Chapter 27 Relativity

Historical Development Late 1600s Newton discovered his laws of mechanics Applied to a wide variety of problems over the next two centuries Worked well Late 1800s Maxwell’s equations explained the physics of electromagnetism and light Early 1900s Relativity Quantum Mechanics Introduction

Types of Relativity Theory Special relativity Concerned with objects and observers moving at a constant velocity Topic of this chapter General relativity Applies to situations when the object or the observer is accelerated Has implications for understanding gravitation Introduction

Relativity The term relativity arises when a situation is described from two different points of view When the railroad car moves with a constant velocity, Ted and Alice see different motions of the ball Section 27.1

Reference Frames A reference frame can be thought of as a set of coordinate axes Inertial reference frames move with a constant velocity The principle of Galilean relativity is the idea that the laws of motion should be the same in all inertial frames For example, adding or subtracting a constant velocity does not change the acceleration of an object If Newton’s Second Law is obeyed in one inertial frame, it is obeyed in all inertial frames Section 27.1

Interpretation by Ted and Alice Ted observes the ball’s motion purely along the vertical direction Alice sees the ball undergo projectile motion with a nonzero displacement in both the x- and y-directions Ted would think the ball’s horizontal velocity is zero, but Alice would disagree Both agree that the ball’s acceleration is downward with a magnitude g Both agree the ball’s horizontal acceleration is zero Both agree that the only force acting on the ball is the force of gravity and that Newton’s second law is obeyed Section 27.1

Galilean Relativity and Light According to Maxwell’s equations, the speed of light is a constant He also showed the speed of light is independent of the motion of both the source and the observer Assume Ted is moving with a constant velocity v relative to Alice when he turns on a flashlight Newton’s mechanics predict that speed of the light wave relative to Alice should be c + v According to Maxwell’s theory, Ted and Alice should both observe the light wave to move with speed c Section 27.1

Galilean Relativity and Light, cont. Galilean Relativity and electromagnetism predict different results for observers in different inertial frames Experiments showed that Maxwell’s theory was correct The speed of light in a vacuum is always c Galilean relativity for how the speed of light depends on the motion of the source is wrong Section 27.1

Special Relativity Einstein developed a theory to analyze the Ted and Alice situation called special relativity His work was not motivated by any particular experiment He suspected the speed of light is the same in all reference frames Maxwell was correct He then worked out what that implies for all the other laws of physics Section 27.2

Postulates of Special Relativity The basics of the theory were stated in two postulates about the laws of physics For fast-moving objects Newton’s theory breaks down and Einstein’s theory gives a correct description of motion in this regime Postulates of Special Relativity All the laws of physics are the same in all inertial reference frames The speed of light in a vacuum is a constant, independent of the motion of the light source and all observers Section 27.2

Postulates – Details First postulate is traced to the ideas of Galileo and Newton on relativity The postulate goes further than Galileo because it applies to all physical laws Not just mechanics The second postulate is motivated by Maxwell’s theory of light This is not consistent with Newton’s mechanics The postulates will lead to a new theory of mechanics that corrects and extends Newton’s Laws Section 27.2

More About Light Our everyday experience with conventional waves cannot be applied to light Light does not depend on having a conventional material medium in which to travel A light wave essentially carries its medium with it as it propagates In the electric and magnetic fields The lack of a conventional medium was surprising and hard to reconcile with conventional intuition Section 27.2

Inertial Reference Frames Inertial reference frames play a crucial role in special relativity A definition of what it means to be inertial is needed The modern definition of an inertial reference is one in which Newton’s First Law holds You can test for an inertial frame by observing the motion of a particle for which the total force is zero If the particle moves with a constant velocity, the reference frame is inertial Newton’s other laws should also apply in all inertial frames Section 27.2

Earth as a Reference Frame Since the Earth spins about its axis as it orbits the Sun, all points on the Earth’s surface have a nonzero acceleration Technically, a person standing on the surface of the Earth is not in an inertial reference frame However, the Earth’s acceleration is small enough that it can generally be ignored In most situations we can consider the Earth to be an inertial reference frame Section 27.2

