Published on March 24, 2008
Chapter 2 Special Theory of Relativity: Chapter 2 Special Theory of Relativity Albert Einstein (1879 – 1955) Nobel, 1921 Einstein’s Miraculous Year (1905): Einstein’s Miraculous Year (1905) All papers published in the German journal Annalen der Physik. Completed Ph.D. Dissertation, University of Zurich, July 20, 1905: A New Determination of Molecular Dimensions. Published August 19, 1905. On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat (Annalen der Physik 17 : 549-560.) On the Electrodynamics of Moving Bodies (Annalen der Physik 17 : 891-921.) On a Heuristic Point of View Concerning the Production and Transformation of Light (Annalen der Physik 17 : 132-148.) Introduction: Introduction Newtonian principle of relativity Galilean invariance Galilean-Newtonian Relativity: Galilean-Newtonian Relativity Galilean-Newtonian relativity is known as a “classical” theory. Einstein’s special theory of relativity is known as a “modern” theory. Galilean Transformation: Galilean Transformation x x´ y´ z y v x´ = x – vt y´ = y z´ = z t´ = t Time is absolute z´ K K´ O´ O Galilean Transformation: Galilean Transformation x x´ y´ y v x´ = x – vt y´ = y z´ = z t´ = t Time is absolute K K´ O´ O vt x´ x EVENT 2.1 Historical Perspective: 2.1 Historical Perspective Light is a wave. Waves require a medium through which to propagate. Medium as called the “ether.” (from the Greek aither, meaning upper air) Maxwell’s equations assume that light obeys the Newtonian-Galilean transformation. 2.2 The Michelson-Morley Experiment: 2.2 The Michelson-Morley Experiment Experiment designed to measure small changes in the speed of light was performed by Albert A. Michelson (1852 – 1931, Nobel ) and Edward W. Morley (1838 – 1923). Used an optical instrument called an interferometer that Michelson invented. Device was to detect the presence of the ether. Outcome of the experiment was negative, thus contradicting the ether hypothesis. A.A. Michelson and E.W. Morley, American Journal of Science, 134 – 333, 1887) Michelson-Morley Experiment: Michelson-Morley Experiment The Either: The Either “Upwind” “Downwind” c + v c - v v v c c c v v´ = (c2 – v2)1/2 Perpendicular to “wind.” 2.3 Einstein’s Postulates: 2.3 Einstein’s Postulates Big problem at the turn of the century: Michelson-Morley showed that the Galilean transformation did not hold for Maxwell’s equation. Maxwell’s equations could not be wrong. Galilean transformation did hold for the laws of mechanics. Einstein proposed a solution. Inertial Reference Frame: Inertial Reference Frame An inertial reference frame is one in which no accelerations are observed in the absence of external forces. that is not accelerating. Newton’s laws hold in all inertial reference frames. Noninertial Reference Frame: Noninertial Reference Frame A noninertial reference frame is one that is accelerating with respect to an inertial reference frame. In a noninertial reference frame, bodies have accelerations in the absence of applied forces. Reference Frames: Reference Frames Platform is accelerating noninertial frame Ground is the inertial frame Platform at rest, tree moving—ball is seen by observers on platform as being deflected, but no force acts on it. Violation of Newton’s second law. Platform moving. Observer on the ground (inertial frame) sees ball move in a straight line, but sees the catcher move away. The Earth: The Earth In many cases the Earth’s surface can be considered an inertial reference frame, even though strictly speaking it is not. For small scale phenomena the Earth is approximately an inertial reference frame. Slide19: Problem of Simultaneity Conclusion: Conclusion An event in a given system must be specified by stating both its space and time coordinates. Result: The principle of simultaneity. Slide21: Four- Dimensional Spacetime Spacetime Coordinates (x, y, z, t): Spacetime Coordinates (x, y, z, t) x = 3.7 rods, y = 1.2 rods, z = 0 rods, t = whatever time appears on the clock nearest to A at the time of the flash. Flash Principle of Simultaneity: Principle of Simultaneity Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K´) moving with respect to the first frame. Recall that in the Galilean transformation time is considered absolute regardless of the relative motion of the reference inertial reference frames. Einstein’s Postulates: Einstein’s Postulates The Principle of Relativity The constancy of the speed of light. The Principle of Relativity: The Principle of Relativity All the laws of physics are the same in all inertial systems.