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LIGHT and SPECIAL
RELATIVITY FRAMES OF REFERENCE
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SUMMARY The
location of an object and its velocity depends upon the frame of reference of an observer. Inertial frame of reference has zero
acceleration. Newton’s Law of Motion are valid. Non-inertial frame of reference has a
non-zero acceleration. Newton’s Laws of motion are not valid. Different
observer can have different views on the motion of an object: space and
time are relative concepts, not absolute ones. Postulate
for the propagation of light through space - ether model. Michelson-Morley
experiment attempted to measure
the relative velocity of the Earth through the ether. Null result. Ether model had to be
rejected. Maxwell
– electromagnetic radiation propagates at the speed of light
independent of the frame of reference. The
ether model discarded. Einstein:
all the laws of physics are valid in inertial frames of reference: · Space and time are relative quantities · Speed of light is a constant and does not depend upon the motion
of the source or observer. |
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OVERVIEW OF
SPECIAL RELATIVITY Relativity is the
study of the relative motions of objects Newton’s
view of the Universe turns out to be wrong. From a Newtonian point of view,
time is an absolute quantity. A better model for the working of the
Universe are Einstein’s theories of Special Relativity and General Relativity. Space and time
are interconnected and different observes can get different measurements for
time intervals and lengths. Einstein’s
Theories of Relativity are one of the greatest intellectual achievements of
the 20th Century. Special Relativity, developed by Einstein in 1905,
deals with systems that are moving at constant velocity (zero acceleration) with
respect to each other. General Relativity proposed in 1916 deals with
gravitation and systems that are accelerating with respect to each
other.
What is
the trajectory of a ball thrown into the air? This is
a simple question. But what is the answer? Mary stands on
the back of a moving cart and throws a ball vertically into the air. Steve standing on the ground watches the event of
throwing and then catching the ball. What is the path of the ball?
Surprisingly, there is no unique answer to this question. The answer depends
upon the person observing the motion of the ball. Mary
claimed the ball travelled in a straight-line path, up then down. Steve claimed the ball travelled in a parabolic
arc. Who is correct? Both are correct descriptions as shown in the following
animation.
Is it day or
night? – it depends
on the location of the observer.
Is the pig on the
right or left? – it depends
upon the direction the observer is facing.
Which is bigger
– the bear or mouse? – depends upon the viewing location
“up”
or “down”?
For
us, it has not been easy to realise that the concept “up” and
“down” are relative concepts. We are inclined to use
“common sense” and ascribe absolute sense to concepts. A description of motion depends upon the frame of reference of
the observer This
is what Einstein’s Theory of Special Relativity is all about.
Difference observes in different frames of reference can measure different
values for length, time intervals and momentum. In
the Theory of Special
Relative both space
and time
are relative concepts and this idea is in conflict with our
notions of space and time based upon “common sense”. But
what is meant by time and space being relative? Consider observes Mary and
Steve in two inertial frames of reference. They both make length and time
interval measurements of the same event. Mary records the length as 623 m and the time interval as 156 s. Steve records the length as 321 m and the time interval as 556
s. Who is correct? Both
can be correct because their measurements of the same event*
depend upon the relative velocity between the two inertial frames of
reference. So, to start out study of Special Relativity we need to define
clearly the frame of reference for an observer. *
An event is
something that happens independently of the frame of reference, for example,
a ball being thrown up into the air or lightning flash. FRAMES OF
REFERENCE A
coordinate system is necessary to describe the position and the velocity and
apply Newton’s Laws of motion. So, the measurements of position,
velocity and force depend upon the frame of reference
of the observer. If
Newton’s Laws are valid in one frame of reference, then they are also
valid in any reference frame moving at a uniform velocity relative to the
first frame. This is known as Newtonian
principle of relativity or Galilean
invariance. Newton showed that it was not possible to perform any
experiment to determine the absolute motion in space. Such reference frames are known as inertial frames of reference. A frame of reference
which is accelerating
and Newton’s Laws do not apply is known as a non-inertial frame of reference. A
truck travelling at a constant velocity can be regarded as an inertial frame
of reference. In figure 1, these is no experiment that Mary or Steve perform
inside their truck to determine their speed. A ball hanging from the ceiling
falls vertical. Mary and Steve conclude that there is a zero net force acting
on the ball. However, Eve is in an accelerating truck and therefore in a
non-inertial frame of reference. Eve incorrectly concluded that there is a
force acting on the ball to cause it to deviate from the horizontal. Such
forces are known as fictitious forces. The ball no longer falls vertically
because of the inertia of the ball (Newton’s 1st Law) and
the greater the acceleration of the truck, the greater the deflection of the
hanging ball.
