Relative Motion Red Car sees Blue Car travel at U Blue Car throws Ball at V' What is V – the speed of the Ball seen by Red Car? Newton’s Principia in 1687. Galilean Relativity Velocities add: V= U +V' V’= 25m/s V = 45m/s U = 20 m/s TheThe GalileanTransformationsGalileanTransformations Consider two reference frames S and S'. The coordinate axes in S are x, y, z and those in S' are x', y', z'. Reference frame S' moves with velocity v relative to S along the x-axis. Equivalently, S moves with velocity −v relative to S'. The Galilean transformations of position are: The Galilean transformations of velocity are: What if instead of a ball, it is a light wave? Do velocities add according to Galilean Relativity? Clocks slow down and rulers shrink in order to keep the speed of light the same for all observers! Time is Relative! Space is Relative! Only the SPEED OF LIGHT is Absolute! James Clerk Maxwell 1860s Light is wave. The medium is the Ether. 1 cxm==3.0 108 / s με0 o Measure the Speed of the Ether Wind Light should travel faster with the wind and slower against it during the year. Michelson-Morely Experiment 1887 The speed of light is independent of the motion and is always c. The speed of the Ether wind is zero. Lorentz Contraction The apparatus shrinks by a factor : 1/− vc22 On the Electrodynamics of Moving Bodies 1905 EinsteinEinstein’’ss PrinciplePrinciple ofof RelativityRelativity • Maxwell’s equations are true in all inertial reference frames. • Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s =300,000km/s = 300m/µs . • Therefore, light travels at speed c in all inertial reference frames. Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other. Special Theory: Inertial Frames: Frames do not accelerate relative to eachother; Flat Spacetime – No Gravity. General Theory: Noninertial Frames: Frames accelerate: Curved Spacetime, Gravity & acceleration. Postulates of Special Relativity 1905 1. The laws of physics are the same in all inertial reference frames. 2. The speed of light in a vacuum is constant in all inertial reference frames, independent of the relative motion of source and observer. We assume a vacuum for this class. 3. Months later……E = mc2 Slower than Light? Light slows down in a medium like glass as a function of wavelength. Who slows down more, red or blue in glass? Faster than Light? Cherenkov radiation is electromagnetic radiation emitted when a charged particle passes through an insulator at a speed greater than the speed of light in that medium. The characteristic "blue glow" of nuclear reactors is due to Cherenkov radiation. Light travels at c for all inertial frames Light takes time to travel: Synchronizing Clocks All clocks in the same frame have the same reading but you have to take into consideration the time it takes for light to travel. Relativity of Simultaneity •Two events occurring simultaneously in one reference frame are not simultaneous in any reference frame moving relative to it. Neither reference frame is ‘correct’: There are no preferred or special reference frames. It’s all relative baby! • Any two observers at rest in the same frame MUST measure events to be simultaneous even if they don’t SEE it that way. WB 37.4 # 11 & 12 Seeing is NOT the same as Being! The time when an event is SEEN is not necessarily when the event actually happens! Example 37.5 on page 1156 Add problem #49 to this weekend’s HW due Tuesday. Δt,v Space Ship 0 Speed = v Δt, v Time Dilation Spaceship Frame Earth Frame: Time Dilation Δt ' Δ=tt =Δγ ' v 2 1− c2 1 γγ=≥, 1 v 2 1− c2 Δt ' Δ=tt =Δγ ' v 2 1− c2 The smallest (proper) time interval Δt0 between two events is measured in the reference frame in which both events occur at the same place (space ship). In another frame moving at speed v with respect to the first, the measured time interval is increased by a factor of gamma (Earth frame): Earth Lab Frame: Dilated Time: Δtt=Δγ ' Space Ship Frame: Proper Time: Δt ' TimeTime DilationDilation The time interval between two events that occur at the same position is called the proper time Δτ. In an inertial reference frame moving with velocity v = βc relative to the proper time frame, the time interval between the two events is The “stretching out” of the time interval is called time dilation. Time Dilation – Generalization • If a clock is moving with respect to you, the time interval between ticks of the moving clock is observed to be longer that the time interval between ticks of an identical clock in your reference frame • All physical processes are measured to slow down when these processes occur in a frame moving with respect to