Relativity. Overview & Postulates Events Relativity of Simultaneity. Relativity of Time. Relativity of Length Relativistic momentum and energy

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1 Relativity Overview & Postulates Events Relativity of Simultaneity Simultaneity is not absolute Relativity of Time Time is not absolute Relativity of Length Relativistic momentum and energy

2 Relativity (1905- ) Hendrik Lorentz ( ) Henry Poincare ( ) Albert Einstein ( ) 2005-Year of Physics (UNESCO) 100 years of relativity

3 Is relativity difficult? Mathematically, it is not difficult as you will soon see Philosophically, it is, because it runs deeply counter to our intuition Henceforth we must be very careful about who measures what about an event, and exactly how that measurement is made

4 Relativity. Motivation. No action at a distance! Any signal needs finite time to propagate. Need to reconsider: Newton s Gravitation Law Coulomb s Electrostatic Law etc. Postulate existence of a maximum speed Turns out to be speed of light. The speed of light in vacuum has the same value c in all directions and in all inertial reference frames (Michelson-Morley, 1887).

5 Simultaneity is not absolute Time is not absolute Length is not absolute Consequences. Corrected expressions for Energy and Momentum E=mc 2

6 Relativity. Overview. Relativity also has to do with how events look from different reference frames Reference frame 1: me standing here Reference frame 2: me walking Inertial reference frames: frames of reference which do not accelerate Key Idea The laws of physics are the same for observers in all inertial reference frames. There is no preferred reference frame. Galileo assumed that only the laws of mechanics were the same in all inertial reference frames

7 An Event Defined. An event is something that happens and which can be assigned 3 spatial and 1 time coordinate Example: Catching of a touchdown pass by a PSU receiver. Non-example: The party after the game. This may be called an event but it is not one in the physics sense of the word

8 Observation of an Event Observation of an Event An event may be recorded by any number of observers, all possibly in different (inertial) reference frames These observers may assign different space-time coordinates to the event Example: camera flash goes off 1km to your left at 9am firework goes off 2km to your right, also at 9am Light takes longer to travel to you from firework 2km away, so to you the camera flash appears to have happened before the firework need to know travel time of light from source to detector to sort it all out

9 Gedanken Experiment Two spaceships each constitute inertial reference frames for their occupants, Sally and Sam Sally and Sam are at the midpoints of their ships Relative velocity is v

10 Gedanken Experiment Two meteorites hit the ships as shown, equidistant from the middles One causes a red flare to go off The other causes a blue flare to go off Both meteorites leave a permanent mark on each ship Suppose that Sam receives the red and blue light pulses at the same time When does Sally receive the light pulses? Sally sees red flash first, then blue Both Sally and Sam are in the middle of their respective ships, so Sam: Meteorites hit my ship simultaneously. Red and blue were simultaneous events Sally: Meteorites hit my ship at different times. Red and blue were not simultaneous events. Sally and Sam do not agree with one another, but both of them are right!

11 Relativity of simultineity. Question: Will two events that occur at the same time according to Joe necessarily occur at the same time according to Jane?

12 Relativity of simultineity. For two events Joe may say they occurred at the same time. Jane may say they did not. Both are correct! Does it only happen in relativity that two observers see the same thing but draw different conclusions? No: recall, for example, the Doppler effect If you are walking on the sidewalk, a passing police car siren changes pitch If you are the police officer in the police car, you hear no change in pitch Faraday s Induction & Lorentz Force Normally, at relative speeds much less than the speed of light, we don t notice departures from simultaneity camera flash and firework look like they go off at the same exact time, since their distances are small on the scale of the speed of light

13 The Relativity of Time The Relativity of Time The time interval between two events depends on their relative motion and on the spatial separation of the events Another Gedanken experiment Two events again, but restricted in a particular way: To one of the observers, the two events appear to happen at the same point in space

14 Sally is on a train train moves with velocity v relative to platform Events: 1) Sally sends a pulse of light from source B 2) Detector at source sees reflected light What time interval does Sally measure? t o = 2D/c only one clock C needed to do this The Relativity of Time The Relativity of Time

15 The Relativity of Time The Relativity of Time What does Sam measure on the platform? events 1&2 occur at different places in Sam s reference frame, so he must use 2 synchronized clocks These clocks measure t = 2L/c (same c as for Sally!!!) L = ((½v t) 2 + D 2 ) ½ Use Sally s t o =2D/c L = ((½v t) 2 +(½c t o ) 2 ) ½ Use Sam s t = 2L/c t = t o /(1-(v/c) 2 ) ½

16 The Relativity of Time The Relativity of Time The meaning of t = t o /(1-(v/c) 2 ) ½ Since v<c always, t > t o always Therefore, Sam measures a greater time interval between the events than Sally. SAME TWO EVENTS. Conclusions: Relative motion between two inertial reference frames can change the rate at which time passes between two events. In other words, identical clocks do not necessarily tick at the same rate!

