Copyright © 2010 Pearson Education, Inc. Lecture Outline Chapter 29 Physics, 4 th Edition James S. Walker

Copyright © 2010 Pearson Education, Inc. Chapter 29 Relativity

Copyright © 2010 Pearson Education, Inc. Units of Chapter 29 The Postulates of Special Relativity The Relativity of Time and Time Dilation The Relativity of Length and Length Contraction The Relativistic Addition of Velocities Relativistic Momentum Relativistic Energy and E = mc 2

Copyright © 2010 Pearson Education, Inc. Units of Chapter 29 The Relativistic Universe General Relativity

Copyright © 2010 Pearson Education, Inc The Postulates of Special Relativity The postulates of relativity as stated by Einstein: 1.Equivalence of Physical Laws The laws of physics are the same in all inertial frames of reference. 2. Constancy of the Speed of Light The speed of light in a vacuum, c = 3.00 x 10 8 m/s, is the same in all inertial frames of reference, independent of the motion of the source or the receiver.

Copyright © 2010 Pearson Education, Inc The Postulates of Special Relativity The first postulate is certainly reasonable; it would be hard to discover the laws of physics if it were not true! But why would the speed of light be constant? It was thought that, like all other waves, light propagated as a disturbance in some medium, which was called the ether. The Earth’s motion through the ether should be detectable by experiment. Experiments showed, however, no sign of the ether.

Copyright © 2010 Pearson Education, Inc The Postulates of Special Relativity Other experiments and measurements have been done, verifying that the speed of light is indeed constant in all inertial frames of reference. With water waves, our measurement of the wave speed depends on our speed relative to the water:

Copyright © 2010 Pearson Education, Inc The Postulates of Special Relativity But with light, our measurements of its speed always give the same result:

Copyright © 2010 Pearson Education, Inc The Postulates of Special Relativity The fact that the speed of light is constant also means that nothing can go faster than the speed of light – it is the ultimate speed limit of the universe.

Copyright © 2010 Pearson Education, Inc The Relativity of Time and Time Dilation To begin to understand the implications of relativity, consider a light clock: The time it takes for light to make a round trip is:

Copyright © 2010 Pearson Education, Inc The Relativity of Time and Time Dilation Now, look at the clock moving at a speed v : The light has to travel farther. Now the round trip time is:

Copyright © 2010 Pearson Education, Inc The Relativity of Time and Time Dilation Therefore, a moving clock will appear to run slowly.

Copyright © 2010 Pearson Education, Inc The Relativity of Time and Time Dilation As the speed gets closer to the speed of light, the clocks run slower and slower:

Copyright © 2010 Pearson Education, Inc The Relativity of Time and Time Dilation This result applies to any kind of clock or process that is time-dependent – if it did not, the first postulate would be violated. Definitions Event: a physical occurrence that happens at a specified location at a specified time. Proper time: the amount of time separating two events that occur at the same location.

Copyright © 2010 Pearson Education, Inc The Relativity of Time and Time Dilation Time dilation has been measured with extremely accurate atomic clocks in airplanes, and also is frequently observed in subatomic particles. Another consequence of time dilation is that different observers will disagree about the simultaneity of events occurring at different places.

Copyright © 2010 Pearson Education, Inc The Relativity of Length and Length Contraction The observer on Earth sees the astronaut’s clock running slow; it takes him 25.6 years to go from Earth to Vega, but only 3.61 years have passed on the astronaut’s clock.

Copyright © 2010 Pearson Education, Inc The Relativity of Length and Length Contraction But how does it appear to the astronaut, who thinks his clock is fine? He sees the distance as contracted instead – for him, Vega is only 3.57 light-years away.

Copyright © 2010 Pearson Education, Inc The Relativity of Length and Length Contraction Proper length, L 0 : The proper length is the distance between two points as measured by an observer who is at rest with respect to them. So in the above example, 25.3 light-years is the proper length. With some arithmetic, we find:

Copyright © 2010 Pearson Education, Inc The Relativity of Length and Length Contraction Length contraction as a function of v :

Copyright © 2010 Pearson Education, Inc The Relativity of Length and Length Contraction Important note: Length contraction occurs only in the direction of motion. Other directions are unaffected.

Copyright © 2010 Pearson Education, Inc The Relativistic Addition of Velocities Suppose two space ships are heading towards each other, each with a speed of 0.6 c with respect to Earth. How fast do the astronauts in one ship see the other ship approach? It can’t be 1.2 c, but what is it? Here we give the answer:

Copyright © 2010 Pearson Education, Inc The Relativistic Addition of Velocities So in the above example, the relative speed would be 0.88 c. Below is a plot of the speed a rocket would have if it increased its speed by 0.1 c every time it fired its rockets.

