Special Theory of Relativity An Introduction

Transcription

1 Speial Theory of Relatiity An Introdution DJM 8/

2 Bakground Most basi onept of relatiity is as old as Galilean and Newtonian Mehanis. Crudely speaking, it is: The Laws of Physis look the same in different frames of referene In Newtonian Relatiity: This applies to Mehanis In Einstein s Relatiity: This applies to all Laws of Physis When applied to all laws, diffiult issues arise. Impliations seem to iolate intuition masses inrease, lengths are shortened, time expands Has generated fasination, has brought glamour... Bottom line: STR is orroborated by experiment Until th entury, Newton s Laws (F dp/dt, F ab -F ba, onseration of energy and momentum) were held true (and still are) and absolutely belieed in (not any more, as written originally)

3 Bakground Reall: Newton s Laws are alid only in an inertial frame An inertial frame of referene is one in whih Newton s st Law holds; it exists only in empty spae; it is at rest w.r.to stars) Basi Newtonian mehanis issue: A state of a system is speified at some time t o by giing position oordinates (x o,y o,z o ) and eloity ( ox, oy, oz ) at t o. Can alulate position (x,y,z) and eloity ( x, y, z ) at a later time t by knowing all fores ating on the system and applying Newton s laws. Often desirable to speify suh a state in terms of a new set of oordinate axes, whih is moing with respet to the first Three important questions arise: How do we transform the state from the old to the new frame? i.e. How do we onert x,y,z,t x,y,z,t? What happens to Newton s Laws under this transformation? What happens to E&M theory (Maxwell s Equations)? Theory of Relatiity onerns itself with these questions

4 Bakground Albert Einstein formulated the modern Theory of Relatiity. He proposed two suh theories: The Speial Theory of Relatiity 95 - Deals with the ase of an inertial frame of referene moing with onstant eloity with respet to another inertial frame Its onsequenes are most important as The General Theory of Relatiity 95 - Deals with the ase of an inertial frame of referene aelerating with respet to another inertial frame. A relationship results between aelerated motion and graitational effets It is the urrent theory desribing the graitational interation Not based on quantum mehanis Requires knowledge of tensor alulus

5 The S and S Inertial Frames y y Eent y y t -t onstant x x x x z S S z x x - t x x t t t

6 Newtonian Relatiity (Some History) Galilean Transformation of spae-time Two obserers (,) are in their own, separate, inertial frame of referene S and S S moes with respet to S with onstant along x-axis Eah obseres the same eent giing position and time as measured in their own frame of referene (x,y,z,t ; x,y,z,t ) Eah has own meter stik and lok When temporarily at rest, stiks same length, loks synhronized i.e. at the instant t, then t and x in S x in S When S is moing with respet to S, the state (x and t) of the eent as seen by the two obserers is related by: x x -t - gies x x t y y (S S ) y y z z z z t t t t

7 Newtonian Relatiity (Some History) Transformation of eloities and aelerations follow diretly by taking deriaties x x - and a x a x y y a y a y z z a z a z This is the Newtonian answer to Question # (intuitie) Important impliit assumptions: spae is absolute, time is absolute dx x Note (definition): x lim et. dt t t Implies that, for meaningful x, measurement of x and t must be made with respet to a single (the same) referene frame When using Galilean Transformations this is not important beause time is absolute and, therefore, t t

8 Newtonian Relatiity (Some History) All laws of mehanis are the same in all inertial frames Example: Newton s nd Law under Galilean transformation When a mass aelerates, let the fore ating on it be measured by S and S. They find, respetiely: F ma and F ma But a a Therefore F F And, therefore, Newton s nd Law retains its mathematial form in both inertial frames (is the same, inariant ). All other laws of mehanis are also inariant, as they are deriable from nd Law Note assumption: mass m is the same in both S and S i.e. m m This is Newtonian answer to Question # Corollary: Impossible to proe by mehanial tests that one frame is at absolute rest rather than in motion with onst. w.r. to another one. ( Mehanis looks the same in both) Corollary: All inertial frames are equialent; Length, time and mass are independent of relatie motion of the obserer

