Introduction to special relativity

1. Introductiontospecialrelativity Bytheend of thistopicyoushould be able to: statethemeaning of thetermframe of reference; statewhatGalileanrelativitymeans; solveproblems of Galileanrelativity; understandthesignificance of thespeed of light; statethetwopostulates of theprinciple of relativity; appreciatethatabsolute time doesnotexist, and thatsimultaneityis a relative concept.

2. Frames of reference Theobserveralongwiththerulers and clocksthat he orshe uses tomeasuredistances and times constitutewhatiscalled a frame of reference. Iftheobserverisnotaccelerated, theframeiscalledaninertialframe of reference.

3. Frames of reference Todiscussmotionwemustfirstspecifytheframe of referencewithrespecttowhichthemotionistobedescribed. Thus, anobserverwithrulers and clockswhoisfirmlyattachedtothesurface of theearthwillmeasuredifferentthingsthananobservertraveling in a train.

4. Absolute Time Assumption: twoobserversalwaysagreeonwhatthe time coordinates are; in otherwords, time iscommontobothobservers. “Absolute, true and mathematical time, of itself, and fromitsownnature, flowsequablywithoutanyrelationtoanythingexternal.”

5. Relativeness of motion Itisimpossibleforone of theobserverstoclaimthat he orsheis “really” at rest and thattheotheris “really” moving. Thereis no experimentthat can beperformedbythetrainobserver, say, thatwillconvinceherthat se “really” moves. Whateverresultsthetrainobservergetsout of herexperiments, thegroundobserveralsogetsout of thesameexperimentsperformed in hisgroundframe of reference.

6. Galileantransformations x’ = x – vt t’ = t Galileantransformations are therelationbetweencoordinates of eventswhenoneframemovespasttheotherwithuniformvelocityon a straight line. Observers in bothframes are equallyjustified in consideringthemselves to be at rest and thedescriptionstheygive are equallyvalid.

7. Non-inertialframes A non-inertialframe, bycontrast, isanacceleratingframe, and in this case itispossibletodistinguishtheobserverwhois “really” moving.

8. Law of addition of velocities u = u’ + v

9. Examplequestion A ballrollsonthefloor of a train at 2 m/s (withrespecttothefloor). Thetrainmoveswithrespecttotheground (a) totheright at 12 m/s, (b) totheleft at 12 m/s. Whatisthevelocity of theballrelativetotheground?

10. Thisapparentlyfoolproofargumentpresentsproblems. Itissaidthat Einstein, as a boy, askedhimselfwhatwouldhappenif he held a mirror in front of him and ran forward at thespeed of light. Withrespecttotheground, themirrorwouldbemoving at thespeed of light. Rays of light leavingyoungEinstein’sfacewouldalsobemoving at thespeed of light relativetotheground. Thismeantthattherayswouldnotbemovingrelativetothemirror, hencethereshouldbe no reflection in it. Thisseemedoddto Einstein. He expectedthatlookingintothemirrorwouldnotrevealanythingunusual. At theend of the 19th century, considerable effortsweremadetodetectvariations in thespeed of light dependingonthestate of motion of thesource of light. The experimental resultwasthat no suchvariationsweredetected!

11. Thespeed of light In 1864, James C. Maxwell synthesizethelaws of electricity and magnetism, and discoveredthatacceleratedelectricchargesproduced a pair of self-sustainingelectric and magneticfields at rightanglestoeachother, whicheventuallydecoupledfromthecharge and moved awayfromit at thespeed of light. Oneprediction of the Maxwell theorywasthatthespeed of light is a universal constant.

12. Thespeed of light Thisresultis in conflictwithGalileanrelativity.

13. Ether Maxwell believedthatelectromagneticwavesrequiered a medium (theether) fortheirpropagation. He wasforcedtoadmitthathisequationswerenot in factvalid in allframes of referencebutonly in a smallsubset, namelythoseinertialframesthatwere at restrelativetotheether. Ifwhilemovingthroughtheetheryouweretoemit a light signal in thedirection of motion, thevelocity of light wasexpectedtobelessthanthevelocity of light wouldbeifyouwere at restrelativetotheether.

14. Specialrelativity Einstein threwawaytheentirenotion of theether. Electromagneticwaves do not requiere a medium and thespeed of light in a vacuumisthesameforallobservers. Thelaws of physics are thesame in allinertialframes of reference. Thismeansimmediatelythatthelaws of Galileanrelativityhavetobemodified. Einstein wasthusledto a modification of thetransformationlawsfrom to

15. Postulates of theoryof specialrelativity • Thelaws of physics are thesame in allinertialframes. • Thespeed of light in a vacuumisthesameforallinertialobservers. Consequence: Absolute time doesnotexist. Observers in motionrelativetoeachothermeasure time differently!!! Space and time are nowinevitablylinked and are notindependent of eachother as theywere in Newtonianmechanics.

16. Relativity of simultaneity Eventsthat are simultaneousforoneobserver and whichtake place at differentpoints in space, are notsimultaneousforanotherobserver in motionrelativetothefirst. Ontheotherhand, iftwoevents are simultaneousforoneobserver and take place at thesamepoint in space, they are simultaneousforallotherobservers as well.

17. Examplequestion Observer T is in themiddle of a traincarriagethatismovingwithconstantspeedtotherightwithrespecttothetrainstation. Two light signals are emitted at thesame time as far as theobserver, T, in thetrainisconcerned. Are theemissionssimultaneousforobserver G ontheground? Thesignalsarrive at T at thesame time as far as T isconcerned. Do theyarrive at T at thesame time as far as G isconcerned? Accordingto G, whichsignalisemittedfirst?

18. Exercises