Light Clock The two postulates lead to a surprising result concerning the nature of time A light clock keeps time by using a pulse of light that travels back and forth between two mirrors The time for the clock to “tick” once is the time needed for one round trip: 2ℓ / c Section 27.3

Moving Light Clock The clock moves with a constant velocity v relative to the ground From Ted’s reference frame, the light pulse travels up and down between the two mirrors Section 27.3

Moving Light Clock, cont. The time for the clock to make one tick as measured by Ted is Alice sees the light pulse travel a longer distance The speed of light is the same for Alice as for Ted Because of the longer distance, according to Alice the light will take longer to travel between the mirrors Section 27.3

Moving Clock, Alice’s Time For Alice, the time for one tick of the clock is The time for Ted is different from the time for Alice The operation of the clock depends on the motion of the observer Section 27.3

Moving Clocks Run Slow Alice’s measures a longer time than Ted Postulate 1 states that all the laws of physics must be the same in all inertial reference frames Therefore the result must hold for any clock Special relativity predicts that moving clocks run slow This effect is called time dilation For typical terrestrial speeds, the difference between Δt and Δt0 is negligible Section 27.3

Time Dilation When the speed is small compared to c, the factor is very close to 1 Approximations given in Insight 27.1 may be used in many terrestrial cases Section 27.3

Speeds Greater the c If the value of the speed is greater than the speed of light, Δt / Δt0 will be imaginary Speeds greater than the speed of light have never been observed in nature Experiments have shown that the time dilation predicted by special relativity is correct The result applies to all clocks, even biological ones Section 27.3

Proper Time The time interval Δt0 is measured by the observer at rest relative to the clock This quantity is called the proper time The time interval measured by a moving observer is always longer than the proper time The proper time is always the shortest possible time that can be measured for a process, by any observer Section 27.3

Muon Decay Muons are subatomic particles created when cosmic rays collide with atoms in the atmosphere These muons typically have speeds of about 0.99c A physicist at rest relative to the muons measures their average lifetime as τ = 2.2µs The muons then decay into other particles This is the proper time since the “clock” is moving with the muon Section 27.3

Muon Decay, cont. Another physicist finds the lifetime of the muons moving relative to the laboratory to be 7.1 x τ The moving muons “live” longer than muons at rest Agrees with experiments Section 27.3

Twin Paradox An astronaut, Ted, visits a nearby star, Sirius, and returns to Earth Sirius is 8.6 light-years from Earth Ted is traveling at 0.90 c Alice, Ted’s twin, stays on Earth and monitors Ted’s trip Section 27.3

Twin Paradox, Times Alice measures the trip as taking 19 years Ted’s body measures the proper time of 8.3 years Alice concludes that Ted will be younger than she is Ted calculates the Earth (and Alice) move away from him at 0.90 c Ted concludes Alice will age 8.3 years while he ages 19 years Ted concludes that Alice will be younger than he is Section 27.3

Twin Paradox, Resolution It appears that time dilation leads to contradictory results Alice’s analysis is correct She remains in an inertial frame and so can apply the results of special relativity Ted is incorrect He accelerates when he turns around at Sirius Special relativity cannot be applied during this time spent in an accelerating frame Section 27.3

Time Dilation and GPS Each GPS satellite contains a very accurate clock The satellite clocks are moving in orbit, so they experience time dilation They run slow by about 7µs per day To accurately determine a position, the effect of time dilation must be accounted for Section 27.3

Simultaneity Two events are simultaneous if they occur at the same time Ted is standing in the middle of his railroad car He moves at a speed v relative to Alice Two lightning bolts strike the ends of the car and leave burn marks on the ground which indicate the location of the two ends of the car where the bolts strike Section 27.4

