Spacetime Diagrams IIA2290-22 1Spacetime Diagrams IIRelativity and AstrophysicsLecture 22Terry HerterA2290-22 Spacetime Diagrams II 2Outline Spacetime diagrams Worldlines Non-constant motion Twin paradox Rotating lines of simultaneity Sending out a signalSpacetime Diagrams IIA2290-22 2A2290-22 Spacetime Diagrams II 3Spacetime diagram (or map) Plots space on one axis and time on the other Payoffs Place space and time on equal footing Review history of events and motions along a given line in spacetime Plot same events in different spacetime diagrams for different inertial reference frames Can find what is different and what is the same between the framesspacetimeOACBDsame timesame placeReference event O, along with other events A, B, C, and D. The dashed horizontal line represents events simultaneous in time while the dashed vertical line represents events at the same location.A2290-22 Spacetime Diagrams II 4Same events, different frames Considers a light flash reflected off a mirror and back as seen from three different frames rocket, lab, and super-rocket The mirrors are at rest in the rocket frame and the super-rocket is traveling in the same direction but faster (relative to lab frame) than the rocket The light paths look like that shown below:labrocketsuper-rocketSpacetime Diagrams IIA2290-22 3A2290-22 Spacetime Diagrams II 5Spacetime interval is conserved The spacetime interval is conserved (the same for each frame).(interval)2= (time)2– (space)2= constant The red hyperbola in the spacetime plots show the line of constant interval. The arrows represent the same arrow in spacetime! Maps show different perspectives of the same arrow in spacetime. Spacetime geometry is Lorentz geometry, not Euclidean.labtimeRlab spaceRrockettimerocket spaceRsuper-rockettimesuper-rocket spaceA2290-22 Spacetime Diagrams II 64Worldline A worldline tracks the passage of a particle through spacetime Every particle has a worldline The worldline is a path in spacetime The spacetime map is an image of the worldline of a particle in a particular inertial frame Clocks don’t move (in x) in lab frame but move along in time – time passes at lattice points so that worldline of clocks are vertical In the lab frame the line of simultaneity is horizontal Note: All particles have worldlines, even “stationary” ones.1235Trajectories of particles in space (not spacetime). Each particle starts at the reference clock (square) and moves with constant velocity.spacetimeO5 4 1 2 3Worldline of particles in spacetime for the lab frame. The world line for particle 1 contains a sample set of event points that make up the worldline.Spacetime Diagrams IIA2290-22 4A2290-22 Spacetime Diagrams II 7spacetimeONon-constant speed particles Accelerated particles Have curved worldlines which represent their non-constant speed (acceleration is due to a force on the particle) This is true even if the particle is moving in a spatially straight line, like a train on a track. Usefulness of spacetime diagrams – Useful for recognizing patterns of events for looking at the laws of nature but useless for influencing the events they represent (since events are recorded and sent back to the (ideal) observer)Worldline of an accelerated particle. The particle stops at Z and then continues to P.PZlimits on worldline slopeAt every point in spacetime the particle is limited by the speed of light which place a limiting slope in the spacetime diagram.A2290-22 Spacetime Diagrams II 8Length along a path Distance along a spatial path We could use a flexible tape or short stick to measure spatial distance along a track or path (Spatial distance)2= (x2- x1)2+ (y2- y1)2for each segment Sum together distances to get the total length The path lengths are the same for all surveyors Different paths give different lengths Straight line Shortest path/distance between two pointsThe total length of the curved/winding path is long that the straight line between the two end points. The increased length for any segment is given byeast2northO4681024681012142/122eastin increasenorthin increaselengthin increaseSpacetime Diagrams IIA2290-22 5A2290-22 Spacetime Diagrams II 9Proper time along a worldline Wristwatch (proper) time Sum spacetime interval along a worldline in Lorentz geometry the same way distance measures path length in Euclidean geometry (interval)2= (t2- t1)2-(x2- x1)2for each segment Sum together the intervals to get the total proper time The proper time is the same for all inertial observers Different world lines give different total proper times Straight worldline (free particle) Longest proper time between two pointsThe total proper time of the curved/winding worldline is shorter than the proper time for the straight worldline between the two end points. The increased proper time for any segment is given byspace2timeO4681024682/122spacein incrementin timeadvanceeproper timin increasePrinciple of Maximal Aging:a free particle follows the worldline of maximal agingA2290-22 Spacetime Diagrams II 10Principle of Maximal Aging A free particle follows the worldline of maximal aging In Special Relativity straight worldlines have the longest proper time between two points Free particles follow straight worldlines All inertial observers agree on which worldline is straight and has the longest proper timeIn spacetime the curved worldline is transversed in shorter proper time.in timeincreasespacein increasein timeincrease2/122northin increaseeastin increasenorthin increase2/122eastnorthOincreasein northincreasein eastPath in SpacedirectpathBWorldline in SpacetimespacetimeOincreasein timeincreasein spacestraightworldlineBSpacetime Diagrams IIA2290-22 6A2290-22 Spacetime Diagrams II 11“Kinked” Worldline For a curved worldline the clock associated with the particle is accelerating We take the limit in which we have a large acceleration over a short time The worldline is
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