallel to the
floor, the photon pulse would follow a curved trajectory falling to the floor just like a
particle with mass!
But the effects of Newtonian gravity and acceleration cannot be distinguished from each
other, therefore photons are affected by gravity! This means that NewtonÕs equations
describing gravity are incomplete, and Einstein formulated more general ones
predicting that light should be bent by gravity fields, which was indeed observed in 1919
and gave great credence to the theory.
A seemingly unrelated topic is that of defining a straight line. Following common
sense one would use a string or wire and connect it tautly between two points. If one had
a long enough string or wire it would eventually go all around the earth and what
seemed a straight line on the surface would look like a circle from space! This is due to
the curvature of the surface of the earth, which, as the astronomers of the Renaissance
showed, is not flat. Mathematically one can
characterize a flat surface as having zero curvature and the surfaces such as the earthÕs
surface, which is essentially a sphere, as positive. The taut string or wire on the surface
of the earth has one property that cannot be denied: it is the shortest distance between two
points on the surface. Such lines are called geodesics.
Engineers have found that the best way to check the straightness of a line is with
light rays. On the surface of the earth light rays will not follow the curvature of the earth
and are therefore straighter than material lines; but over short distances of the
magnitude usually encountered the differences are indiscernible. However, according to
the general theory of relativity light rays are bent by gravitational fields, so they are not
straight either! But is there anything better than light to define a straight line? No, all
other beams of objects that have non-zero rest mass would be bent more. Thus we have no
option but to define the trajectory of a light beam in a vacuum as a straight line. If we do
this we must take the viewpoint that light travels along geodesics of space-time, and it is
space-time that has curvature, while light is not affected by gravity, and high
concentrations of mass warp space-time (places of high curvature are associated with
large mass, i.e., gravity). It turns out that the curvature of space-time is negative, and
the simplest object that has negative curvature is a saddle, which increases in height in
one direction but decreases in another:
Einstein first thought that space-time had, on the average, positive curvature and that if
one travelled far enough in a constant direction one would return to oneÕs starting place
and the universe could be said to be closed! However, this hypothesis does not fit observed
facts as well as space-time having on the average negative curvature. If one does not like
the view that space-time is curved, one can throw out the concept of a straight line and
consider that light rays are bent by gravity fields produced by large masses. The former
view, curved space time, is a more geometric and static view of nature, while the latter
one is more dynamic; however, both views are absolutely equivalent and each is just as
valid as the other. Conclusions reached from one are as valid as conclusions reached
from the other. This is in fact another form of the principle of equivalence.
The general theory of relativity shows that light is affected by gravity. What
implications does this have other than the bending of light? If objects interact with
gravity fields, then they have gravitational potential energy; thus light in the
presence of a gravity field must have gravitational potential energy. As a photon falls
towards the source of a gravity field, its gravitational potential energy decreases and its
kinetic energy increases; as it leaves the gravity field, its potential energy increases and
its kinetic energy decreases. PlanckÕs relationship, \X(E=hu), shows that as the potential
energy decreases, so does the frequency of the photon, and conversely as the potential
energy increases, so does the frequency. The gravitational increase of frequency is called
the gravitational blue shift, while the gravitational decrease of frequency is called the
gravitational red shift. Thus on leaving a source of gravity the frequency of a photon
undergoes a red shift, and while falling into a source of gravity a photon undergoes a
blue shift. This means an observer in a low gravity environment sees photons coming
from a high gravity environment as red shifted, while observers in a high gravity
environment see photons coming from a low gravity one as blue shifted. The red and blue
gravitational shifts have much more profound implications than just color schemes, as
the most accurate clocks we have are oscillating photons emitted either by cesium or
hydrogen atoms. If the oscillations of photons are slowed down (red shift in frequency),
the period of oscillation increases and therefore produces a gravitational time dilation;
conversely if the oscillations are speeded up and the period decreases, a gravitational
time contraction occurs. Hence the rate of processes decreases, the higher the gravity
they are in, relative to the observer. Thus in high gravity fields time slows down (to
outside observers)! If the gravity field were infinitely strong, processes in the gravity field
viewed from outside would be infinitely slow and time would be observed to have stopped!
Please look over the following review questions and exercises. If you're unsure of the
answers, check your text and then the appendix of this study guide for the information.
Do not submit these to your instructor.