Chapter 36 Relativistic Mechanics

Transcription

1 Chapter 36 Relatiistic Mechanics What is relatiit? Terminolog and phsical framework Galilean relatiit Einstein s relatiit Eents and measurements imultaneit Time dilation Length contraction Lorentz transformations pacetime interal Position and elocit transformations Relatiistic dnamics Momentum Energ

2 What is relatiit? The theor of relatiit in fact refers to two somewhat distinct theories first suggested b Albert Einstein that epanded Newtonian mechanics and assume a redefined perspectie about space and time: The special theor of relatiit (905): a reconstruction of the concepts of space and time considered in inertial frames E: From the er beginning, the relatiistic conjectures proed useful in the noel fields of elementar-particle phsics, such as in the calculation of energ trades in nuclear reactions The general theor of relatiit (9-96): an etension of perspectie to non-inertial (that is, accelerated) frames E: The general theor is ultimatel a theor of grait, such as it is especiall important in understanding astronomical phenomena: planetar orbits, black holes, cosmic epansion, etc. In our discussion we ll focus on some elements and consequences of the special relatiistic theor based on an onl apparentl innocuous obseration: the laws of phsics are the same in all inertial sstems o, let s look into the repercussions of this idea upon our understanding of space and time b first reiewing some terms and the classical perspectie

3 What is relatiit? Terminolog and phsical framework The concepts of relatiit are tpicall deeloped b comparing phsical eents (occurrences defined b points in space and time) that take place in inertial frames of reference (frames moing with constant elocit relatie to other inertial frames) t E: Consider an eent (like a spark) occurring at position and time (', ', z', t') in frame. Thence, in frame, the eent occurs at position and time ',, z, t t,, z, t Eent The frames hae parallel aes and are associated with proper times (time interals between eents at the same location) starting at time t = 0 when the origins coincide a that frame (,, z) moes with constant elocit relatie to frame (', ', z'), parallel to -direction; in turn moes with elocit relatie to Recall that, in Newtonian mechanics, Newton s first law is alid onl in inertial frames, so the hold in both frames short of the difference in position; the time is absolute and independent of motion ' '

4 What is relatiit? Galilean relatiit According to the Galileo s relatiit (63), the laws of classical mechanics are the same in all inertial reference frames Thus, if we consider an object in two frames, its position and elocit are gien b the Galilean transformations: u u Position: Velocit: r t z z u u u u z u u z t t In turn, because acceleration and force are gien b the deriatie of the elocit, the do not depend on the inertial frame: r ' t z z u u u u u u z F z ' F a a

5 Quiz : According to Galilean relatiit, which of the following represents the kinetic energ of an object in the inertial frames described on the preious slides? a) Kinetic energ is inariable (that is, it is the same in both frames): b) K K m c) K K m u K K

6 What is relatiit? Einstein s relatiit At the beginning of the XX-th centur, Albert Einstein noticed that the Galilean relatiit is at odds with Mawell s model for electromagnetic waes according to which light traels in acuum with a speed epected to be independent of the reference frame: 8 m c s This inference had been also proen b the Michelson Morle eperiment (887) To eplain this behaior, he suggested two postulates which ield a noel conception of space and time:. All laws of phsics are the same in all inertial frames of reference. The speed of light in acuum is the same in all inertial frames, and it is the maimum speed possible Thus, whereas Newton conceied space and time as absolute and separate, Einstein s relatiit approached them based on operatorial definitions that demonstrated them as depending on each other and on the motion of the obserer Consequentl, the new theor made some startling predictions which since then hae been proed eperimentall: for instance, simultaneit, time interals and lengths hae been shown to depend on the relatie motion of obserers that measure them in different inertial frames

