Elements of Modern Physics
2
We begin our discussion of modern physics with the theory of relativity which
aims at relating the observations made by observers in relative motion with
respect to each other. Here only the restrictive case of the special theory of
relativity is analysed, in which the observers are moving with constant velocity
with respect to each other. This will help in choosing appropriate frames of
reference and in presenting the later topics in a unified manner. After a brief
consideration of the drawbacks of the classical theory, the main results of the
special theory of relativity are obtained, and applied to describe some specific
physical situations.
1.1 INERTIAL FRAMES OF REFERENCE
Most physical observations describe the behaviour of certain objects in space
as a function of time. Since the position of a body can be stated only relative to
some other bodies, the description of these observations requires a frame of
reference which is a technical term for the combination of a set of spatial
coordinate axes and a time variable.
It was realised by Galileo and others, that the form of the laws of nature
depends on the choice of the frame of reference. Among all the possible frames
of reference, there exists a class called the inertial frames of reference, in
which these laws take a simple form. Inertial frames of reference are those in
which a body that is not acted upon by external forces, moves with constant
velocity. It is implicit here that if two reference frames move with constant
velocity with respect to each other, and one of them is inertial, the other also is
an inertial frame. It was found that the laws of mechanics take on the same
form in all inertial frames of reference.
1.2 GALILEAN TRANSFORMATIONS
Consider to inertial frames of reference F and F′, such that their coordinate
axes coincide at t = 0, and F′ moves with velocity v along the x-axis with
respect to F. Then, it may be expected that the coordinates in the two frames
are related by the equations
t′ = t
x′ = x – yt
(1.1)
y′ = y
z′ = z
called Galilean transformations. In writing these relations, it is assumed that
(i) it is possible to define a time t which is the same for all inertial frames of
reference, and (ii) the distance between two points is independent of the frames
of reference.
2
We begin our discussion of modern physics with the theory of relativity which
aims at relating the observations made by observers in relative motion with
respect to each other. Here only the restrictive case of the special theory of
relativity is analysed, in which the observers are moving with constant velocity
with respect to each other. This will help in choosing appropriate frames of
reference and in presenting the later topics in a unified manner. After a brief
consideration of the drawbacks of the classical theory, the main results of the
special theory of relativity are obtained, and applied to describe some specific
physical situations.
1.1 INERTIAL FRAMES OF REFERENCE
Most physical observations describe the behaviour of certain objects in space
as a function of time. Since the position of a body can be stated only relative to
some other bodies, the description of these observations requires a frame of
reference which is a technical term for the combination of a set of spatial
coordinate axes and a time variable.
It was realised by Galileo and others, that the form of the laws of nature
depends on the choice of the frame of reference. Among all the possible frames
of reference, there exists a class called the inertial frames of reference, in
which these laws take a simple form. Inertial frames of reference are those in
which a body that is not acted upon by external forces, moves with constant
velocity. It is implicit here that if two reference frames move with constant
velocity with respect to each other, and one of them is inertial, the other also is
an inertial frame. It was found that the laws of mechanics take on the same
form in all inertial frames of reference.
1.2 GALILEAN TRANSFORMATIONS
Consider to inertial frames of reference F and F′, such that their coordinate
axes coincide at t = 0, and F′ moves with velocity v along the x-axis with
respect to F. Then, it may be expected that the coordinates in the two frames
are related by the equations
t′ = t
x′ = x – yt
(1.1)
y′ = y
z′ = z
called Galilean transformations. In writing these relations, it is assumed that
(i) it is possible to define a time t which is the same for all inertial frames of
reference, and (ii) the distance between two points is independent of the frames
of reference.