Review Modern Physics, Ph 311 It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative and would, therefore, show that there is still a difficulty in the theory itself… - Albert Michelson, 1907 1/3 to 2/3of our modern economy !!!
Inertial Reference Frame • A reference frame is called an inertial frame if Newton laws are valid in that frame. • Such a frame is established when a body, not subjected to net external forces, is observed to move in rectilinear motion at constant velocity.
Newtonian Principle of Relativity • If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. • This is referred to as the Newtonian principle of relativity or Galilean invariance/relativity.
The Galilean Transformation For a point P • In system K: P = (x, y, z, t) • In system K’: P = (x’, y’, z’, t’) P x K K’ x’-axis x-axis
Conditions of the Galilean Transformation • Parallel axes • K’ has a constant relative velocity in the x-direction with respect to K • Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers
The Inverse Relations Step 1. Replace with . Step 2. Replace “primed” quantities with “unprimed” and “unprimed” with “primed.”
Results of Maxwell's electrodynamics • Visible light covers only a small range of the total electromagnetic spectrum • All electromagnetic waves travel in a vacuum with a speed c given by: (where μ0 and ε0 are the respective permeability and permittivity of “free” space)
Need for Ether • The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether. • Ether had to have such a low density that the planets could move through it without loss of energy • It also had to have an enormous elasticity/toughness to support the high velocity of light waves
An Absolute Reference System • Ether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made. • The Michelson-Morley experiment was an attempt to figure out Earth’s relatives movement through (with respect to) the ether so that Maxwell’s equations could be corrected for this effect.
0 The Michelson Interferometer 1. AC is parallel to the motion of the Earth inducing an “ether wind”2. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions3. After reflection the beams recombine at A slightly out of phase due to the “ether wind” as viewed by telescope E.
The Lorentz-FitzGerald Contraction • Another hypothesis proposed independently by both H. A. Lorentz and G. F. FitzGerald suggested that the length ℓ1, in the direction of the motion was contracted by a factor of …thus making the path lengths equal to account for the zero phase shift. • This, however, was an ad hoc assumption that could not be experimentally tested.
Length contracted for the moving muon, it’s own life time just 2.2 micro seconds Life time of the muon delayed for observer on Earth so that it can travel the whole distance as observed from Earth
Lorentz Transformation Equations So there is four-dimensional space time !!!
Mary has a light clock. A suitable clock is just any periodic process, the time it takes for one cycle of the process is the period, its inverse is the frequency. Tom watching Mary go by figures that her time is delayed due to her moving in a straight line with a constant high velocity.
Atomic Clock Measurement Figure 2.20: Two airplanes took off (at different times) from Washington, D.C., where the U.S. Naval Observatory is located. The airplanes traveled east and west around Earth as it rotated. Atomic clocks on the airplanes were compared with similar clocks kept at the observatory to show that the moving clocks in the airplanes ran slower.
No simultaneity if not also at the same position, just a consequence of the Lorentz transformaitons
The Lorentz Velocity Transformations In addition to the previous relations, the Lorentz velocitytransformations for u’x, u’y, and u’zcan be obtained by switching primed and unprimed and changing v to –v:
Einstein’s Two Postulates With the belief that Maxwell’s equations (and with it all of the known physics of the time) must be valid in all inertial frames, Einstein proposes the following postulates: • The principle of relativity: 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. • The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.
Relativistic Momentum • Rather than abandon the conservation of linear momentum, let us look for a modification of the definition of linear momentum that preserves both it and Newton’s second law. • To do so requires reexamining mass to conclude that: Relativistic dynamics can be derived by assuming that mass is increasing with velocity. The Lorentz factor gets larger when velocities get larger and so does mass apparently as we can see from the relativistic momentum equation. Einstein derived relativistic dynamics that way. His derivations are sure correct, but the foundations are somewhat shaking as there is no really good definition for mass.
Relativistic Energy • Due to the new idea of relativistic mass, we must now redefine the concepts of work and energy. • Therefore, we modify Newton’s second law to include our new definition of linear momentum, and force becomes:
Relativistic Kinetic Energy Equation (2.58) does not seem to resemble the classical result for kinetic energy, K = ½mu2. However, if it is correct, we expect it to reduce to the classical result for low speeds. Let’s see if it does. For speeds u << c, we expand in a binomial series as follows: where we have neglected all terms of power (u/c)4 and greater, because u << c. This gives the following equation for the relativistic kinetic energy at low speeds: which is the expected classical result. We show both the relativistic and classical kinetic energies in the next Figure. They diverge considerably above a velocity of 0.1c. Best to use relativistic dynamics as soon as the speed of something is larger than 1 % of the speed of light.
