Chapter 36 The Special Theory of Relativity
Units of Chapter 36 Galilean Newtonian Relativity The Michelson Morley Experiment Postulates of the Special Theory of Relativity Simultaneity Time Dilation and the Twin Paradox Length Contraction Four-Dimensional Space Time
Units of Chapter 36 Galilean and Lorentz Transformations Relativistic Momentum The Ultimate Speed E = mc 2 ; Mass and Energy Doppler Shift for Light The Impact of Special Relativity
36-1 Galilean Newtonian Relativity INFO: Relativity Tutorial Definition of an inertial reference frame: One in which Newton s first law is valid. Earth is rotating and therefore not an inertial reference frame, but we can treat it as one for many purposes. A frame moving with a constant velocity with respect to an inertial reference frame is itself inertial.
36-1 Galilean Newtonian Relativity Relativity principle: The basic laws of physics are the same in all inertial reference frames.
36-1 Galilean Newtonian Relativity This principle works well for mechanical phenomena. However, Maxwell s equations yield the velocity of light; it is 3.0 x 10 8 m/s. So, which is the reference frame in which light travels at that speed? Scientists searched for variations in the speed of light depending on the direction of the ray, but found none.
36-2 The Michelson Morley Experiment This experiment was designed to measure the speed of the Earth with respect to the ether. The Earth s motion around the Sun should produce small changes in the speed of light, which would be detectable through interference when the split beam is recombined.
36-2 The Michelson Morley Experiment The Michelson interferometer is sketched here, along with an analogy using a boat traveling in a river.
36-2 The Michelson Morley Experiment This interferometer was able to measure interference shifts as small as 0.01 fringe, while the expected shift was 0.4 fringe. However, no shift was ever observed, no matter how the apparatus was rotated or what time of day or night the measurements were made. The possibility that the arms of the apparatus became slightly shortened when moving against the ether was considered, but a full explanation had to wait until Einstein came into the picture.
36-3 Postulates of the Special Theory of Relativity 1. The laws of physics have the same form in all inertial reference frames. 2. Light propagates through empty space with speed c independent of the speed of source or observer. This solves the ether problem the speed of light is in fact the same in all inertial reference frames.
36-4 Simultaneity One of the implications of relativity theory is that time is not absolute. Distant observers do not necessarily agree on time intervals between events, or on whether they are simultaneous or not. Why not? In relativity, an event is defined as occurring at a specific place and time. Let s see how different observers would describe a specific event.
36-4 Simultaneity Thought experiment: lightning strikes at two separate places. One observer believes the events are simultaneous the light has taken the same time to reach her but another, moving with respect to the first, does not.
Here, it is clear that if one observer sees the events as simultaneous, the other cannot, given that the speed of light is the same for each. 36-4 Simultaneity
36-5 Time Dilation and the Twin Paradox A different thought experiment, using a clock consisting of a light beam and mirrors, shows that moving observers must disagree on the passage of time.
36-5 Time Dilation and the Twin Paradox Calculating the difference between clock ticks, we find that the interval in the moving frame is related to the interval in the clock s rest frame:
36-5 Time Dilation and the Twin Paradox The factor multiplying t 0 occurs so often in relativity that it is given its own symbol, γ: We can then write.
36-5 Time Dilation and the Twin Paradox Example 36-1: Lifetime of a moving muon. (a) What will be the mean lifetime of a muon as measured in the laboratory if it is traveling at v = 0.60c = 1.80 x 10 8 m/s with respect to the laboratory? Its mean lifetime at rest is 2.20 μs = 2.2 x 10-6 s. (b) How far does a muon travel in the laboratory, on average, before decaying?
36-5 Time Dilation and the Twin Paradox To clarify: The time interval in the frame where two events occur in the same place is t 0. The time interval in a frame moving with respect to the first one is Δt.
