2 Review of Scientific “Theories” Recall discussion from the first day of classA scientific “theory” is a proposed explanation/description for observed factsIt is possible for a theory to be a good approximation or have some usefulness even if it is not fully correctOne of the best examples is “Newtonian” Physics vs. Relativity & Quantum Mechanics
3 Newtonian Physics Physics principles as explained by Newton and others Newton’s 3 Laws and Law of GravityMaxwell’s Equations of ElectromagnetismEquations for motion, momentum, kinetic energy, etc. discussed earlier in this classUnderlying foundations of space and time as absolute
4 Relativity and Quantum Mechanics New physics as described by Einstein and others, most of the work done in the early 1900sTime dilation, length contractionUncertainty principleBohr Theory of the AtomDifferent fundamental assumptions about the Universe
5 The Special Theory of Relativity Aimed to answer some burning questions:Could Maxwell’s equations for electricity and magnetism be reconciled with the laws of mechanics?Where was the aether?
6 The ConflictNewtonian physics seems to describe the world as we are used to itHowever, several experiments as well as some hypothetical arguments signaled some problemsRelativity and Quantum Mechanics improve upon Newtonian physics
7 Newtonian PhysicsNewtonian physics accurately describes the Universe when…Speeds are not too largeGravity is not too strongYou are at a macroscopic level, i.e. not dealing with individual molecules/atoms
8 Newtonian Physics, cont. Under the conditions of the previous slide, there is no reason to use anything other than Newtonian physicsEquations give the same results to high accuracyExample: Trajectories of satellites and space probes use Newtonian physics
9 RelativityRelativity is a set of physics concepts and laws deduced by primarily by Albert EinsteinSpecial RelativityPublished by Einstein in 1905“Special” case with no forces/accelerationGeneral RelativityPublished by Einstein in 1915Extension of previous theory to include forces
11 What ISN’T Relativity?Relativity does not simply mean “everything is relative”On the contrary, relativity says certain things are relative, and other things are absoluteRelativity also tells us by how much those certain things are relative and in what way
12 Experimental and Theoretical Need for Relativity Michelson-Morley ExperimentSpeed of light is the same regardless of the Earth’s motion through the aether (“absolute space”)Maxwell’s Equations of ElectromagnetismPredict very unusual things, like magnetic fields with “loose ends”, when speeds are extremely large
13 Michelson-Morley Experiment For a long time, scientists believed in an “aether”—absolute spaceIn the Michelson-Morley experiment, the speed of light was measured “with” and “across” the “flow of the aether” as the Earth moved through it
14 Michelson-Morley Experiment FlashContrary to expectations, however, the speed of light was the same both “with” and “across”!
15 Theoretical Foundations of Relativity To explain all of these things, Einstein came up with new laws of physics based on two assumptionsThe laws of physics are the same in all inertial (non-accelerating) framesThe speed of light is the same as measured by all observers in all inertial framesEinstein took these principles “on faith”The principles and their implications have passed subsequent experimental testing
16 Relatively Speaking What do Einstein’s two assumptions imply? All motion is relativeRelativity of simultaneityRelativistic velocity additionTime dilationLength contractionRelativistic mass increaseE = mc2
18 All Motion is Relative, cont. You and your friend Jackie like to travel in bizarre spherical spaceshipsWho is moving?Who is stationary?
19 All Motion is Relative, cont. In spite of our everyday intuition, the only velocities that can be measured are relative velocitiesExamples:Relative to the surface of the EarthRelative to the SunRelative to a distant galaxy
20 Galilean Relativity 1,000,000 ms-1 1,000,000 ms-1 How fast is Spaceship A approaching Spaceship B?Both Spaceships see the other approaching at 2,000,000 ms-1.This is Galilean or Classical Relativity.VT = V1 + V2Mention the Ether
21 Einstein’s Special Relativity 0 ms-1300,000,000 ms-1Moving with respect to the Ether1,000,000 ms-1Both spacemen measure the speed of the approaching ray of light.How fast do they measure the speed of light to be?
23 Addition of Velocities In normal circumstances, if you are moving and throw an object, an outside observer will see the object at a different velocityStraight-forward velocity additionBut all observers measure the speed of light to be the same
24 Velocity Additions Do Not Apply to Light Even if you are moving away from your friend at a very high velocity, you will both see a light beam moving at c.Nice to know formula
25 Relativistic Velocity Additions A formula for adding velocities exists, but it is not required for the course.The formula works such that you can never get velocities greater than cFor small velocities, is approximately the same as just adding the velocities
28 Relativity of Simultaneity Two lights an equal distance from M go offA passing train carries M’M’ sees the light from B firstM see the light flashes at the same timeM’ is moving in the direction of BThis relativity is determined by the speed of light and the relative motion of the objects/observers
29 Relativity of Simultaneity Events which are simultaneous in one frame may not be in another!Each observer is correct in their own frame of reference
30 The Lorentz FactorCalculating length contraction, time dilation, and other quantities requires calculating the Lorentz factor = v/cIf v = 99% of c, then = 0.99 is always < 1 1
31 The Lorentz Factor, cont. Some examples:v = 0.1% of c =v = 1% of c =v = 10% of c = 1.005v = 50% of c = 1.155v = 90% of c = 2.294v = 99% of c = 7.089v = 99.9% of c = 22.37
32 Time DilationDistance = Velocity x TimeA clock using light pulses to keep time. Every time the pulse returns, a unit of time has passedDNote:Consider a clock that uses a light pulse to tick . . .This time is known as Proper Time. Because the clock is rest the frame of the occurring event. The Proper Time interval between two events is always the time interval measured by an observer for whom the two events take place at the same position.
