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1、Simple, Interesting, and Unappreciated Facts about Relativistic AccelerationJohn MallinckrodtCal Poly Pomona2003 AAPT Winter MeetingAustin Texas江西不孕不育医院 srsgyyQuestionnDoes relativity allow an object to “accelerate as a rigid body?nAnswer: Yes. “Simply apply an appropriate amount of force to every p

2、iece of the object.FollowupnIf an object is accelerating as a rigid body, how does the acceleration of its “front end compare to that of its “rear end?na) Obviously, they would have to be the same so that all points move with the same speed at all times.nb) Obviously, points nearer the “front end wo

3、uld have to have smaller accelerations so that the object Lorentz contracts properly.nc) Both of the above. (If not, why not?)Outline of TalknAnalyze a simple worldline that will turn out to be that of a point object undergoing constant proper acceleration.nSummarize its geometric characteristics on

4、 a spacetime diagram.nConsider pairs of separated particles that start from rest, maintain constant proper accelerations, and either a) have identical accelerations, or b) maintain their proper separation.nExtend the analysis to continuous bodies.nFind a constraint on the length of a rigidly acceler

5、ating object and understand its connection to the existence of an event horizon for rigidly accelerating reference frames.nConsider the behavior of clocks on a rigidly, but otherwise arbitrarily moving object.nWatch a simulation of a rod that accelerates from rest to a maximum speed and decelerates

6、back to rest.nSummarize main points.Claims, Disclaims, and AcknowledgementsI believe this material is accessible, surprising, uncontroversial, but nevertheless not well known.On the other hand, there are closely related questions that I feel less competent to discuss.A few (incomplete) references:Ha

7、milton (AJP, 46, 83)Desloge and Philpott (AJP, 55, 252)Desloge (AJP, 57, 598)Desloge (AJP, 57, 1121)Nikolic (AJP, 67, 1007)Taylor and Wheeler “Spacetime PhysicsMould, “Basic Relativity (Especially Chapter 8).(If Im lucky, they wont come up.) Classical Features of the Motion of Interest Consider the

8、motion described in an inertial frame byIn units where c = 1 and with s = “vertex distance = constant. NoImageUsing nothing more than the definitions of v and a it is easy to show that NoImageNoImageNoImageNoImageNoImageRelativistic Features of the Motion of Interest 1How much does the moving object

9、 “feel its speed change when its speed observed in the inertial frame changes from v to v + dv? (Lorentz velocity addition)NoImageHow much time elapses in the frame of the object during a time dt in the inertial frame? (Time dilation)NoImageThus, the proper (“felt) acceleration is given byNoImageTha

10、t is, the proper acceleration is constant and inversely proportional to the vertex distance.Relativistic Features of the Motion of Interest 2How does the proper time change along the trajectory?Note that this formula for the elapsed proper time depends only on the velocity as measured in the inertia

11、l frame (which is monotonically changing) and the vertex distance and that it is directly proportional to the vertex distance.NoImageWith tP(v = 0) = 0, we can integrate to findNoImageNoImageNoImageNoImageGeometric ConsequencesnAn event horizon exists and prevents any causal connection to events on

12、the “dark side of that horizon.NoImageNoImageNoImageNoImagenThe worldline for an object undergoing constant proper acceleration is a hyperbola that corresponds to the locus of all events having a constant spacelike separation from “the origin. (x2 - t2 = 2 = constant)nThe proper acceleration is simp

13、ly the inverse of that separation.(ap = 1/nAs the object accelerates, the “line of instantaneous simultaneity always passes through “the origin. (dx/dt = t/x)Generalization toArbitrary WorldlinesThe instantaneous position, velocity, and acceleration of an arbitrarily moving object associates it with

14、 a unique “instantaneous constant (proper) acceleration worldline.NoImageTwo Objects with Identical Constant AccelerationEach object moves along a hyperbolic path having its own asymptotic light coneThe separation is constant in the inertial frameA and B disagree on matters of simultaneityIn fact, B

15、 might even say that As trajectory is time-reversed except for the fact that that portion of As worldline is hidden behind Bs event horizonNoImageTwo Objects with Identical Asymptotic Light ConesNoImageSince the vertex distances are different, so are the accelerationsThe “front object, B, has a smal

16、ler proper acceleration than the “rear object, AA and B agree at all times on matters of simultaneityA and B agree at all times on their common velocityA and B agree that their proper separation is constantA and B agree that Bs clock runs faster in direct proportion to their respective vertex distan

17、ces.A Rigidly Accelerating Rod that Flashes SynchronouslyNoImageConsider the sequence of events in both the inertial and accelerating frames.Note that, even within the frame of the rod, flashing synchronously is not the same as flashing at a definite time interval because the clocks run at different

18、 rates.Extension to the Dark Side of the Event HorizonConsider a family of hyper-bolic worldlines sharing the same focus (vertex).Any spacelike line through the focus is a “line of instan-taneous simultaneity and intersects all worldlines at positions of identical slope (velocity).Positive accelerat

19、ions on the right, negative on the left.Why cant a rigid body straddle the vertex?NoImageAccelerating as a Rigid Body up to a Final SpeedNoImageHow do we get a Lorentz contracted rod in the inertial frame?Must accelerate to a uniform velocity in the inertial frame.Requires the rear end to stop accel

20、erating before the front.Clocks are not synchronized in the moving frame.Note subsequent penetration of the former “event horizon.Generalized Rigid Body MotionNoImageUnder uniform acceleration the worldlines of the front and rear are identical but shifted. The body does not Lorentz contract.Under “r

21、igid body acceleration the bodys motion is arbitrary as long as the acceleration of the “front end never exceeds 1/L.The worldline for the “front end is always less curved (smaller a) than that of the “rear end.Clocks return to synchronization whenever they return to the velocity at which they were

22、synchronized.SimulationHere is an Interactive Physics simulation of the end points of a rigid body (of adjustable length) whose rear end accelerates with constant proper acceleration from rest to an (adjustable) maximum velocity and then back to rest with the opposite proper acceleration. Summary of Main PointsnUniformly accelerating the various parts of a body increases their proper separations and develops tensile stresses within the body.nBodies can accelerate rigidly, but only by accelerating nonuniformly.nThe worldlines of different positions (as observed in an inertial frame) follow a f