9 Failure of Newtonian Relatiity (Some More History) Maxwell s (4) equations desribe eletromagnetism suessfully Maxwell s equations predit the existene of e.m. waes propagating through free spae with speed 3. x 8 m/s Question #: With respet to what frame is to be measured? Question #: Through what medium do e.m. waes propagate? 9th entury answer: Ether hypothesis - existene of a massless but elasti substane permeating all spae Eletromagneti waes propagate through the ether is to be measured with respet to the ether An experiment is needed to test the presene of ether! Mihelson-Morley Experiment (88) Premise: If one beliees in the ether (and that is 3x 8 w.r. to it), and in Maxwell s Eqns, and in Galilean transformations for E&M, then if one measures the speed of light in a frame moing w.r. to ether, one should be able to measure of this moing frame. Plan: Measure earth s speed as it moes through ether Result: Null; an not detet any motion of the earth through ether

10 Failure of Newtonian Relatiity (Some More History) Impliations: Question (remember #3?): Do Maxwell s equations obey Newtonian Relatiity? (i.e. do they stay inariant under Galilean transformations, under the ether hypothesis?) Short Answer!: No! If they did, equations would produe for the speed of light (and in a moing spae ship, eletrial or optial phenomena would be different than in a stationary spae ship; then, some one on the spaeship ould measure the speed of the spae ship) Contradited by Mihelson-Morley experiment. Another problem: Maxwell s equations seem to predit the same alue for for all inertial frames. Therefore: If the soure of the e.m. waes is moing, the e.m. wae is predited to moe through spae with the same. But: This is ontradited by Galilean transformations. Experiments support 3. x 8 m/s in all frames

11 What to Do? Are Maxwell s equations not the same in all inertial frames? Maxwell s equations ould be wrong (only years old!) If so, Galilean transformations would be retained and Maxwell s equations should be fixed to obey Galilean transformation and A preferred inertial frame must exist whih should produe different alues for as one moes through it. Keep looking! Are Galilean transformations wrong? Then, keep Maxwell s equations as they are; experiments OK Either hypothesis would hae to be abandoned Absolute spae and absolute time would hae to be abandoned What about? All moing frames seem to gie same alue: 3. x 8 m/s Poinare: A total onspiray of nature is itself a law of nature Einstein to the resue: Fruitless to ontinue to show experimentally that a speial inertial frame exists. Proposes Two Postulates.

12 Speial Theory of Relatiity Einstein s Two Postulates (95) Postulate I (Priniple of Relatiity) All laws of physis must be the same ( inariant ) in all inertial referene frames (Can not detet absolute uniform motion) Postulate II (Constany of ) The speed of light in auum is onstant (same alue, 3. x 8 m/s) in all inertial referene frames, regardless of motion of soure or obserer

13 Some Obserations Problems soled by STR: Keep Maxwell s equations Aept measurements of, all produing 3. x 8 m/s No need for either build a theory on experimental results, not the other way around! New issues arise: Galilean transformations are abandoned need new transformation equations Newton s laws don t obey Postulate I any longer hae to be hanged Absolute spae and absolute time must be abandoned Constany of mass (and onseration of mass) must be abandoned Laws are OK the way they are as long as << New and non-intuitie preditions appear Experimental erifiation needed Aeptane by sientifi ommunity is slow

14 Consequenes - Simultaneity Usual Meaning: Two eents ourring at the same time Simultaneity is not an absolute onept anymore It depends on motion of the obserer Taken for granted in Newtonian physis But it doesn t matter, sine << in Newtonian world Preise new definition needed: Two eents in an inertial frame of referene are simultaneous if light signals from the loations of these eents reah an obserer positioned halfway between them at the same time Consequenes: Two eents simultaneous in one inertial frame are not simultaneous in another frame in motion w.r.to the first t(x); t t ; t and t are intimately intertwined with x,x

15 Consequenes - Time Dilation Cloks slow down when seen by obserer moing w.r. to them! Consider the duration (time interal T) of an eent Let eent be: a light ray goes from the origin of S to a mirror at y d, is refleted there, and returns to the origin. If S and S are momentarily at rest, both agree, using their own loks and meter stiks, the duration of the round trip ( proper time, i.e. measured in the rest frame of the eent, at a fixed loation) is: y d Mirror T T Next: let S moe along x-axis of S with onstant. d S will measure the time interal again. Call it T x x z S

16 Consequenes - Time Dilation As seen by S, light sent by S follows path shown, taking time t to get from the origin of S to the mirror and bak New time interal T, measured by S, is T t, where ( t) d ( t) sole for t : t, T or y t t y t t y onstant x x d x d T γt where γ - Sine γ always, z S z z S T T (Relatiisti T > Proper T)