33 Time DilationL=vDt½ cDtD½ vDtVWe are now watching the clock move horizontally with velocity v. We will examine one cycle, more specifically one-half of one cycle. During a cycle of the light photon the clock will have moved horizontally a distance L, and if we calculate the distance travelled by the light in this one cycle (a upside down V), the distance would be c times the time we measured for the cycle, that is ct. So for one-half cycle the distance travelled by the light is ½ ct.Consider a clock that uses a light pulse to tick . . .Now sinceThis says a moving clock run slow. If ts =1 then you watching it move would notice it taking more than 1s (tm >1) on your clock, so ts runs slow.
34 Time Dilation ExampleYou and a friend are having a eating contest. Your friend is on a train traveling at speed v=0.9 c. By her watch, she finishes her food in 5 seconds. Determine the time you measure, if you are standing still at the train station.Since eating is happening on the train, that is the “proper” time, ts=5.
35 Time Dilation Example 2 Both people think they won! Now it is your turn to eat. According to your watch you finish your food in 5 seconds. How long does your friend think it took you to finish the food?Now eating is happening at the station, so that is the “proper” time, again ts=5.Your friend would consider you to be moving. Remember the proper time is where the event and clock are togetherBoth people think they won!
36 Space TravelAlpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri). How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.?How long do people on the ship think it takes?People on ship have ‘proper’ time since they see earth leave, and Alpha Centauri arrive. DtsDts = 1.4 years
37 Space TravelAnother approach that solves any special relativity problem by treating space and time as spacetime.The only requirement is that both separated units are recorded in the same units. (i.e.: light seconds, light minutes, light years, …)
38 Space TravelAlpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri).How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.?How long do people on the ship think it takes?An amazing technique is to place time and space in the same units then use the following relativistic formula:
39 Time Dilation ReviewTime flows more slowly in a moving frame as observed by an outside observerBut remember motion is relativeIf you and I are moving past each otherI see your clock moving more slowlyBut you also see mine moving more slowly…!!!
40 Length ContractionObjects moving relative to an outside observer appear contracted in the direction of their motion as measured by the observer
41 Length Contraction, cont. If you and I move past each other in some sweet sports carsI measure your sports car as being shorterYou measure my sports car as being shorterOnly applies to the direction of motionWe see our sports cars as still being the same height
43 Length Contraction Example People on ship and on earth agree on relative velocity v = 0.95 c. But they disagree on the time (4.5 vs 1.4 years). What about the distance between the planets?Earth/Alpha: d0 = v t= .95 (3x108 m/s) (4.5 years)= 4x1016m (4.3 light years)Ship: d = v t= .95 (3x108 m/s) (1.4 years)= 1.25x1016m (1.3 light years)Length in moving frameLength in object’s rest frame
44 Twin Paradox Traveling twin will be younger! Twins decide that one will travel to Alpha Centauri and back at 0.95c, while the other stays on earth. Compare their ages when they meet on earth.Earth twin thinks it takes 2 x 4.5 = 9 yearsTraveling twin thinks it takes 2 x 1.4 = 2.8 yearsTraveling twin will be younger!Note: Traveling twin is NOT in inertial frame!
45 QuestionYou’re eating a burger at the interstellar café in outer space - your spaceship is parked outside. A speeder zooms by in an identical ship at half the speed of light. From your perspective, their ship looks:(1) longer than your ship(2) shorter than your ship(3) exactly the same as your shipLs > LMIn the speeder’s reference frameIn your reference frameAlways <1
46 Comparison: Time Dilation vs. Length Contraction Dto = time in same reference frame as eventi.e. if event is clock ticking, then Dto is in the reference frame of the clock (even if the clock is in a moving spaceship).Lo = length in same reference frame as objectlength of the object when you don’t think it’s moving.Time seems longerfrom “outside”Dt > DtoLength seems shorterfrom “outside”L0 > L
47 Relativistic Mass Increase Einstein made two other surprising discoveries…Mass must increase with speed, as viewed by an outside observerDue to conservation of momentumThere is “leftover” energy even when the object is at restDue to conservation of energyE = mc2
48 Relativistic MassRest massRest massActually written
49 E = mc2 E = mc2 = m0c2 This E is the total energy of an object When the object is at rest…v = 0 = 1E = m0c2 (“rest mass energy”)The reason that energy can be released through fusion/fission
50 Total EnergyRelativistic kinetic energy is the extra energy an object with mass has as a result of its motion:We can solve this for the Kinetic energy of an object:
51 Relativistic Momentum Note: for v<<c p=mvNote: for v=c p=infinityRelativistic EnergyNote: for v=0 E = mc2Note: for v<<c E = mc2 + ½ mv2Note: for v=c E = infinity (if m<> 0)Objects with mass can’t go faster than c!
