General Physics IGeneral Physics I

Lecture 20: More Twists on Lecture 20: More Twists on Space and TimeSpace and Time

Prof. WAN, Xin （万歆）

xinwan@zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

Summary on Relativity Summary on Relativity

● The two basic postulates of the special theory of relativity are

− The laws of physics must be the same in all inertial reference frames.

− The speed of light in vacuum has the same value

in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light.

c = 3.00×108 m /s

Summary on Relativity Summary on Relativity

● To satisfy the postulates of special relativity, the Galilean transformation equations must be replaced by the Lorentz transformation equations:

t ' = γ(t − vx /c2)

x ' = γ(x − vt )

y ' = y

z ' = z

γ =1

√1 − v2 /c2where

ct

xx '

ct '

q

q

tanθ = β

Summary on Relativity Summary on Relativity

● Three consequences of the special theory of relativity are− Events that are simultaneous for one observer are

not simultaneous for another observer who is in motion relative to the first.

− Clocks in motion relative to an observer appear to be slowed down by a factor . This phenomenon is known as time dilation.

− The length of objects in motion appears to be contracted in the direction of motion by a factor . This phenomenon is known as length contraction.

γ = (1−v2/c2)−1/2

1/ γ = (1−v2/c2

)1/2

Summary on Relativity Summary on Relativity

● The relativistic form of the velocity transformation equation is

γ =1

√1 − v2 /c2where

ux is the speed of an object as measured in the K frame and

u'x is its speed measured in the K' frame.

Summary on Relativity Summary on Relativity

● The relativistic expression for the linear momentum of a particle moving with a velocity u is

● The relativistic expression for the kinetic energy of a particle is

Not relative v of K & K'

Mass is an invariant

OutlineOutline

● Relativistic Doppler effect● Twin paradox● The equivalence principle● Global navigation satellite systems

It All Started with...It All Started with...

Really, The Big Bang TheoryReally, The Big Bang Theory

● The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. It states that the universe was expanded from a very high density state.

● This theory was born of the observation of E. Hubble (1929) that other galaxies are moving away from our own at great speed, in all directions, as if they had all been propelled by an ancient explosive force.

● The glow of cosmic microwave background radiation discovered by Arno Penzias and Robert Wilson (1964), is found throughout the universe and thought to be a tangible remnant of leftover light from the big bang.

Galactic RedshiftsGalactic Redshifts

How redshifts are related to the velocities of the galaxies?

Classical Doppler EffectClassical Doppler Effect

● Finally, if both source and observer are in motion, we find the following general relationship for the observed frequency:

f ' =v+vOv−vS

f

The word toward is associated with an increase in observed frequency. The words away from are associated with a decrease in observed frequency.

Analyze Moving SourceAnalyze Moving Source

● For observer A, the wavelength is shortened to

● The frequency heard by observer A is

● For observer B, simply use a negative vS.

Relativistic Doppler EffectRelativistic Doppler Effect

x

y

z

x '

y '

z '

v

c

Assume during T measured by Bob, the source emits n waves. The length of the wave train

cT − vT

The wave length

λ =cT − vT

n

According to the source

n = f 0T ' = f 0(sT )

f =cλ

=s

1 − v /cf 0

= √ 1 + v /c1 − v /c

f 0

CommentsComments

● When the source and receiver are receding from each other, we can simply replace v by -v.

● When the source is approaching, the (visible) wavelength is shifted toward shorter wavelengths (blueshift). When the source is receding, the wavelength is shifted toward longer wavelengths (redshift).

● The result is also valid when the source is fixed and the receiver approaches it with speed v.

● For small v/c, the result conforms with the classical Doppler effect.

Twin ParadoxTwin Paradox

● Consider Twin Mary (astronaut) and Frank (stockbroker). At age of 30, Mary leaves on a mission to Alpha Centauri B, the nearest planet about the size of Earth outside of our solar system, 4 light years away. Mary will travel at a high speed to the planet and return.

Mary's biological clock will tick more slowly during her trip, so she will come back younger.

Frank will appear to be moving rapidly with respect to my system. Frank will be younger when I return.

This is the twin paradox. Who will be younger?

Calculation on the Wall StreetCalculation on the Wall Street

● Assume Mary's spaceship blasts off and quickly reaches a coasting speed of v = 0.8c.

● Mary will travel 5 years to Alpha Centauri B, and another 5 years back. So Frank will be 30 + 5 + 5 = 40 years old.

● Mary's clock is ticking slower and each leg of the trip takes only

Mary will return at the age of 30 + 3 + 3 = 36.

5√1 − 0.82= 3 years

Who Is Right?Who Is Right?

● Mary will experience acceleration, turning around, and deacceleration during her trip. Her system is, therefore, not an inertial frame.

● There is no paradox at all. ● But can they check their time lapse by sending

radio signals during the years?● Yes. But the frequency will change due to the

Doppler effect. Note that this is not time dilation, but how one perceives the frequency of the signals.