Simultaneity, cont. Did the two lightning bolts strike simultaneously? According to Alice She is midway between the burn marks The light pulses reach her at the same time She sees the bolts as simultaneous According to Ted The light pulse from at B struck before the bolt at A Since he is moving toward B The two bolts are not simultaneous in Ted’s view Section 27.4

Simultaneity, final All observers agree on the order of the events Simultaneity is relative and can be different in different reference frames This is different from Newton’s theory, in which time is an absolute, objective quantity It is the same for all observers Section 27.4

Order of Events – Example Observers in different reference frames will disagree on the length of an object and the simultaneity of events The observers will agree on the order of two events that occur at the same place Example A moving observer has two clocks, one sensitive to red and one sensitive to blue One clock stops when the red light beam hits it and the other stops when the blue light beam hits it Both the moving observer and a stationary observer will agree on the order of the events Section 27.4

Length Contraction Alice marks two points on the ground and measures length L0 between them Ted travels in the x-direction at constant velocity v and reads his clock as he passes point A and point B This is the proper time interval of the motion Section 27.5

Length Contraction, cont. Distance measured by Alice = L0 = v Δt Distance measured by Ted = L = v Δt0 Since Δt ≠ Δt0, L ≠ L0 The difference is due to time dilation and The length measured by Ted is shorter than Alice’s length Section 27.5

Proper Length Ted is at rest Alice moves on the meterstick with speed v relative to Ted Ted measures a length shorter than Alice Moving metersticks are shortened The proper length, Lo, is the length measured by the observer at rest relative to the meterstick Section 27.5

Length Contraction Equation Length contraction is described by When the speed is very small, the contraction factor is very close to 1 This is the case for typical terrestrial speeds Section 27.5

Proper Length and Time, Review Proper time is measured by an observer who is at rest relative to the clock used for the measurement Proper length is measured by an observer who is at rest relative to the object whose length is being measured Section 27.5

Experimental Support A large number of experiments have shown that time dilation and length contraction actually do occur At ordinary terrestrial speeds the effects are negligibly small For objects moving at speeds approaching the speed of light, the effects become significant Section 27.5

Addition of Velocities Ted is traveling on a railroad car at constant speed vTA with respect to Alice He throws an object with a speed relative to himself of vOT What is the velocity vOA of the ball relative to Alice? Alice is at rest on the ground Section 27.6

Newton’s Addition of Velocities Newton would predict that vOA = vOT + vTA The velocity of the object relative to Alice = the velocity of the object relative to Ted + the velocity of Ted relative to Alice This result is inconsistent with the postulates of special relativity when the speeds are very high For example, if the object’s speed relative to Ted is 0.9 c and the railroad car is moving at 0.9 c, then the object would be traveling at 1.8 c relative to Alice Newton’s theory gives a speed faster than the speed of light Section 27.6

Relativistic Addition of Velocities The result of special relativity for the addition of velocities is The velocities are: vOT – the velocity of an object relative to an observer vTA – the velocity of one observer relative to a second observer vOA – the velocity of the object relative to the second observer Section 27.6

Relativistic Addition of Velocities, cont. When the velocities vOT and vTA are much less than the speed of light, the relativistic addition of velocities equations gives nearly the same result as the Newtonian equation For speeds less than approximately 10% the speed of light, the Newtonian velocity equation works well For the example, with each speed being 0.9 c, the relativistic result is 0.994 c Compared to 1.8 c from Newton’s prediction Experiments with particles moving at very high speeds show that the relativistic result is correct Section 27.6

Relativistic Velocities and the Speed of Light A slightly different result occurs when the velocities are perpendicular to each other Again when vOT and vTA are both less than c, then vOA is also less than c In general, if an object has a speed less than c for one observer, its speed is less than c for all other observers Since no experiment has ever observed an object with a speed greater than the speed of light, c is a universal “speed limit” Section 27.6

Relativistic Velocities, final Assume the object leaving Ted’s hand is a pulse of light Then vOT = c From the relativistic velocity equation, Alice observes the pulse is vOA = c Alice sees the pulse traveling at the speed of light regardless of Ted’s speed If an object moves at the speed of light for one observer, it moves at the speed of light for all observers Section 27.6