Copyright © 2010 Pearson Education, Inc Relativistic Momentum If adding more and more energy to a rocket only brings its speed closer and closer to c, how can energy and momentum be conserved? The answer is that momentum is no longer given by p = mv.

Copyright © 2010 Pearson Education, Inc Relativistic Momentum As the speed gets closer and closer to c, the momentum increases without limit; note that the speed must be close to the speed of light before the difference between classical and relativistic momentum is noticeable:

Copyright © 2010 Pearson Education, Inc Relativistic Energy and E = mc 2 If the momentum increases without limit, the energy must increase without limit as well:

Copyright © 2010 Pearson Education, Inc Relativistic Energy and E = mc 2 The rest energy of ordinary objects is immense! In nuclear reactors, only a fraction of a percent of the mass of fuel becomes kinetic energy, but even that is enough to create enormous amounts of power.

Copyright © 2010 Pearson Education, Inc Relativistic Energy and E = mc 2 Every elementary particle, such as the electron, has an antiparticle with the same mass but opposite charge. The antiparticle of the electron is called the positron. Mass: Charge:

Copyright © 2010 Pearson Education, Inc Relativistic Energy and E = mc 2 When an electron and a positron collide, they completely annihilate each other, emitting only energy in the form of electromagnetic radiation.

Copyright © 2010 Pearson Education, Inc Relativistic Energy and E = mc 2 We can find the relativistic kinetic energy by subtracting the rest energy from the total energy:

Copyright © 2010 Pearson Education, Inc Relativistic Energy and E = mc 2 At ordinary speeds, the relativistic kinetic energy and the classical kinetic energy are indistinguishable.

Copyright © 2010 Pearson Education, Inc The Relativistic Universe It may seem as though relativity has nothing to do with our daily lives. However, medicine makes use of radioactive materials for imaging and treatment; satellites must take relativistic effects into account in order to function properly; and space exploration would be a disaster if relativistic effects were not handled properly.

Copyright © 2010 Pearson Education, Inc General Relativity Einstein thought about the distinction between gravitational force and acceleration, and concluded that within a closed system one could not tell the difference.

Copyright © 2010 Pearson Education, Inc General Relativity This leads to the principle of equivalence: All physical experiments conducted in a uniform gravitational field and in an accelerated frame of reference give identical results. Therefore, the people in the elevators on the previous page cannot, unless they are able to see outside the elevators, tell if they are in a gravitational field or accelerating uniformly.

Copyright © 2010 Pearson Education, Inc General Relativity When the elevator is moving at a constant speed, the light from the flashlight travels in a straight line. When the elevator accelerates, the light bends.

Copyright © 2010 Pearson Education, Inc General Relativity The principle of equivalence then tells us that light should bend in a gravitational field as well.

Copyright © 2010 Pearson Education, Inc General Relativity This gravitational bending of light can be observed during a solar eclipse, when stars appearing very close to the Sun can be seen.

Copyright © 2010 Pearson Education, Inc General Relativity If the gravitational field is strong enough, light may be bent so much that it cannot escape. An object that is this dense is called a black hole. Calculations show that the radius of a black hole of a given mass will be: Plugging in the numbers shows us that the Earth would have to have a radius of about 0.9 cm in order to be a black hole.

Copyright © 2010 Pearson Education, Inc General Relativity One way to visualize the bending of light around massive objects is to imagine that space itself is bent (there is a deeper truth to this as well). The region around a black hole then might look like this:

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 29 The laws of physics are the same in all inertial frames of reference. The speed of light in a vacuum is the same in all inertial frames of reference, independent of the motion of the source or the receiver. Clocks moving with respect to one another keep time at different rates. An observer sees a moving clock running slowly:

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 29 Length in the direction of motion appears contracted: Relativistic velocity addition: It is impossible to increase the speed of an object from less than c to greater than c.

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 29 Relativistic momentum: Total relativistic energy:

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 29 Rest energy: Relativistic kinetic energy: Principle of equivalence: All physical experiments conducted in a gravitational field and in an accelerated frame of reference give identical results.

Copyright © 2010 Pearson Education, Inc. Summary of Chapter 29 For an object of mass M and radius R to be a black hole, its radius must be less than the Schwarzschild radius: