Presentation on theme: "Physics 334 Modern Physics"— Presentation transcript:
1Physics 334 Modern Physics Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were based on the textbook “Modern Physics” by Thornton and Rex. Many of the images have been used also from “Modern Physics” by Tipler and Llewellyn, others from a variety of sources (PowerPoint clip art, Wikipedia encyclopedia etc), and contributions are noted wherever possible in the PowerPoint file. The PDF handouts are intended for my Modern Physics class, as a study aid only.
2CHAPTER 4 Special Theory of Relativity 1 4-1 Foundations of Special RelativityThe Experimental Basis of RelativityEinstein’s Postulates4-2 Relationship between Space and TimeThe Lorentz TransformationTime Dilation and Length ContractionThe Doppler EffectThe Twin Paradox and Other SurprisesAlbert Michelson ( )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
3Newtonian (Classical) Relativity Newton’s laws of motion must be implemented with respect to (relative to) some reference frame.x’z’y’xzyA 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.
4Difference Between Inertial and Non-Inertial Reference Frame
5Newtonian 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.If the axes are also parallel, these frames are said to be Inertial Coordinate Systems
6The Galilean Transformation For a point P:In one frame S: P = (x, y, z, t)In another frame S’: P = (x’, y’, z’, t’)The Inverse Relations1. Parallel axes2. S’ has a constant relative velocity (here in the x-direction) with respect to S.3. Time (t) for all observers is a Fundamental invariant, i.e., it’s the same for all inertial observers.
7A need for etherIn 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 constantAether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made.Properties of Aether:Low densityElasticityTransverse wavesGalilean transformationvMaxwell’s equations are not invariant under Galilean transformations.The Michelson-Morley experiment was an attempt to show the existence of aether.
8Michelson-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 path length and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether.Perpendicular propagationParallel and anti-parallel propagationSupposed velocity of earth through the aether
9Michelson-Morley Experimental Analysis Exercise 4-1: Show that the time difference between path differences after 90° rotation is given by:Recall that the phase shift is w times this relative delay:or:The Earth’s orbital speed is: v = 3 × 104 m/s , and the interferometer size is: L = 1.2 m, So the time difference becomes: 8 × 10−17 s, which, for visible light, is a phase shift of: rad = periodsThe Michelson interferometer should’ve revealed a fringe shift as it was rotated with respect to the aether velocity. MM expected 0.4 of the width of a fringe, and could only see 0.01 equal to the uncertainty in the measurement.Interference fringes showed no change as the interferometer was rotated.Thus, aether seems not to exist!
10Einstein’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 Michelson and Morley’s work…)Albert Einstein ( )It involved a fundamental new connection between space and time and that Newton’s laws are only an approximation.
11Einstein’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: The laws of physics are the same in all inertial reference frames.The constancy of the speed of light: The speed of light in a vacuum is equal to the value c, independent of the motion of the source.
12Relativity of Simultaneity In Newtonian physics, we previously assumed that t’ = t.With synchronized clocks, events in S and S’ 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 S may not be in S’.Also, time may pass more slowly in some systems than in others.
14Lorentz Transformation Exercise 4-2: The equations for a spherical wavefronts in S isx2+y2+z2=c2t2 , Show that the equation for the spherical wavefronts in S’ cannot be x’2+y’2+z’2=c2t’2 in the Galilean transformation.Exercise 4-3: Show that x’ = g (x – vt) so that x = g’ (x’ + vt’) , yields the g factoidand that for small velocities
15Lorentz Transformation Exercise 4-4: Use x’ = g (x – vt) and x = g’ (x’ + vt’) , to find t’ = g (t – v x /c2)Exercise 4-5: Use x’ = g (x – vt) and t’ = g (t – v x /c2) to show that the equations for spherical wave fronts in S and S’ are the same.
16Lorentz Transformation Equations A more symmetrical form:
17Properties 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)
18The 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.
19Relativistic Velocity Transformation Exercise 4-6: Suppose a shuttle takes off quickly from a space ship already traveling very fast (both in the x direction). Imagine that the space ship’s speed is v, and the shuttle’s speed relative to the space ship is u’. What will the shuttle’s velocity (u) be in the rest frame?
20The Inverse Lorentz Velocity Transformations If we know the shuttle’s velocity in the rest frame, we can calculate it with respect to the space ship. This is the Lorentz velocity transformation for u’x, u’y , and u’z. This is done by switching primed and unprimed and changing v to –v:
21Lorentz velocity transformation Example: As the outlaws escape in their really fast getaway ship at 3/4c, the police follow in their pursuit car at a mere 1/2c, firing a bullet, whose speed relative to the gun is 1/3c. Question: does the bullet reach its target a) according to Galileo, b) according to Einstein?vpg = 1/2cvbp = 1/3cvog = 3/4cpolicebulletoutlawsvpg = velocity of police relative to groundvbp = velocity of bullet relative to policevog = velocity of outlaws relative to ground
22Galileo’s addition of velocities In order to find out whether justice is met, we need to compute the bullet's velocity relative to the ground and compare that with the outlaw's velocity relative to the ground.In the Galilean transformation, we simply add the bullet’s velocity to that of the police car:
23Einstein’s addition of velocities Due to the high speeds involved, we really must relativistically add the police ship’s and bullet’s velocities:
24Gedanken (Thought) experiments It was impossible to achieve the kinds of speeds necessary to test his ideas (especially while working in the patent office…), so Einstein used Gedanken experiments or Thought experiments.Einstein pic:Lightning and space ship:Young Einstein
25The complete Lorentz Transformation Length contractionSimultaneity problemsTime dilationIf 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.
