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SPECIAL RELATIVITY(Einstein 1905)Motivations:Explaining the results of the Michelson-Morleyexperiment without invoking a “force” exertedon bodies moving through the aether.Make the equations that describe electromagnetism (calledMaxwell’s equations) simple and symmetrical in all referenceframes, independent of whether the frames are moving or not.[Note: some material in these notes is courtesy of D. Watson and M. Begelman]

SPECIAL RELATIVITY(Einstein 1905)Based on two postulates:The RELATIVITY PRINCIPLE:the laws of physics are the same inall inertial frames.The CONSTANCY OF THE SPEED OF LIGHT: the speedof light, c 299,792 km/s, is the same for all inertial observers,independent of their velocity of motion relative to thesource of light.SPECIAL only applies to inertial reference frames, those for whichthe state of motion is not influenced by external forces

From the two principles of special relativity, some importantconsequences are derived.Relativity of SimultaneityTo inside observer,light beams hit topand bottomsimultaneouslyTo outside observer, downward beam hits first(car catches up) - upward beam hits next(chases car).

Time DilationA light pulse goes from the floorto the ceiling and back. Sincec const but the distance is longerin case B (moving frame):Time intervals seen inmoving reference framesappear longer than the sameinterval seen at rest.A[Image from wing by M. Fowler]

ChuckBeverlyThe clock is at rest with respect to the reference frame 2

Dt1 gDt 2where g 121-Vc2(Lorentzfactor)†Chuck†Dt1 Dt 2BeverlyThe clock is at rest in Frame 2 (Chuck’s frame)

Some numerical examples of time dilation

Lorentz contractionObjects seen in moving referenceframes appear shorter along thedirection of motion than the sameobject seen at rest.[Image from http://www.mncs.k12.mn.us/physics/relativity][I will not derive the mathematical expression for time dilation and Lorentzcontraction in class, but I have made handwritten notes with the detailedderivations for whoever would like to see them. The derivations will not beon any question or exam (but just for your own pleasure )]

ChuckBeverly

Dx1 Dx 2 / gDy1 Dy 2Chuck†where g 1V21- 2c(Lorentzfactor)†Beverlyx: direction of motion -- y: direction perpendicular to motion

Some numerical examples of length contraction

Special-relativistic velocity addition (in the direction of motion)and NOT v1 v2 Vas in classical mechanics.Note: v1 never exceeds c.ChuckBeverlyNote: velocities can be positive (towards east in this example) or negative (towards west)

Special-relativistic velocity addition: an exampleChuckBeverly

Special-relativistic velocity addition: an example (continued)

Mass is relativeAn object seen in a movingreference frame appears tobe more massive than thesame object seen at rest.The relation between the two masses is given byTwo consequences:m gm0where m 0 is the mass measured at rest.When trying to bring an object to a velocity vc, its mass†m appears infinite and therefore youwould need an infiniteamount of†energynothing can move faster than light!E Mc2Mass and energy are equivalent

SOME EXPERIMENTAL TESTS OF SPECIAL RELATIVITYHigh-energy particleaccelerators: radioactiveparticles are seen tolive much longer whenmoving at speed closethan that of light thanwhen at rest (directprobe of time dilation).No matter how muchthey are accelerated,they can never reach thespeed of light.High-energy particle accelerator at Fermi Lab(near Chicago)Nuclear reactors/bombs:mass is converted inenergy (E mc2)

SUMMARY OF THE PREDICTIONS OF THE THEORY OFSPECIAL RELATIVITYRelativity of SimultaneityTime dilationDt gDt 0[subscript “0” refers to the frame in which clock andmeter are at rest]Length contraction along the direction of motion†L L0 / gSpace and Time are relative†light, and add up in such aVelocities are relative, except for that ofway that they never exceed the velocity of light.There is no reference frame in which light can appear tobe at rest.Mass is relative and mass and energy are equivalent.

Warping (or “mixing” of Spacetime)Space in one reference frame is a “mixture” of space and timefrom another reference frame.Similarly, time in one reference frame is a “mixture” of space andtime from another reference frame.The “mixture” is called SPACETIMEThe Minkowski (absolute) interval is the same in all frames:Absolute interval (distance in Frame 1) 2- c 2 ( time interval in Frame 1) 2(distance in Frame 2) 2- c 2 ( time interval in Frame 2) 2

Geometric analogy for the absolute interval:the example of the Mledinans (Thorne pp. 88-90)Absolute distance (on amap) covered is thesame for men andwomen, even thoughthey take differentpaths and have differentcoordinate systems.Figure from Thorne,Black holes and time warpsThe direction the mencall North is a mixture ofthe women’s north andeast. The direction thewomen call North is partnorth, part west,according to the men.North and east can be seen as the equivalent of space and time, and the absolutedistance as the equivalent of the absolute (Minkowski) spacetime interval.

Aberration: an example from “everyday” life Looking out of the window of . a train at rest . a train moving at 30 km/hr a train moving at 60 km/hr . a train moving at 250 km/hr[Images from http://www.fourmilab.ch/cship/aberration.html]

Relativistic aberration of Light : when the velocity ofthe elevator is close to that of lightDirection of light beam is relative:the angle that the beam makeswith the wall of the elevatordepends on the velocity of theelevator.

Traveling at relativistic velocities: movie clips by Daniel Weiskopf - Copyright 1997-2001

of light, c 299,792 km/s, is the same for all inertial observers, independent of their velocity of motion relative to the source of light. SPECIAL only applies to inertial reference frames, those for which