52 QuestionCalculate the rest energy of an electron (m=9.1x10-31 kg) in joules.Calculate the electron’s Kinetic energy if it is moving at 0.98c.
53 Simultaneous? Simultaneous depends on frame! A flash of light is emitted from the exact center of a box. Does the light reach all the sides at the same time?At RestYESMovingNOSimultaneous depends on frame!
54 Simultaneous?Many times, questions are concerned with the determination of the spatial interval and/or the time interval between two events. In this case a useful technique is to subtract from each other the appropriate Lorentz contraction describing each event.
55 Three Other Effects 3 strange effects of special relativity Lorentz TransformationsRelativistic Doppler EffectHeadlight EffectWe will consider three consequences of a constant speed of light
56 Lorentz Transformations But what happens if the beam is moving?
57 Lorentz Transformations Light from the top of the bar has further to travel.It therefore takes longer to reach the eye.So, the bar appears bent.Weird!
58 Doppler Effect The pitch of the siren: The same applies to light! Rises as the ambulance approachesFalls once the ambulance has passed.The same applies to light!Approaching objects appear blue (Blue-shift)Receding objects appear red (Red-shift)When the ambulance is approaching, the sound waves are bunched up.Likewise, once the ambulance has passed, the waves are spread out.
59 Headlight effect Beam becomes focused. VCounter intuitive resultBeam becomes focused.Same amount of light concentrated in a smaller areaTorch appears brighter!
60 Warp Program used to visualise the three effects Notes for the demo. Demonstrate Doppler shift.Demonstrate Lorentz Transformations – including being able to see the back of objects when they are in front of you!Demonstrate Headlight effect by turning it on/off.
63 UnderstandingAn observer has a pendulum that has a period of 3.00 seconds. His friend who happens to own a spaceship (with cool engines), zooms by the stationary pendulum. If the speedometer of the spaceship says 0.95c, what will the friend measure are the period of the pendulum?Since I am with the pendulum, my measured time is the Proper Time.This makes sense because a moving clock would run slower from my perspective. So the pendulum would have a period of 9.6s.
64 Understanding Vega is 25 light-years away Travel to Vega at 0.999c The length would appear contracted to youAbout 1 light-yearMake the trip in ~1 light-year (each way) as measured by youEarth would measure 25 years each wayYou would spend 2 years (your time) travelling and arrive 50 years in the future Earth time.
65 Understanding Strange but True! You throw a photon (3x108 m/s). How fast do I think it goes when I am:Standing stillRunning 1.5x108 m/s towardsRunning 1.5x108 m/s away3x108 m/s3x108 m/s3x108 m/sStrange but True!
66 UnderstandingA 1.0 m long object with a rest mass of 1.0 kg is moving at 0.90c. Find its relative length and massUse length contraction formula:Mass increase formula:
67 UnderstandingFor a 1.0 kg mass moving at 0.90c. Find the rest energy and kinetic energy of the objectFor rest energy, Use energy formula:For Kinetic energy, Use relativistic energy formula:Now:Therefore
68 Understanding A person’s pulse rate is 65 beats per minute. If the person is on a spaceship moving at 0.10c, what is the pulse rate as measured by a person on Earth?What would the pulse rate be if the ship were moving at 0.999c?a) Use time dilation:a) Use time dilation:We need time for a heart beat
69 Understanding A muon at rest has an average lifespan of 2.20 x 10 -6 s What will an observer on Earth measure as its lifespan if it travels at 0.990c?What distance would we observe it travel before disintegrating?What distance would it travel if relativistic effects were not taken into account?a) Time dilation:b) Distance formulac) Distance formulaTo Show consistencyThis is the distance the muon measures it travels before disintegrating
70 UnderstandingYou measure the length of an object as 100m when it passes you at 0.90c. What is its length when at rest?Use length contraction formula:
71 UnderstandingAs a rocket ship sweeps past the Earth with speed v, it sends out a light pulse ahead of it. How fast does the light pulse move according to the people sitting on the Earth?
72 UnderstandingA train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train?Units:We are given that tb-ta=0 and what we want to determine is tb’-ta’
73 UnderstandingA train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train?The negative sign reminds us that even a occurred after event b