36-5 Time Dilation and the Twin Paradox Example 36-2: Time dilation at 100 km/h. Let us check time dilation for everyday speeds. A car traveling covers a certain distance in 10.00 s according to the driver s watch. What does an observer at rest on Earth measure for the time interval?
36-5 Time Dilation and the Twin Paradox Example 36-3: Reading a magazine on a spaceship. A passenger on a high-speed spaceship traveling between Earth and Jupiter at a steady speed of 0.75c reads a magazine which takes 10.0 min according to her watch. (a) How long does this take as measured by Earth-based clocks? (b) How much farther is the spaceship from Earth at the end of reading the article than it was at the beginning?
36-5 Time Dilation and the Twin Paradox It has been proposed that space travel could take advantage of time dilation if an astronaut s speed is close enough to the speed of light, a trip of 100 light-years could appear to the astronaut as having been much shorter. The astronaut would return to Earth after being away for a few years, and would find that hundreds of years had passed on Earth.
36-5 Time Dilation and the Twin Paradox This brings up the twin paradox if any inertial frame is just as good as any other, why doesn t the astronaut age faster than the Earth traveling away from him? The solution to the paradox is that the astronaut s reference frame has not been continuously inertial he turns around at some point and comes back.
36-5 Time Dilation and the Twin Paradox Conceptual Example 36-4: A relativity correction to GPS. GPS satellites move at about 4 km/s = 4000 m/s. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to 1 part in 10 13.
36-6 Length Contraction If time intervals are different in different reference frames, lengths must be different as well. Length contraction is given by or Length contraction occurs only along the direction of motion.
36-6 Length Contraction Example 36-5: Painting s contraction. A rectangular painting measures 1.00 m tall and 1.50 m wide. It is hung on the side wall of a spaceship which is moving past the Earth at a speed of 0.90c. (a) What are the dimensions of the picture according to the captain of the spaceship? (b) What are the dimensions as seen by an observer on the Earth?
36-6 Length Contraction Example 36-6: A fantasy supertrain. A very fast train with a proper length of 500 m is passing through a 200-m-long tunnel. Let us imagine the train s speed to be so great that the train fits completely within the tunnel as seen by an observer at rest on the Earth. That is, the engine is just about to emerge from one end of the tunnel at the time the last car disappears into the other end. What is the train s speed?
36-6 Length Contraction Conceptual Example 36-7: Resolving the train and tunnel length. Observers at rest on the Earth see a very fast 200-m-long train pass through a 200-m-long tunnel (as in Example 36 6) so that the train momentarily disappears from view inside the tunnel. Observers on the train measure the train s length to be 500 m and the tunnel s length to be only 78 m. Clearly a 500-m-long train cannot fit inside a 78-m-long tunnel. How is this apparent inconsistency explained?
36-7 Four-Dimensional Space-Time Space and time are even more intricately connected. Space has three dimensions, and time is a fourth. When viewed from different reference frames, the space and time coordinates can mix; the result is called fourdimensional space time. Rather than the space and time intervals being invariant separately under transformations, as is the case for Galilean relativity, the invariant quantity in fourdimensional space time is the interval (Δs) 2 = (c Δt) 2 (Δx) 2
36-8 Galilean and Lorentz Transformations A classical (Galilean) transformation between inertial reference frames involves a simple addition of velocities:
36-8 Galilean and Lorentz Transformations For an object that is moving at speed u parallel to the x axis in frame S, its velocity in S is again given by addition.
36-8 Galilean and Lorentz Transformations Relativistically, the conversion between one inertial reference frame and another must take into account length contraction: For the moment, we will leave γ as a constant to be determined; requiring that the speed of light remain unchanged when transforming from one frame to the other shows γ to have its previously defined value.
36-8 Galilean and Lorentz Transformations We can then solve for t, and find that the full transformation involves both x and t. This is called the Lorentz transformation:.
36-8 Galilean and Lorentz Transformations The Lorentz transformation gives the same results we have previously found for length contraction and time dilation. Velocity transformations can be found by differentiating x, y, and z with respect to time:
36-8 Galilean and Lorentz Transformations Example 36-8: Adding velocities. Calculate the speed of rocket 2 with respect to Earth.
