Special Relativity

1
Special Relativity

• The Failure of Galilean Transformations
• The Lorentz Transformation
• Time and Space in Special Relativity
• Relativistic Momentum and Energy

2
The Galilean Transformations

• Consider the primed coordinate system moving
  along the x-axis at speed u. Consider events
  where clocks record the time at the location of
  the event.
• Galilean Transformation between coordinate
  systems
• xx-ut
• yy
• zz
• tt

vxvx-u vyvy vzvz
3
The Ether and the Michelson-Morley Experiment

• What about light? Waves must be waving something
• The EtherAn absolute reference frame?
• Should be able to detect and measure Earths
  motion through the ether by detecting an ether
  wind that should modify the speed of light along
  and transverse to the wind direction.

4
The Ether and the Michelson-Morley Experiment

• Michelson-Morley detected no change in speed of
  light

Galilean Transformations not correct for light
!!!!
5
Einsteins Postulates of Special Relativity
A reference frame in which a mass point thrown
from the same point in three different (non
co-planar) directions follows rectilinear paths
each time it is thrown, is called an inertial
frame. L. Lange (1885) as quoted by Max von
Laue in his book (1921) Die Relativitätstheorie,
p. 34, and translated by Iro).

• The Principle of Relativity. The laws of physics
  are the same in all inertial reference frames.
• The Constancy of the Speed of Light. Light moves
  through vacuum at a constant speed c that is
  independent of the motion of the light source.

6
The Lorentz Transformations
7
The Lorentz Transformations

• Constancy of speed of light can be satisfied if
  space-time coordinates satisfy the linear
  Lorentz-Transformation equations

Lengths perpendicular to are unchanged
Coefficients determined by invoking symmetry
arguments and Einsteins postulates .
8
The Lorentz TransformationsTime coordinate
Invoking the constancy of speed of light.
Consider flash of light set off at tt0 at
common origins. At a later time t an observer in
frame S will measure a spherical wavefront of
light with radius ct, moving away from the origin
and satisfying Similarly, at a time t, an
observer in frame S will measure a spherical
wavefront of light with radius ct, moving away
from the origin O with speed c and satisfying
Inserting equations 4.10-4.13 into 4.15 and
comparing with 4.14

• t should be the same if y--gt-y or z--gt-z.
• Consider motion of the origin O of frame S.
  Synchronized at tt0. x-coordinate of O is
  given by xut in frame S, and x0 in frame S.

At this point
9
The Lorentz Transformations

• Reveal that

Lorentz Factor
Thus the Lorentz transformations linking the
space and time coordinates (x,y,z,t) and
(x,y,z,t) of the same event measured from S
and S are
When ,relativistic formulas must agree
with Newtonian equations.
10
The Lorentz Transformations
Four-dimensional space-time Space and time mix
!!!! for light wave xct,xct,xct,.

• Array Representation

Minkowski Diagram
11
Electromagnetic Wave Equation
Galilean Transformation
Lorentz Transformation
12
Space-Time Interval
(interval)2(separation in time)2-(separation in
space)2
Space-time interval is invariant

• Time-like interval
• Space-like interval
• Light-Like Interval
• Causality

13
Space-Time Diagrams
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15
Relativity of Simultaneity

• Observer in S measures two flash-bulbs going off
  at same time t but at different locations x1and
  x2. Then an observer in S would measure the time
  interval between the two flashes as
• Events that occur in simultaneously in one
  inertial frame do NOT occur simultaneously in all
  other inertial reference frames

16
Proper Time And Time Dilation
17
Proper Length and Length Contraction

• Measure positions at endpoints at same time in
  frame S and in frame S, Lx2-x1

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Example of Time Dilation and Length Contraction
Cosmic Ray Muons