17 Time Dilation Example: Muon (µ) Lifetime When at rest in the laboratory (i.e. in its rest frame ) the µ has (aerage) lifetime τ o. x -6 s, before it deays. At the speed of light it would trael ~6 m before deaying We know µ are produed high in the atmosphere, ~ km up, from ollisions of osmi rays with the atmosphere How are we able to obsere them on the earth? As measured by an earth based obserer, the life time of the µ is dilated ( non-proper time interal): τ γτ o Sine a high energy (relatiisti) µ has speed then γ >> and τ >> τ o Therefore it has enough time to reah the earth and be obsered

18 Consequenes - Length Contration A length gets shortened when measured by obserer in motion with respet to it! Consider the length between the earth and a star. A spae ship (S ) traels from the earth to the star moing with onstant speed along x-axis of an obserer at rest on the earth (S ) Earth obserer measures the distane to the star L ( proper length ) and omputes the time it takes for the spae ship (the moing frame) to get there as T L /; (T is non-proper T). Spae ship obserer, at rest in S, sees the star moing with -; measures time T ( proper time, at fixed plae) it takes to reah the star and omputes distane ( non-proper L) to be: L T The two time measurements are related by T γt Therefore, substituting T in L : L T /γ, or: L L γ Sine γ > always, L < L (Relatiisti L < Proper L)

19 Length Contration Example: Human Spae Trael A 5 year-old astronaut wishes to trael from earth to Alpha Pegasus (distane L lt yr away), using a fast spaeship Will he/she get there in his/her lifetime? Use.95 for spaeship speed w.r. to earth (γ 3.) Astronaut measures T L / (L /γ) /(3.*.95) 3.8 yr (measures proper time interal, omputes ontrated length) Therefore, astronaut will be and feel years old upon arrial, will get there alie and well. Earth obserer omputes: T L / yr/.95yr/ltyr 5 yr for astronaut to get there, (measures proper length, omputes dilated time interal), T γt 3.*3.8 5 yr Biologial, (e.g. heart beat) and other proesses on spaeship will slow down as seen by earth obserer

20 Lorentz Transformations Reall, Galilean transformations are not alid when Need new equations to transform (x,y,z,t ) (x,y,z,t ) Assume relatiisti transformation for x x is the same as the lassial one exept for an unknown onstant γ, therefore, x γ(x -t ) () and x γ(x t ) () y y y y z z z z t? t? γ must be the same in both equations (Postulate I). Substitute x from () in () and sole for t x γ ( x t ) x x x t (3). Form ; Set (Postulate II) γ t t t γ Sole for γ. Substitute γ in (), (), (3) and get final result:

21 Note: Lorentz Transformations are alid for any. As, they redue to the Galilean Transformations, Therefore, G. T. are good enough for Newtonian Mehanis > produes imaginary, non-physial numbers. Therefore beomes a limiting speed for physial phenomena Lorentz Transformations x t t z z y y t x x x t t z z y y t x x or

22 Lorentz Transformation of Veloities By differentiation, x y z x x y γ z γ If x, then x x x or x x y γ z γ i.e. if in one frame, then in any frame These equations refer to measurements done in different frames. Vetor addition of eloities in the same frame remain the same x y z x x

23 Lorentz Transformation of Veloities Example Two spaeships A and B approah the Earth from opposite diretions Eah ship moes with.8 with respet to the Earth How fast is eah ship moing with respet to the other? i.e. what is the speed of one spaeship measured in the rest frame of the other? Must establish S and S : Let S be the earth, let S be spaeship A, moing w.r.to S with speed.8 Eent is spaeship B, moing with speed B -.8 in S Speed B of spaeship B (eent) relatie to A (S ) is: B B B (-.8)(.8).98 Note: B (Galilean) B

24 Relatiisti Dynamis Newton s Laws are inariant under Galilean transformations but not under Lorentz transformations. They must be hanged Experiment shows that Newton s Laws: Fore laws (with p m) Conseration of energy Conseration of momentum may retain their (mathematial) form IF the lassial onept of mass is modified: m onstant, independent of relatie motion between objet and obserer. Conseration of mass is abandoned! In relatiisti physis, let: m m(), that is: A mass m moing with speed with respet to an obserer is different than the mass m of the same objet when at rest with respet to the obserer ( proper mass or rest mass ) Can show: m m or, m m γ Sine γ > always, m > m (Relatiisti m > Proper m)

25 Relatiisti Dynamis Newton s Laws (now relatiistially orret): r p r m m γm r r dp F dt New Definition of Relatiisti Mass, Seond Law of Momentum, where then : Motion s K W F ds Definition of Work, K.E. r r [ Σp ] [ Σp ] Conseration of Momentum i initial r i r final Note: As and m >> m, een a small requires a large F to produe it. Example: Going from to. (.), p.m. Going from.98 to.99 (same.), p.m (or times more). Therefore F is times more in the seond ase What about Energy/Conseration of Energy?