Momentum According to Newton’s mechanics, a particle of mass m0 moving with speed v has a momentum given by p = m0 v Conservation of momentum is one of the fundamental conservation rules in physics and is believed to be satisfied by all the laws of physics, including the theory of special relativity The momentum of a single particle can also be written as Section 27.7

Relativistic Momentum From time dilation and length contraction, measurements of both Δx and Δt can be different for observers in different inertial reference frames Should proper time or proper length be used? Einstein showed that you should use the proper time to calculate momentum Uses a clock traveling along with the particle The result from special relativity is Section 27.7

Relativistic Momentum, cont. Einstein showed that when the momentum is calculated by using the special relativity equation, the principle of conservation of momentum is obeyed exactly This is the correct expression for momentum and applies even for a particle moving at high speed, close to the speed of light When a particle’s speed is small compared to the speed of light, the relativistic momentum becomes p = m0 v which is Newton’s momentum Section 27.7

Newton’s vs. Relativistic Momentum As v approaches the speed of light, the relativistic result is very different than Newton’s There is no limit to how large the momentum can be However, even when the momentum is very large, the particle’s speed never quite reaches the speed of light Section 27.7

Mass Newton’s second law gives mass, m0, as the constant of proportionality that relates acceleration and force This works well as long as the object’s speed is small compared with the speed of light At high speeds, though, Newton’s second law breaks down Section 27.8

Relativistic Mass When the postulates of special relativity are applied to Newton’s second law, the mass needs to be replaced with a relativistic factor At low speeds, the relativistic term approaches m0 and the two acceleration equations will be the same When v ≈ c, the acceleration is very small even when the force is very large Section 27.8

Rest Mass When the speed of the mass is close to the speed of light, the particle responds to a force as if it had a mass larger than m0 The same result happens with momentum where at high speeds the particle responds to impulses and forces as if its mass were larger than m0 Rest mass is denoted by m0 This is the mass measured by an observer who is moving very slowly relative to the particle The best way to describe the mass of a particle is through its rest mass Section 27.8

Mass and Energy Relativistic effects need to be taken into account when dealing with energy at high speeds From special relativity and work-energy, For v << c, this gives KE ≈ ½ m0 v2 which is the expression for kinetic energy from Newton’s results Section 27.9

Kinetic Energy and Speed For small velocities, KE is given by Newton’s results As v approaches c, the relativistic result has a different behavior than does Newton Although the KE can be made very large, the particle’s speed never quite reaches the speed of light Section 27.9

Rest and Total Energies The kinetic energy can also be thought of as the difference between the final and initial energies of the particle The initial energy, m0c2, is a constant called the rest energy of the particle A particle will have this much energy even when it is at rest The total energy of the particle is the sum of the kinetic energies and the rest energy Section 27.9

Mass as Energy The rest energy equation implies that mass is a form of energy It is possible to convert an amount of energy (m0c2) into a particle of mass m0 It is possible to convert a particle of mass m0 into an amount of energy (m0c2) The principle of conservation of energy must be extended to include this type of energy Section 27.9

Speed of Light as a Speed Limit Several results of special relativity suggest that speeds greater than the speed of light are not possible The factor that appears in time dilation and length contraction is imaginary if v > c The relativistic momentum of a particle becomes infinite as v → c This suggests that an infinite force or impulse is needed for a particle to reach the speed of light Section 27.9

Speed Limit, cont. The total energy of a particle becomes infinite as v → c This suggests that an infinite amount of mechanical work is required to accelerate a particle to the speed of light The idea that c is a “speed limit” is not one of the postulates of special relativity Combining the two postulates of special relativity leads to the conclusion that it is not possible for a particle to travel faster than the speed of light Section 27.9

