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Physics 334Modern 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.
CHAPTER 4Special Theory of Relativity 1 • 4-1 Foundations of Special Relativity • The Experimental Basis of Relativity • Einstein’s Postulates • 4-2 Relationship between Space and Time • The Lorentz Transformation • Time Dilation and Length Contraction • The Doppler Effect • The Twin Paradox and Other Surprises 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 the theory itself… - Albert Michelson, 1907
y y’ x z x’ z’ Newtonian (Classical) Relativity Newton’s laws of motion must be implemented with respect to (relative to) some reference frame. 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.
Difference Between Inertial and Non-Inertial Reference Frame
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 relativityor Galilean invariance. If the axes are also parallel, these frames are said to be Inertial Coordinate Systems
The 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 Relations 1. Parallel axes 2. 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.
v A need for ether • 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 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. • Properties of Aether: • Low density • Elasticity • Transverse waves • Galilean transformation Maxwell’s equations are not invariant under Galilean transformations. The Michelson-Morley experiment was an attempt to show the existence of aether.
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 path length and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether. Perpendicular propagation Parallel and anti-parallel propagation Supposed velocity of earth through the aether
Michelson-Morley Experimental Analysis Exercise 4-1: Show that the time difference between path differences after 90° rotation is given by: • 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: 0.2 rad = 0.03 periods Recall that the phase shift is w times this relative delay: or: The 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!
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 Michelson and Morley’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 • 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.
Relativity 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.
Lorentz Transformation • Exercise 4-2: The equations for a spherical wavefronts in S is • x2+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 factoid • and that for small velocities
Lorentz 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.
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)
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.
Relativistic 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?
The 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:
Lorentz 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? vbp = 1/3c vog = 3/4c vpg = 1/2c outlaws police bullet vpg = velocity of police relative to ground vbp = velocity of bullet relative to police vog = velocity of outlaws relative to ground
Galileo’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:
Einstein’s addition of velocities Due to the high speeds involved, we really must relativistically add the police ship’s and bullet’s velocities:
Gedanken (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. Young Einstein
Length contraction Simultaneity problems Time dilation 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.
Time 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.
We 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.
Proper Time • To 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) occurringat the same position (i.e., at rest) in a system as measured by a clock at that position. Same location Proper 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?
Melinda Mary S’ Time 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. Mary 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? S Frank
Time Dilation • Mary 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:
Time Dilation • 1) ∆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!
Mirror Mirror c∆t/2 L v∆t/2 v Time Dilation Example: Reflection S’ S Mary Frank Fred Exercise 4-7: Show that the event in its rest frame (S’) occurs faster than in the frame that’s moving compared to it (S).
Time 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.
Proper Length When 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.
Lp Frank Sr. where Mary’s and Melinda’s measured length is: Length Contraction • Frank Sr., at rest in system S, measures the length of his somewhat bulging waist: • Lp = xr -xℓ • Now, Mary and Melinda S’, measure it, too, making simultaneous measurements (t’l = t’r) of the left, x’l, and the right x’rendpoints • Frank Sr.’s measurement in terms of Mary’s and Melinda’s: ← Proper length Moving objects appear thinner!
Length 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.
v = 80% c v = 99% c v = 99.9% c Lorentz Contraction v = 10% c A fast-moving plane at different speeds.
With relativistic correction Experimental 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 correction Top of the atmosphere Now 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!
Detecting 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 107 Experiments have confirmed this relativistic prediction
Space-time • When 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 1908 and are often called Minkowski diagrams. Paths in Minkowski space-time are called world-lines.
Particular 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).
The Light Cone The past, present, and future are easily identified in space-time diagrams. And if we add another spatial dimension, these regions become cones.
Space-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 Δs2between two events is invariant (the same) in any inertial frame. • Δsis 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 alight-likeseparation. 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 aspace-like separation. Δs2 < 0: Δx2 < c2 Δt2, and the two events can be causally connected. The interval is said to be time-like.
The Doppler Effect The 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. • 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! Christian Andreas Doppler (1803-1853)
Waves from a source at rest Viewers at rest everywhere see the waves with their appropriate frequency and wavelength.
Recall theDoppler 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.
v∆t c∆t The Relativistic Doppler Effect • So 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 from • Where f0 is the proper frequency • Exercise 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.
Using the Doppler shift to sense rotation The Doppler shift has a zillion uses.
Using 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?
“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.
