Chapter 28 – Special Relativity

1. A physical activity that takes place at a definite point in space and time is called: a sport. an event. a happening. a locale. 2. Which of the following was not discussed in the first 4 sections of Ch 28? Wave-particle duality Proper time interval Time dilation Length contraction Answer: C

What’s special about special relativity? In 1905, Einstein’s first paper on relativity dealt only with inertial reference frames (constant velocity). 10 years later, he published a more encompassing theory of relativity that considered accelerated motion and it’s connection to gravity. This was a discussion of “general” relativity. His earlier work was special in that it was more specific

28.1 Events and Inertial Reference Frames An event is a physical “happening” that occurs at a certain place and time. To record the event, each observer uses a reference frame that consists of a coordinate system (space) and a clock (time). An inertial reference frame is one in which Newton’s law of inertia is valid. An inertial reference frame can be moving, but it cannot be accelerating. In spite of its centripetal acceleration, the Earth is considered an inertial reference frame, because the effects of its rotation and orbit are relatively minor.

Reference Frames Extend infinitely far in all directions. You can be thousands of miles away, yet still in the same reference frame Observers are at rest in their reference frames A reference frame is not the same as a point of view. Therefore all observers at rest relative to each other share the same reference frame and will view time and length the same way.

3.4 Relative Velocity without relativity One example of an inertial reference frame is on the ground. This observer is at rest in his reference frame. The train traveling at a constant speed is another frame of reference. Anybody sitting down in the train is an observer at rest in his reference frame.

3.4 Relative Velocity Let’s say the train is traveling at 9m/s relative to the ground: vTG = +9 m/s, where + indicates to the right. An observer at rest in the trains sees the man on the ground traveling at vGT = -9m/s, where – indicates to the left. Is one of these reference frames more “real” than the other?

3.4 Relative Velocity (See Chapter 3) A passenger inside the train starts to walk up to the bar car in the front of the moving train. Observers in their seats note that he is traveling at vPT = 2 m/s. What is his velocity (v PG ) according to the ground-based observer? A. + 2m/s, B. +7 m/s C. +9 m/s D. +11m/s

3.4 Relative Velocity Same situation, but now imagine the passenger moving to the rear of the train (that’s left). What is his speed vPG now? A. + 2m/s, B. +7 m/s C. +9 m/s D. -7 m/s

3.4 Relative Velocity Same situation, but now imagine the passenger moving to the rear of the train (that’s left). What is his speed vPG now? Answers: when going towards the front, the ground-based observer sees him going at 11 m/s to the right. When going towards the rear of the train, it’s 7 m/s, still to the right.

Ocean waves are approaching the beach at 10 m/s (vWG ) Ocean waves are approaching the beach at 10 m/s (vWG ). A boat heading out to sea travels at 6 m/s (vBG ). How fast (speed only, not direction) are the waves moving in the boat’s reference frame (vWB )? 4 m/s 6 m/s 16 m/s 10 m/s Answer: C

Using the same logic as with the train: VWG = VWB + VBG and therefore: Ocean waves are approaching the beach at 10 m/s (vWG ). A boat heading out to sea travels at 6 m/s (vBG ). How fast (speed only, not direction) are the waves moving in the boat’s reference frame (vWB )? 4 m/s 6 m/s 16 m/s 10 m/s Using the same logic as with the train: VWG = VWB + VBG and therefore: VWB = VWG – VBG where these must be added/subtracted as vectors. STT36.2

The event and the observation of the event do not always occur at the same time

Mr. A, at rest in the reference frame, stands at the origin, looking in a positive direction. Mr. B, at rest in the same reference frame, stands at x = 900 m, looking in a negative direction. A firecracker explodes somewhere between them. Mr. B sees the light flash at t = 3µs. Mr. A sees the flash at t = 4µs. Light travels at 300 m/µs. In this reference frame: a. Where did the explosion happen? b. When did the explosion happen?

Where did the explosion happen? - Light travels at 300 m/µs. It took 1µs longer for the light to get to Mr. A, implying Mr. A’s position is 300 m farther away from the explosion that that of Mr. B. If x is the position of the explosion: (x – 0m) = (900m – x) + 300m, so x = 600m When did the explosion happen? - It took 2 µs for the light to go from the explosion to Mr. A at 600 m from the explosion. So t = (4-2) µs = 2 µs.

Mr. A and Mr. B observed the event at different times, but after some calculations, they can agree upon the time and place it occurred.

Simultaneity An experimenter in an inertial reference frame stands at the origin looking in the positive x direction. At t = 3.0 µs she sees firecracker 1 explode at x = 600 m. At t = 5.0 µs, she sees firecracker 2 explode at x = 1200m. Light travels at 300 m/ µs. Are the two explosions simultaneous? If not, which one happens first?

