Modern physics is a branch of science that explains how the world works at a very small level. It's like looking at tiny building blocks that make up everything around us.
Two main ideas form the foundation of modern physics:
Relativity: This idea explains how time and space can change depend...
It was found that there was no displacement of the interference fringes,
so that the result of the experiment was negative and would, therefore,
show that there is still a difficulty in the theory itself. . . .
Albert Michelson, Light Waves and Their Uses, 1907
O ne of the great theories of physics appeared early in the twentieth century
when Albert Einstein presented his special theory of relativity in 1905. We
learned in introductory physics that Newton’s laws of motion must be measured
relative to some reference frame. A reference frame is called an inertial frame if Inertial frame
Newton’s laws are valid in that frame. If a body subject to no net external force
moves in a straight line with constant velocity, then the coordinate system at-
tached to that body defines an inertial frame. If Newton’s laws are valid in one
reference frame, then they are also valid in a reference frame moving at a uni-
form velocity relative to the first system. This is known as the Newtonian prin-
ciple of relativity or Galilean invariance. Galilean invariance
Newton showed that it was not possible to determine absolute motion in
space by any experiment, so he decided to use relative motion. In addition, the
Newtonian concepts of time and space are completely separable. Consider two
inertial reference frames, K and K!, that move along their x and x œ axes, respec-
tively, with uniform relative velocity v as shown in Figure 2.1. We show system K!
moving to the right with velocity v with respect to system K, which is fixed or
y y′ Figure 2.1 Two inertial systems
are moving with relative speed v
v along their x axes. We show the
K′ system K at rest and the system K!
K moving with speed v relative to
the system K .
O′ x′
O x
z′
z
19
,20 Chapter 2 Special Theory of Relativity
stationary somewhere. One result of the relativity theory is that there are no
fixed, absolute frames of reference. We use the term fixed to refer to a system that
is fixed on a particular object, such as a planet, star, or spaceship that itself is
moving in space. The transformation of the coordinates of a point in one system
to the other system is given by
x œ " x # vt
yœ " y (2.1)
zœ " z
Similarly, the inverse transformation is given by
x " x œ $ vt
y " yœ (2.2)
œ
z"z
œ
where we have set t " t because Newton considered time to be absolute. Equa-
Galilean transformation tions (2.1) and (2.2) are known as the Galilean transformation. Newton’s laws of
motion are invariant under a Galilean transformation; that is, they have the same
form in both systems K and K!.
In the late nineteenth century Albert Einstein was concerned that although
Newton’s laws of motion had the same form under a Galilean transformation,
Maxwell’s equations did not. Einstein believed so strongly in Maxwell’s equa-
tions that he showed there was a significant problem in our understanding of
the Newtonian principle of relativity. In 1905 he published ideas that rocked the
very foundations of physics and science. He proposed that space and time are
not separate and that Newton’s laws are only an approximation. This special
theory of relativity and its ramifications are the subject of this chapter. We begin
by presenting the experimental situation historically—showing why a problem
existed and what was done to try to rectify the situation. Then we discuss
Einstein’s two postulates on which the special theory is based. The interrelation
of space and time is discussed, and several amazing and remarkable predictions
based on the new theory are shown.
As the concepts of relativity became used more often in everyday research
and development, it became essential to understand the transformation of mo-
mentum, force, and energy. Here we study relativistic dynamics and the relation-
ship between mass and energy, which leads to one of the most famous equations
in physics and a new conservation law of mass-energy. Finally, we return to elec-
tromagnetism to investigate the effects of relativity. We learn that Maxwell’s
equations don’t require change, and electric and magnetic effects are relative,
depending on the observer. We leave until Chapter 15 our discussion of Einstein’s
general theory of relativity.
2.1 The Apparent Need for Ether
Thomas Young, an English physicist and physician, performed his famous ex-
periments on the interference of light in 1802. A decade later, the French physi-
cist and engineer Augustin Fresnel published his calculations showing the de-
tailed understanding of interference, diffraction, and polarization. Because all
known waves (other than light) require a medium in which to propagate (water
waves have water, sound waves have, for example, air, and so on), it was naturally
, 2.2 The Michelson-Morley Experiment 21
assumed that light also required a medium, even though light was apparently
able to travel in vacuum through outer space. This medium was called the lumi-
niferous ether or just ether for short, and it must have some amazing properties. The concept of ether
The ether had to have such a low density that planets could pass through it,
seemingly for eternity, with no apparent loss of orbit position. Its elasticity must
be strong enough to pass waves of incredibly high speeds!
