Lecture Notes
Relativity - Special Theory
(part of Classical Mechanics (II)
PH33003/PH43017)
S. Murugesh
Last update: February 18, 2009
,Contents
1 The Issue 3
1.1 Galelian transformations and the Galeian group . . . . . . . . . . . . . . . . . . . . 3
1.2 Maxwell’s equations do not agree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Possible resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Special Theory 6
2.1 The Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Space-time, Events and World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Two Events in two different inertial frames . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Objects with velocity c are special . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Time like, space like and light like intervals: Causality . . . . . . . . . . . . 9
2.4 Time dilation: Proper time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Length contraction: Proper length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Four Vectors and the ∗ product 13
3.1 The four-velocity vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 The ∗ product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The four-accelaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 The case of uniform accelaration . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Velocity addition and Lorentz transformations 16
4.1 Velocity addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Relativistic Mechanics 19
5.1 Momentum, as we know it, of an isolated system of particles is not conserved if we
look at it from different reference frames . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 The four-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Four-momentum: Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3.1 Momentum, mass and force . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Decay process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.5 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
,6 Waves 25
6.1 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1.1 Longitudinal Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1.2 Transverse Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
, Chapter 1
The Issue
In this chapter we review the relevant aspects of classical mechanics, and the issues that led to the
Special Theory of Relativity.
1.1 Galelian transformations and the Galeian group
Consider two observers in two reference frames, S and S ′ , designated by coordinates (x, y, z) and
time t, and (x′ , y ′ , z ′ ) and time t′ , respectively. Let the two reference frames be in uniform relative
motion with respect to each other with constant velocity V = {vx , vy , vz }. Also, let the two reference
frames differ in their orientation (which remains a constant), and differ in their origin in space and
time. Then the two space and time coordinates are in general related by
′
x vx x x0
y′
′ = R3×3 vy
y + y0 .
(1.1)
z vz z z0
t′ 0 0 0 1 t t0
Here, R3×3 is a constant 3-parameter rotation (orthogonal 3×3) matrix representing the orientation
of the S ′ frame with respect to S frame, and (x0 , y0 , z0 ) and t0 are the difference in their spacial and
temporal origins. In all, there are 10 parameters that relate the two space and time coordinates
(3 for constant spacial rotations in R3×3 , 3 boost parameters -(vx , vy , vz ), 3 spacial displacements
-(x0 , y0 , z0 ), and one temporal displacement-t0 ).
Eq. (1.1) constitutes the most general Galelian transformation connecting two coordinate sys-
tems. Any non-accelerating reference frame is defined as an inertial reference frame. Any reference
frame related to any other inertial reference frame by a Galelian transformation will also be inertial.
Indeed, the significance of such transformations arises from the fact - Forces (or accelerations) seen
in one inertial reference frame is the same in all inertial reference frames. More strictly, the form
of the equations of mechanics are invariant with respect to such transformations. I.e.,
mi r̈i = mi r̈′i . (1.2)
Further,
mi r̈i = Fi ([r], t) (1.3)
will go over to
mi r̈′i = Fi ([r(r′ )], t(t′ )) = F′i ([r′ ], t′ ). (1.4)
Relativity - Special Theory
(part of Classical Mechanics (II)
PH33003/PH43017)
S. Murugesh
Last update: February 18, 2009
,Contents
1 The Issue 3
1.1 Galelian transformations and the Galeian group . . . . . . . . . . . . . . . . . . . . 3
1.2 Maxwell’s equations do not agree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Possible resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Special Theory 6
2.1 The Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Space-time, Events and World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Two Events in two different inertial frames . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Objects with velocity c are special . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Time like, space like and light like intervals: Causality . . . . . . . . . . . . 9
2.4 Time dilation: Proper time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Length contraction: Proper length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Four Vectors and the ∗ product 13
3.1 The four-velocity vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 The ∗ product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The four-accelaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 The case of uniform accelaration . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Velocity addition and Lorentz transformations 16
4.1 Velocity addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Relativistic Mechanics 19
5.1 Momentum, as we know it, of an isolated system of particles is not conserved if we
look at it from different reference frames . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 The four-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Four-momentum: Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3.1 Momentum, mass and force . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Decay process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.5 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
,6 Waves 25
6.1 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1.1 Longitudinal Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1.2 Transverse Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
, Chapter 1
The Issue
In this chapter we review the relevant aspects of classical mechanics, and the issues that led to the
Special Theory of Relativity.
1.1 Galelian transformations and the Galeian group
Consider two observers in two reference frames, S and S ′ , designated by coordinates (x, y, z) and
time t, and (x′ , y ′ , z ′ ) and time t′ , respectively. Let the two reference frames be in uniform relative
motion with respect to each other with constant velocity V = {vx , vy , vz }. Also, let the two reference
frames differ in their orientation (which remains a constant), and differ in their origin in space and
time. Then the two space and time coordinates are in general related by
′
x vx x x0
y′
′ = R3×3 vy
y + y0 .
(1.1)
z vz z z0
t′ 0 0 0 1 t t0
Here, R3×3 is a constant 3-parameter rotation (orthogonal 3×3) matrix representing the orientation
of the S ′ frame with respect to S frame, and (x0 , y0 , z0 ) and t0 are the difference in their spacial and
temporal origins. In all, there are 10 parameters that relate the two space and time coordinates
(3 for constant spacial rotations in R3×3 , 3 boost parameters -(vx , vy , vz ), 3 spacial displacements
-(x0 , y0 , z0 ), and one temporal displacement-t0 ).
Eq. (1.1) constitutes the most general Galelian transformation connecting two coordinate sys-
tems. Any non-accelerating reference frame is defined as an inertial reference frame. Any reference
frame related to any other inertial reference frame by a Galelian transformation will also be inertial.
Indeed, the significance of such transformations arises from the fact - Forces (or accelerations) seen
in one inertial reference frame is the same in all inertial reference frames. More strictly, the form
of the equations of mechanics are invariant with respect to such transformations. I.e.,
mi r̈i = mi r̈′i . (1.2)
Further,
mi r̈i = Fi ([r], t) (1.3)
will go over to
mi r̈′i = Fi ([r(r′ )], t(t′ )) = F′i ([r′ ], t′ ). (1.4)
