Relativity

1. Review! Relativity

2. Relativity • 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 Section 27.1

3. 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

4. Simultaneity • Two events are simultaneous if they occur at the same time • The two bolts are not simultaneous in Ted’s view • 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 Section 27.4

5. Time Dilation • Proper Time - The time interval Δto is measured by the observer at rest relative to the clock Section 27.3

6. Length Contraction • The proper length, Lo, is the length measured by the observer at rest relative to the meterstick Section 27.5

7. New Stuff! Ch 27 Relativity

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

9. 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

10. 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

11. 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 0.9c, the relativistic result is 0.994 c • Experiments with particles moving at very high speeds show that the relativistic result is correct Section 27.6

12. 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

13. 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

14. Momentum • According to Newton’s mechanics, a particle of mass mo moving with speed v has a momentum given by p = mo 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 can also be written as Section 27.7

15. 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

16. 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 = mo v which is Newton’s momentum Section 27.7

17. 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

18. Mass • Newton’s second law gives mass, mo, 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

19. 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 mo 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

20. 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 mo • This “enhancement of the mass” depends on the particle’s speed • The same result happens with momentum where at high speeds the particle responds to impulses and forces as if its mass were larger than mo • Rest mass is denoted by mo • 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

21. 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 m v2 which is the expression for kinetic energy from Newton’s results Section 27.9

22. 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

23. 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, moc2, 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

24. Mass as Energy • The rest energy equation implies that mass is a form of energy • It is possible to convert an amount of energy (moc2) into a particle of mass mo • It is possible to convert a particle of mass mo into an amount of energy (moc2) • The principle of conservation of energy must be extended to include this type of energy Section 27.9

25. 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

26. Speed Limit, cont. • The total energy of a particle becomes infinite as v → c, suggesting 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

27. 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 Section 27.9

28. 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

29. 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

30. 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

31. 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 • Experiments in 1919 verified light passing near the Sun during an eclipse was deflected by the predicted amount Section 27.10

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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