* There is no way to detect absolute motion, and no preferred inertial system exists. *Particular quantities (velocity, momentum, kinetic energy, …) have different values in different inertial reference frames, but the laws of physics (conservation of energy and momentum, …) are the same. The Constancy of the Speed of Light: The Constancy of the Speed of Light Observers in all inertial systems measure the same value for the speed of light in a vacuum. (c = 2.9979 x 108 m/s) The Ultimate Speed: The Ultimate Speed The speed of light has been defined to be exactly: c = 299 792 458 m/s Light travels at this ultimate speed, as do any massless particles. No entity that carries energy or information can exceed this speed limit. No particle that does have a mass, can actually reach c. Electrons have been accelerated to at least 0.999 999 999 95 times the speed of light—still less than c. 2.4 The Lorentz Transformation: 2.4 The Lorentz Transformation Einstein needed to find a new transformation (the old one being the Galilean transformation). It must fit both the laws of mechanics and Maxwell’s electrodynamic equations. It must allow time to be relative. Historical Note: Historical Note The Lorentz transformation was derive in 1890 by Hendrik A. Lorentz (1853 – 1928, Nobel 1902) to explain the null result in the Michelson-Morely experiment. He proposed that one of the arms of the interferometer would contract by a factor of (1-v2/c2)1/2 explaining the null result. Only basis was to fit the experimental data. Galilean Transformation: Galilean Transformation x x´ y´ y v x´ = x – vt y´ = y z´ = z t´ = t Time is absolute K K´ O´ O vt x´ x EVENT Lorentz Transformation Equations: Lorentz Transformation Equations x´ = y´ = y z´ = z t´ = x - vt 1- b 2 1- b 2 t - vx c2 where b = v/c Lorentz Transformation Equations: Lorentz Transformation Equations x´ = g ( x – vt ) y´ = y z´ = z t´ = g ( ) 1- b 2 t - vx c2 where b = v/c where g = 2.5 Time Dilation and Length Contraction: 2.5 Time Dilation and Length Contraction Slide34: Frank measures proper time interval. Why? Time Dilation: Time Dilation Dt´ = Dt0 1 – v2/c2 Clocks moving relative to an observer are measured by that observer to run more slowly, as compared to the clock at rest. Time where clock is at rest relative to the observer. Proper time. Time where clock is moving relative to the observer. Time Dilation: Time Dilation T´ = g T0 Where: T´ = Dt´ and T0 = Dt0 Example 2.1: Example 2.1 Show that Frank in the fixed system will also determine the time dilation result by having the sparkler be at rest in the system K'. Time Dilation: Time Dilation T = g T0´ For clock in the K´ frame Example 2.2: Example 2.2 It is the year 2050 and the United Nations Space Federation has perfected an antimatter propulsion system. The manned starship is preparing to leave for Alpha Centauri, about 4 lightyears away. Provisions have been placed aboard to allow a trip 16 years’ total duration. How fast must the spacecraft travel is the provisions are to last? Neglect acceleration, turnaround, and visiting times, because they are negligible compared to the actual travel times. Length Contraction: Length Contraction L0 = g L L0 > L Length Contraction: Length Contraction L = L0 1- v2/c2 The length of an object is measured to be shorter when it is moving relative to the observer than when it is at rest. Length were observer is at rest relative to the length being measured. Length were observer is moving relative to the length being measured. Example 2.3: Example 2.3 Consider the solution of Example 2.2 from the standpoint of length contraction. 2.6 Addition of Velocities: 2.6 Addition of Velocities ux = ux´ + v 1 + (v/c2)ux´ uy = uz = g [1 + (v/c2)ux´ ] g [1 + (v/c2)ux´ ] uy´ uz´ Motion on the x direction Notation: Notation v is the relative velocity of reference frames. u is the velocity of an object w.r.t. a particular reference frame. Example 2.4: Example 2.4 The commander of the spaceship just discussed is holding target practice for junior officers by shooting protons at small asteroids and space debris off to the side as the spaceship passes by. What speed will an observer in the space station measure for the protons? Example 2.5: Example 2.5 In 1851, H. L. Fizeau measured the “either drag” coefficient for light passing in opposite directions through flowing water. Let a moving system K´ be at rest w.r.t. the flowing water and let v be the speed of the flowing water w.r.t. a fixed observer in K. The speed of light in the water at rest (i.e., in system K´) is u´, and the speed of light as measured in K is u. If the index of refraction of the water is n, Fizeau found experimentally that Example 2.5 (Cont.): Example 2.5 (Cont.) u = u´ + ( 1- 1/n2 )v which was in agreement with Fresnel’s prediction. This result was considered confirmation of the either concept. Show that this result can be obtained easily using relativity. 