Simultaneity An experimenter in an inertial reference frame stands at the origin looking in the positive x direction. At t = 3.0 µs she sees firecracker 1 explode at x = 600 m. At t = 5.0 µs, she sees firecracker 2 explode at x = 1200m. Light travels at 300 m/ µs. Are the two explosions simultaneous? If not, which one happens first? Ans: We see that the first explosion occurred at t = I µs, and then took 2 µs to travel to the experimenter. The light from the second firecracker took 1200 m/300 µs/s or 4 µs to travel. It too occurred at t = 1 µs. The events therefore are simultaneous.

28.2 The Postulates of Special Relativity The Relativity Postulate. The laws of physics are the same in every inertial reference frame. All inertial reference frames are equally valid. The Speed of Light Postulate. The speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c, no matter how fast the source of light and the observer are moving relative to one another. Number 2 seems to contradict our idea of inertial reference frames. Why should moving light be different from a moving train or a boat?

Einstein’s Principle of Relativity Maxwell’s equations are true in all inertial reference frames. Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s. Therefore, light travels at speed c in all inertial reference frames. Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other.

28.3 The Relativity of Time: Time Dilation If you are in the same reference frame as the light clock, Δt0 = 2D/c where D is the distance between mirror and receiver, and c is the speed of light.

28.3 The Relativity of Time: Time Dilation An observer on the earth sees the light pulse travel a greater distance between ticks (2s as opposed to 2D). But she still measures the speed of light in the spaceship as c (postulate #2). Therefore, her clock reads a Δt > Δt0

28.3 The Relativity of Time: Time Dilation This is the time interval the earth-based observer would read. Since v< c, the denominator is less than 1, so Δt (Earth-based observer time interval) is be greater than Δt0 (astronaut observer time interval). It can be shown that:

28.3 The Relativity of Time: Time Dilation PROPER TIME INTERVAL The time interval between two events that occur at the same position is called the proper time interval In general, the proper time interval between events is the time interval measured by an observer who is at rest relative to the events. In the light clock example, the proper time interval was measured by the astronaut, because from his reference frame, both events (light leaving source, light hitting detector) happened at the same position. leaving but not by the earth-bound observer. Proper time interval = Dilated time interval =

From the Sun to Saturn Who measures the proper time interval?

From the Sun to Saturn

From the Sun to Saturn This implies it is the astronaut who measures the proper time interval, Δt0 .

From the Sun to Saturn

From the Sun to Saturn

From the Sun to Saturn to find the proper time interval,Δt0 Δt0 = 2310 s

From the Sun to Saturn to find the proper time interval, Δt0 Δt0 = 2310s

From the Sun to Saturn

A tree and a pole are 3000 m apart A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is at rest under the tree. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” In Nancy’s frame of reference, does event 1 occur before, after or at the same time as event 2? Answer: C before event 2 after event 2 at the same time as event 2

A tree and a pole are 3000 m apart A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is at rest under the tree. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2? before event 2 after event 2 at the same time as event 2 Nancy sees event 1 before event 2, but she would agree they were simultaneous since she and Mark are in the same reference frame. STT36.4

28.4 The Relativity of Length: Length Contraction The shortening of the distance between two points is one example of a phenomenon known as length contraction. Length contraction: L0 is the proper length, the length between 2 points measured by an observer at rest with respect to them. Note that the observer who experiences the proper time interval, is not the one who measures the proper length

28.4 The Relativity of Length: Length Contraction Example 4 The Contraction of a Spacecraft An astronaut, using a meter stick that is at rest relative to a cylindrical spacecraft, measures the length and diameter to be 82 m and 21 m respectively. The spacecraft moves with a constant speed of 0.95c relative to the earth. What are the dimensions of the spacecraft, as measured by an observer on earth.

28.4 The Relativity of Length: Length Contraction For this problem, the earthbound observer would determine the distance to Alpha Centaur to be the proper length, L0, as shown in the picture, and the observer in the spaceship would see the contracted length. However, it is the astronaut that sees the proper length of the spaceship she is traveling in, while experimenters will measure it at the contracted length.

The distance from the sun to Saturn

The distance from the sun to Saturn

The distance from the sun to Saturn L = 0.62 x 1012 m

Beth and Charles are at rest relative to each other Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC. LA = LB = LC LB = LC > LA LA > LB = LC LA > LB > LC LB > LC > LA Answer: C

Beth and Charles are at rest relative to each other Beth and Charles are at rest relative to each other. Anjay running at close to the speed of light, runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC. LA = LB = LC LB = LC > LA LA > LB = LC LA > LB > LC LB > LC > LA STT36.7

Simultaneity An experimenter in an inertial reference frame stands at the origin looking in the positive x direction. At t = 3.0 µs she sees firecracker 1 explode at x = 600 m. At t = 5.0 µs, she sees firecracker 2 explode at x = 1200m. Light travels at 300 m/ µs. Are the two explosions simultaneous? If not, which one happens first?

