Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 26: Relativity The Postulates of Special Relativity Simultaneity Time Dilation Length Contraction Relativistic Velocity Addition Relativistic Momentum Relativistic Rest Mass Energy, Kinetic Energy, and Total Energy

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 §26.1 Postulates of Relativity Postulate 1: The laws of physics are the same in all inertial reference frames (the principle of relativity). An inertial reference frame is one in which no accelerations are observed in the absence of external forces. (Recall Newton’s first law).

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 Scientists in the 19 th century believed that light was a vibration in an invisible medium called the ether. The Michelson-Morley experiment found that there was no observable change in the speed of light due to the Earth’s motion around the Sun. The conclusion was that the ether did not exist.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 Postulate 2: The speed of light in vacuum is the same in all inertial reference frames. The value is independent of the motion of the source of light or of the observer.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 What follows does not show that Newtonian physics is wrong; it is just not complete. Newtonian physics breaks down at speeds near c; this is where special relativity is needed. All of the familiar relationships from Newtonian physics are contained within special relativity in the limit of low speeds. This is known as the correspondence principle.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 §26.2 Simultaneity and Ideal Observers The location of an event can be specified by four coordinates: the three spatial coordinates (x,y,z) and a time coordinate t. Taken together (x,y,z,t) are the four coordinates of space-time.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 Consider a high speed train with AC=BC & A’C’=B’C’. The marks ACB are on a stationary train platform and the marks A’C’B’ are on the moving train. A B C A’ B’ C’ v At t=0: AA’, BB’, and CC’ are lined up. At this instant two lightning bolts strike at AA’ and BB’.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 An observer standing at point C (on the platform) will see both strikes simultaneously. The observer at C’ (on the train) will see the strike at AA’ first followed by the strike at BB’. The observers do not agree on what happened, but both are correct.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 This experiment shows that events that are simultaneous in one frame (the platform) are not simultaneous in another frame (the train).

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 §26.3 Time Dilation Consider a light clock. A light pulse reflects back and forth between two mirrors. One complete trip can be considered one “tick” on the clock. The time interval for a round trip by the light pulse is Mirror L Light pulse Mirror

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 As long as an observer is stationary with respect to the clock, he will measure a time interval of  t 0 between clock ticks. Now put the light clock on a moving train. What does a stationary observer outside the train see?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 The motion of the train is to the right with speed v. Know The light pulse appears to have the path shown below. How long does it take the light pulse to return to the bottom mirror? L vtvt

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Eliminating L gives Let, (Lorentz Gamma factor) The person outside the train will measure a longer time interval between ticks compared to the observer in the train.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 The relative motion between the observers (train and outside the train) changes the rate at which time passes. The effect is known as time dilation. Moving clocks run slower. The quantity  t 0 is known as the proper time. This is time interval between two events that occur in the same place.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 Example (text problem 26.6): An unstable particle called the pion has a mean lifetime of 25 ns in its own frame. A beam of pions travels through the laboratory at a speed of 0.60c. (a) What is the mean lifetime of the pion as measured in the laboratory? Given:  t 0 = 25 ns and v = 0.60c (  =0.60)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 Example continued: (b) How far does a pion travel (as measured by laboratory observers) during this time?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 §26.4 Length Contraction To measure the length of an object, its ends must be located simultaneously. The proper length is measured when an object is at rest relative to you.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 A person on a train platform will measure the length of the platform to be L 0. A moving object will pass by the platform in a time  t so L 0 = v  t (note:  t is not the proper time). A person riding on a train will measure the length of the platform to be L = v  t 0 (this person measures the proper time). where L 0 is the proper length.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 The two observers measure different lengths for the platform. Moving meter sticks are shorter. This effect is known as length contraction and only applies to lengths parallel to the direction of motion.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Example (text problem 26.15): A cosmic ray particle travels directly over a football field from one goal line to the other, at a speed of 0.50c. (a) If the length of the field between goal lines is 91.5 m, what is the length measured in the rest frame of the particle? Given: L 0 = 91.5 m and v = 0.50c (  =0.50)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 Example continued: (b) How long does it take the particle to go from one goal line to the other according to earth observers? (c) How long does it take in the rest frame of the particle?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 Example (text problem 26.17): Two spaceships are moving directly toward each other with a relative velocity of 0.90c. If an astronaut measures the length of his own spaceship to be 30.0 m, how long is the spaceship as measured by an astronaut in the other ship? Given: L 0 = 30.0 m and v = 0.90c (  =0.90)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 §26.5 Velocities in Different Reference Frames What is the velocity of the probe as measured by Abe? According to Galilean relativity, it is v PA = v PB +v BA.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Once the speeds get large enough, the relativistic velocity transformation formula must be used.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Example (text problem 26.25): The rogue starship, Galaxa, is being chased by the battlecruiser, Millenia. The Millenia is catching up to the Galaxa at a rate of 0.55c when the captain of the Millenia decides it is time to fire a missile. First the captain shines a laser range finder to determine the distance to the Galaxa, and then he fires a missile that is moving at a speed of 0.45c with respect to Millenia. What speed does the Galaxa measure for (a) the laser beam and (b) the missile as they both approach the starship? The laser beam will be measured to have speed c.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Example continued: The velocity of Millenia relative to Galaxa is v MG =0.55c. The velocity of the missile relative to Millenia is v mG = 0.45c. The velocity of the missile relative to Galaxa is

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 §26.6 Relativistic Momentum Note: it is true that  p =  F  t, but  F  ma.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 Example: Show that the expression for relativistic momentum reduces to its classical form when v<<c. When v<<c,  <<1 To first order in v/c

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 Example (text problem 26.36): A body has a mass of 12.6 kg and a speed of 0.87c. (a) What is the magnitude of the momentum? Given: m = 12.6 kg and v = 0.87c (  =0.87)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 Example continued: (b) If a constant force of N acts in the direction opposite to the body’s motion, how long must the force act to bring the body to rest?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 §26.7 Mass and Energy The rest mass energy of a particle is its energy measured in its rest frame.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l eV (electron volt) is the change in energy that a charge e (the fundamental unit of charge) experiences when accelerated through a 1 volt potential difference.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 A proton has m=1.67  kg; its rest mass energy is The mass of the proton can be written as 938 MeV/c 2.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 A quantity is conserved if its value is unchanged in a given reference frame. A quantity is invariant if its value is the same in all reference frames.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 §26.8 Relativistic Kinetic Energy Kinetic energy Total energy

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 Example: Show that the expression for relativistic kinetic energy reduces to its classical form when v<<c. When v<<c,  <<1

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 The relationship between relativistic momentum and total energy is Other useful relationships are

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 Example (text problem 26.48): When an electron travels at 0.60c, what is its total energy in MeV? Given: v = 0.60c (  =0.60)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 Example (text problem 26.53): An electron has a total energy of 6.5 MeV. (a) What is its momentum (in MeV/c)? The energy-momentum relationship is Solving for p:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 Example continued: (b) What is the value of  for this electron? (c) What is the kinetic energy of this electron?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 41 Summary The Postulates of Relativity Simultaneity Time Dilation (moving clocks run slower) Length Contraction (moving meter sticks are shorter) Addition of Relativistic Velocities Relativistic Momentum Rest Mass Energy Relativistic Kinetic Energy