36-9 Relativistic Momentum The expression for momentum also changes at relativistic speeds:
36-9 Relativistic Momentum Example 36-9: Momentum of moving electron. Compare the momentum of an electron when it has a speed of (a) 4.00 x 10 7 m/s in the CRT of a television set, and (b) 0.98c in an accelerator used for cancer therapy.
36-9 Relativistic Momentum The formula for relativistic momentum can be derived by requiring that the conservation of momentum during collisions remain valid in all inertial reference frames.
36-9 Relativistic Momentum Gamma and the rest mass are sometimes combined to form the relativistic mass: Care must be taken not to use the relativistic mass inappropriately; it is valid only as a component of the relativistic momentum.
36-10 The Ultimate Speed A basic result of special relativity is that nothing can equal or exceed the speed of light. Such an object would have infinite momentum not possible for anything with mass and would take an infinite amount of energy to accelerate to light speed.
36-11 E = mc 2 ; Mass and Energy At relativistic speeds, not only is the formula for momentum modified; that for energy is as well. The total energy can be written: In the particle s own rest frame,
36-11 E = mc 2 ; Mass and Energy Example 36-10: Pion s kinetic energy. A π 0 meson (m = 2.4 x 10-28 kg) travels at a speed v = 0.80c = 2.4 x 10 8 m/s. What is its kinetic energy? Compare to a classical calculation.
36-11 E = mc 2 ; Mass and Energy Example 36-11: Energy from nuclear decay. The energy required or released in nuclear reactions and decays comes from a change in mass between the initial and final particles. In one type of radioactive decay, an atom of uranium (m = 232.03714 u) decays to an atom of thorium (m = 228.02873 u) plus an atom of helium (m = 4.00260 u), where the masses given are in atomic mass units (1 u = 1.6605 x 10 27 kg). Calculate the energy released in this decay.
36-11 E = mc 2 ; Mass and Energy Example 36-12: A 1-TeV proton. The Tevatron accelerator at Fermilab in Illinois can accelerate protons to a kinetic energy of 1.0 TeV (10 12 ev). What is the speed of such a proton?
36-11 E = mc 2 ; Mass and Energy Combining the relations for energy and momentum gives the relativistic relation between them:
36-11 E = mc 2 ; Mass and Energy All the formulas presented here become the usual Newtonian kinematic formulas when the speeds are much smaller than the speed of light. There is no rule for when the speed is high enough that relativistic formulas must be used it depends on the desired accuracy of the calculation.
36-12 Doppler Shift for Light The Doppler shift for light can be derived using the fact that light travels at the same speed in all inertial reference frames. In the figure, when the source and observer are approaching each other, the observer sees the time between wave fronts as being dilated.
36-12 Doppler Shift for Light Hence, one can derive the observed frequency and wavelength: If the source and observer are moving away from each other, v changes sign.
36-12 Doppler Shift for Light Example 36-13: Speeding through a red light. A driver claims that he did not go through a red light because the light was Doppler shifted and appeared green. Calculate the speed of a driver in order for a red light to appear green.
36-13 The Impact of Special Relativity The predictions of special relativity have been tested thoroughly, and verified to great accuracy. The correspondence principle says that a more general theory must agree with a more restricted theory where their realms of validity overlap. This is why the effects of special relativity are not obvious in everyday life. Relativity Animations
Inertial reference frame: one in which Newton s first law holds. Principles of relativity: the laws of physics are the same in all inertial reference frames; the speed of light in vacuum is constant regardless of speed of source or observer. Time dilation: Summary of Chapter 36.
Summary of Chapter 36 Length contraction: Gamma:
Summary of Chapter 36 Lorentz transformations:. Relativistic momentum:
Summary of Chapter 36 Mass energy relation: Kinetic energy: Total energy: Relationship between energy and momentum: