Special Relativity (Math)  Reference from Tipler chapter 39-1 to 39-3  Newtonian relativity  Einstein’s postulates  Lorentz transformation  Time dilation  Length contraction  The relativistic Doppler effect 1

Light is Wave! 2 The puzzle was deepened and broadened by the end of the 19th century through the assimilation of optics into Maxwell's theory of electric and magnetic fields. So. What is the transportation media for Light?

Principle of Relativity 3

Inertial reference frame  Coordinate systems that move at constant velocity relative to each other are called inertial reference frames.  The special theory of relativity, developed by Einstein describes measurements made in different inertial reference frames.  The general theory of relativity describes accelerated reference frames and gravity. 4

Principle of Newtonian relativity  If physics experiments (projectile motion experiment or measuring the period of a pendulum or whatever) were performed in an inertial (non-accelerated) reference frame, the result of the experiment would be the same as ones performed in a stationary reference frame.  Imagine you were on a train traveling with a perfectly constant velocity. You would not be able to tell that you were moving if there were no windows to look out.  Every nonaccelerated observer observes the same laws of nature. 5 Absolute motion cannot be detected.

Ether (before late 19 th century )  Scientists could not imagine that waves can travel without medium.  Scientists postulated the existence of the ether  which is an unknown form of matter.  which permeates everywhere and everything.  which is the medium for electromagnetic waves. 6

No ether 7  Experiments looked for this speed difference, but found nothing.  In 1887, Michelson and Morley experimentally showed that ether does not exist.

Question 1  The Michelson-Morley experiment showed that. a. the speed of light is not relative. b. the speed of light is relative. c. light changes frequency with repeated reflections. d. light can be split into parts.

Einstein’s postulates 9  Einstein published “On the electrodynamics of moving bodies.”  In it, he postulated:  If Maxwell’s equations describes the laws of nature, the speed of light, the direct consequence of them, should be the same for all non-accelerating observers. Postulate 1: Absolute uniform motion cannot be detected. Postulate 2: The speed of light is independent of the motion of the source. Postulate 2 (alternate): Every observer measures the same value for c for the speed of light.

Question 2 Einstein stated that the laws of physics are a) different in different situations. b) common sense applied to microscopic and macroscopic things. c) he same in all frames of reference. d) the same in all uniformly moving frames of reference

Galilean relativity 11  Suppose a girl on the train throws a ball toward the front of a train with speed u. The train is moving at a constant velocity v relative to the platform. A boy is observing it from a platform. u v Speed of Ball moves speed v+u

Observing speed of light  Now suppose the girl zips by in her spaceship at 0.25 c. The light from her flashlight travels at speed c away from her c c Galilean relativity: speed of light = 1.25c Einstein relativity: speed of light = c

Let’s define two reference frames 13  A reference frame, S, with coordinate system (x,y,z), time, t, and origin O.  Another reference frame, S’, with coordinate system (x’,y’,z’), time, t’, and origin O’ and moves with a constant velocity v relative to the S frame.  Coincides O=O’ while t=t’

Galilean transformation 14  The classical transformation, Galilean transformation is a “common sense” transformation that you learned in Phys 121. When v is much smaller than c, Galilean transformation is valid.  Galilean transformation is given by  The inverse transformation is  If a particle has velocity u x = dx/dt in frame S, its velocity in S’ is u’ x = dx’/dt’ = u x – v.  The accelerations of the particle in both frames are the same, a x = a’ x.

Failure of Galilean transformation 15  According to Galilean transformation, if light moves along the x axis with speed u’ x = c in S’, the speed in S is u x = c + v.  This is inconsistent with u x = c according to Einstein’s alternate postulate 2.  Assume that relativistic transformation equation for x is a slight modification.  Let’s find what  should be.

Lorentz factor  16  Consider a light pulse that starts at coincident O and O’ at t = t’ = 0.  The x component of the wave front of the light pulse is x = ct and x’ = ct’. x = ct x’ = ct’

Lorentz transformation 17  When v is close to c, Lorentz transformation must be used since c should be the same in both frames. It is given by  Lorentz factor is close to 1 when v << c, and Lorentz transformation reduces to Galilean transformation.  The inverse transformation is