Radio CommunicationRadio Communication

● During the space trip to Alpha Centauri B, Mary will send radio signals to Frank at annual intervals. Frank, likewise, will send signals annually.

● The frequency of the signals n0 = 1 signal/year from the source.

● The receiver, moving away at a relative speed v = 0.8c, will read the frequency

● On the way back, the frequency is 3n0, or 1 signal every four month.

ν =√1 − 0.8

√1 + 0.8ν0 =

ν0

3(1 signal per 3 years)

Communication DiagramCommunication Diagram

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6

The farthest signal takes 4 year to arrive at the earth.

Frank receives 6 messages, so he knows Mary has aged 6 years.

Mary receives 10 messages, so she knows Frank has aged 10 years.

There is no way Frank knows that Mary has turned around until 9 year later.

The Happy EndingThe Happy Ending

● When Mary returns home, Mary will have been paid 10-year salary that will take her only 6 years!

● Frank has invested her 10 years of salary, making her a rich woman at the age of 36.

The General Theory of RelativityThe General Theory of Relativity

● Mass has two seemingly different properties: a gravitational attraction for other masses and an inertial property that resists acceleration.

It appears that gravitational mass and inertial mass may indeed be exactly proportional. (The value for the gravitational constant G was chosen to make the magnitudesof the two mass numerically equal.) But why?

The Equivalence PrincipleThe Equivalence Principle

● Einstein proposed that no experiment, mechanical or otherwise, could distinguish between the two cases.

FGravity ＝ Acceleration

The PostulatesThe Postulates

● The two postulates of Einstein’s general theory of relativity are

− (The principle of relativity) All the laws of nature have the same form for observers in any frame of reference, whether accelerated or not.

− (The principle of equivalence) In the vicinity of any point, a gravitational field is equivalent to an accelerated frame of reference in the absence of gravitational effects.

Free FallFree Fall

● In a free-falling elevator, gravity is cancelled by the inertial force.

An inertial frame of reference.

g

Light in a Falling ElevatorLight in a Falling Elevator

● In a free-falling elevator, according to the special theory of relativity, light propagates along a straight line in the speed of light.

g

Light in a Falling ElevatorLight in a Falling Elevator

● To a outside observer (t = 0)

g x

Light in a Falling ElevatorLight in a Falling Elevator

● To a outside observer (t = 1)

g x x

Light in a Falling ElevatorLight in a Falling Elevator

● To a outside observer (t = 2)

gx

xx

Light in a Falling ElevatorLight in a Falling Elevator

● To a outside observer (t = 3)

g

x

x

xx

An Alternative ViewAn Alternative View

● Suppose that a light pulse is sent horizontally across the elevator, while the elevator accelerates upward. This causes the light to arrive at a location lower on the wall than the spot it would have hit if the elevator were not accelerating. Thus, in the frame of the elevator, the trajectory of the light pulse bends downward as the elevator accelerates upward to meet it.

● Because the accelerating elevator cannot be distinguished from a nonaccelerating one located in a gravitational field, Einstein proposed that a beam of light should also be bent downward by a gravitational field.

Clock in a Falling ElevatorClock in a Falling Elevator

● In a free-falling elevator, the special theory of relativity is valid. Time elapses with constant intervals.

g

Synchronize with Alice Synchronize with Alice

gAlice

Bessie

Charlie

Charlie starts at the same height with Alice. Their clocks are adjusted to tick at the same speed.

A very tall tower

Compare with BessieCompare with Bessie

g

Alice

BessieCharlie

When Charlie meets Bessie, she is moving up relatively with a finite speed. So Bessie’s clock is slower than Charlie’s.

A very tall tower

Clocks Upstairs and DownstairsClocks Upstairs and Downstairs

● By comparison, Alice’s clock is faster than Bessie’s clock.● When Charlie is at the same height as Bessie

● Bessie’s clock is slower by a factor

If you live in a second-story apartment, and the thought of the universe stealing a split second of your life keeps you awake at night, science offers a solution. Just run in circles to keep your watch in synch with clocks on ground floor.

12mv2 = ΔU = mgh

s = √1 − v2/c2 ≈ 1 −

ΔUmc2

Clock Runs Slow in a G-FieldClock Runs Slow in a G-Field

● The interesting effect predicted by the general theory is, therefore, that time scales are altered by gravity. A clock in the presence of gravity runs more slowly than one located where gravity is negligible.

● [Gravitational Redshift] Consequently, the frequencies of radiation emitted by atoms in the presence of a strong gravitational field are red-shifted to lower frequencies when compared with the same emissions in the presence of a weak field.

Spacetime CurvatureSpacetime Curvature

The curvature of spacetime influences the motion of massive bodies within it; in turn, as massive bodies move in spacetime, the curvature changes and the geometry of spacetime is in constant evolution. Gravity then provides a description of the dynamic interaction between matter and spacetime.

