PHYS 342: More info The TA is Meng-Lin Wu: His is His office hour is 10:30am to 12pm on Mondays His office is Physics 105 Due this Friday in class: Problem set 1: From chapter 2; problems 1 through 9 and 11

Quick quiz! Does a frame of reference attached to spaceship moving in deep space with a constant linear velocity V relative the distant stars represent a Inertial Frame? How about a frame of reference attached to a child enjoying a ride on a merry-go-round that moves with constant angular velocity? Since the laws of physics are the same in every inertial frame, if you perform identical experiments in two different inertial frames, you should get exactly the same results, T or F? Explain.

2.1 The Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein’s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction 2.6 Addition of Velocities 2.7 Experimental Verification 2.8 Twin Paradox 2.9 Space-time 2.10 Doppler Effect 2.11 Relativistic Momentum 2.12 Relativistic Energy 2.13 Computations in Modern Physics 2.14 Electromagnetism and Relativity Special Theory of Relativity It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative and would, therefore, show that there is still a difficulty in the theory itself… - Albert Michelson, 1907 Albert Michelson ( )

The Galilean Transformations Time (t) for all observers is a Fundamental invariant, i.e., it’s the same for all inertial observers.

Einstein’s Two Postulates With the belief that Maxwell’s equations must be valid in all inertial frames, Einstein proposed the following postulates: The principle of relativity: All the laws of physics (not just the laws of motion) are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists. The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

Re-evaluation of Time! In Newtonian physics, we previously assumed that t ’ = t. With synchronized clocks, events in K and K ’ can be considered simultaneous. Einstein realized that each system must have its own observers with their own synchronized clocks and meter sticks. Events considered simultaneous in K may not be in K ’. Also, time may pass more slowly in some systems than in others.

The constancy of the speed of light Consider the fixed system K and the moving system K ’. At t = 0, the origins and axes of both systems are coincident with system K ’ moving to the right along the x axis. A flashbulb goes off at both origins when t = 0. According to postulate 2, the speed of light will be c in both systems and the wavefronts observed in both systems must be spherical. K K’

The constancy of the speed of light is not compatible with Galilean transformations. Spherical wavefronts in K: Spherical wavefronts in K’: Note that this cannot occur in Galilean transformations: There are a couple of extra terms (-2xvt + v 2 t 2 ) in the primed frame.

Finding the correct transformation which yields: What transformation will preserve spherical wave-fronts in both frames? Try x ’ =  (x – vt) so that x = ’ (x ’ + vt ’ ), where  could be anything. By Einstein’s first postulate: ’ =  The wave-front along the x ’ - and x -axes must satisfy: x ’ = ct ’ and x = ct Thus: ct ’ =  (ct – vt) or t ’ =  t (1– v/c) and: ct = ’ (ct ’ + vt ’ ) or t = ’ t ’ (1 + v/c) Substituting for t in t ’ =  t (1– v/c) :

Finding the transformation for t “prime” Now substitute x ’ =  ( x – v t ) into x =  ( x ’ + v t ’ ) : x =  [  (x – v t) + v t ’ ] Solving for t ’ we obtain: x    (x – v t) =  v t ’ or: t ’ = x /  v   ( x / v – t ) or: t ’ =  t + x /  v   x / v or: t ’ =  t + (  x / v) ( 1 /    1 ) or: 1 /    1 =  v 2 /c 2

Lorentz Transformation Equations

A more symmetrical form:

Properties of  Recall that  = v / c < 1 for all observers.  equals 1 only when v = 0. In general: Graph of  vs.  : (note v < c )

The Binomial Approximation (1+x) a ≈1+ax iff x<<1 We want to approximate:  =(1-v 2 /c 2 ) -1/2 So we have:  ≈1+1/2v 2 /c 2 This is just one example of the Taylor expansion of a function f(x)

 Factoids which yields: Some simple properties of  : When the velocity is small :

The complete Lorentz Transformation If v << c, i.e., β ≈ 0 and  ≈ 1, yielding the familiar Galilean transformation. Space and time are now linked, and the frame velocity cannot exceed c.