Physics 116 Session 29 Relativity Nov 17, 2011 R. J. Wilkes Email: - - PowerPoint PPT Presentation

• R. J. Wilkes

Email: ph116@u.washington.edu

Physics 116

Session 29

Relativity

Nov 17, 2011

Announcements:

• Updated quiz score totals will be posted on WebAssign tomorrow
• Nice series on PBS covering topics we will discuss in class:

Brian Greene’s Fabric of the Cosmos http://www.pbs.org/wgbh/nova/physics/fabric-of-cosmos.html

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Lecture Schedule

(up to exam 3)

Today

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Paradox?

• Each one says the other’s clock is slow: contradiction
• Einstein says: both are correct
• Paradox does not occur if we reject relativity

Galileo/Newton/Maxwell “classical physics” universe: – Time is absolute, and universal (same in all reference frames) – Ether rest frame = absolute universal reference frame

• Speed of vehicle adds/subtracts from light speed
• So, Q: Why believe in such a silly concept as relativity?

1899, Henri Poincare:

“…The simultaneity of two events, or the order of

their succession, as well as the equality of two time intervals, must be defined in such a way that the statements of the natural laws be as simple as possible.” Einstein took Poincare seriously! Last time:

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Einstein’s insight: we live in 4 dimensions, not 3

A: worse contradictions and complications arise if we don’t!

• The world makes sense only if we treat time in the same way as

space coordinates - then

– Maxwell’s equations work in any (non-accelerated) frame – Michelson’s experiment is explained (And many, many other things…)

• 3 space dimensions (up-down, N-S, E-W) + time = 4 dimensions

– Universe occupies a 4-dimensional space-time continuum – Time is also relative: it’s just another coordinate

• We don’t find it peculiar that Bill and Phil measure different location

coordinates for the same event - why not different times? – Objects trace out “world-lines” in 4D spacetime

• “Event” = something that occurs at some point in spacetime

– Emission of light pulse, detection of light pulse: some interaction

• Events are what physics observations must agree upon

– Not their coordinates! Any coordinate frame is OK

Last time:

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Example of worldlines

• Can’t draw pictures in 4D, but can sketch motion in 1 space coord:

Plots of time vs position = worldlines

Jill’s picture of the same situation: She’s riding a bike (slower than train). She is at rest relative to her bike. Notice: slopes of worldlines = relative speed Vertical = 0 speed More slanted = faster speed x coordinate (earth) time x’ coordinate (train) time Bill (moving backward) Me (at rest

• n earth)

Me (at rest

• n train)

Phil (moving forward) x’’ coordinate (bike) time Bill (moving backward) Me (at rest

• n bike)

Phil (moving forward)

0.1 c P B J

Jill Bill Phil

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Your worldline in spacetime

• Every object has a worldline – for example, you!

“Light cone” = worldline of light moving outward from here/now Nothing can go faster! position time

Future

Recall: slope of worldline ~ speed

Past “Elsewhere” Here, right now

You can’t influence events here – and they can’t affect you!

coming to class leaving class Sitting in class YOU

Your worldline, in rest frame of room A-118

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Time dilation

Use light clock to analyze difference in time between frames:

• In Earth frame:

Train frame viewed from Earth:

0.1 c Earth

Train P B P

A B

d d D D

This is what Bill says about Phil’s clock – it ticks every Notice: Exactly the same calculation for Earth frame viewed from train: Phil says Bill’s clock ticks at 10.05 ns intervals

Cosmic ray muons: time dilation confirmed

• Cosmic rays = high energy protons and nuclei that circulate

in our Galaxy for millions of years.

• When they strike our atmosphere, they smash air nuclei and

make pion particles which in turn produce muons.

• Muons decay after 2.2 microsec in their own rest frame.

