The structure of science and common sense The speed of ...phys16/lectures/sl09.pdfThe structure of...

transcript

The structure of science and common sense

The speed of light

Time dilation

The twin paradox

The Doppler effect

The twin paradox and inertial frames

The structure of science and commonsenseWhat I tell you today is true, but you won’t believeit or feel it in your bones

no reason to expect Newtonian mechanics orcommon sense to work at large velocities

what we feel in our bones depends on what we areused to

discipline of modern science allows to say sensiblethings even when our common sense doesn’t work

from www.raremaps.com

science as exploration of parameter space

working way out from the known to the unknown

one direction is increasing speed

what does the world look like at large velocities?

the reason you don’t understand is that you are slow

when the rules change - there be dragons

The speed of light

c ≡ 299, 792, 458 m/s

exactly

the meter is defined in terms of the second

c is the speed of light “in vacuum” - “empty space”

c is fast, but not infinitely fast

with modern electronics, we see the effect of thespeed of light often

The speed of light

c ≡ 299, 792, 458 m/s

exactly

The speed of light

c ≡ 299, 792, 458 m/s

exactly

the surprising thing about light in a vacuum is thatthe speed of light that we measure doesn’t depend onthe velocity of the object that produced the light andit doesn’t depend on the velocity of the measuringapparatus - light-speed-meter always reads the same

c=299792458 v →

c=299792458← v

c=299792458................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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this doesn’t make any sense at all!!!

absolutely crazy but built into the way the worldworks

the surprising thing about light in a vacuum is thatthe speed of light that we measure doesn’t depend onthe velocity of the object that produced the light andit doesn’t depend on the velocity of the measuringapparatus - light-speed-meter always reads the same

c=299792458 v →

c=299792458← v

c=299792458................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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this doesn’t make any sense at all!!!

absolutely crazy but built into the way the worldworks

the principle of relativity - uniform motion is relative

there is no absolute sense in which I can say I ammoving

there is no preferred notion of standing still

we can tell if our motion is not uniform - we can feelacceleration in our bones

but uniform motion is not detectable, so long aseverything else we need is moving along with us

assume, with Einstein, that it remains true atrelativistic speeds - then my light-speed-meter mustwork as advertised

Inertial framesinertial frame of reference already plays an importantrole in Newtonian mechanics

formalize the notion that motion is relative in anoperational way - define what velocity means bydescribing precisely what we need to measure it

speed of light - measure it by dividing the distancetraveled by the time taken

But first, you have to synchronize your clocks!

inertial frame is a real or imaginary collection ofclocks that are fixed with respect to one another andsynchronized

Inertial framesinertial frame of reference already plays an importantrole in Newtonian mechanics

formalize the notion that motion is relative in anoperational way - define what velocity means bydescribing precisely what we need to measure it

speed of light - measure it by dividing the distancetraveled by the time taken

But first, you have to synchronize your clocks!

inertial frame is a real or imaginary collection ofclocks that are fixed with respect to one another andsynchronized

two fundamental assumptions together are amazinglypowerful

A. The laws of physics are the same in all inertialframes.

B. One of the laws of physics is that the velocity oflight is a constant — with the same value in allinertial frames.

Time dilationobservations of a single clock moving with speed vdone with a number of clocks fixed in an inertialframe show the ticking of the moving clock slowed

down by a factor of√

1− v2/c2

mirror

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idealized “light clock” made out of two parallelmirrors and a pulse of light bouncing back and forthbetween them

mirror

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each tick of the clock is the time for the pulse to getfrom one mirror to the other = L/c

mirror

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now suppose the two mirrors are mounted on paralleltracks a distance L apart and moved down the trackswith velocity v

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from the point of view of the many clocks in theinertial frame, the light pulse has to go farther whenthe single light clock is moving

mirror

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if light always travels at the same velocity, the ticksof the light clock take longer when it is moving

mirror

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call the factor by which the ticks are longer γ - eachvertical transit of the light from one mirror to theother in the moving frame takes time γL/c becauseL/c is the time for a tick in the rest frame

mirror

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because light travels at the same speed, c, the lengthof the path from one mirror to the next is thereforeγL/c - vertical distance is L - horizontal distance isv · γL/c

mirror

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(γL)2 = (γLv/c)2 + L2

γ =1√

1− v2/c2

this can’t just be a special property of light clocks,because if we used some other kind of clock tomeasure the time, and got a different result, then wewould be able to distinguish between the movingframe and the frame in which the light clock is fixed,violating the principle relativity - every kind of clockmust tick out seconds at the same rate

γ > 1 - remember the light clock

This is not science fiction!

