lOMoARcPSD|6353920
SPA4402 MODERN PHYSICS
INTRODUCTION TO SPECIAL RELATIVITY
Albert Einstein introduced his Special Theory of Relativity in 1905. To understand the
theory, we will first review the background, the theoretical and experimental developments
since Newton.
‘Special’ means ‘not general’. The theory is only about inertial frames of reference.
Remember that, for Galileo and Newton, such frames had a central importance in
mechanics and cosmology because the Principle of Relativity states that no laboratory-
based measurement could establish one rather than another as “at rest”, or “in motion”.
1st axiom: Principle of Relativity
The laws of physics are the same in every inertial frame of reference.
The Galilean Transform
Galileo and Newton were familiar with the relationships among inertial frames. If in the
frame S the coordinates of an event are (𝑥, 𝑡), then, in a frame 𝑆′ moving at relative
velocity 𝑣 in the 𝑥 -direction its coordinates are (𝑥′, 𝑡′). If we let the origins coincide at 𝑥 =
𝑥 ′ = 0, 𝑡 = 𝑡 ′ = 0 then the coordinates are related by
(Image adapted from Kane)
𝑥 ′ = 𝑥 − 𝑣𝑡
𝑡′ = 𝑡
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, lOMoARcPSD|6353920
SPA4402 MODERN PHYSICS
These equations are the Galilean transform (adding equations for 𝑦 and 𝑧 is trivial).
All of mechanics is invariant under the generalised Galilean transform. This means that the
same formulae, for example the collisions of billiard balls or for the Coriolis force, are
recovered if we substitute 𝑥 ′ for 𝑥, 𝑡 ′ for 𝑡, and then simplify. The velocity 𝑣 simply drops
out. Mechanics is the same in any inertial laboratory, whatever its velocity.
James Clerk Maxwell took the experimental findings of Ampère, Faraday, and Coulomb on
electricity and magnetism, and synthesised them into a beautiful system of four inter-linked
equations. His Treatise on Electricity and Magnetism was published in 1873. Light turned
out to be an electromagnetic wave, with a speed of:
1
𝑐= ≈ 3 × 108 m/s
√𝜀0 𝜇0
It soon became clear that Maxwell’s equations are not invariant under the Galilean
Transform. One interpretation was that there is a “stationary” rest frame, in which light has
the velocity 𝑐. This would be the “luminiferous ether”, a postulated medium for light to
‘wave’ in. However, this interpretation did not agree with experiment (stellar aberration,
the Michelson-Morley attempt to measure the speed of the Earth through the ether, etc).
Other people sought ways to retain the Principle of Relativity. Consider the more general
linear transform,
𝑥 ′ = 𝑎𝑥 + 𝑏𝑡
𝑡 ′ = 𝛼𝑥 + 𝛽𝑡
with the coefficients 𝑎, 𝑏, 𝛼 and 𝛽 being functions of 𝑣. It turns out that the simple
requirement that the back transform (𝑥 and 𝑡 expressed in terms of 𝑥 ′ and 𝑡 ′ ) restricts the
transform to the form
𝑥 ′ = 𝑎𝑥 + 𝑎𝑣𝑡
′
1 − 𝑎2
𝑡 = 𝑥 + 𝑎𝑡
𝑎𝑣
with the single parameter 𝑎 appearing in each of the four coefficients.
Note that the Galilean Transform is obtained by putting 𝑎 = 1.
Downloaded by youn sam ()
SPA4402 MODERN PHYSICS
INTRODUCTION TO SPECIAL RELATIVITY
Albert Einstein introduced his Special Theory of Relativity in 1905. To understand the
theory, we will first review the background, the theoretical and experimental developments
since Newton.
‘Special’ means ‘not general’. The theory is only about inertial frames of reference.
Remember that, for Galileo and Newton, such frames had a central importance in
mechanics and cosmology because the Principle of Relativity states that no laboratory-
based measurement could establish one rather than another as “at rest”, or “in motion”.
1st axiom: Principle of Relativity
The laws of physics are the same in every inertial frame of reference.
The Galilean Transform
Galileo and Newton were familiar with the relationships among inertial frames. If in the
frame S the coordinates of an event are (𝑥, 𝑡), then, in a frame 𝑆′ moving at relative
velocity 𝑣 in the 𝑥 -direction its coordinates are (𝑥′, 𝑡′). If we let the origins coincide at 𝑥 =
𝑥 ′ = 0, 𝑡 = 𝑡 ′ = 0 then the coordinates are related by
(Image adapted from Kane)
𝑥 ′ = 𝑥 − 𝑣𝑡
𝑡′ = 𝑡
Downloaded by youn sam ()
, lOMoARcPSD|6353920
SPA4402 MODERN PHYSICS
These equations are the Galilean transform (adding equations for 𝑦 and 𝑧 is trivial).
All of mechanics is invariant under the generalised Galilean transform. This means that the
same formulae, for example the collisions of billiard balls or for the Coriolis force, are
recovered if we substitute 𝑥 ′ for 𝑥, 𝑡 ′ for 𝑡, and then simplify. The velocity 𝑣 simply drops
out. Mechanics is the same in any inertial laboratory, whatever its velocity.
James Clerk Maxwell took the experimental findings of Ampère, Faraday, and Coulomb on
electricity and magnetism, and synthesised them into a beautiful system of four inter-linked
equations. His Treatise on Electricity and Magnetism was published in 1873. Light turned
out to be an electromagnetic wave, with a speed of:
1
𝑐= ≈ 3 × 108 m/s
√𝜀0 𝜇0
It soon became clear that Maxwell’s equations are not invariant under the Galilean
Transform. One interpretation was that there is a “stationary” rest frame, in which light has
the velocity 𝑐. This would be the “luminiferous ether”, a postulated medium for light to
‘wave’ in. However, this interpretation did not agree with experiment (stellar aberration,
the Michelson-Morley attempt to measure the speed of the Earth through the ether, etc).
Other people sought ways to retain the Principle of Relativity. Consider the more general
linear transform,
𝑥 ′ = 𝑎𝑥 + 𝑏𝑡
𝑡 ′ = 𝛼𝑥 + 𝛽𝑡
with the coefficients 𝑎, 𝑏, 𝛼 and 𝛽 being functions of 𝑣. It turns out that the simple
requirement that the back transform (𝑥 and 𝑡 expressed in terms of 𝑥 ′ and 𝑡 ′ ) restricts the
transform to the form
𝑥 ′ = 𝑎𝑥 + 𝑎𝑣𝑡
′
1 − 𝑎2
𝑡 = 𝑥 + 𝑎𝑡
𝑎𝑣
with the single parameter 𝑎 appearing in each of the four coefficients.
Note that the Galilean Transform is obtained by putting 𝑎 = 1.
Downloaded by youn sam ()
