CHAPTER 2 Special Theory of Relativity 1 2

CHAPTER 2 Special Theory of Relativity 1 2. 1 The Need for Aether 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 Prof. Rick Trebino, Georgia Tech, www. frog. gatech. edu Albert Michelson (1852 -1931) 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 theory itself… - Albert Michelson, 1907

Newtonian (Classical) Relativity Newton’s laws of motion must be implemented with respect to (relative to) some reference frame. y z x A reference frame is called an inertial frame if Newton’s laws are valid in that frame. Such a frame is established when a body, not subjected to net external forces, moves in rectilinear motion at constant velocity.

Newtonian Principle of Relativity If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. This is referred to as the Newtonian principle of relativity or Galilean invariance. y z x

The Galilean Transformation For a point P: In one frame K: P = (x, y, z, t) In another frame K’: P = (x’, y’, z’, t’) K y P = (x, y, z, t) P = (x’, y’, z’, t’) z x Newton derived rules to transform from K to K’ and back from K’ to K.

Conditions of the Galilean Transformation 1. Parallel axes 2. K’ has a constant relative velocity, vfr (here in the x-direction) with respect to K. 3. Time (t) for all observers is a Fundamental invariant, i. e. , it’s the same for all observers. More generally:

The Inverse Relations Step 1. Replace -vfr with +vfr. Step 2. Replace “primed” quantities with “unprimed” and “unprimed” with “primed. ” More generally:

2. 1: The Need for Aether The wave nature of light seemed to require a propagation medium. It was called the luminiferous ether or just ether (or aether). Aether had to have such a low density that the planets could move through it without loss of energy. It had to have an elasticity to support the high velocity of light waves. And somehow, it could not support longitudinal waves. And (it goes without saying…) light waves in the aether obeyed the Galilean transformation for moving frames.

Maxwell’s Equations & Absolute Reference Systems In Maxwell’s theory, the speed of light, in terms of the permeability and permittivity of free space, was given by: Thus the velocity of light is a constant. Aether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made. The Michelson-Morley experiment was an attempt to measure the earth’s velocity with respect to the aether and hence to prove the existence of aether.

2. 2: Michelson-Morley Experiment Michelson and Morley realized that the earth could not always be stationary with respect to the aether. And light would have a different travel time and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether. Parallel and anti-parallel propagation Mirror Perpendicular propagation Beamsplitter Mirror Supposed velocity of the aether

The Michelson Interferometer and Spatial Fringes x z Input beam Mirror Recall that the Michelson Interferometer can yield spatial fringes. If the input beam is a plane wave, the intensity cross term becomes: Beamsplitter Fringes Mirror Changing the light travel time of one beam with respect to the other by T shifts the phase of the fringes by w. T. This is the key to the MM experiment.

Michelson-Morley Experiment: Details If light requires a medium, then its velocity relative to an observer on earth depends on the velocity of the medium. Velocity vectors add. Parallel velocities Anti-parallel velocities

Michelson-Morley Experiment: Details 2 In the other arm of the interferometer, the total velocity must be perpendicular, so light must propagate at an angle. Perpendicular velocity to mirror Perpendicular velocity after mirror

Michelson-Morley Experiment: Details 3 Let c be the speed of light, and v be the velocity of the aether. Parallel and anti-parallel propagation Perpendicular propagation The delays for the two arms depend differently on the velocity of the aether! Note the square root! This yields a fringe phase shift:

Michelson-Morley Exp’t: More Details But the fringe shift must be compared to a reference fringe phase. So Michelson and Morley did the measurement again, the second time after rotating the apparatus by 90 , which yields the opposite fringe shift. The fringe shift between the two experiments will be double the singlemeasurement fringe shift:

Michelson-Morley Experiment Analysis… So the difference in fringe shift between the initial measurement and that of the 90 -rotated apparatus is: Assuming v << c:

Michelson-Morley Experimental Prediction But w/c = k = 2 p/l: v/c = 0. 0001, and it gets squared! The Earth’s orbital speed is: v = 3 × 104 m/s and the interferometer size is: L = 1. 2 m So the time difference Fortunately, L/l is large! is: 8 × 10− 17 s For visible light, this is a phase shift of: 0. 2 rad = 0. 03 periods Although the time difference was a very small number, the phase shift was well within the experimental range of measurement for visible light in the Michelson interferometer.