Global Positioning System - based on atomic clocksin satellites moving at about 4000 m/s - relativisticeffects are small but crucial

QM clocks in unstable elementary particles likemuons - we can watch these internal clocks ticking(at least in an average sense) by watching theparticles decay - compare moving muons with muonsat rest

The twin paradoxmaybe in time dilation we have just confused theissue with a bizarre definition of measurement

perhaps even the GPS and our experiment on decaysof elementary particles are just a bad definition ofwhat we mean by the ticking of their internal clocks

maybe there is some alternative way of constructingour light-speed meters so that the speed of light isnot constant and the bizarre features of relativity goaway

Think again! Round trip travel at relativistic speedsproduces the twin paradox.

Twin 1 travels to planet X and back to earth atrelativistic speeds. Twin 2 stays behind. When Twin1 returns, he is younger than Twin 2. This is a directcomparison of identical clock in the same place. Itcannot be just definition. This is a real effect. Veryaccurate experiments have been done showingexactly this effect with the internal clocks of unstableelementary particles.

the twin paradox is so peculiar that I will work outan example of how it looks to the two twins who areaging differently - we must first first understand therelativistic Doppler effect

The Doppler effectthe speed of a light beam does not change when wego from one inertial frame to another - verysurprising - but its frequency does change - notsurprising at all, since the same thing happens forsound or any other wave

called the Doppler effect and shows up, for example,in the change in sound of a train whistle when thetrain goes by

Doppler effect for sound

suppose a train is moving towards me at speed v andits whistle emits sound waves which, when the trainis at rest, have frequency ν0, and wavelength λ0

speed of sound, Vs, is the product of the frequencyand the wavelength

Vs (m/s) = λ0 (m/cycle) · ν0 (cycles/s)

period = T0 = ν−10 = time between crests

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Vs−vν0

sound meter ............................................................................................................................................ | t = 0train

sound meter ............................................................................................................................................ | ............................................................................................................................................ | t = 1/ν0train

wave crests are closer together than if the train wereat rest - λv is the wavelength for the train at speed v

λv =Vs − v

λ0 =Vs

Vs − v

Doppler effect for light

a rocket is moving towards me at speed v and itsbeacon emits light waves which, when the rocket isat rest, have frequency ν0, and wavelength λ0

speed of light, c, is the product of the frequency andthe wavelength

c (m/s) = λ0 (m/cycle) · ν0 (cycles/s)

period = T0 = ν−10 = time between crests

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Vs−vν0

rocket looks same as train except for time dilation.........................................................................................

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γcν0

γvν0

γ(c−v)ν0

light meter ............................................................................................................................................ | t = 0rocket

light meter ............................................................................................................................................ | ............................................................................................................................................ | t = γ/ν0rocket

t = γν0

> 1ν0

- we see the rocket clocks ticking slowly

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Vs−vν0

rocket looks same as train except for time dilation.........................................................................................

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γcν0

γvν0

γ(c−v)ν0

t = γν0

> 1ν0

- we see the rocket clocks ticking slowly

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γcν0

γvν0

γ(c−v)ν0

wave crests closer together than for a rocket at rest -λv is the wavelength for the rocket at speed v

λv =γ(c− v)

λ0 =c

γ(c− v)=

√1− v2/c2

1− v/c=

√√√√1 + v/c

1− v/c

√√√√1 + v/c

1− v/c

and the light has higher frequency when the rocket ismoving towards us - called blue-shift because raisingfrequency in the optical spectrum is a shift towardsthe blue

again, if the rocket is moving away, the argument isexactly the same — we just have to replace v → −v -called red-shift

√√√√1 + v/c

1− v/c

There is one very important distinction to noteabout the relativistic Doppler effect versus thenonrelativistic version. In the relativistic version, itdoesn’t matter whether the rocket is approaching theobserver at speed v or the observer is approachingthe rocket at speed v. It can’t, because of theprinciple of relativity. This is not true for thenonrelativistic Doppler effect because the air inwhich sound moves defines a special frame.