2.7 Experimental Verification: 2.7 Experimental Verification Muon decay Atomic clock measurement Velocity addition Muon Decay: Muon Decay Cosmic rays enter the upper atmosphere and interact with particles in the upper atmosphere creating p mesons (pions), decay into other particles called muons. Obey radioactive decay law N = N0e-(0.693t/t ) 1/2 No. muons left at time t No. muons at t = 0 Half life Muon Decay: Muon Decay t1/2 = 1.56 x 10-6 s Atomic Clock Measurement: Atomic Clock Measurement 1972 two physicists J.C. Hafele and Richard E. Keating used four 133Cs atomic clocks. Flew one clock eastward, one clock westward. Two stayed fixed on earth. Travel Predicted Observed Eastward -40 ± 23 ns -59 ± 10ns Westward 257 ± 21 ns 273 ± 7 ns J.C. Hafele and R.E. Keating, Science 177, 166 - 170 Example 2.6: Example 2.6 Assuming a jet airplane travels 300 m/s and the circumference of the Earth is about 4 x 107 m, calculate the time dilation effect expected for a round-the-world trip exclusive of the Earth’s rotation and gravitational correction. Addition of Velocities: Addition of Velocities Pion decay experiment at CERN. 2.8 Twin Paradox: 2.8 Twin Paradox One twin stays at home. One twin travels on a spaceship at very high speeds. Relativity says traveling twin will age more slowly. But one can say the twin on Earth is traveling w.r.t. the twin in the spaceship and should be the younger. This is the paradox. Who is really younger. Answer: Traveling twin because of accelerations for the traveling twin—non inertial frame.. 2.9 Spacetime: 2.9 Spacetime Four dimensional spacetime can be represented by spacetime diagrams. First used by Herman Minkowski in 1908 and often called Minkowski diagrams. A line connecting two points in spacetime is called a worldline. Example 2.7: Example 2.7 Draw the spacetime diagram for the motion of the twins discussed in Section 2.8. Draw light signals being emitted from each twin at annual intervals and count the number of light signals received by each twin from the other. 2.10 Doppler Effect: 2.10 Doppler Effect Difference between relativistic Doppler effect for light and the Doppler effect for sound. With sound, the effect depends on which is moving, the source or the receiver. With light it is only the relative motion of the source and receiver that counts. Example 2.8: Example 2.8 Analyze the light signals sent out by Frank and Mary by using the relativistic Doppler effect. 2.11 Relativistic Momentum: 2.11 Relativistic Momentum Know Newton’s laws are invariant under a Galilean. But, we know that the correct relativistic transformation is the Lorentz transformation. Must correct Newton’s laws. Conservation of linear momentum does not change. Relativistic Momentum: Relativistic Momentum p = g mu Where: g = 1 1- u2/c2 Example 2.9: Example 2.9 Show that linear momentum is conserved for the collision just discussed and shown in the previous figure. 2.12 Relativistic Energy: 2.12 Relativistic Energy Consider concepts of energy and force. Trying to keep as many concepts from classical physics as necessary. K= mc2 (g – 1) Example 2.10: Example 2.10 Electrons in a television set are accelerated by a potential difference of 25,000 volts before striking the screen. Calculate the velocity of the electrons and determine the error in using the classical kinetic energy result. Total Energy and Rest Energy: Total Energy and Rest Energy Total Energy and Rest Energy: Total Energy and Rest Energy E g mc2 (Total Energy) E0 mc2 (Rest Energy) Equivalence of Mass and Energy: Equivalence of Mass and Energy E0 = mc2 Even when a particle has no velocity and therefore no kinetic energy, it still has energy by virtue of its mass. The laws of conservation of energy and conservation of mass must now be combined into one law: Law of conservation of mass-energy. Mass and energy are equivalent. Momentum and Mass: Momentum and Mass m´ = m0 1 – u2/c2 The mass of an object is measured to increase as its speed increases. Mass measured when object is at rest relative to the observer—rest mass. Mass measured by an observer moving relative to the mass. Relationship of Energy and Momentum: Relationship of Energy and Momentum Linear momentum appears to be a more fundamental concept than kinetic. No law of conservation of kinetic energy. Conservation of linear momentum is inviolate as far as is known. Relativistic Energy and Momentum: Relativistic Energy and Momentum E2 = p2c2 + E0 E2 = p2c2 + m2c4 2 Massless Particles: Massless Particles For a particle having no mass: E = pc Therefore, for a massless particle: u = c 2.13 Computations in Modern Physics: 2.13 Computations in Modern Physics Common units Energy: Electron volts (eV) Mass: MeV/c2 Unified (atomic) mass unit (u) Momentum: eV/c Binding energy Mole: Mole 1 mole (mol) of a substance is defined as the amount of a substance that contains as many atoms or molecules as there are in 12.00 grams of Carbon 12. Mass (12C) = 12 u. 1 mol is the number of grams of a substance numerically equal to the molecular mass of a substance. Mole: