Relativity II. The laws of physics are identical in all inertial frames of reference. equivalently

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1 Relatiity II I. Henri Poincare's Relatiity Principle In the late 1800's, Henri Poincare proposed that the principle of Galilean relatiity be expanded to inclde all physical phenomena and not jst mechanics. This iew was spported experimentally by the Michelson-Morely experiment. Poincare's work was ery important becase it had a great inflence pon yong Albert Einstein. The laws of physics are identical in all inertial frames of reference. eqialently All reference frames in niform linear motion are eqialent. Yo can not physically determine absolte motion. II. Albert Einstein's Postlates We hae preiosly seen that H.A. Lorentz deeloped the basic eqations of special relatiity based pon a theory of the electron and determining the transformation eqations nder which Maxwell's eqations were coariant. We now consider the man who transformed physics in the twentieth centry. Einstein considered the problem in a radically different way. First, he began with two fndamental postlates: 1) The laws of physics are the identical in all inertial reference frames. 2) The speed of electromagnetic radiation in acm is constant, independent of any motion of the sorce. To Lorentz, the fact that all obserers measre the speed of light was a conseqence (reslt) of his theory based on the electron. It had no deeper meaning. To Einstein, this fact abot light was too niqe to be a coincidence. Ths, it was the nderlying principle pon which natre operated. Likewise, Einstein felt that Poincare's relatiity principle was spported by experimental eidence and mst be a fndamental principle pon which to bild a theory. Lorentz theory was based pon nerified properties of electrons and the ether while Einstein saw no

2 need for the ether at all. His theory was based pon experimentally erified facts! Einstein obtained the same eqations as Lorentz bt they had a far different meaning. The interferometer didn't contract de to being bilt ot of matter. Space itself is contracted. It makes no sense to talk abot the tre length of an object. It is the length of the object as seen in this reference frame. Or ery concepts abot the natre of space and time mst be modified! III. Eents A physical eent is defined by spatial and time coordinates (x,y,z,t) or eqialently (x',y',z',t'). IV. Simltaneity Eents that appear simltaneos to one obserer may not be simltaneos to a second obserer! Example: Consider two light beams emitted in opposite directions from the middle of a train traeling at 0.3 c as shown below. According to a person on the train, the light beams strike the detectors at the end of the train simltaneosly at t = L/c. Howeer, a person standing by the train track beliees the beam at the back of the train strikes first! 0.3 c Space-Time Diagram ct ct' x' x

3 The green line shows that the train obserer sees the two black eents simltaneosly. Howeer, simltaneos eents according to the track obserer occr along the prple line with the line flowing toward the pper right corner for increasing time. Black dashed line denotes the light cone. V. General Space Time Problems In Newtonian Mechanics, space and time are independent. Einstein showed that time and space are intertwined. Ths, it is not possible to separate space and time in most problems. (Problems in Chapter 6 of Scham s Otline shold help clarify this.) VI. Classical Doppler Shift (See Uniersity Physics Textbook) Anyone who has watched ato racing on TV is aware of the Doppler shift. As a race car approaches the camera, the sond of its engine increases in pitch (freqency). After the car passes the camera, the pitch of the car s engine decreases. We cold se this pitch to determine the relatie speed of the car. This techniqe is sed in many real world applications inclding ltrasond imaging, Doppler radar, and to determine the motion of stars. A. Moing Obserer Assme that we hae a stationary adio sorce that prodces sond waes of freqency f and speed. A stationary obserer shown below sees the time between each wae as y Sorce Obserer x Distance Time Speed 1 T f If the obserer is now moing at a elocity relatie to the sorce then the speed of the waes as seen by the obserer is

4 Speed where the positie sign is when the obserer is moing toward the sorce. The time between waes is now T' 1 f '. Taking the ratio of or two reslts we get that T T' f '. f f ' f B. Moing Sorce We now consider the case in which the sorce is moing toward the obserer. In this case, the wae's speed is nchanged bt the distance between wae fronts (waelength) is redced (increased) for the sorce moing toward (away) from the obserer as shown below: y Sorce T ' Obserer x From the diagram, we find the new waelength as ' T

5 ' T 1 ' 1 f T ' 1 c ' c f f ' Ths, the freqency seen by the obserer for a moing sorce is gien by f ' f. Note: The motion of the obserer and sorce create different effects. For sond, this difference is explained de to motion relatie to the preferred reference frame! This preferred frame is the reference frame stationary to the medim propagating the sond (air)! Problem: Or classical deriation wold imply that by measring the Doppler shift of light, yo cold determine if the obserer or sorce was in motion (ie measre absolte motion). Ths, or work is not consistent with relatiity that reqires the effects de to motion by the obserer and sorce to be the same. This problem cased Einstein to discard the ether theory of light and thereby inflenced his deelopment of Relatiity.

6 VII. Relatiistic Doppler Shift Since light has no medim, there shold be no difference between moing the sorce and the obserer. The problem with or preios deriation when dealing with light was that we didn't considered that space and time coordinates are different for the sorce and obserer. Ths, we mst accont for the contraction of space and dilation of time de to motion. After acconting for differences in time and space, we get the following reslt for both moing sorce and moing obserer f ' c c f See Chapter 3 of Modern Physics by Bersetin, Fishbane, and Gasiorowicz for the proof. VIII. Newton Second Law and Linear Momentm A. Newton Second Law Newton's Second Law has the same form in Special Relatiity that it does in Classical Physics. This garantees that Special relatiity prodces the same reslts as Newtonian mechanics at slow speeds. F ext dp dt B. Classical Linear Momentm According to the first postlate of relatiity, the laws of physics are the same for all inertial reference frames. Ths, all inertial obserers shold agree that linear momentm is consered in collisions! Einstein discoered that for collisions the classical linear momentm, dr p mo mo, was not consered for all obserers nder the Lorentz dt transformation!! Ths, there was a problem with calclating the linear momentm!

7 Einstein discoered that the following qantity was consered dring collisions: dr p mo where to is the proper time. o dt Howeer, this formla doesn't conceptally make sense as it inoles the measrement of position and time by different obserers. Srely, the linear momentm of an object doesn't reqire two different obserers to hae reality!! Using Calcls, we can rearrange the eqation as follows: p m o dr dt dt dt o m o c. p m o 1- c 2 m Ths, yo cold consider that it was the mass and not the elocity whose calclation had to be modified!! Einstein actally only considered the concept of rest mass to be sefl. I hae fond that this philosophical debate has no pedagogical ale at the leel of this corse. Most ndergradate stdents are better sered both in nderstanding the connection between relatiity and classical physics and in soling practical engineering problems by considering m to be the tre mass. Ths, I will teach the material with this philosophical approach. II. Rest Mass and Relatiistic Mass The mass of an object measred by an obserer depends on the motion of the object relatie to the obserer in relatiity. A. If the object is at rest with respect to an obserer, its mass will be the lowest and is called its rest mass, mo. This is an intrinsic property of the object and is the mass that we sed in classical physics.

8 B. An object moing at a speed with respect to an obserer will hae a larger apparent mass. This relatiistic mass, m, is related to the object's rest mass by the following formla for objects with non-zero rest masses: m m o 1 c 2 C. Linear Momentm Newton's Second Law is still correct bt yo mst se the correct mass (relatiistic mass) in calclating the linear momentm.