17 The Relativity of Time The Relativity of Time Since Sally s frame was somewhat special the two events occurred at the same location we give her time measurement a special name: the proper time Measurements of this same time from other inertial reference frames are always greater This effect is known as time dilation Defining γ = 1/(1-(v/c) 2 ) ½ = 1/(1-β 2 ) ½ t = γ t o

18 The Relativity of Time The Relativity of Time For speeds v = 0.01c, γ = So we will only see time dilation effects when things are moving very fast, near the speed of light Experimental proof of time dilation muon decay muons decay change form on average in 2.2µs. This has been measured with stopped muons. Can do this with equipment upstairs. When muons are moving fast, though like in the accelerator at Brookhaven National Laboratory on Long Island they decay after a much longer time. With v = c, γ = 29 and hence the internal clock carried by the little muon ticks 29 times more slowly then our clocks. We measure, for these fast-moving muons, an average decay time of 63µs. atomic clocks highly accurate atomic clocks flown on regular airplanes report times which are dilated and consistent with the predictions of relativity

19 Relativity of Time. Twin paradox. Relativity of Time. Twin paradox. One twin makes a 5-year round trip on a space ship with a speed of 0.98c, while the other one stays on Earth. How much older will the Earth twin have become upon his brother s arrival?

20 Relativity of Length Relativity of Length To measure the length of an object which is at rest in our reference frame is easy If it is moving, though, what do we do? We must make sure that we measure the two endpoints simultaneously Since simultaneity is relative, it should not be a surprise that length is too

21 L = L o /γ Relativity of Length Relativity of Length The length of an object is always measured to be longest when the measurement is done in the rest frame of the object. Measurements made by frames moving with respect to the object measure shorter lengths. See text for proof

22 The Lorentz Transformation The Lorentz Transformation y S y' S' v z Newton: "Galilean transformation" x' = x - vt y' = y z' = z t' = t V' = V-v ( relative velocity) x z' x' Einstein: "Lorentz transformation" x' = γ(x - vt) y' = y z' = z t' = γ (t - vx/c 2 ) V = V'+v 1+ V' v c 2 β = v c << 1 γ = Note that with the Lorentz transformation, the speed of EM waves is INDEPENDENT of reference frame!! 1 1 β 2 1

23 Lorentz Example Lorentz Example A clock moves along the x axis at a speed of 0.600c and reads zero as it passes through the origin. (a) calculate the clock s Lorentz factor (b) what time does the clock read as it passes x = 180 m? γ = 1 1 β x ' t = = γ ( x x γv ' = v t) = 0.8µ s

24 Lorentz Example Lorentz Example Show that the interval s 2 =t 2 -x 2 remains unchanged under a Lorentz transformation.

25 Lorentz challenge problem Lorentz challenge problem In the velocity transformation formula, show the relative speed is always less not greater than c.. And it is equal to c,, if at least one of the added speeds equals c. V V ' + v = V ' v 1+ 2 c 25

26 Relativistic Energy and Momentum. Suppose that K=mv 2 /2, then v=(2k/m) 1/2 (*) 1/2 (*) In principle, we can do work on the particle and increase its (kinetic) energy indefinitely. If (*) were true, the speed would also grow unboundedly and would become greater than c. In fact, we want v<c,, and v c,, when E.. Therefore Similarly for momentum: Recall that

27 Relativistic Energy and Momentum. Sample problem. Derive the classical limit of the relativistic energy and momentum.

28 Relativistic Energy and Momentum. Sample problem. Quasars are thought to be the nuclei of active galaxies in the early stages of formation. A typical quasar radiates energy at the rate of W(!). At what rate is the mass of this quasar being reduced to supply this energy? Express your answer in solar mass units per year (1 smu = 2x10 30 kg/y).

29 Relativistic Energy and Momentum. Challenge problem. Find out the annual U.S. energy production. How many kilograms of matter would have that much rest energy?

30 Relativity The Postulates & Events Relativity of Simultaneity Time dilation Lorentz contraction Relativistic momentum and energy p = γmv Recap. Lorentz transformations t = γ t o L = L o /γ K = mc 2 (γ-1)

31 Subatomic physics Photons, quanta & quantum physics Photoelectric effect Photon momentum Probability waves Matter waves Next Lecture. Schrödinger's equation