the observer – These processes can be chemical and biological as well as physical The faster you move, the longer you liver – relative to people at rest! Relative to yourself, you age at a normal rate! 1 γγ=≥, 1 v v 2 1− ββ= , <1 c2 c Time Dilation – Verification • Time dilation is a very real phenomenon that has been verified by various experiments • These experiments include: – Airplane flights – Muon decay –Twin Paradox Time Dilation Verification – Muon Decays • Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron • Muons have a half-life of Δtp = 2.2 µs when measured in a reference frame at rest with respect to them (a) • Relative to an observer on the Earth, muons should have a lifetime of γ Δtp (b) • A CERN experiment measured lifetimes in agreement with the predictions of relativity Sample Problem Superwoman can travel at 0.75c in her glass space ship. She has to fly to Beta Diva, 25.0 light years away as measured from Earth in the Earth frame, to battle alien evil guys. a) What is the total time for the trip for Superwoman? b) What is the total time you measure on Earth? Beta Diva As you approach c, lengths contract. Length Contraction The proper length of an object is longest in the reference frame in which it is at rest. In another frame moving parallel to the object, its length is shortened by a factor of gamma. v 2 LL' ==−/1γ L PPc2 LP Ruler (Proper) Frame Earth Frame Measures length LP Measures length L LengthLength ContractionContraction The distance L between two objects, or two points on one object, measured in the reference frame S in which the objects are at rest is called the proper length ℓ. The distance L' in a reference frame S' is NOTE: Length contraction does not tell us how an object would look. The visual appearance of an object is determined by light waves that arrive simultaneously at the eye. Length and length contraction are concerned only with the actual length of the object at one instant of time. Sample Problem Superwoman can travel at 0.75c in her glass space ship. She has to fly to Beta Diva, 25.0 light years away as measured from Earth in the Earth frame, to battle alien evil guys. a) What is the total time for the trip for Superwoman? b) What is the total time you measure on Earth? c) How far is Beta Diva as measured by Superwoman? d) As Superwoman leaves Earth, you measure her length as she flies overhead at 0.75c. What is her length you measure? At rest she measures 2.75 m tall. Beta Diva Length Contraction: Distances v 2 LL' ==−/1γ L PPc2 Proper Length of Ship Rest (Proper) Frame Distance Moving Frame Distance Earth Frame measures L0 Rocket Frame measures L Length Contraction Paradox? Twin Paradox? The Spacetime Interval has the same value in all inertial reference frames. The spacetime interval is invariant. It is a value that all experimenters, in all reference frames agree on. It is analagous to the invariance of distance in geometry: the distance between two points is independent of the coordinates system. The interval has the units of meters. Addition of Velocities Galilean Relativity Addition of Velocities vu+ ' .5cc+ u = = = c vu ' .5cc 1+ 1+ c2 c2 If v = .5c TheThe LorentzLorentz VelocityVelocity TransformationsTransformations Consider two reference frames S and S'. An object moves at velocity u along the x-axis as measured in S, and at velocity u' as measured in S' . Reference frame S' moves with velocity v relative to S, also along the x-axis. The Lorentz velocity transformations are: NOTE: It is important to distinguish carefully between v, which is the relative velocity between two reference frames, and u and u' which are the velocities of an object as measured in the two different reference frames. TheThe LorentzLorentz TransformationsTransformations Consider two reference frames S and S'. An event occurs at coordinates x, y, z, t as measured in S, and the same event occurs at x', y', z', t' as measured in S' . Reference frame S' moves with velocity v relative to S, along the x-axis. The Lorentz transformations for the coordinates of one event are: Lorentz Transformation Equations Sxyxt:( , , , ) S ':(xyxt ', ', ', ') Transform coordinates from S to S’ : ⎛⎞v x''''=−γ ()xvty = y z = zt =−γ ⎜⎟tx2 ⎝⎠c Transform coordinates from S’ to S : ⎛⎞v x =+γ ()xvty'' ' = yz ' ==+ ztγ ⎜⎟tx '2 ' ⎝⎠c Lorentz Transformations, Pairs of Events The Lorentz transformations can be written in a form suitable for describing pairs of events For S to S’ For S’ to S Δ=x' γ ( Δ−Δxvt) Δ=x γ ( Δ+Δxvt'') ⎛⎞v ⎛⎞v Δ=t γ Δ+tx'' Δ Δ=t' γ ⎜⎟ Δ−tx2 Δ ⎜⎟2 ⎝⎠c ⎝⎠c Problem Superwoman can travel at 0.75c in her glass space ship.