Special Relativity

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

The Galilean TransformationsConsider 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

x’=x-ut

y’=y

z’=z

t’=t

vx’=vx-uvy’=vy

vz’=vz

“The Ether” and the Michelson-Morley Experiment

• What about light? Waves must be waving something…

• The Ether…An absolute reference frame?

• Should be able to detect and measure Earth’s motion through the ether by detecting an “ether wind” that should modify the speed of light along and transverse to the “wind” direction.

“The Ether” and the Michelson-Morley Experiment

•Michelson-Morley detected no change in speed of light…

Galilean Transformations not correct for light !!!!

Einstein’s Postulates of Special Relativity

• 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.

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 Lorentz Transformations

The Lorentz Transformations• Constancy of speed of light can

be satisfied if space-time coordinates satisfy the linear Lorentz-Transformation equations…

Coefficients determined by invoking symmetry arguments and Einstein’s postulates ….

Lengths perpendicular to are unchanged

The Lorentz TransformationsTime coordinate

t’ should be the same if y-->-y or z-->-z.

Consider motion of the origin O’ of frame S’. Synchronized at t=t’=0. x-coordinate of O’ is given by x=ut in frame S, and x’=0 in frame S’.

Invoking the constancy of speed of light. Consider flash of light set off at t’=t=0 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

At this point:

The Lorentz Transformations• Reveal that

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

Lorentz Factor

When ,relativistic formulas must agree with Newtonian equations….

The Lorentz TransformationsArray Representation: Four-dimensional space-time

Space and time “mix” !!!!for light wave x=ct,x’=ct’,x”=ct”,….

Minkowski Diagram

Electromagnetic Wave EquationGalilean Transformation Lorentz Transformation

Space-Time Interval

Time-like interval

Space-like interval

Light-Like Interval

Causality

Space-time interval is invariant

(interval)2=(separation in time)2-(separation in space)2

Space-Time Diagrams

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

Proper Time And Time Dilation

Proper Length and Length Contraction

• Measure positions at endpoints at same time in frame S’ and in frame S, L’=x2’-x1’

Example of Time Dilation and Length Contraction: Cosmic Ray Muons