The electromagnetic theory of light (1860s) of the Scottish mathematical
physicist James Clerk Maxwell shows that the speed of light in different media
depends only on the electric and magnetic properties of matter. In vacuum, the
speed of light is given by v " c " m0P0, where m0 and P0 are the permeability
and permittivity of free space, respectively. The properties of the ether, as pro-
posed by Maxwell in 1873, must be consistent with electromagnetic theory, and
the feeling was that to be able to discern the ether’s various properties required
only a sensitive enough experiment. The concept of ether was well accepted by
1880.
When Maxwell presented his electromagnetic theory, scientists were so con-
fident in the laws of classical physics that they immediately pursued the aspects
of Maxwell’s theory that were in contradiction with those laws. As it turned out,
this investigation led to a new, deeper understanding of nature. Maxwell’s equa-
tions predict the velocity of light in a vacuum to be c. If we have a flashbulb go
off in the moving system K!, an observer in system K! measures the speed of the
light pulse to be c. However, if we make use of Equation (2.1) to find the relation
between speeds, we find the speed measured in system K to be c $ v, where v is
the relative speed of the two systems. However, Maxwell’s equations don’t dif-
ferentiate between these two systems. Physicists of the late nineteenth century
AIP/Emilio Segrè Visual Archives.
proposed that there must be one preferred inertial reference frame in which the
ether was stationary and that in this system the speed of light was c. In the other
systems, the speed of light would indeed be affected by the relative speed of the
reference system. Because the speed of light was known to be so enormous, 3 %
108 m/s, no experiment had as yet been able to discern an effect due to the rela-
tive speed v. The ether frame would in fact be an absolute standard, from which
other measurements could be made. Scientists set out to find the effects of the
ether. Albert A. Michelson (1852–
1931) shown at his desk at the
University of Chicago in 1927. He
was born in Prussia but came to
2.2 The Michelson-Morley Experiment the United States when he was
two years old. He was educated
The Earth orbits around the sun at a high orbital speed, about 10#4c, so an obvi- at the U.S. Naval Academy and
ous experiment is to try to find the effects of the Earth’s motion through the later returned on the faculty.
ether. Even though we don’t know how fast the sun might be moving through Michelson had appointments at
several American universities in-
the ether, the Earth’s orbital velocity changes significantly throughout the year
cluding the Case School of
because of its change in direction, even if its orbital speed is nearly constant. Applied Science, Cleveland, in
Albert Michelson (1852–1931) performed perhaps the most significant 1883; Clark University, Worcester,
American physics experiment of the 1800s. Michelson, who was the first U.S. citi- Massachusetts, in 1890; and the
zen to receive the Nobel Prize in Physics (1907), was an ingenious scientist who University of Chicago in 1892
built an extremely precise device called an interferometer, which measures the until his retirement in 1929.
phase difference between two light waves. Michelson used his interferometer to During World War I he returned
to the U.S. Navy, where he devel-
detect the difference in the speed of light passing through the ether in different
oped a rangefinder for ships. He
directions. The basic technique is shown in Figure 2.2. Initially, it is assumed that spent his retirement years in
one of the interferometer arms (AC) is parallel to the motion of the Earth Pasadena, California, where he
through the ether. Light leaves the source S and passes through the glass plate continued to measure the speed
at A. Because the back of A is partially silvered, part of the light is reflected, of light at Mount Wilson.
, 22 Chapter 2 Special Theory of Relativity
Mirror M2
D
Ether drift
v
Optical path!
Partially silvered!
length !2
mirror
S
Mirror!
A B C M1
Monochromatic!
Figure 2.2 A schematic diagram light source Compensator
of Michelson’s interferometer ex-
periment. Light of a single wave- Optical path length !1
length is partially reflected and
partially transmitted by the glass
at A. The light is subsequently
reflected by mirrors at C and D,
and, after reflection or transmis-
E
sion again at A, enters the tele-
scope at E. Interference fringes
are visible to the observer at E.
eventually going to the mirror at D, and part of the light travels through A on to
the mirror at C. The light is reflected at the mirrors C and D and comes back to
the partially silvered mirror A, where part of the light from each path passes on
to the telescope and eye at E. The compensator is added at B to make sure both
light paths pass through equal thicknesses of glass. Interference fringes can be
found by using a bright light source such as sodium, with the light filtered to
make it monochromatic, and the apparatus is adjusted for maximum intensity of
the light at E. We will show that the fringe pattern should shift if the apparatus
is rotated through 90° such that arm AD becomes parallel to the motion of the
Earth through the ether and arm AC is perpendicular to the motion.
We let the optical path lengths of AC and AD be denoted by /1 and /2, re-
spectively. The observed interference pattern consists of alternating bright and
dark bands, corresponding to constructive and destructive interference, respec-
tively (Figure 2.3). For constructive interference, the difference between the two
From L. S. Swenson, Jr., Invention and Discovery 43 (Fall 1987).
Figure 2.3 Interference fringes
as they would appear in the eye-
piece of the Michelson-Morley
experiment.
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