7 Eents and measurements imultaneit Because the relatiistic time is associated with each inertial frame, one ma image each such frame as containing snchronized clocks in each point Then, if two eents, and, occur in two different positions within the same frame when the reading on the clocks in the respectie points are the same, we call the eents to be simultaneous imultaneit does not mean simultaneous perception, because the information will trael with a finite speed between the eent locations E: Within the same frame, simultaneit can be easil erified b an obserer at the location of Eent at distance Δ from Eent b measuring the time interal Δt between his Eent and the moment when he sees the light from Eent. If Δ/Δt = c, then the eents were simultaneous. Quiz : Which is true if the obserer in the eample measures Δ/Δt > c? a) The eents are still simultaneous, because speed of light is absolute b) Eent occurred before Eent c) Eent occurred after Eent Eent light Eent Δ = cδt Note that this procedure can be used to snchronize all the necessar clocks within an gien inertial sstem, that is, to define a local time

8 Eents and measurements Relatiit of simultaneit Howeer, if two eents are obsered from different inertial frames, if the are simultaneous for one obserer, the ma not be simultaneous for the other obserer: this is termed the relatiit of simultaneit E: Consider two eents obsered from two different inertial frames, and moing with speed relatie to parallel with the line connecting the two eents. Then, if the eents are simultaneous for, the cannot be simultaneous for. This is because the signals will moe with the same speed in both frames, but, while is at rest with respect to the eent locations, moes toward one of the sources Quiz 3: a that in the eample aboe the eents are simultaneous for the obserer. Thence, for obserer, when did Eent occur? a) imultaneous with Eent b) Before Eent c) After Eent t 0 t t t 3 Eent Eent Wae front traeling with speed of light for both obserers encounters the light from Eent is encounters simultaneousl the light from both eents. Yet, will see Eent onl later.

9 Eents and measurements Time dilation In relatiistic mechanics, time flows differentl in different inertial frames To see how this works, consider a special clock in frame measuring a proper time Δt' using a mirror, a light source and a detector at the same location as the source The clock is also obsered in a reference frame giing a time Δt The source emits light pulses which are reflected b the mirror so the return to the detector which triggers a new pulse; the clock ticks eer time the pulse returns Light traels a longer distance in frame, to it should take a longer time Reference frame ct c t mirror Light speed is the same in both frames ct ' c t Rest frame of the clock ' mirror source t detector Therefore, the tick-interal measured in frame is gien b: source detector c c c t t c t t t t t

10 Eents and measurements Time dilation. Comments Consequentl, the time interal between two eents t (proper time) measured in two separate frames is different: t the time Δt measured in the reference frame dilates compared to the proper time Δt' measured in the rest frame o, moing clocks run slow, but this becomes obserable onl as long as is comparable to c or the time interals must be measured with er high precision E: GP. The global positioning sstem uses orbiting satellites with atomic clocks that keep time differentl from clocks on the ground. To determine an accurate position, the software in our GP receier must carefull correct for time dilation effects. Eercise : The twin parado. Luke leaes his home planet Tatooine to isit Coruscant, a planet at a distance d = 9 l from Tatooine. Meanwhile, his twin sister Leia waits for him on Tatooine. For most of his trael, Luke traels with a speed of 0.9c. a) What is the age difference between Luke and Leia when Luke returns home? t b) Recall that inertial frames are equialent: with respect to Luke, Leia moes with a elocit 0.9c. How come Leia gets older than Luke?

11 Eents and measurements Length contraction In relatiistic mechanics, length ma be different in different inertial frames To see how this works, consider a ruler in frame of proper length L. The ruler is also obsered in a reference frame giing a length L The length of the ruler is measured in using a light pulse emitted and then detected at one of its ends after it is reflected b a mirror at the other end Emission here Reference frame L t ct t source ' Rest frame of the ruler (proper length) ' mirror Detection here t L t ct detector L ct L L Because the ruler moes in frame, the signal traels different distances to and from the mirror, so the times are different (because the speed is the same c). Therefore, if we calculate the total time, we can obtain L: L L L t t t c c c t t t c L L L

12 Eents and measurements Length contraction. Comments Consequentl, the length L of an object measured in a frame where the object is at moing with constant elocit is shorter than the length L' of the object in its own frame (proper length): this is not an optical illusion! L L L Note that the length contracts as seen from the reference frame, but onl parallel with the direction of motion E: Particle accelerators. The tanford Linear Accelerator is a -mile-long electron accelerator as we measure it from our frame. Yet, due to the relatiistic speeds of the accelerated electrons, the accelerator s length is less than m in their proper frame. Eercise : Muon deca. Muons are unstable particles created when cosmic particles interact with the Earth s upper atmosphere. ubsequentl, muons moe with speeds of the order of = 0.99c, and deca with a mean lifetime in their proper frame of about Δt M = s. What is the muon mean deca distance in the frame of the Earth?

