2 Topics l Recap l Length Contraction l Cosmic Ray Muons l Spacetime l Summary
3 Recap l Postulates of Relativity l Laws of physics are the same in all inertial (that is, non-accelerating) frames l The speed of light in vacuum is independent of the motion of source and observer l Relativity of Simultaneity l Events simultaneous for one observer need not be simultaneous for another moving relative to the first
4 Recap -Time Dilation If t V is the elapsed time measured by a clock moving at speed v relative to a clock that measures an elapsed time of t R and is at rest relative to an observer then where
5 Recap -Time Dilation Very Important Point (VIP): By Principle 1, the time dilation effect must be perfectly symmetrical: If Betty and Ann are moving at a constant velocity relative to each other, Betty sees Ann’s clock run slower and Ann sees Betty’s clock run slower
6 Length Contraction You : d Y = v t Y d P = v t P Prof Consider the measurement of the length of a table by You and your professor who moves at speed v from A to B A B
7 Length Contraction Prof The ratio of your elapsed time t Y to the elapsed time t P measured by your professor is t Y / t P = , so A B You : d Y = v t Y d P = v t P
8 Summary: Moving clocks tick more slowly Moving lengths contract Again, it is important to understand that these statements must hold true for all inertial observers: every inertial observer will see moving clocks run more slowly and moving lengths contracted longitudinally Time Dilation & Length Contraction
9 Cosmic Ray Muons ~ 9 km Mean muon lifetime 2 ~ 2 microsecond Time to the ground 9km / 300,000 km/s 30 ~ 30 microsecond 15 For every 15 microseconds of our clock, the muon’s “clock” advances 1 only 1 microsecond!
10 Muon Decay – Detailed Example Muons decay according to the statistical law of radioactivity where N 0 is the initial number of muons, N(t) is the number remaining after an elapsed time t and is the mean muon lifetime of 2.2 s
11 Muon Decay – Detailed Example The time to reach the ground is about 15 times the mean muon lifetime. Therefore, according to Newtonian physics only a fraction will reach the ground. But because of time dilation and length contraction the fraction reaching the ground is predicted to be about
12 Space + Time = Spacetime The distinction between past, present and future is only an illusion, even if a stubborn one. Albert Einstein Space: is a (3-D) hyper-surface through spacetime on which all events are simultaneous, for someone Time: is the direction perpendicular to this hyper-surface
13 Earth’s Time Axis 3008 AD 2008 AD 2508 AD y x now Event: A place at a given time Spacetime: The set of all events (t,x,y,z) A B CD O worldline
14 Summary l Clocks and Rulers l Moving clocks run more slowly l Moving lengths contract longitudinally l Spacetime l Space is a hyper-surface of simultaneous events and time is the direction perpendicular to it l Worldline l A trajectory through spacetime
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