Mass-Energy Conversions Conversion of mass into energy is important in nuclear reactions, but also occurs in other cases A chemical reaction occurs when a hydrogen atom is dissociated The mass of a hydrogen atom must be less than the sum of the masses of an electron and proton The energy is lower by 13.6 eV when bound in the atom Mass is not conserved when a hydrogen atom dissociates Δm0 = 2.4 x 10-35 kg This is much less than the mass of a proton and can be ignored Section 27.9

Conservation Principles Conservation of mass Mass is a conserved quantity in Newton’s mechanics The total mass of a closed system cannot change Special relativity indicates that mass is not conserved The principle of conservation of energy must be extended to include mass Momentum is conserved in collisions Use the relativistic expression for momentum Electric charge is conserved It is possible to create or annihilate charges as long as the total charge does not change Section 27.9

General Relativity A noninertial reference frame is one that has a nonzero acceleration Physics in noninertial frames is describe by general relativity General relativity is based on a postulate known as the equivalence principle The equivalence principle states the effects of a uniform gravitational field are identical to motion with constant acceleration Section 27.10

Equivalence Example Ted stands in an elevator at rest (A) He feels the normal force exerted by the floor on his feet He concludes that he is in a gravitational field The elevator is not in a gravitational field and has an acceleration of g (B) Since there is an acceleration, Ted feels the same force on his feet Section 27.10

Equivalence Principle, cont. According to the equivalence principle, there is no way for Ted to tell the difference between the effects of the gravitational field and the accelerated motion The equivalence principle has the following consequences Inertial mass and gravitational mass are equivalent Light can be deflected by gravity Section 27.10

Light and Gravity The light beam travels through the elevator while the elevator is in distant space When a = 0, Ted sees the light beam travel in a straight line (A) When a ≠ 0, the light beam travels in a curved line relative to the elevator (B) In a gravitational field, the light beam also curves (C) Section 27.10

Deflection of Light by Sun The gravitational field of the Sun should deflect light from a star Easiest to see during a solar eclipse Experiments in 1919 verified light passing near the Sun during an eclipse was deflected by the predicted amount Section 27.10

Black Holes Black holes contain so much mass that light is not able to escape from their gravitational attraction A black hole can be “seen” by its effect on the motion of nearby objects Stars near a black hole move in curved trajectories and so the mass and location of the black hole can be determined Section 27.10

Gravitational Lensing If the black hole is between the star and the Earth, light from the star can pass by either side of the black hole and still be bent by gravity and reach the Earth The black hole acts as a gravitational lens Light from a single star can produce multiple images Analysis of the images can be used to deduce the mass of the black hole Section 27.10

Relativity and Electromagnetism Alice is at rest with the charged line and the point charge Ted sees the line of charge and the point charge in motion The moving charged line acts as a current Section 27.11

Relativity and EM, cont. Ted says that there is an electric force and a magnetic force on the particle Alice says there is only an electric force Both are correct They will agree on the total force acting on the particle The larger electric force seen by Ted due to his motion is canceled by the magnetic force produced Maxwell’s equations were already consistent with special relativity Section 27.11

Importance of Relativity The relation between mass and energy and the possibility that mass can be converted to energy (and energy to mass) mean that mass is not conserved Instead we have a more general view of energy and its conservation The three conservation principles in physics are Conservation of energy Conservation of momentum Conservation of charge It is believed that all the laws of physics must obey these three conservation principles Section 27.12

Importance of Relativity, cont. The rest energy of a particle is huge This has important consequences for the amount of energy available in processes such as nuclear reactions Relativity changes our notion of space and time Time and position are two primary quantities in physics but it is not possible to give precise definitions of such quantities Our everyday intuition breaks down when applied to special relativity Section 27.12

Importance of Relativity, final Relativity plays a key role in understanding how the universe was formed and how it is evolving Black holes can’t be understood without relativity Relativity shows that Newton’s mechanics is not an exact description of the physical world Instead, Newton’s laws are only an approximation that works very well in some cases, but not in others We shouldn’t discard Newton’s mechanics, but understand its limits Section 27.12