Note: time in your own rest frame = your “proper time”

• The muons are “relativistic” – they have v ~ c
• They are typically produced at 15 km altitude
• They are detected abundantly at sea level

How far could a muon travel in 2.2 microsec of Earth time? Suppose a muon has v = 0.999c in our (Earth = rest frame) coordinates: But actually, relative to our frame, 2.2 microsec in the muon’s proper time is dilated to a much longer time, so it can travel much further: We would detect almost no muons at sea level if this were true

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Postulates of Special Relativity

• 1. The laws of physics are the same in any inertial reference frame
• Inertial frame = coordinate system where Newton’s Laws apply
• In general: coordinate origin has no acceleration
• New idea: this applies not just to mechanics but to all physics
• 2. The speed of light in vacuum c is the same in all inertial frames,

independent of motion of source or observer

• Really new idea! Seems bizarre at first glance
• Not so crazy if rate of passage of time differs between frames

That’s all ! Simple, but revolutionized our picture of the universe

• Light clock comparison shows that operational definition of time does

differ between observers in different inertial frames

• Remember the meaning of what we found:
• !t = a time interval in frame where clock is at rest (its proper time)
• !t’ = !t as measured from a frame with speed v relative to clock frame

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Time dilation: Lorentz factors

Time dilation factor depends on v relative to c: So if 1 year passes in light clock’s frame, 7 years have passed in frame where v = 0.990 c relative to clock

Remember, !t = a time interval in frame where clock is at rest (its proper time) !t’ = !t as measured from a frame with speed v relative to clock frame

So if 1 year passes in frame where v = 0.990 c, only 0.14 years have passed in light clock’s frame We can go the other way:

• We can also use light as a measuring stick: the distance from A to B is

measured in units of (light speed)*(time for light to go from A to B): Distance light travels in 1 year = 1 light-year (ly) ~ 1015 meters

• “Proper length” of an object = its length measured in a frame where the
• bject is at rest
• What is L’ = length of a meter stick as measured by observer moving with

speed v relative to the rest frame of the object?

Notice: v (relative speed) is the same in both frames Length in meter stick’s rest frame (observer 1) is L1 =xB - xA = c !t1 But observer 2 (on train) sees meter stick moving toward him with speed v Points A and B move past him in time L2 =v !t2 , but he knows time ticks more slowly for observer 1, and c is the same for both

Length contraction too! Fitzgerald contraction

2

v

1 A B

• Famous example – but there is no real “paradox”!
• Twins 1 and 2 are 20 years old when 2 travels to a star 25.3 light-years

away, with constant speed* v=0.990c

• How long does it take by twin 1’s clock?
• How long does it take according to twin 2’s clock?

we know !t’ = 25.6 yr (time according to twin 1 who is moving backward at speed v=0.990 c relative to spaceship), we want to find !t (in spaceship clock’s rest frame) Notice: v (relative speed) is the same in both frames

“Twin paradox”

1 2

v

Earth Star

Q: How can it take only 3.6 yrs? A: the star is closer for twin 2 ! 25.3 ly in Earth frame =

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Relativistic momentum and energy

• We find that Newtonian momentum p = mv also needs a Lorentz factor, if

the particle is moving at significant speed compared to c

• We can view this as meaning that mass, in effect, grows with v
• We can apply Newtonian mechanics calculations using this version of m

– Notice: a = F/m -- this means we need an ever-increasing force to maintain constant acceleration of an object – and can never reach v=c if m > 0

• Einstein showed that the total energy of an object is given by

– Einstein’s most famous result! Notice contradiction of classical physics: any object has non-zero energy at rest, and mass itself is a form of energy Then since E = rest energy + kinetic energy, must have Notice that p ~ Newtonian for small v/c p blows up as v gets closer to c !

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General relativity

• Einstein, 1915: extended relativity to accelerated frames: general relativity

– GR really describes the geometry of spacetime: gravity of massive objects warps spacetime in their vicinity – Equivalence Principle: Observations cannot distinguish a uniformly accelerated frame from a uniform gravity field – Eddington, 1919: GR predictions matched observed anomalies in orbit of Mercury, Newtonian predictions do not – Einstein is right*

• More predictions and consequences of GR:

– Gravitational time dilation and redshift – Deflection of light by gravity – Gravitational waves – Black holes

• Applications confirming GR today

– GPS satellite orbits: precision needed requires GR calculations – Gravitational lensing, black holes: astronomical observations confirm – Gravitational wave astronomy: see http://www.ligo-la.caltech.edu/LLO/overviewsci.htm

– Notice: LIGO is a variety of Michelson apparatus!

• We’re still looking for unexplained anomalies: UW is a center for this work

– See http://www.npl.washington.edu/eotwash/index.html *“If relativity is proved right, the Germans will call me a great German, the Swiss will call me a great Swiss, and the French will call me a great citizen of the world. If relativity is proved wrong, the French will call me a Swiss, the Swiss will call me a German, and the Germans will call me a Jew.” -Einstein