Michelson’s and Morley’s Set Up They folded the path to increase the total path of each arm to 11 m.

Michelson-Morley Experiment: Results The Michelson interferometer should’ve revealed a fringe shift as it was rotated with respect to the aether velocity. MM expected 0. 4 periods of shift and could resolve 0. 005 periods. They saw none! Interference fringes showed no change as the interferometer was rotated. Their apparatus Michelson and Morley's results from A. A. Michelson, Studies in Optics

In several repeats and refinements with assistance from Edward Morley, he always saw a null result. Michelson’s Conclusion The aether hypothesis seemed to be selfcontradictory. But the complete explanation would require additional insight from Einstein—over 20 years later. Albert Michelson (1852 -1931) Edward Morley (1838 -1923)

Possible Explanations for MM’s Null Result Many explanations were proposed, but the most popular was the aether drag hypothesis. This hypothesis suggested that the Earth somehow “dragged” the aether along as it rotates on its axis and revolves about the sun. This was contradicted by stellar abberation wherein telescopes had to be tilted to observe starlight due to the Earth’s motion. If aether were dragged along, this tilting would not occur.

Lorentz-Fitz. Gerald Contraction Another idea, proposed independently by Lorentz and Fitz. Gerald, suggested that the length, L, in the direction of the motion contracted by a factor of: Hendrik A. Lorentz (1853 -1928) George F. Fitz. Gerald (1851 -1901) So: This forced the path lengths to be equal and the phase shift to be zero. But there was no insight as to why such a contraction should occur. And if you tried to measure it, your ruler contracted, too!

2. 3: Einstein’s Postulates Albert Einstein was only two years old when Michelson and Morley reported their results. At age 16, Einstein began thinking about Maxwell’s equations in moving inertial systems. In 1905, at the age of 26, he published his startling proposal: the Principle of Relativity. It nicely resolved the Michelson and Morley experiment (although this wasn’t his intention and he maintained that in 1905 he wasn’t aware of MM’s work…) Albert Einstein (1879 -1955) It involved a fundamental new connection between space and time and that Newton’s laws are only an approximation.

Einstein’s Two Postulates Einstein simply said that light is a wave that does not require a medium. Waves could be fundamental quantities, just as particles are! And 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 (nonaccelerating) 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 so 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 wave-fronts observed in both systems must be spherical. K

The constancy of the speed of light is not compatible with Galilean transformations. Spherical wave-fronts in K: x 2 + y 2 + z 2 = c 2 t 2 Spherical wave-fronts in K’: x’ 2 + y’ 2 + z’ 2 = c 2 t’ 2 This cannot occur in Galilean transformations: There a couple of extra terms (-2 xvt + v 2 t 2) in the primed frame. Even looking only along the x-axis, it’s wrong:

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

Finding the Transformation for t’ Now substitute x’ = g ( x – v t ) into x = g ( x’ + v t’ ): x’ x = g [g (x – v t) + v t’ ] Isolating the t’ term: Dividing by g v: Rearranging: or: g v t’ = x - g 2 (x – v t) t’ = x / g v - g ( x / v – t ) t’ = g t - (g x / v) (1 - 1 / g 2) 1 - 1 / g 2 = 1 - (1 - v 2/c 2 ) = v 2/c 2

Lorentz Transformation Equations

Lorentz Transformation Equations A more symmetrical form:

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

g Factoids Some simple properties of g: which yields: When the velocity is small: When the velocity is large (v = c – e, where e is small): And when v c, g ∞

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