Physics 116
Session 29 Relativity Nov 17, 2011
R. J. Wilkes Email: [email protected] 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 Lecture Schedule (up to exam 3)
Today
3 Last time: Paradox? •! Each one says the other’s clock is slow: contradiction •! Einstein says: both are correct 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! •! 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?
4 Einstein’s insight: we live in 4 dimensions, not 3 Last time: 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
5 Example of worldlines
•! Can’t draw pictures in 4D, but can sketch motion in 1 space coord: time Bill Plots of time vs position = worldlines time Phil Me Bill Me (at rest (moving (at rest on earth) backward) on train) Phil (moving forward) x coordinate (earth) x’ coordinate (train)
P Jill time Me B J
(at rest 0.1 c on bike) Bill Jill’s picture of the same situation: (moving Phil She’s riding a bike (slower than train). backward) (moving She is at rest relative to her bike. forward) Notice: slopes of worldlines = relative speed Vertical = 0 speed x’’ coordinate (bike) More slanted = faster speed 6 Your worldline in spacetime
•! Every object has a worldline – for example, you! Your worldline, in rest frame of room A-118 time Recall: slope of worldline ~ speed Future leaving class
“Elsewhere” “Light cone” = worldline of light YOU You can’t influence events here moving outward from here/now – and they can’t affect you! Nothing can go faster!
Here, right now position Sitting in class
Past
coming to class
7 Time dilation
Use light clock to analyze difference in time between frames:
D D P d P B d
0.1 c Train Earth A B
•! In Earth frame: Train frame viewed from Earth:
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
8 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: We would detect almost no muons at sea level if this were true 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: 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 10 Time dilation: Lorentz factors
Time dilation factor depends on v relative to c:
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 light clock’s frame, 7 years have passed in frame where v = 0.990 c relative to clock
We can go the other way:
So if 1 year passes in frame where v = 0.990 c, only 0.14 years have passed in light clock’s frame
11 Length contraction too! Fitzgerald contraction
•! 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 object 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
2 1 A B v “Twin paradox”
•! 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
Q: How can it take only 3.6 yrs? Star A: the star is closer for twin 2 ! 1 2 25.3 ly in Earth frame = v
Earth 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 Notice that p ~ Newtonian for small v/c p blows up as v gets closer 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
14 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: *“If relativity is proved right, the Germans will –! Gravitational time dilation and redshift call me a great German, the Swiss will call me a –! Deflection of light by gravity great Swiss, and the French will call me a great –! Gravitational waves citizen of the world. If relativity is proved wrong, the French will call –! Black holes me a Swiss, the Swiss will call me a German, •! Applications confirming GR today and the Germans will call me a Jew.” -Einstein –! 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 15