26Time Dilation and Length Contraction More very interesting consequences of the Lorentz Transformation:Time Dilation:Clocks in S’ run slowly with respect to stationary clocks in S.Length Contraction:Lengths in S’ contract with respect to the same lengths in stationary S.
27We must think about how we measure space and time. In order to measure an object’s length in space, we must measure its leftmost and rightmost points at the same time if it’s not at rest.If it’s not at rest, we must ask someone else to stop by and be there to help out.In order to measure an event’s duration in time, the start and stop measurements can occur at different positions, as long as the clocks are synchronized.If the positions are different, we must ask someone else to stop by and be there to help out.Ruler:Clock:
28Proper TimeTo measure a duration, it’s best to use what’s called Proper Time.The Proper Time, t, is the time between two events (here two explosions) occurring at the same position (i.e., at rest) in a system as measured by a clock at that position.Same locationProper time measurements are in some sense the most fundamental measurements of a duration. But observers in moving systems, where the explosions’ positions differ, will also make such measurements.What will they measure?
29Time Dilation and Proper Time Frank’s clock is stationary in S where two explosions occur.Mary, in moving S’, is there for the first, but not the second. Fortunately, Melinda, also in S’, is there for the second.S’MaryMelindaMary and Melinda are doing the best measurement that can be done. Each is at the right place at the right time.If Mary and Melinda are careful to time and compare their measurements, what duration will they observe?SFrank
30Time DilationMary and Melinda measure the times for the two explosions in system S’ as t’1 and t’2 . By the Lorentz transformation:This is the time interval as measured in the frame S’. This is not proper time due to the motion of S’:Frank, on the other hand, records x2 – x1 = 0 in S with a (proper) time: t = t2 – t1, so we have:
31Time Dilation1) ∆t ’ > ∆t:(γ >1) the time measured between two events at different positions is greater than the time between the same events at one position: this is time dilation.2) The events do not occur at the same space and time coordinates in the two systems.3) System S requires 1 clock and S’ requires 2 clocks for the measurement.4) Because the Lorentz transformation is symmetrical, time dilation is reciprocal: observers in S see time travel faster than for those in S’. And vice versa!
32Time Dilation Example: Reflection MirrorMirrorc∆t/2Lv∆t/2S’vSMaryFrankFredExercise 4-7: Show that the event in its rest frame (S’) occurs faster than in the frame that’s moving compared to it (S).
33Time stops for a light wave Because:And, when v approaches c:For anything traveling at the speed of light:In other words, any finite interval at rest appears infinitely long at the speed of light.
34Proper LengthWhen both endpoints of an object (at rest in a given frame) are measured in that frame, the resulting length is called the Proper Length.We’ll find that the proper length is the largest length observed. Observers in motion will see a contracted object.
35Length Contraction Lp Lp = xr - xℓ Frank Sr.Length ContractionFrank Sr., at rest in system S, measures the length of his somewhat bulging waist:Lp = xr - xℓNow, Mary and Melinda S’, measure it, too, making simultaneous measurements (t’l = t’r ) of the left, x’l , and the right x’r endpointsFrank Sr.’s measurement in terms of Mary’s and Melinda’s:← Proper lengthPot belly from:where Mary’s and Melinda’s measured length is:Moving objects appear thinner!
36Length contraction is also reciprocal. So Mary and Melinda see Frank Sr. as thinner than he is in his own frame.But, since the Lorentz transformation is symmetrical, the effect is reciprocal: Frank Sr. sees Mary and Melinda as thinner by a factor of g also.Length contraction is also known as Lorentz contraction.Also, Lorentz contraction does not occur for the transverse directions, y and z.