The twin paradox and inertial framesDoppler effect to understand time dilation and thetwin paradox by keeping track of every tick of themoving clock

twin 1 takes a trip to planet X, at distance L fromearth, but stays in constant communication with twin2 behind on earth using radio waves, or some otherelectromagnetic waves that have fixed frequency in1’s frame (the rocket frame). We will assume thetypical form of the twin paradox that we discussedabove, where 1 goes out to planet X at speed v,quickly turns around, and returns at the same speed.

1 leaves 2 at t = 0 traveling at constant v – stayingin constant radio communication

t = 02

1 v →X

1 arrives at planet X at t = L/v, turns around(quickly) and sends a turn-around signal to 2,indicating that he has reached the planet.

t = L/v2

←↩ v 1/X

1 leaves 2 at t = 0 traveling at constant v – stayingin constant radio communication

t = 02

1 v →X

1 arrives at planet X at t = L/v, turns around(quickly) and sends a turn-around signal to 2,indicating that he has reached the planet.

t = L/v2

←↩ v 1/X

turn-around signal reaches 2 at t = tX = L/v + L/c

t = tX2/

← v 1

1 and 2 are reunited at t = 2L/v

t = 2L/v2

a radio is a clock – # of ticks (cycles) is t ν – # ofticks sent by 1 is T ν, where T is the total time heaged on the trip

t = tX2/

← v 1

t = 2L/v2

a radio is a clock – # of ticks (cycles) is t ν – # ofticks sent by 1 is T ν, where T is the total time heaged on the trip

t = tX2/

← v 1

t = 2L/v2

a radio is a clock – # of ticks (cycles) is t ν – # ofticks sent by 1 is T ν, where T is the total time heaged on the trip - we would expect ν × 2L/v

t = tX2/

← v 1

t = 2L/v2

a radio is a clock – # of ticks (cycles) is t ν – # ofticks sent by 1 is T ν, where T is the total time heaged on the trip - we would expect ν × 2L/v

t = 02

1 v →X

t = L/v2

←↩ v 1/X

t = tX tX = L/v + L/c2/

← v 1

t = 2L/v 2L/v − tX = L/v − L/c2

# of cycles received by the 2 is

√√√√1− v/c

1 + v/c

√√√√1 + v/c

1− v/c

v− L

√√√√1− v/c

1 + v/cL

(1 + v/c

√√√√1 + v/c

1− v/cL

(1− v/c

= ν2L

√1− v2/c2 = ν T or T =

√1− v2/c2

time dilation and the twin paradox!

# of cycles received by the 2 is

√√√√1− v/c

1 + v/c

√√√√1 + v/c

1− v/c

v− L

√√√√1− v/c

1 + v/cL

(1 + v/c

√√√√1 + v/c

1− v/cL

(1− v/c

√1− v2/c2 + ν

√1− v2/c2

note that 1 sent the same # of ticks on the way andthe way back

But now, you say, why isn’t the situationsymmetrical? After all, from 1’s point of view, 2(along with the rest of the earth) has moved away atspeed v and then come back at the same speed. Whyis it that 1 is younger at the end, rather than 2?Let’s look at the trip from 1’s point of view,assuming that it is 2 who is sending out radio signalsthe whole time. Now things look a bit different.

at t = 0, 1 watches 2 and the earth recede atconstant v -

t = 0← v 2

planet X reaches 1 at t = T/2, turns around(quickly) and starts to recede

t = T/22

X v ↪→

1 and 2 are reunited at t = T

t = T2

Now we can check that the two pictures areconsistent. The number of cycles that 1 receives is

(√1− v/c

1 + v/c+

√1 + v/c

1− v/c

[√(1− v/c

1 + v/c

)(1− v/c

1− v/c

√(1 + v/c

1− v/c

)(1 + v/c

1 + v/c

(1− v/c√1− v2/c2

+1 + v/c√1− v2/c2

)= Tν

1√1− v2/c2

Relativity may be strange, but it is consistent.

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