Physics 116 Session 29 Relativity Nov 17, 2011 R. J. Wilkes Email: ph116@u.washington.edu
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
Paradox? Last time: • � 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: Bill Plots of time vs position = worldlines Phil time time Me Bill Me (at rest (moving (at rest on earth) backward) on train) Phil (moving forward) x coordinate (earth) x’ coordinate (train) P Jill Me time J B (at rest 0.1 c on bike) Jill’s picture of the same situation: Bill Phil She’s riding a bike (slower than train). (moving (moving She is at rest relative to her bike . backward) forward) Notice: slopes of worldlines = relative speed Vertical = 0 speed More slanted = faster speed x’’ coordinate (bike) 6
Your worldline in spacetime • � Every object has a worldline – for example, you! Your worldline, in rest frame of room A-118 Recall: slope of time worldline ~ speed Future leaving class “Elsewhere” “Light cone” = worldline of light YOU moving outward from here/now You can’t influence events here Nothing can go faster! – and they can’t affect you! position Here, right now 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 d B 0.1 c Train A B Earth • � 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) ~ 10 15 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 L 1 =x B - x A = c ! t 1 But observer 2 (on train) sees meter stick moving toward him with speed v Points A and B move past him in time L 2 =v ! t 2 , 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
Recommend
More recommend