Total Energy and Rest Energy We rewrite in the form The term mc2 is called the rest energy and is denoted by E0. This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by E and is given by
Momentum and Energy We square this result, multiply by c2, and rearrange the result. We use for β2 and find
Momentum and Energy (continued) The first term on the right-hand side is just E2, and the second term is E02. The last equation becomes We rearrange this last equation to find the result we are seeking, a relation between energy and momentum. or is a useful result to relate the total energy of a particle with its momentum. The quantities (E2 – p2c2) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, “accelerator Equation” correctly gives E0 as the particle’s total energy. There can be mass less particles that still have momentum. These can collide with massive particles. For such a collision one needs to invoke special relativity!
Binding Energy The binding energy is the difference between the rest energy of the individual particles and the rest energy of the combined bound system. A couple of eV for chemical reactions. A couple of MeV for nuclear reactions.
0 A Conducting Wire
Principle of Equivalence The principle of equivalence is an experiment in non-inertial reference frames. Consider an astronaut sitting in a confined space on a rocket placed on Earth. The astronaut is strapped into a chair that is mounted on a weighing scale that indicates a mass M. The astronaut drops a safety manual that falls to the floor. • Now contrast this situation with the rocket accelerating through space. The gravitational force of the Earth is now negligible. If the acceleration has exactly the same magnitude g on Earth, then the weighing scale indicates the same mass M that it did on Earth, and the safety manual still falls with the same acceleration as measured by the astronaut. The question is: How can the astronaut tell whether the rocket is on earth or in space? • Principle of equivalence: There is no experiment that can be done in a small confined space that can detect the difference between a uniform gravitational field and an equivalent uniform acceleration.
Gravitational Time Dilation • Since the frequency of the clock decreases near the Earth, a clock in a gravitational field runs more slowly according to the gravitational time dilation. This is because 4D space-time is “bend” – non-Euclidian, so there are no Euclidian straight lines to follow but Geodesics in a space whit Riemann’s coordinates • A very accurate experiment was done by comparing the frequency of an atomic clock flown on a Scout D rocket to an altitude of 10,000 km with the frequency of a similar clock on the ground. The measurement agreed with Einstein’s general relativity theory to within 0.02%.
Tests of General Relativity Bending of Light • During a solar eclipse of the sun by the moon, most of the sun’s light is blocked on Earth, which afforded the opportunity to view starlight passing close to the sun in 1919. The starlight was bent as it passed near the sun which caused the star to appear displaced. • Einstein’s general theory predicted a deflection of 1.75 seconds of arc, and the two measurements found 1.98 ± 0.16 and 1.61 ± 0.40 seconds. • Since the eclipse of 1919, many experiments, using both starlight and radio waves from quasars, have confirmed Einstein’s predictions about the bending of light with increasingly good accuracy.
Light Retardation • As light passes by a massive object, the path taken by the light is longer because of the spacetime curvature. • The longer path causes a time delay for a light pulse traveling close to the sun. • This effect was measured by sending a radar wave to Venus, where it was reflected back to Earth. The position of Venus had to be in the “superior conjunction” position on the other side of the sun from the Earth. The signal passed near the sun and experienced a time delay of about 200 microseconds. This was in excellent agreement with the general theory.
Spacetime Curvature of Space • Light bending for the Earth observer seems to violate the premise that the velocity of light is constant from special relativity. Light traveling at a constant velocity implies that it travels in a straight line. • Einstein recognized that we need to expand our definition of a straight line. • The shortest distance between two points on a flat surface appears different than the same distance between points on a sphere. The path on the sphere appears curved. We shall expand our definition of a straight line to include any minimized distance between two points. • Thus if the spacetime near the Earth is not flat, then the straight line path of light near the Earth will appear curved.
Perihelion Shift of Mercury • The orbits of the planets are ellipses, and the point closest to the sun in a planetary orbit is called the perihelion. It has been known for hundreds of years that Mercury’s orbit precesses about the sun. Accounting for the perturbations of the other planets left 43 seconds of arc per century that was previously unexplained by classical physics. • The curvature of spacetime explained by general relativity accounted for the 43 seconds of arc shift in the orbit of Mercury.