Simultaneity An experimenter in an inertial reference frame stands at the origin looking in the positive x direction. At t = 3.0 µs she sees firecracker 1 explode at x = 600 m. At t = 5.0 µs, she sees firecracker 2 explode at x = 1200m. Light travels at 300 m/ µs. Are the two explosions simultaneous? If not, which one happens first? Ans: We see that the first explosion occurred at t = I µs, and then took 600 m/300 µs/m or 2 µs to travel to the experimenter. The light from the second firecracker took 1200 m/300 µs/m or 4 µs to travel. It too occurred at t = 1 µs. The events therefore are simultaneous. The experimenter would agree to that, even though she sees them at different times.

Ryan (stationary) and Peggy (moving with the fireworks) do relativity Ryan and Peggy are both half-way between the firecrackers, which are attached to the train. The signal box on the train has 2 light detectors. If the light flash is seen by the right detector first, the signal light will turn green. If the light flash is seen simultaneously, it will turn red.

Ryan’s frame of reference Ryan observed the firecrackers go off simultaneously He was halfway between the firecrackers The light traveled at equal speeds from the different directions. He concludes that firecrackers exploded simultaneously.

Note that although light waves reach Ryan simultaneously, he would think that since Peggy and the signal box move right, the light wave from the right explosion is detected before the light wave from the left. This would mean that the signal box light turns green.

Peggy’s frame of reference Are the explosions simultaneous in Peggy’s frame of reference? If so, since explosions occurred equidistant from her, the light would travel at equal speeds from the different directions, so both flashes would reach her at the same time. Signal box light would turn red. But we already saw the light go green!

Sequence of events in Peggy’s reference frame The signal light is seen as green in both reference frames and so must show the same color. For the light to be green in Peggy’s reference frame, the right firecracker has to go off before the left one. This is the only way that light coming from equally distant points would arrive at different times. Signal box light would turn green, which agrees with Ryan.

at the same time as event 2 A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2? STT36.5 before event 2 after event 2 at the same time as event 2

at the same time as event 2 A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2? If Nancy had been midway between the tree and pole, this would have been equivalent to Peggy and Ryan. But events occur in the same order for all observers in the reference frame. Nancy’s time frame for the two events is shorter before event 2 after event 2 at the same time as event 2 STT36.5

Conservation of momentum The Newtonian momentum of an object is defined as the product of its mass and velocity (mv). Conservation of momentum of a system of objects before and after they interact, is a law of physics that is valid in all inertial reference frames.

28.5 Relativistic Momentum When the speed of an object is close to c, the effects of relativity must be taken into account. The calculation for relativistic momentum is: The graph shows the effects of relativity are not significant until the objects moves at some fraction of c The relativistic momentum is always larger than its non-relativistic counterpart.

28.5 Relativistic Momentum Often it is convenient to ask for the ratio of relativistic to non-relativistic momenta for an object. This number can be thought of as the “relativistic factor” and can be calculated as: This number is always larger than one!

A collision of an electron with a target in a particle accelerator produces a muon that moves forward with a speed of 0.95c relative to the laboratory. The muon’s mass is 1.90 x 10-28 kg. Determine the “relativity factor”, the factor by which the relativistic momentum is greater than the classical momentum.

A collision of an electron with a target in a particle accelerator produces a muon that moves forward with a speed of 0.95c relative to the laboratory. The muon’s mass is 1.90 x 10-28 kg. The relativistic factor is: = 3.20 The relativistic momentum is 3.20 times that of the non-relativistic momentum.

28.6 The Equivalence of Mass and Energy In a short addendum to his original paper, Einstein showed that the total energy, and not only the kinetic energy, of an object was dependent on its speed and mass: If the object is at rest relative to its reference frame, the denominator becomes 1 and we get the the most famous equation in the world: The ratio of E/E0 is the relativistic factor:

Relativistic Kinetic Energy It can be shown, by use of a binomial expansion of the square root term that when v<<c, KE = ½ mv2.

Kinetic energy and total energy

Kinetic energy and total energy

Kinetic energy and total energy Note the difference between the values for resting energy and kinetic energy for an object moving at non-relativistic speeds.

Kinetic energy and total energy For the electron, m = 9.11 x 10-31 kg, start by calculating the relativistic factor E/E0:

EXAMPLE 37.12 Kinetic energy and total energy

28.6 The Equivalence of Mass and Energy Example 8 The Sun is Losing Mass The sun radiates electromagnetic energy at a rate of 3.92x1026W. What is the change in the sun’s mass during each second that it is radiating energy? What fraction of the sun’s mass is lost during a human lifetime of 75 years.

28.6 The Equivalence of Mass and Energy