NJIT Physics 320: Astronomy and Astrophysics – Lecture IV Carsten Denker Physics Department Center for Solar–Terrestrial Research

September 24, 2003NJIT Center for Solar-Terrestrial Research The Theory of Special Relativity  The Failure of the Galilean Transformations  The Lorentz Transformation  Time and Space in Special Relativity  Relativistic Momentum and Energy

September 24, 2003NJIT Center for Solar-Terrestrial Research Wave Theory and Ether  Luminiferous Ether  transport light waves, no mechanical resistance   Science of early Greek: earth, air, water, and fire  heavens composed of fifth element = ether  Maxwell: There can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform body of which we have any knowledge.  Measuring absolute velocity?  Inertial reference systems (Newton’s 1 st law)

September 24, 2003NJIT Center for Solar-Terrestrial Research Galilean Transformation Equations  Michelson–Morley experiment:  c = 3  10 8 m/s = const.  velocity of Earth through ether is zero  Crisis of Newtonian paradigm for v/c << 1 Newton’s laws are obeyed in both inertial reference frames!

September 24, 2003NJIT Center for Solar-Terrestrial Research The Lorentz Transformations  Einstein 1905 (Special Relativity): On the Electrodynamics of Moving Bodies  Einstein’s postulates:  The Principle of Relativity: The laws of physics are the same in all inertial reference frames  The Constancy of the Speed of Light: Light travels through a vacuum at a constant speed of c that is independent of the motion of the light source.  Linear transformation equations between space and time coordinates (x, y, z, t) and (x, y, z, t ) of an event measured in two inertial reference frames S and S.

September 24, 2003NJIT Center for Solar-Terrestrial Research Linear Transformation Equations Principle of Relativity

September 24, 2003NJIT Center for Solar-Terrestrial Research Linear Transformation Equations (cont.) Rotational symmetry Boundary conditions at origin Galilean Transformations

September 24, 2003NJIT Center for Solar-Terrestrial Research Linear Transformation Equations (cont.) Spherically symmetric wave front in S and S Lorentz Transform Inverse Lorentz Transform

September 24, 2003NJIT Center for Solar-Terrestrial Research Time and Space in Special Relativity  Intertwining roles of temporal and spatial coordinates in Lorentz transformations  Hermann Minkowski: Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union between the two will preserve an independent reality.  Clocks in relative motion will not stay synchronized  Different observers in relative motion will measure different time intervals between the same two events

September 24, 2003NJIT Center for Solar-Terrestrial Research Time Dilation  The shortest time interval is measured by a clock at rest relative to the two events. This clock measures the proper time between the two events.  Any other clock moving relative to the two events will measure a longer time interval between them. Flashbulbs at x 1 and x 2 at same time t Strobe light every  t at x 1 = x 2

September 24, 2003NJIT Center for Solar-Terrestrial Research Length Contraction  The longest length, called the rod’s proper length, is measured in the rod’s rest frame.  Only lengths or distances parallel to the direction of the relative motion are affected by length contraction.  Distance perpendicular to the direction of the relative motion are unchanged. Rod along x–axis at rest in S

September 24, 2003NJIT Center for Solar-Terrestrial Research Group Assignment Problem 4.4  A rod moving relative to an observer is measured to have its length L moving contracted to one–half of its original length when measured at rest. Find the value of u/c for the rod’s rest frame relative to the observer’s frame of reference.

September 24, 2003NJIT Center for Solar-Terrestrial Research Doppler Shift Sound speed v s and radial velocity v r Relativistic Doppler shift

September 24, 2003NJIT Center for Solar-Terrestrial Research Redshift Source of light is moving away from the observer: Source of light is moving toward the observer: Redshift parameter: Redshift Blueshift Radial motion!

September 24, 2003NJIT Center for Solar-Terrestrial Research Group Assignment Problem 4.9  Quasar 3C 446 is violently variable. Its luminosity at optical wavelength has been observed to change by a factor of 40 in as little as 10 days. Using the redshift parameter z = measured for 3C 446 determine the time for the luminosity variation as measured in the quasar’s rest frame.

September 24, 2003NJIT Center for Solar-Terrestrial Research Relativistic Velocity Transformations

September 24, 2003NJIT Center for Solar-Terrestrial Research Relativistic Momentum and Energy The mass m of a particle has the same value in all reference frames. It is invariant under a Lorentz tranformation. Relativistic momentum vector Relativistic kinetic energy

September 24, 2003NJIT Center for Solar-Terrestrial Research Relativistic Energy Total relativistic energy Rest energy Total energy of a system of n particles Total momentum of a system of n particles

September 24, 2003NJIT Center for Solar-Terrestrial Research Group Assignment Problem 4.16  Find the value of v/c when a particle’s kinetic energy equals its rest energy.

September 24, 2003NJIT Center for Solar-Terrestrial Research Class Project Exhibition Science Audience

September 24, 2003NJIT Center for Solar-Terrestrial Research Homework Class Project  Read the Storyline hand–out  Prepare a one–page document with suggestions on how to improve the storyline  Choose one of the five topics that you would like to prepare in more detail during the course of the class  Homework is due Wednesday October 1 st, 2003 at the beginning of the lecture!

September 24, 2003NJIT Center for Solar-Terrestrial Research Homework Solutions Problem 2.3

September 24, 2003NJIT Center for Solar-Terrestrial Research Homework Solutions Problem 2.9 A geosynchronous satellite must be parked over the equator and orbiting in the direction of Earth’s rotation. This is because the center of the satellite’s orbit is the center of mass of the Earth–satellite system (essentially Earth’s center).

September 24, 2003NJIT Center for Solar-Terrestrial Research Homework Solutions Problem 2.11

September 24, 2003NJIT Center for Solar-Terrestrial Research Homework  Homework is due Wednesday October 1 st, 2003 at the beginning of the lecture!  Homework assignment: Problems 4.5, 4.13, and 4.18  Late homework receives only half the credit!  The homework is group homework!  Homework should be handed in as a text document!