Gravitational Wave Experiments • Taylor and Hulse discovered a binary system of two neutron stars that lose energy due to gravitational waves that agrees with the predictions of general relativity. • LIGO is a large Michelson interferometer device that uses four test masses on two arms of the interferometer. The device will detect changes in length of the arms due to a passing wave. • NASA and the European Space Agency (ESA) are jointly developing a space-based probe called the Laser Interferometer Space Antenna (LISA) which will measure fluctuations in its triangular shape. No success so far, perhaps general relativity (and special relativity with it) are not really true, just very very good approximations to something else?
Dual nature of light (electromagnetic radiation) both/neither wave and/nor particle http://usatoday30.usatoday.com/tech/science/genetics/2008-05-08-platypus-genetic-map_N.htm “Australia's unique duck-billed platypus is part bird, part reptile and part mammal according to its gene map. The platypus is classed as a mammal because it has fur and feeds its young with milk. It flaps a beaver-like tail. But it also has bird and reptile features — a duck-like bill and webbed feet, and lives mostly underwater. Males have venom-filled spurs on their heels.”
Light according to Maxwell Fig. 3-2, p. 67
Wien’s Displacement Law • The intensity (λ, T) is the total power radiated per unit area per unit wavelength at a given temperature. • Wien’s displacement law: The maximum of the distribution shifts to smaller wavelengths as the temperature is increased.
3.5: Blackbody Radiation • When matter is heated, it emits radiation. • A blackbody is a cavity in a material that only emits thermal radiation. Incoming radiation is absorbed in the cavity. • Blackbody radiation is theoretically interesting because the radiation properties of the blackbody are independent of the particular material. Physicists can study the properties of intensity versus wavelength at fixed temperatures.
Rayleigh-Jeans Formula • Lord Rayleigh used the classical theories of electromagnetism and thermodynamics to show that the blackbody spectral distribution should be • It approaches the data at longer wavelengths, but it deviates badly at short wavelengths. This problem for small wavelengths became known as “the ultraviolet catastrophe” and was one of the outstanding exceptions that classical physics could not explain. k: Boltzmann’s constant 8.614 10-5 eV/K
Planck’s Radiation Law • Planck assumed that the radiation in the cavity was emitted (and absorbed) by some sort of “resonators” that were contained in the walls. These resonators were modeled as harmonic oscillators. He effectively invented new physics in the process. His result cannot be explained with classical Boltzmann-Maxwell statistics. • Planck made two modifications to the classical theory: • The oscillators (of electromagnetic origin) can only have certain discrete energies determined by En = nhf, where n is an integer, f is the frequency, and h is called Planck’s constant. h = 6.6261 × 10−34 J·s. • The oscillators can absorb or emit energy in discrete multiples of the fundamental quantum of energy given by Planck’s radiation law, only one fundamental constant h left that can explain Wien’s and Stephan’s constants …, significant progress
Experimental Results Only if the energy threshold to get electrons out of the metal (work function) is exceeded.
Einstein’s Theory • Einstein suggested that the electromagnetic radiation field is quantized into particles called photons. Each photon has the energy quantum: where f is the frequency of the light and h is Planck’s constant. • The photon travels at the speed of light in a vacuum, and its wavelength is given by
Einstein’s Theory • Conservation of energy yields: where is the work function of the metal. Explicitly the energy is • The retarding potentials measured in the photoelectric effect are the opposing potentials needed to stop the most energetic electrons.
X-Ray Production • An energetic electron passing through matter will radiate photons and lose kinetic energy which is called bremsstrahlung, from the German word for “braking radiation.” Since linear momentum must be conserved, the nucleus absorbs very little energy, and it is ignored. The final energy of the electron is determined from the conservation of energy to be • An electron that loses a large amount of energy will produce an X-ray photon. Current passing through a filament produces copious numbers of electrons by thermionic emission. These electrons are focused by the cathode structure into a beam and are accelerated by potential differences of thousands of volts until they impinge on a metal anode surface, producing x rays by bremsstrahlung as they stop in the anode material.
Inverse Photoelectric Effect. • Conservation of energy requires that the electron kinetic energy equal the maximum photon energy where we neglect the work function because it is normally so small compared to the potential energy of the electron. This yields the Duane-Hunt limit which was first found experimentally. The photon wavelength depends only on the accelerating voltage and is the same for all targets. Lets have 10 – 50 keV, very short wavelenth, very energetic photons