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**Physics 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.

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**CHAPTER 4 Special 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 ( ) 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

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**Newtonian (Classical) Relativity**

Newton’s laws of motion must be implemented with respect to (relative to) some reference frame. x’ z’ y’ x z y 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.

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**Difference Between Inertial and Non-Inertial Reference Frame**

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**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. If the axes are also parallel, these frames are said to be Inertial Coordinate Systems

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**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.

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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 v Maxwell’s equations are not invariant under Galilean transformations. The Michelson-Morley experiment was an attempt to show the existence of aether.

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**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

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**Michelson-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 = periods 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!

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**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 ( ) It involved a fundamental new connection between space and time and that Newton’s laws are only an approximation.

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**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.

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**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.

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**The constancy of the speed of light**

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**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

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**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.

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**Lorentz Transformation Equations**

A more symmetrical form:

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**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)

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**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.

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**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?

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**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:

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**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? vpg = 1/2c vbp = 1/3c vog = 3/4c police bullet outlaws vpg = velocity of police relative to ground vbp = velocity of bullet relative to police vog = velocity of outlaws relative to ground

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**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:

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**Einstein’s addition of velocities**

Due to the high speeds involved, we really must relativistically add the police ship’s and bullet’s velocities:

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**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. Einstein pic: Lightning and space ship: Young Einstein

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**The complete Lorentz Transformation**

Length contraction Simultaneity problems Time dilation 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.

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**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.

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**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. Ruler: Clock:

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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) occurring at 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?

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**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. S’ Mary Melinda 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

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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:

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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!

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**Time Dilation Example: Reflection**

Mirror Mirror c∆t/2 L v∆t/2 S’ v 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).

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**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.

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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.

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**Length Contraction Lp Lp = xr - xℓ**

Frank Sr. 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’r endpoints Frank Sr.’s measurement in terms of Mary’s and Melinda’s: ← Proper length Pot belly from: where Mary’s and Melinda’s measured length is: Moving objects appear thinner!

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**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.

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**Lorentz Contraction v = 10% c v = 80% c**

A fast-moving plane at different speeds. v = 99% c v = 99.9% c

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**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 With 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! Image and some text taken from Warren Rogers Modern Physics lectures

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**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

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**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 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.

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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 and are often called Minkowski diagrams. Paths in Minkowski space-time are called world-lines.

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**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).

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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.

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**Christian Andreas Doppler (1803-1853)**

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. 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!

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**Waves from a source at rest**

Viewers at rest everywhere see the waves with their appropriate frequency and wavelength. Circular fringes:

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**Recall 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.

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**The Relativistic Doppler Effect**

v∆t 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.

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**Using the Doppler shift to sense rotation**

The Doppler shift has a zillion uses.

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**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?

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“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.

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**Lorentz-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”

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**The 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 Problem Frank 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 Paradox Who, in fact, is younger upon Mary’s return?

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**The 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. x t Image from

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Twin Paradox Exercise 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.)

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