AS109_lec13(2).pptx - AS 109 Special Relativity Motion Near...

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Unformatted text preview: AS 109 3/20/2018 Special Relativity: Motion Near the Speed of Light Everyday Relativity If speeds are much less than the speed of light c: - Velocities add Time & space are the same for all reference frames where phenomena take place & observers make measurements Time and Space Newton assumed that time and space are absolute - Universal system of coordinates for measurements - Measurements of lengths and times should not depend on motion of the observer or presence of massive objects Einstein: this is not the case Measurement of velocity depends on reference frame of the observer, so constant-velocity motion is relative (same is true in everyday relativity) Measurements of time and length are also relative! Constant Speed of Light Michelson-Morley experiment (1887): the speed of light in a vacuum is constant (c = 3x108 m/s) Does not depend on motion of either the source of light or the observer Only wavelength (or frequency) of light is affected by relative motion - Doppler effect Einstein: nothing physical, nor any information, can travel faster than light - Otherwise, it would be possible to observe a phenomenon to happen before observing the event that caused it Einstein’s Special Relativity (1905) Special Relativity: compare measurements made from reference frames moving at constant velocity relative to each other [concept: relative velocity] Rest frame: platform where observer (or object or phenomenon of interest) is not moving Einstein’s Assumptions: 1. Speed of light does not depend on relative motion of source & observer (wavelength does – Doppler effect) 2. Measurements made of slow phenomena occurring in observer’s rest frame give normal results - Einstein worked out consequences using high-school algebra Special Relativity: Einstein’s Results For measurements of objects & events occurring in a reference frame moving at a constant velocity relative to observer: 1. Time in the moving system passes more slowly when measured in the frame of the observer 2. The observer measures lengths to be shorter in the dimension parallel to the direction of motion than when object is at rest (other two dimensions are not affected) Lorentz Factor: Quantifies Effects of Relativity Lorentz factor (v = relative velocity of frames): v = 0: =1 v = 0.98c: = 5 v = 0.995c: = 10 v = c: = Special Relativity: Quantitative Results 1. Time (as measured by observer) passes more slowly in the moving system - time interval t is longer: t = trest “time dilation” Why? Video clip: In rest frame (inside spaceship): Time for flash of light to reach ceiling trest = H/c In observer’s frame: t = d/c Trigonometry: So, it must take longer in observer’s frame Special Relativity: Quantitative Results 2. Lengths & distances are shorter along direction of velocity than in rest frame L = Lrest/ “length contraction” velocities are to the right Spaceship traveling at 0.98c (=5) is measured by observer on Earth to be only 20% as long as usual Length returns to normal when spaceship lands Time on spaceship passes 5 times more slowly than on Earth Passage of time returns to normal when spaceship lands But the passengers have aged 5 times less during the trip than people who remained on Earth! Time Sequence of Events Einstein insisted that causes of events should always precede the events they cause Causality is preserved But how about simultaneity of two events that have independent causes? Video clip: Simultaneity of non-causally-related events is not preserved Why? Speed of light by which events are observed is constant and light does not travel infinitely fast Time when we observe an event depends on both when it happened & its distance from the observer Question for Discussion Does our common notion of reality remain intact if two observers in motion relative to each other measure time, length, & whether events are simultaneous differently? Does this mean that reality can be relative and therefore subjective rather than objective? The Twin Paradox Spaceship goes to another star system at near-light speed ex.: star is 9.80 lt-yr away, v = 0.980c (=5.0) [1 lt-yr = c x 1 year] Upon arrival, spaceship immediately turns around & returns to Earth at same speed Passenger on spaceship has a twin on Earth Time for round-trip t: Earth’s frame: t = 2(9.80 lt-yr)/(0.980c) = 20.0 yr [Note: lt-yr/c = yr] but we measure that passengers age by t/ = (20.0 yr)/5.0 = 4.0 yr Spaceship’s frame: distance to star is d/ = (9.80 lt-yr)/5.0 = 1.96 lt-yr so, t = 2(1.96 lt-yr)/(0.98c) = 4.0 yr Space-traveler is now (20.04.0) yr = 16 yr younger than twin! The Twin Paradox: Why Not Symmetric? Paradox: If motion is relative, why doesn’t twin on spaceship see Earth speed off at 0.98c and therefore observe the twin on Earth to age less? Answer: The situation is not symmetric because the spaceship accelerates up to 0.98c, decelerates to zero upon arrival at the star system, accelerates back to 0.98c back toward Earth, then decelerates again to stop at Earth Only non-accelerating (constant velocity) reference frames are equivalent Does Time “Dilation” Really Happen? Direct measurement with atomic clock flown on a jet: Clock measured shorter time than identical clock on the ground Muon decays: at rest, a muon particle (which is a lepton) decays into an electron (plus a muon neutrino and an electron anti-neutrino) in 2.2 microseconds If the muon is produced with a speed of, e.g., 0.995c (=10), it survives longer (22 microsec in this example) before decaying - Every experiment agrees with predictions of Einstein’s formulas! Energy Einstein combined the rest-mass energy (mc2) with kinetic energy: E = mc2 Kinetic energy Ekin = (-1)mc2 [If v << c, -1 ≈ (1/2) (v/c)2 so Ekin ≈ (1/2)mv2] At rest, =1, so E = mc2 - formula for rest-mass energy Impossible for Matter to Travel at c E = mc2 But = if v = c Infinite energy needed to accelerate an object or particle with mass up to v = c Only massless particles like photons can move at c How about faster-than-light speed? Only hypothetical particles (“tachyons”) that always move faster than light - There is no solid evidence in support of existence of tachyons 4-Dimensional Space-Time In Einstein’s equations, there are 3 dimensions of space (x,y,z) and a 4th dimension proportional to time (ct) Space-time Time, lengths/distances, & motions are intimately tied together Space-time and the particles & fields imbedded in it are the fabric of the universe Space Travel Advantage of travel near light speed: Distances are times shorter time required is less Disadvantages: 1. Time passes times faster on Earth than in spaceship Kiss your friends & family goodbye 2. Energy requirement is enormous: mc2 3. Space debris hits at very high relative velocity Deadly [Energy of impact = m(debris)c2] Relativistic (near light-speed) Trip: The View Since length contraction only works parallel to direction of motion, view from spaceship would be distorted Most of view is collapsed toward forward direction Video clip: View of Orion at 3 different velocities toward center of belt: ‘39 (by Brian May; band: Queen; album: “Night at the Opera”) In the year of ‘39 assembled here the volunteers in the days when lands were few Here the ship sailed out into the blue and sunny morn, sweetest sight ever seen And the night followed day and the story tellers say that the score brave souls inside For many a lonely day sailed across the milky seas, ne'er looked back, never feared, never cried Chorus: Don't you hear my call though you're many years away, don't you hear me calling you? Write your letters in the sand for the day I take your hand in the land that our grand-children knew In the year of ‘39 came a ship in from the blue - the volunteers came home that day And they bring good news of a world so newly born, though their hearts so heavily weigh For the earth is old and gray, little darlin' we'll away, but my love this cannot be Oh so many years have gone though I'm older but a year, your mother's eyes from your eyes cry to me [Chorus] Don't you hear my call, though you're many years away, don't you hear me calling you All your letters in the sand cannot heal me like your hand, for my life, still ahead, pity me ...
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