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Special RelativitySpecial Relativity22

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TopicsTopics

Recap

Length Contraction

Cosmic Ray Muons

Spacetime

Summary

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RecapRecap

Postulates of Relativity Laws of physics are the same in all inertial

(that is, non-accelerating) frames The speed of light in vacuum is

independent of the motion of source and observer

Relativity of Simultaneity Events simultaneous for one observer

need not be simultaneous for another moving relative to the first

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Recap -Recap - Time DilationTime Dilation

If tV is the elapsed time measured by a

clock moving at speed v relative to a clock that measures an elapsed time of tR

and is at rest relative to an observer then

R Vt t

where 2

2

1

1vc

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Recap -Recap - Time DilationTime Dilation

Very Important Point (VIP):

By Principle 1, the time dilation effectmust be perfectly symmetrical:

If Betty and Ann are moving at a constantvelocity relative to each other, Betty sees Ann’s clock run slower and Ann sees Betty’sclock run slower

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Length ContractionLength Contraction

You:dY = v tY

dP = v tP

Prof

YY Y

PP

P P

ddd

t t

tt

Consider the measurement of thelength of a table by You and

your professor who moves at speed v from A to B

AB

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Length ContractionLength Contraction

Prof

The ratio of your

elapsed time tY to the elapsed

time tP measured by your

professor is tY / tP = , so

AB You:dY = v tY

dP = v tP

YY P

P

dt

td

PYd d

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Summary: Moving clocks tick more slowly

Moving lengths contractAgain, 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 Time Dilation & Length ContractionContraction

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Cosmic Ray MuonsCosmic Ray Muons

~ 9 km

Mean muon lifetime ~ 22 microsecond

Time to the ground 9km / 300,000 km/s ~ 3030 microsecond

For every 1515microseconds ofour clock, themuon’s “clock” advancesonly 11 microsecond!

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Muon Decay – Detailed Muon Decay – Detailed ExampleExample

/0( ) tN t N e

Muons decay according to the statistical law of radioactivity

where N0 is the initial numberof muons, N(t) is the numberremaining after an elapsed timet and is the mean muon lifetimeof 2.2 s

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Muon Decay – Detailed Muon Decay – Detailed ExampleExample

15 73 10e

The time to reach the ground is about 15 times the mean muonlifetime. Therefore, accordingto Newtonian physics only a fraction

will reach the ground. But because of time dilation andlength contraction the fractionreaching the ground is predictedto be about 1 0.4e

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Space + Time = SpacetimeSpace + 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 tothis hyper-surface

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Earth’s Time Axis

3008 AD

2008 AD

2508 AD

yx

now

Event: A place at a given time Spacetime: The set of

all events

(t,x,y,z) A

B

CD

O

worldline

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SummarySummary

Clocks and Rulers Moving clocks run more slowly Moving lengths contract longitudinally

Spacetime Space is a hyper-surface of

simultaneous events and time is the direction perpendicular to it

Worldline A trajectory through spacetime