Mole 1 mol of CO2 has a mass of [12 + (2x16)] = 44 grams/mol. n (mol) = The number of mols in 132 grams of CO2 is: n = = 3.0 mol 132g 44g/mol mass (grams) molecular mass (g/mol) Ideal gas Law in Terms of Molecules: Avogadro’s Number: Ideal gas Law in Terms of Molecules: Avogadro’s Number Amedeo Avogadro (1776 – 1856) Avogadro’s hypothesis states: equal volumes of gas at the same pressure and temperature contain equal numbers of molecules. Avogadro’s number: NA = 6.02 x 1023 molecules/mole. That is, Avogadro was the first to clearly realize that the volume of a gas depends on the number of molecules it contains. Or, 1 mole of any substance contains NA numbers of molecules (or atoms as the case may be). In Other Words: In Other Words A mole of oxygen (32 g) is 16 times as massive as a mole of hydrogen gas (2 g), but is made of molecules that are 16 times as massive as hydrogen. Therefore, one mole of oxygen contains the same number of molecules as one mole of hydrogen. Calculation of Avogadro’s Number: Calculation of Avogadro’s Number 1 atom of mass m of 12C = 1.9926 x 10-26 kg/molecule The atomic mass of 12C = 12 g/mol (12 x 10-3 kg/mol) = Avogadro’s Number (NA ) 12 x 10-3 kg/mol 1.9926 x 10-26 kg/molecule = 6.0214 x 1023 molecules/mol Example 2.11: Example 2.11 A 2-GeV proton hits another 2-GeV proton in a head-on collision. (a) Calculate v, b, p, K, and E for each of the protons. (b) What happens to the kinetic energy? Binding Energy: Binding Energy The potential energy associated with the force keeping a system together is called the binding energy EB. The binding energy of a system is the work required to pull the particles out of the system into separate, free particles at rest Binding Energy: Binding Energy EB = S mic2 – Mbound systemc2 Example 2.12: Example 2.12 What is the minimum kinetic energy the protons must have in the head-on collision of Equation (2.78), p + p p+ + d, in order to produce the positively charged pion and deuteron? The mass of the p+ is 139.6 MeV/c2. Example 2.13: Example 2.13 The atomic mass of the 4He atom is 4.002603 u. Find the energy of the 4He nucleus. Example 2.14: Example 2.14 The molecular binding energy is called the dissociation energy. It is the energy required to separate the atoms in a molecule. The dissociation energy of the NaCl molecule is 4.24 eV. Determine the fractional mass increase of the Na and Cl atoms separate from NaCl. What is the mass increase for a mole of NaCl? Example 2.15: Example 2.15 A positively charged S+ particle produced in a particle physics experiment decays very quickly into a neutron and positively charged pion before either its energy or momentum can be measured. The neutron and the pion are observed to move in the same direction as the S+ was originally moving, with momenta of 4702 MeV/c and 169 MeV/c, respectively. What was the kinetic energy of the S+ and its mass? 2.14 Electromagnetism and Relativity: 2.14 Electromagnetism and Relativity Einstein was convinced that magnetic fields were nothing other than electric fields observed in another inertial frame. That is the key to electromagnetism and relativity. Homework Problem 13: Homework Problem 13 Two events occur in an inertial system K as follows: Event 1: x1= a, t1= 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2= 3a/2c, y1 = 0, z1 = 0 In what frame K´ will these events appear to occur at the same time? Describe the motion of the system K´. Homework Problem 19: Homework Problem 19 A rocket ship carrying passengers blasts off to go from New York to Los Angles, a distance of 5000 km. How fast must the rocket ship go to have its length shortened by 1%? Homework Problem 22: Homework Problem 22 The Apollo astronauts returned from the moon under Earth’s gravitational force and reached speeds of almost 25,000 mi/h with respect to the Earth. Assuming (incorrectly) they had this speed for the entire trip from the moon to the Earth, what was the time difference for the trip between their clocks and clocks on Earth? Homework Problem 31: Homework Problem 31 A spaceship is moving at a speed of 0.80 c away from an observer at rest. A boy in the spaceship shoots a proton gun with protons having a speed of 0.70 c. What is the speed of the protons measured by the observer at rest when the gun is shot (a) away from the observer and (b) toward the observer? Homework Problem 59: Homework Problem 59 A particle having a speed of 0.92 c has a momentum of 10-16 kg•m/s. What is its mass? Homework Problem 67: Homework Problem 67 Calculate the momentum, kinetic energy, and total energy of an electron traveling at a speed of (a) 0.01 c, (b) 0.1 c, and (c) 0.9 c. Homework Problem 80: Homework Problem 80 A free neutron is an unstable particle and beta decays into a proton with the emission of an electron. How much kinetic energy is available in the decay?