Published on November 6, 2007
Slide1: 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 CHAPTER 2 Special Theory of Relativity 2 Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein Albert Einstein (1879-1955) Gedanken (Thought) experiments: 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 The complete Lorentz Transformation: 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. Simultaneity: Simultaneity Timing events occurring in different places can be tricky. Depending on how they’re measured, different events will be perceived in different orders by different observers. Due to the finite speed of light, the order in which these two events will be seen will depend on the observer’s position. The time intervals will be: Fred: -2L/c; Frank: 0; Fil: +2L/c But this obvious position-related simultaneity problem disappears if Fred and Fil have synchronized watches. 0 L -L Synchronized clocks in a frame: Synchronized clocks in a frame It’s possible to synchronize clocks throughout space in each frame. This will prevent the position-dependent simultaneity problem in the previous slide. But there will still be simultaneity problems due to velocity. Simultaneity: Simultaneity So all stationary observers in the explosions’ frame measure these events as simultaneous. What about moving ones? 0 L -L Compute the interval as seen by Mary using the Lorentz time transformation. Mary experiences the explosion in front of her before the one behind her. And note that Dt’ is independent of Mary’s position! 2.5: Time Dilation and Length Contraction: 2.5: Time Dilation and Length Contraction Time Dilation: Clocks in K’ run slowly with respect to stationary clocks in K. Length Contraction: Lengths in K’ contract with respect to the same lengths in stationary K. More very interesting consequences of the Lorentz Transformation: We must think about how we measure space and time.: 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: Proper Time To measure a duration, it’s best to use what’s called Proper Time. The Proper Time, T0, 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? Time Dilation and Proper Time: Time Dilation and Proper Time If Mary and Melinda are careful to time and compare their measurements, what duration will they observe? Frank’s clock is stationary in K where two explosions occur. Mary, in moving K’, is there for the first, but not the second. Fortunately, Melinda, also in K’, is there for the second. K Frank Mary and Melinda are doing the best measurement that can be done. Each is at the right place at the right time. Time Dilation: Time Dilation Mary and Melinda measure the times for the two explosions in system K’ as and . By the Lorentz transformation: This is the time interval as measured in the frame K’. This is not proper time due to the motion of K’: . Frank, on the other hand, records x2 – x1 = 0 in K with a (proper) time: T0 = t2 – t1, so we have: Time Dilation: 1) T ’ > T0: 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 K requires 1 clock and K’ requires 2 clocks for the measurement. 4) Because the Lorentz transformation is symmetrical, time dilation is reciprocal: observers in K see time travel faster than for those in K’. And vice versa! Time Dilation Time Dilation Example: Reflection: Frank Mary Time Dilation Example: Reflection Fred K’ K Let T be the round-trip time in K Reflection (continued): Reflection (continued) The time in the rest frame, K, is: or or or or So the event in its rest frame (K’) occurs faster than in the frame that’s moving compared to it (K). Time stops for a light wave: 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: 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. Length Contraction: Length Contraction ← Proper length Moving objects appear thinner! Frank Sr., at rest in system K, measures the length of his somewhat bulging waist: L0 = xr - xℓ Now, Mary and Melinda measure it, too, making simultaneous measurements ( ) of the left, , and the right endpoints, Frank Sr.’s measurement in terms of Mary’s and Melinda’s: Length contraction is also reciprocal.: 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. Lorentz Contraction: Lorentz Contraction A fast-moving plane at different speeds. v = 10% c 2.6: Addition of Velocities: 2.6: Addition of Velocities Taking differentials of the Lorentz transformation [here between the rest frame (K) and the space ship frame (K’)], we can compute the shuttle velocity in the rest frame (ux = dx/dt): 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 Lorentz Velocity Transformations: The Lorentz Velocity Transformations Defining velocities as: ux = dx/dt, uy = dy/dt, u’x = dx’/dt’, etc., we find: with similar relations for uy and uz: Note the g’s in uy and uz. The Inverse Lorentz Velocity Transformations: 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: Relativistic velocity addition plot: Relativistic velocity addition plot Example: Lorentz velocity transformation: vpg = velocity of police relative to ground vbp = velocity of bullet relative to police vog = velocity of outlaws relative to ground vpg = 1/2c vog = 3/4c vbp = 1/3c police outlaws bullet Example: Lorentz velocity transformation 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? 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: Galileo’s addition of velocities Einstein’s addition of velocities: Einstein’s addition of velocities Due to the high speeds involved, we really must relativistically add the police ship’s and bullet’s velocities: Example: Addition of velocities: Example: Addition of velocities We can use the addition formulas even when one of the velocities involved is that of light. At CERN, neutral pions (p0), traveling at 99.975% c, decay, emitting g rays in opposite directions. Since g rays are light, they travel at the speed of light in the pion rest frame. What will the velocities of the g rays be in our rest frame? (Simply adding speeds yields 0 and 2c!) Parallel velocities: Anti-parallel velocities: “Aether Drag”: “Aether Drag” 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. “Aether Drag”: “Aether Drag” which was what Fizeau found. Armand Fizeau (1819 - 1896) Let K’ be the frame of the water, flowing with velocity, v. We’ll treat the speed of light in the medium ( u, u’ ) as a normal velocity in the velocity-addition equations. In the frame of the flowing water, u’ = c / n 2.7: Experimental Verification of Time Dilation: 2.7: 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: Detecting muons to see time dilation It takes 6.8 ms for the 2000-m path at 0.98c, about 4.5 times the muon lifetime. So, without time dilation, we expect only 1000 x 2-4.5 = 45 muons at sea level. In fact, we see 542, in agreement with relativity! And how does it look to the muon? Lorentz contraction shortens the distance! Since 0.98c yields g = 5, instead of moving 600 m on average, they travel 3000 m in the Earth’s frame. 2.8: The Twin Paradox: 2.8: 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? The Twin-Paradox Resolution: 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. Atomic Clock Measurement: Two airplanes traveled east and west, respectively, around the Earth as it rotated. Atomic clocks on the airplanes were compared with similar clocks kept at the US Naval Observatory to show that the moving clocks in the airplanes ran slower. Atomic Clock Measurement Travel Predicted Observed Eastward -40 ± 23 ns -59 ± 10 ns Traveling twin Westward 275 ± 21 ns 273 ± 7 ns Stay-at-home twin vrotation > vplane There have been many rigorous tests of the Lorentz transformation and Special Relativity.: There have been many rigorous tests of the Lorentz transformation and Special Relativity. Particle Accuracy Electrons 10-32 Neutrons 10-31 Protons 10-27 Quantum Electrodynamics also depends on Lorentz symmetry, and it has been tested to 1 part in 1012. 2.9: Space-time: 2.9: 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: Particular Worldlines x Stationary observers live on vertical lines. A light wave has a 45º slope. Slope of worldline is c/v. Worldlines and Time: Worldlines and Time Observers at x1 and x2. see what’s happening at x = x3 at t = 0 simultaneously. Alternatively, an event occurring at x3 can be used to synchronize clocks at x1 and x2. Moving Clocks: Moving Clocks Observers in a frame moving at velocity, v, will see the event happening at x = x3 at t = 0 at different times. The Light Cone: 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 Interval and Metric: Space-time Interval and Metric Recall that, since all observers see the same speed of light, all observers, regardless of their velocities, must see spherical wave fronts. s2 = x2 + y2 + z2 – c2t2 = (x’)2 + (y’)2 + (z’)2 – c2 (t’)2 = (s’)2 This interval can be written in terms of the space-time metric: Space-time Invariants: Space-time Invariants 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.