In general relativity, spacetime is not 'flat' but is curved by the presence of massive bodies. The distortion caused by each sphere is proportional to its mass.

Experimental VerificationExperimental Verification

In 1979, John Wheeler summarized Einstein’s general theoryof relativity in a single sentence: “Space tells matter how to move and matter tells space how to curve.”

(Update 12 February 2016: high-frequency gravitational waves (i.e. ripples in the spacetime continuum), emitted by a pair of merging black holes, were directly detected for the first time with the Advanced Laser Interferometer Gravitational-Wave Observatory.)

Satellite NavigationSatellite Navigation

● Motivation and the Beginning− The USSR launched the first satellite Sputnik in 1957. − Two American physicists, William Guier and George

Weiffenbach, at Johns Hopkins's Applied Physics Laboratory (APL), decided to monitor Sputnik's radio transmissions. Within hours they realized that, because of the Doppler effect, they could pinpoint where the satellite was along its orbit.

− The next spring, Frank McClure, the deputy director of the APL, asked Guier and Weiffenbach to investigate the inverse problem— pinpointing the user's location given that of the satellite.

GPS: Some MilestonesGPS: Some Milestones

● The Gulf War from 1990 to 1991 was the first conflict in which the military widely used GPS.

● In 1991, a project to create a miniature GPS receiver successfully ended, replacing the previous 50 pound military receivers with a 2.75 pound handheld receiver.

● November 2004, Qualcomm announced successful tests of assisted GPS for mobile phones.

US Global Positioning System: http://www.gps.gov/systems/gps/

http://www.beidou.gov.cn/

● According to the China National Space Administration, the development of the system would be carried out in three steps:

− 2000–2003: experimental BeiDou navigation system consisting of 3 satellites

− By 2012: regional BeiDou navigation system covering China and neighboring regions

− By 2020: global BeiDou navigation system

On March 30, 2015, the first new-generation BeiDou Navigation satellite (and the 17th overall) was successfully set to orbit by a Long March 3C rocket.

http://www.beidou.gov.cn/

Why do we need the corrections from the theory of relativity in GPS navigation?

www.beidou.gov.cn/2012/01/05/20120105b32bf90e15e94e26a119fd799788104e.html

狭义相对论认为高速移动物体的时间流逝得比静止的要慢。每个 GPS 卫星时速为 1.4 万千米，根据狭义相对论，它的星载原子钟每天要比地球上的钟慢 7 微秒。另一方面，广义相对论认为引力对时间施加的影响更大， GPS 卫星位于距离地面大约 2 万千米的太空中，由于 GPS 卫星的原子钟比在地球表面的原子钟重力位高，星载时钟每天要快 45 微秒。两者综合的结果是，星载时钟每天大约比地面钟快 38 微秒。一般说来， GPS 接受器准确度在 30 米之内就意味着它已经利用了相对论效应。

Corrections for GPSCorrections for GPS

● The GPS space segment consists of a constellation of satellites transmitting radio signals to users. The United States is maintaining the availability of at least 24 operational GPS satellites, 95% of the time.

● A GPS satellite orbits the earth with a period of 12 hours. ● According to Lecture 7, the orbit radius (you should work

it out!)

r = ( gRe2T 2

4π2 )

1 /3

= 2.66×107 m

v = ωr = ( 2πgRe2

T )1/3

= 3.87×103 m /s

每个 GPS 卫星时速为 1.4 万千米

The Special Relativity EffectThe Special Relativity Effect

● According to time dilation (e.g., from Lecture 18, Slide Ex: Time Dilation)

● After one day, the time difference is

Δ τs

Δ t= √1−v2 /c2 ≈ 1 −

v2

2c2

Δ τs ≈ −v2

2c2Δ t = −7.18×10−6 s

根据狭义相对论，它的星载原子钟每天要比地球上的钟慢 7 微秒。

The General Relativity EffectThe General Relativity Effect

● Due to the difference in gravity

● After one day, the time difference is

Δ τg

Δt= 1 +

ΔUmc2

= 1 +gRe

2

c2 (1Re

−1r )

Δ τg ≈gRe

2

c2 (1Re

−1r ) Δt = 4.55×10−5 s

由于 GPS 卫星的原子钟比在地球表面的原子钟重力位高，星载时钟每天要快 45 微秒。

Importance of the CorrectionsImportance of the Corrections

● The total effect is that the satellite clock is faster that a clock on the earth's surface

● The error in position due to the effect of relativity is

Δ τ = Δ τg + Δ τs = 3.83×10−5 s

Δ L = cΔ τ = 1.15×104 m = 11.5 km

两者综合的结果是，星载时钟每天大约比地面钟快 38 微秒。一般说来，GPS 接受器准确度在 30 米之内就意味着它已经利用了相对论效应。 (?)

The End of RelativityThe End of Relativity