Fig. 1. Inertial and
non-inertial frames of reference. Mary and Steve cannot perform an experiment
to determine the absolute motion of the truck. Ancient
Greek philosopher Aristotle: it was obvious that objects would
assume a preferred state of rest unless some external force propelled them - concepts
of absolute space and absolute time – that is that both space
and time exist in their own right, independently of each other and of other
material things. So, it is possible to assign absolute values of position and
time to events. Aristotle’s work was held in such high regard that it
remained basically unchallenged until the end of the sixteenth century, when
Galileo showed that it was incorrect. Galileo · Motion must be relative · Motion involves displacements of objects relative to some
reference system · Principle of Galilean Relativity: the laws of mechanics are the
same for a body at rest and an object moving at constant velocity. Isaac Newton · Laws of Motion and his Law of Universal Gravitation - only
possible to determine the relative velocity of one reference frame with
respect to another and not the absolute velocity of either frame · No preferred or absolute reference frame exists. The Principle
of Newtonian Relativity: the laws
of mechanics must be the same in all inertial reference frames. Due to Galileo and
Newton, the concept of absolute space became redundant since there could be
no absolute reference frame with respect to which mechanical measurements
could be made. However, Galileo and Newton retained the concept of absolute
time, or the ability to establish that two events that happened at different
locations occurred at the same time - if an observer in one reference frame
observed two events at different locations as occurring simultaneously, then
all observers in all reference frames would agree that the events were
simultaneous. The
Newtonian concept of the structure of space and time remained unchallenged
until the development of the electromagnetic theory in the nineteenth
century, principally by Michael Faraday and James Clerk Maxwell. Maxwell
showed that electromagnetic waves in a vacuum ought to propagate at a speed
of c
= 3x108 m.s-1 (the speed of light). To 19th
Century physicists this presented a problem. If EM waves were supposed to
propagate at this fixed speed c,
what
was this speed measured relative to? How could you measure it relative to a
vacuum? Newton had done away with the idea of an absolute reference
frame. We now must do away with the concept of absolute time. When
19th Century physicists selected the ether
as the medium for the propagation of electromagnetic waves they were merely
borrowing and adapting an existing concept. The fact that certain physical
events propagate themselves through space led to the hypothesis that space is
not empty but is filled with an extremely fine substance, the ether, which is
the carrier or medium of these phenomena. The ether was the
frame of reference in which the velocity of light was measured with respect
to. However, experiments to detect the ether carried out by Michelson
and Morley gave a null result (1887). It
was not until 1905 that Einstein that the “true” nature of the
propagation of light was resolved. His theory of Special Relativity is based
upon two postulates in which the speed of light in free space is independent
of any inertial frame of reference. 1.
The laws of
physics may be expressed in equations having the same form in all frames of
reference moving at a constant velocity (inertial frame of references) with
respect to one another. This implies that there is no universal frame of
reference. Thus, there is no distinction between “stationary” or
“moving with a constant velocity” in nature. 2.
The speed
of light in free space has the same value for all observes, regardless of
their state of motion. At
first sight, these postulates hardly seem radial. However, the consequences
of these two postulates are amazing. When event occurs, different observers may not agree of time
interval or distance measurements or whether events occurs simultaneously.
The consequences subvert almost all our intuitive concepts of time and space
we form on the basis of our daily experiences. Two
simple examples are discussed below which illustrate how strange nature is !!! Consider
two boats at sea on a clear day. A stone is dropped into the water when Mary’s boat passes Steve’s
boat. The stone dropped causes a circular pattern of ripples to spread out in
the water (water wave spreads uniformly in all directions). However, the
pattern of ripples will appear different for our two observers Steve and Mary. Steve can tell that he is stationary w.r.t. the
water as he observers that he is at the centre of the circular ripple
pattern. Mary knows that she is moving through
the water because she observes an asymmetrical ripple pattern. Water itself
is the frame of reference, and Mary can
measure the speed of the ripples in different directions and calculate the
speed v which
her boat is moving through the water. The
observations of Steve and Mary is clearly demonstrated in the animation shown
in figure 2.
Fig. 2. The
observers Steve and Mary
observe different ripple patterns when a stone was dropped into the water.
The water acts as the frame of reference. From the circular ripple pattern,
Steve knows that he is stationary w.r.t to the water. Mary knows that she is moving
through the water from the left to the right because of the asymmetrical
ripple pattern. Now
consider the two boats at sea in a dense fog so that neither Mary or Steve have
any idea which is moving. At the instant Mary’s
boat is abreast of Steve’s boat, a green flare is fired. The light from the flare
travels uniformly in all directions according to the second postulate of
special relativity. Both Mary and Steve must see a sphere of expanding light with
themselves at the centre, according to the first postulate, even though one
of the boats is changing its position with respect to the point where the
flare went off. The observers cannot detect which one of them is moving
through the water since the fog eliminates any frame of reference other than
the boat itself, and so, since the speed of light is the same for both, they
must see identical phenomena as shown in figure 3.
Fig. 3. Mary and Steve both
see a sphere of light expanding with themselves at the centre even though
they may be changing their position with respect to the point where the flare
went off. They must both see identical phenomena according to the 1st
and 2nd postulates of special relativity. It
is important to recognize that the motion of waves in water are entirely
different from the motion of electromagnetic waves in free space. Water is
itself a frame of reference. The wave speed in water varies with the motion
of the observer. Space is not a frame of reference for electromagnetic waves.
The speed of electromagnetic waves in free space does not depend upon any
frame of reference, the speed of light is the same for all observers. As
a consequence of Einstein’s postulates, many peculiar effects are
predicted. One of the triumphs of modern physics is the experimental
confirmation of these effects. |