13 Quiz 4: This credible gu obseres an UFO moing with constant speed. He measures the time T that the object takes to pass him from one end to the other. Which is true about the time interal T measured b Elis, who s actuall driing the UFO? a) T = T b) T > T c) T < T Quiz 5: a that the gu also measured a length L of the UFO. What about the length L measured b Elis? a) L = L b) L > L c) L < L Problems:. Time dilation of airplanes: A jetliner flies coers the distance D = 4800 km from an Francisco to New York at a constant speed = 300 m/s. a) Find the time of the trip according to an obserer from the surface. b) Find the time of the trip according to an obserer in the plane.. Length contractions of airplanes: Consider again the jetliner in the preious problem, and sa that its length is L = 00 m. a) Find the length of the trip according to an obserer in the plane. b) Find the length of the airplane measured b two obserers on Earth, one right under the plane, and one in a tall tower at the same altitude as the plane.

14 Lorentz transformations pacetime interal We saw before that Galilean relatiit considered time and length inariants in different inertial frames. How about in Einstein s relatiit where these are not absolute? Are there an inariants in the special relatiit? The answer is es: space and time are still inariants, but onl when combined as components spanning a spacetime continuum This geometrical framework was deised b Minkowski to replace the Euclidian space appropriate for Newtonian mechanics where time was absolute The points in the spacetime continuum are eents characterized both b spatial and time coordinates, such that a spacetime interal will be gien b Time interal s c t r patial separation While Galilean relatiit prescribes the inariance under the change of inertial frame for displacements between points in Euclidian space, Einstein s relatiit prescribes the inariance of spacetime interal between eents in the Minkowskian space Problem 3. pacetime inariance: Major Tom is traeling in his tin can with a speed = 0.8c relatie to Earth, when he obseres two short radio pulses emitted on Earth s surface from two points.0 km apart, at a time interal of s. Meanwhile, the same two pulses are measured on Earth to be separated b.0 km. What is the time interal between the two pulses in the Earth s frame?

15 Lorentz transformations Position Furthermore, if Galilean transformation describes how positions and elocit change with the change of reference frame, how about the change of spacetime coordinates (,, z, t) (', ', z', t') between two inertial frames and? The job is done b Lorentz transformations which presere the spacetime interals in Minkowski space Consider two inertial frames with the same initial time and location, t = t' = 0, r 0 = r' 0 Then the spacetime coordinates of an eent in each of the two frames transform as gien b the adjacent formulas: Quiz 6: Galilean transformations are a particular case of Lorentz transformations. When is classical mechanics applicable? a) When << c b) When β 0 c) When γ d) All of the aboe r eent t z z t t c r ' t z z t t c '

16 Eercise 3: Coordinate transformation. Use Lorentz transformations to demonstrate the relatiistic length contraction of a ruler moing with a constant speed with respect to a frame. Time t ' L Time t' Reference frame Rest frame of the ruler (proper length) '

17 Lorentz transformations Velocit How how about the change of elocit u u' between two inertial frames and according to Lorentz transformations? ' u u For simplicit, assume that an object moes with elocit u in the frame parallel to -ais (and so with ) Then the elocit in each of the two frames transform as gien b the adjacent nonlinear formulas: u u u c u u u c Eercise 3: Velocit transformation. A spaceship flies past earth at 0.5c. As he goes b, it launches a bullet forward at 0.5c. If, on earth, the credible gu obseres the sk again, what is the elocit of the bullet in his frame? a) First classicall. ' 0.7c 0.7c b) Then relatiisticall.