26 Relatiisti Energy Kineti Energy (in one dimension): K dp m Fds ds dp d dt 3 m d m m Or: K m -m (m - m ) γm -m () Note: m m( ) ; m m() Interpret m() as Rest Energy E, E m No lassial equialent, but not in onflit with Newtonian physis Can add rest energy on both sides of any Newtonian energy balane without destroying it Interpret K E as Total Energy E, E m from (). i.e. E K E m γm γe

27 Examples Find the eloity of a 6 GeV eletron K 6 GeV 3 E K m K 6x MeV Find γ: γ.7468x E m m.5mev Find : γ (.7468x ) Find the rate at whih the sun loses mass if it radiates energy at the rate of 3.85 x 6 W P E/s 3.85 x 6 W M/s (E/ )/s [(3.85 x 6) /(3. x 8 ) ]/s 4.8 x 9 kg/s 4

28 Conseration of Energy The rest energy m is of more than asual interest Proesses exist where the total rest mass is not onsered Experiment shows that a hange E is ompensated by a hange K so that the total energy hange E, therefore: The Total Energy E is the onsered energy quantity now. Law of Conseration of Energy applies to total energy E In relatiisti physis there are no separate laws of onseration of mass and energy If there is Potential Energy inoled, then E K V E K is equal to the inrease in the mass of the objet when it moes with speed As, K / m, as it should, (binomial expansion) Note: K an not be obtained from K / m by using m γm

29 Energy - Momentum Relationship In Newtonian physis: K p /m relates K to p To find orresponding expression in relatiisti physis, use: E γm and p γm with γ ( - / ) -/ to derie: E p (m ) (ery important/useful) E -p (m ) is inariant (same alue in all frames) Interesting limiting ases: If << : Get K p /m E m m... m m m If : E >> E, therefore E p and K E If m, E p, E K and must hae otherwise E m γm If (and E ), must hae m (and E p) otherwise m γm (/ is OK!) p p p m

30 Units: Mass, Energy, Momentum E p (m ) In nulear/partile physis preferred unit of energy is MeV MeV.6 x -3 J Sine m or m represent energy, their units are MeV Therefore, mass m an be gien in units of MeV/ Example: Rest mass of a proton is m p 938MeV/ ; This means: rest energy of a proton is m p 938 MeV Similarly: p has units of MeV Therefore, p an be gien in units of MeV/ Example: A proton with (total) E GeV has momentum p: p E ( m ) ( MeV) 938MeV MeV ( 938 ) 766MeV/

31 Example Compute the momentum of a 5 MeV proton From E p (m ) p E And from E K m Compute: p E E (K m ) (m ) (5 MeV 938 MeV) (938 MeV) 9 MeV/ Redo, using system of units where Conenient sine, now, p, E, m will all hae units of energy Here E p m and E K m Therefore: p ( K m ) m (5MeV 938MeV) 938 MeV 9 MeV in the regular system of units, would write p 9 MeV/

32 Relatiisti Two-Body Kinematis Elasti Sattering K Gien: m, m, K, θ Find: K, K, θ m K m p m θ θ P Conseration of Energy (using ): E m E E () Conseration of Momentum: p p osθ p osθ () p sinθ -p sinθ (3) Reall: E K m and E p m E K m and E p m E K m and E p m Sole (), (), (3) for K, K, θ m P K

33 Relatiisti Two-Body Kinematis Elasti Sattering Results () Where A, A, ρ, ρ, γ are gien by: ( ) ( ) [ ] ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ' ' ' ' ' ' ' ' ' ' ' ρ θ ot ρ γ ρ ρ otθ ρ ρ otθ θ os p m E θ os p m E 4m A p osθ m E A E θ os p m E θ os p m E 4m A p osθ m E A E ± ± ±

34 Relatiisti Two-Body Kinematis Elasti Sattering Results () ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) m m E m m m E γ p m E 4m A m E A p ρ p m E 4m A m E A p ρ m K m m m A m K m m m A

35 Relatiisti Two-Body Kinematis Elasti Sattering - Obserations Note: Not any θ, θ are possible in two-body kinematis Square root in E, E must be > sin sin θ θ ' ' A A 4m 4m [( ) ] m m m K 4m p [( ) ] m m m K 4m p Note: If m m then θ θ < π/ (unlike lassial mehanis)