Modern Physics At the beginning of the twentieth century, two new theories revolutionized our understanding of the world and modified old physics that had existed for over 200 years: Relativity: Describes objects moving close to or at the speed of light (spaceships, photons, electrons ) Quantum Mechanics: Describes microscopic objects (atoms, electrons, photons.) Both theories explained phenomena that existing physics was unable to deal with. Both theories challenged our fundamental intuition and perception of the world: space-time, mass-energy, locality, causality.
Relativity
Reference Frames A reference frame is a coordinate system in which an experimenter makes measurements of position and time. It is at rest with respect to the experimenter. Experimenters at rest with respect to each other share the same reference frame. Examples: Students in a classroom, drivers driving their cars at the same velocity. An event is labeled by coordinates (x,y,z,t)
Reference Frames An inertial frame of reference is one which a free body (isolated particle) has no acceleration: it remains at rest or moving in a straight line. (Newton s law of inertia applies: Sum of external forces = 0). A reference frame moving with constant velocity relative to an inertial frame is also an inertial frame. Which of the following can be considered as inertial frames of reference? A physics laboratory A skydiver A train moving at constant velocity A plane while taking-off Newtonian (Galilean) Relativity The laws of mechanics are the same in all inertial reference frames.
Galilean Transformations Consider two inertial reference frames S and S that are moving with a constant velocity relative to each other. The axes of both frames are parallel to each other. The origins of both frames coincide at t=0. Event P
Galilean Transformations An observer in frame S measures the coordinates of an event to be (x, y,z,t) An observer in frame S measures the coordinates of an event to be ( x, y, z, t ) The relative velocity between the two frames is v How are the positions and velocities in the two frames related? Event P
Galilean Transformations Galilean transformation of position Event P In general, x = x + vt y = y z = z r = r + vt
Galilean Transformations Galilean transformation of velocity x = x + vt y = y z = z dx = d x + dvt = d x + v u x = u x + v u y = u y u z = u z In general, u = u + v
Galilean Transformations Example: Find the rounrip times taken by the two boats as measured by an observer on the shore if the boats travel at speed c with respect to the river. t 1 t 2 t 1 = 2L c 2 v 2 = 2L c $ 1 v2 c 2 % ' 1 2 ( 2L c $ 1+ 1 2 v 2 c + % 2 ' t 2 = L c + v + L c v = 2Lc c 2 v 2 = $ 2L % c ' $ 1 v2 c 2 % ' 1 ( $ 2L% c ' $ 1+ v2 c + % 2 '
Galilean Transformations Galilean transformation of acceleration du x du y du z u x = u x + v u y = u y u z = u z = d u x = d u y = d u z + dv a x = a x dv = 0 a a = a y = a y a z = Observers in inertial frames (moving at constant velocity) measure the same acceleration and hence the same force. Newtonian (Galilean) Relativity: Newton s laws are the same in all inertial frames. a z
The speed of light Assumption: Light waves travel in a medium called the ether The speed of light in the rest frame of the ether is c = 300,000,000 m/s Using the Galilean transformations, the speed of light moving parallel to a reference frame that is moving with speed v with respect to the ether would be c = c ± v Question: Does the Galilean velocity transformation apply to the speed of light? If the earth is moving through the ether, then changes in the speed of light in the earth s reference frame would indicate its motion through the ether.
Michelson-Morley Experiment Recall boats in the river example: Rounrip time of light in arm 1 (from M 0 to M 2 ) t 2 = L c + v + ( Rounrip time of light in arm 2 from (M 0 to M 1 ) t 1 = 2L% $ c ' $ ( 2L c 2L c 2 v 2 L c v = 1+ v2 c + % 2 ' = 2L c $ 1+ 1 v 2 2 c + % $ 2 ' 2L % $ c ' $ 1 v2 c 2 % ' 1 v2 c 2 1 2 % ' 1
Michelson-Morley Experiment Time difference: t = t 2 t 1 Lv2 c 3 Time difference results in an interference pattern at the telescope. If the apparatus is rotated by 90 degrees, the role of the two arms is interchanged causing a shift in the interference pattern. If the wavelength of the light is λ, the shift is Shift 2c t Null result: No shift was observed
Einstein s Postulates of Special Relativity 1. Principle of relativity: The laws of physics are the same in all inertial reference frames 2. The speed of light is constant in all inertial frames and is independent of the observer s and the source s velocity.