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Special Relativity

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

Special Relativity

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

x’=x-ut

y’=y

z’=z

t’=t

vx’=vx-u

vy’=vy

vz’=vz

- 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.

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

Galilean Transformations not correct for light !!!!

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.

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

Lengths perpendicular toare unchanged

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

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

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’.

At this point:

Lorentz Factor

- 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

When ,relativistic formulas must agree with Newtonian equations….

Four-dimensional space-time

Space and time “mix” !!!!

for light wave x=ct,x’=ct’,x”=ct”,….

Array Representation:

Minkowski Diagram

Electromagnetic Wave Equation

Galilean Transformation

Lorentz Transformation

(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

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

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