37Lorentz Contraction v = 10% c v = 80% c A fast-moving plane at different speeds.v = 99% cv = 99.9% c
38Experimental Verification of Time Dilation Cosmic Ray Muons: Muons are produced in the upper atmosphere in collisions between ultra-high energy particles and air-molecule nuclei. But they decay (lifetime = 1.52 ms) on their way to the earth’s surface:No relativistic correctionWith relativistic correctionTop of the atmosphereNow time dilation says that muons will live longer in the earth’s frame, that is, t will increase if v is large.And their average velocity is 0.98c!Image and some text taken from Warren Rogers Modern Physics lectures
39Detecting muons to see time dilation At 9000 m it takes muons (9000/0.998c ~ 30 μs) about 15 lifetimes to reach earth. If No = 108 and t = 15τ, N ~ 31 muons should reach earth.From relativistic approach, the distance traveled is only 600m at that speed in 1 lifetime (2 μs) and therefore N = 3.68 x 107Experiments have confirmed this relativistic prediction
40Space-time Invariants This is a quantity that is invariant under Lorentz transformation. It is defined in the following way;(∆s)2 = (c2∆t2) - [∆x2 + ∆y2 + ∆z2]The quantity Δs2 between two events is invariant (the same) in any inertial frame.Δs is known as the space-time interval between two events.There are three possibilities for Δs2:Δs2 = 0: Δx2 = c2 Δt2, and the two events can be connected only by a light signal. The events are said to have a light-like separation.Δs2 > 0: Δx2 > c2 Δt2, and no signal can travel fast enough to connect the two events. The events are not causally connected and are said to have a space-like separation.Δs2 < 0: Δx2 < c2 Δt2, and the two events can be causally connected. The interval is said to be time-like.
41Space-timeWhen describing events in relativity, it’s convenient to represent events with a space-time diagram.In this diagram, one spatial coordinate x, specifies position, and instead of time t, ct is used as the other coordinate so that both coordinates will have dimensions of length.Space-time diagrams were first used by H. Minkowski in and are often called Minkowski diagrams. Paths in Minkowski space-time are called world-lines.
42Particular Worldlines Stationary observers live on vertical lines.A light wave has a 45º slope.Worldline is the record of the particle’s travel through spacetime, giving its speed (1/slope) and acceleration (=1/rate of change of slope).
43The Light ConeThe past, present, and future are easily identified in space-time diagrams. And if we add another spatial dimension, these regions become cones.
44Christian Andreas Doppler (1803-1853) The Doppler EffectThe Doppler effect for sound yields an increased sound frequency as a source such as a train (with whistle blowing) approaches a receiver and a decreased frequency as the source recedes.Christian Andreas Doppler ( )A similar change in sound frequency occurs when the source is fixed and the receiver is moving.But the formula depends on whether the source or receiver is moving.The Doppler effect in sound violates the principle of relativity because there is in fact a special frame for sound waves. Sound waves depend on media such as air, water, or a steel plate in order to propagate. Of course, light does not!
45Waves from a source at rest Viewers at rest everywhere see the waves with their appropriate frequency and wavelength.Circular fringes:
46Recall the Doppler Effect A receding source yields a red-shifted wave, and an approaching source yields a blue-shifted wave.A source passing by emits blue- then red-shifted waves.
47The Relativistic Doppler Effect v∆tSo what happens when we throw in Relativity?Exercise 4-8: Consider a source of light (for example, a star) in system S’ receding from a receiver (an astronomer) in system S with a relative velocity v. Show that the frequency can be obtained fromWhere f0 is the proper frequencyExercise 4-9: What would be the frequency if the source was approaching?Exercise 4-10: Use the results from exercise 8 and 9 to deduce the expressions for non-relativistic velocities.
48Using the Doppler shift to sense rotation The Doppler shift has a zillion uses.
49Using the Doppler shift to sense rotation Example: The Sun rotates at the equator once in about 25.4 days. The Sun’s radius is 7.0x108m. Compute the Doppler effect that you would expect to observe at the left and right limbs (edges) of the Sun near the equator for the light of wavelength l = 550 nm = 550x10-9m (yellow light). Is this a redshift or a blueshift?
50“Aether Drag”Exercise 4-12: In 1851, Fizeau measured the degree to which light slowed down when propagating in flowing liquids.Fizeau found experimentally:This so-called “aether drag” was considered evidence for the aether concept.Derive this equation from velocity addition equations.
51Lorentz-FitzGerald Contraction Exercise 4-13: Lorentz and FitzGerald, proposed that the null test of Michelson Morley’s experiment can be explained by using the concept of length contraction to explain equal path lengths and zero phase shift. Show that this proposition can work.Image of FitzGerald”
52The Twin Paradox The Set-up Mary and Frank are twins. Mary, an astronaut, leaves on a trip many lightyears (ly) from the Earth at great speed and returns; Frank decides to remain safely on Earth.The ProblemFrank knows that Mary’s clocks measuring her age must run slow, so she will return younger than he. However, Mary (who also knows about time dilation) claims that Frank is also moving relative to her, and so his clocks must run slow.The ParadoxWho, in fact, is younger upon Mary’s return?
53The Twin-Paradox Resolution Frank’s clock is in an inertial system during the entire trip. But Mary’s clock is not. As long as Mary is traveling at constant speed away from Frank, both of them can argue that the other twin is aging less rapidly.But when Mary slows down to turn around, she leaves her original inertial system and eventually returns in a completely different inertial system.Mary’s claim is no longer valid, because she doesn’t remain in the same inertial system. Frank does, however, and Mary ages less than Frank.xtImage from
54Twin ParadoxExercise 4-14: A clock is placed in a satellite that orbits Earth with a period of 108 min. (a) By what time interval will this clock differ from an identical clock on Earth after 1 year? (b) How much time will have passed on Earth when the two clocks differ by 1.0 s? (Assume special relativity applies and neglect general relativity.)