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LECTURE # 26

LECTURE # 26. RELATIVITY III LENGTH CONTRACTION SPACE TIME INTERVALS LORENTZ TRANSFORMATIONS. PHYS 270-SPRING 2010 Dennis Papadopoulos. APRIL 14, 2010. 1. Special Relativity. Einstein’s postulates X Simultaneity X Time dilation X Length contraction Space-Time intervals

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LECTURE # 26

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  1. LECTURE # 26 RELATIVITY III LENGTH CONTRACTION SPACE TIME INTERVALS LORENTZ TRANSFORMATIONS PHYS 270-SPRING 2010 Dennis Papadopoulos APRIL 14, 2010 1

  2. Special Relativity Einstein’s postulates X Simultaneity X Time dilation X Length contraction Space-Time intervals Lorentz transformation Examples Mr. Tompkins by George Gamow

  3. EINSTEIN’S POSTULATES OF RELATIVITY Postulate 1 – The laws of physics are the same in all inertial frames of reference Postulate 2 – The speed of light in a vacuum is the same in all inertial frames of reference. Simultaneity relative to the observer; causality restricts speeds to below c; Time dilation – proper time shortest time interval can be measured by a single clock.

  4. A physical activity that takes place at a definite point in space and time is called • a gala. • a sport. • an event. • a happening. • a locale.

  5. A physical activity that takes place at a definite point in space and time is called • a gala. • a sport. • an event. • a happening. • a locale.

  6. Proper time is • the time calculated with the correct relativistic expression. • the longest possible time interval between two events. • a time interval that can be measured by a single clock. • the time measured by a light clock

  7. Proper time is • the time calculated with the correct relativistic expression. • the longest possible time interval between two events. • a time interval that can be measured by a single clock. • the time measured by a light clock

  8. Change Point of View s=c(t/2) y=D Dt=2Ds/c An astronaut will measure Dto=2D/c You can easily show that Dt/Dto=Ds/D =g

  9. Time Dilation The time interval between two events that occur at the same positionis called the proper time Δτ.In an inertial reference frame moving with velocity v = βc relative to the proper time frame, the time interval between the two events is The “stretching out” of the time interval is called time dilation.

  10. Molly v Nick 1. Nick measures the time it takes for Molly’s rocket from nose to tail to fly past him as dtN 2.Molly measures the time it takes for the rocket to fly past Nick as dtM 3. Do they measure the same time and if not which is shorter? Nick’s time is the proper time since his clock does not change position. Molly on the other hand had to use two clocks one in the nose and one in the tail. dtM=gdtN. dtN is proper time and the shortest interval. It is the one and only reference frame that the clock is at rest.

  11. Lorentz factor

  12. Length Contraction C1 C2 single clock Two clocks Proper length l is distance between two points measured in a reference frame at rest to the objects 12

  13. Length Contraction The distance L between two objects, or two points on one object, measured in the reference frame S in which the objects are at rest is called the proper length ℓ. The distance L' in a reference frame S' is

  14. So, moving observers see that objects contract in the direction of motion. Length contraction… also called Lorentz contraction FitzGerald contraction

  15. Length Contraction C1 C2 single clock Two clocks S exps: L=1.43x1012 m, it takes 5300 sec to travel the distance L->v=.9c=2.7x108 m/sec. S’ exps: It takes only 2310 s to reach Saturn after they passed the Sun. No conflict because the distance is only .62x1012 m. Saturn’s speed towards them is again .9 c. 15

  16. Stationary observer – moving with the stick Observer that sees stick moving past her Fig. 1-13, p. 19

  17. Contraction ONLY in the direction of motion Transverse direction do not change Fig. 1-15, p. 21

  18. Fig. 1-14, p. 20

  19. Everything is slowed/contracted by a factor of: in a frame moving with respect to the observer. Time always runs slower when measured by an observer moving with respect to the clock. The length of an object is always shorter when viewed by an observer who is moving with respect to the object.

  20. Coordinate systems Let’s look at coordinate systems: thinking classically… x y y x The same cat, the same cat toy, different (arbitrary) choice of coordinate systems. You rolls a cat toy across the floor towards her cat. By Pythagorean’s theorem, the toy rolls: Easier visualization, easier calculation—You are happy (although her cat probably doesn’t care as long as she gets his toy.) This is an example of rotating your coordinate axes in space.

  21. Space-time Intervals D2invariant. Same value in any Cartesian coordinate system. Rotation and translation preserve it. 23

  22. Invariant space-time interval Take the space and time distance of two events. The value of h is the same in both frames Spacetime interval- INVARIANT – Same for all inertial frames

  23. Molly v C2 C1 Nick • Nick measures the time it takes for Molly’s rocket from nose to tail to fly past him as dtN while he is in the same position. • Molly measures the time it takes for the rocket to fly past Nick as dtM while her position changed by L. • 3. Do they measure the same time and if not which is shorter? Nick’s time is the proper time since his clock does not change position. Molly on the other hand had to use two clocks one in the nose and one in the tail. dt2=gdt1. dt1 is proper time and the shortest interval.

  24. The Lorentz Transformations Consider two reference frames S and S'. An event occurs at coordinates x, y, z, t as measured in S, and the same event occurs at x', y', z', t' as measured in S' . Reference frame S' moves with velocity v relative to S, along the x-axis. The Lorentz transformationsfor the coordinates of one event are: • Rules in deriving LT: • Agree with GT for v<<c • Transform not only space but time • Ensure that speed of light is constant

  25. Velocity Transformation In GT u’=u-v

  26. The derivation is for your info only S S’ Derivation of LT v x’ Galilean transformation from S to S’: x Guess that the relativistic version has a similar form but differs by some dimensionless factor G: (we know that this must reduce to the Galilean transformation as v/c ->0 and G->1) The transformation from S’ to S must have the same form: (v->-v) From first postulate of relativity-laws of physics must have the same form in S and S’ substitute into solve for t’, you get:

  27. Derivation of LT-cont. Now we need an expression for the velocity dx’/dt’ in the moving frame: Take derivatives of: where u=dx/dt From second postulate of relativity- the speed of light must be the same for an observer in S and S’ u=dx/dt=c u’=dx’/dt’=c Plug this into: and get: Solve for G:

  28. The Lorentz transformations! S S’ The transformation: v x’ x To transform from S’ back to S:

  29. Definition of length in moving frames

  30. Find speed of B wr to A Take S’ attached to A -> v=.75 c. uB=-.85 c Speed of B wr A=u’= Fig. 1-19, p. 30

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