The Life and Science of Albert Einstein Lecture 6 - Applications of Special Relativity (1) Lifetimes of muons (time dilation) t_1/2 = 1.52e-6 s (half-life at rest) Assume muons form at altitude of 2.23 km and travel to Earth's surface at a speed of 0.98 c. (a) lab frame viewpoint (someone on Earth's surface) time of flight = L_0 / v = 2.23e3 m /(0.98 * 3e8 m/s) = 7.6e-6 s half-flight (including time dilation) : T = T_0 / sqrt(1-(v/c)^2) = 5 T_0 = 7.6e-6 s so according to the observer on Earth, the travel time is equal to the half-life so only 1/2 of the muons reach the ground. This result is consistent with measurements. (b) muon viewpoint time of flight (distance traveled is contracted) : = L / v = L_0 sqrt(1-(v/c)^2) / (0.98 c) = 1.52e-6 s (note contracted length L = 0.446 km) half-life = t_1/2 = 1.52e-6 s (half-life at rest) so according to the muons, again the travel time is equal to the half-life so only 1/2 of the muons reach the ground (c) wrong viewpoint (no relativity) time of flight = L_0 / v = 2.23e3 m /(0.98 * 3e8 m/s) = 7.6e-6 s half-life = t_1/2 = 1.52e-6 s (half-life at rest) so time of flight is 5 half-lives, which means that only (1/2)^5 = 1/32 of the muons reach the ground. This calculation is wrong because it uses the time of flight from the lab frame and the half-life from the muon reference frame. Mixing reference frames like this is deadly! (2) Atomic clocks carried on jet airplanes (time dilation) (3) Radioactive decay (E=mc^2) (4) Particle accelerators (satisfy relativistic conservation laws) (5) Pair production (E=mc^2) - gamma-ray converts into electron and positron in the vicinity of atomic nucleus. Also the inverse process of electron and positron annihilation. (6) Annihilation of electron and positron in flight shows same time coincidence as an electron and positron that annihilate at rest (speed of light does not depend on the speed of source) (7) Relativistic beaming - jets from AGN are often one-sided or much fainter on one side than the other. Relativistic effects cause emission of radiation to become forward peaked in the direction of motion. (8) Superluminal motion (speed of light does not depend on the speed of source) Introduced the properties of Active Galactic Nuclei (AGN). . Quasars (a class of AGN) : point-like sources of radio emission discovered in the mid-20th century . Very bright : 10^15 times the luminosity of the sun or 10,000 times the total luminosity of the Milky Way galaxy. . Broadband spectra : extending from the radio through infrared and optical (visible) to the X- and gamma-raybands. . Now generally believed to be the result of accretion onto a supermassive black hole in the nucleus of a galaxy. . Often show jets of radio/optical/X-ray that extend 10^6 light years. Examples of AGN with one-sided jets include Centaurus A, Pictor A, and Cygnus A. An example of an AGN showing superluminal motion is 3C345. Comparison between sound and light - propagation through air (sound) and ether (light) (A) Sound is carried through a medium, such as the air. In general the measured speed of a sound pulse depends on the relative speed obtaining between the observer and the medium through which the pulse travels. Only when the observer is at rest with respect to the medium will he find the measured speed to be the same in all directions. In addition the speed of the pulse does not depend on the speed of the source. (B) Nineteenth-century physicists believed that electromagnetic radiation also required a medium to propagate through. They invented the concept of the ether to carry electromagnetic disturbances and in which the speed of light was equal to c. Any inertial system in which the ether was at rest (and only in such a system alone) the speed of light would also be c. Numerous experiments were done to search for evidence of the ether. Maxwell suggested using the timing of eclipses of Jupiter's moons. Here's how it works: Let's assume that the solar system (S.S.) is moving with respect to the ether toward some point A. When Jupiter is on the side of its orbit in the direction toward A, then the eclipses of its moons are delayed by the time t_1 = l/(c+v), where l is the diameter of the Earth's orbit, c is the speed of light, and v is the speed of the S.S. with respect to the ether. In this case the light moves contrary to the motion relative to the ether and its velocity therefore appears increased. When Jupiter is on the opposite of its orbit, light moves in the same direction as the S.S. and the velocity appears less. Here the eclipses are delayed by the time t_2 = l/(c-v). The difference between these two times is t_2 - t_1 = 2 l v / c^2 / (1-(v/c)^2) There is only an upper limit of ~ 1 s to t_2 - t_1, which sets a limit of <150 km/s for the motion of the S.S. relative to the ether. This is not out of the question - it is possible that the S.S. could be moving at this speed through the Galaxy (and therefore could be moving this rapidly with respect to the ether). Michelson-Morley experiment Relies on interferometry for high precision. Measures the difference in travel time of light rays moving in two perpendicular arms. If we assume the interferometer's x arm is parallel to the direction of motion relative to the ether and moving at speed v. Then the round-trip travel time for a light ray entering the arm, bouncing off the other side and returning to the starting point is t_x = l/(c-v) + l/(c+v) t_x = 2 l/c/(1-(v/c)^2) The round trip travel time for a light ray moving in the y arm is given by t_y = 2 l/c/sqrt(1-(v/c)^2) In general t_x and t_y are different (they are only the same if v=0, i.e., if the experiment is at rest with respect to the ether.) Comparing these times we find ( t_x - t_y ) / t_x = v^2/2/c^2 (to 2nd order in v/c). Maximum speed of the Earth around the sun is 3e4 m/s so v/c = 1e-4 and v^2/2/c^2 = 5e-9. The M-M experiment had an intrinsic sensitivity a factor of 100 times lower than this, meaning that a very sensitive detection (or non-detection) was possible. M-M floated their interferometer in a pool of mercury and rotated it slowly about its center while observing the interference fringes in the telescope. If there were real motion with respect to the ether then one would expect a varying travel time as different arms of the interferometer became the x or y direction. They repeated the experiment over the course of a year to ensure that at one time the Earth was not coincidentally at rest relative to the ether. They always got negative results. Subsequent experiments with even greater precision all got negative results. There is no evidence for a unique inertial system, i.e., no ether.