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Lecture 17 Relativity A2020 Prof. Tom Megeath What Are the Major Ideas

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4/1/10 Lecture 17 Relativity What are the major ideas of A2020 Prof. Tom Megeath special relativity? Einstein in 1921 (born 1879 - died 1955) Einstein’s Theories of Relativity Key Ideas of Special Relativity • Special Theory of Relativity (1905) • No material object can travel faster than light – Usual notions of space and time must be • If you observe something moving near light speed: revised for speeds approaching light speed (c) – Its time slows down – Its length contracts in direction of motion – E = mc2 – Its mass increases • Whether or not two events are simultaneous depends on • General Theory of Relativity (1915) your perspective – Expands the ideas of special theory to include a surprising new view of gravity 1 4/1/10 Inertial Reference Frames Galilean Relativity Imagine two spaceships passing. The astronaut on each spaceship thinks that he is stationary and that the other spaceship is moving. http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ Which one is right? Both. ClassMechanics/Relativity/Relativity.html Each one is an inertial reference frame. Any non-rotating reference frame is an inertial reference frame (space shuttle, space station). Each reference Speed limit sign posted on spacestation. frame is equally valid. How fast is that man moving? In contrast, you can tell if a The Solar System is orbiting our Galaxy at reference frame is rotating. 220 km/s. Do you feel this? Absolute Time Absolutes of Relativity 1. The laws of nature are the same for everyone In the Newtonian universe, time is absolute. Thus, for any two people, reference frames, planets, etc, 2. The speed of light is the same for everyone time marches along at the same rate. If we the time between two events, that time is the same All of relativity follows from these two ideas! for every observer. However, we have to abandon the idea that time is absolute! 2 4/1/10 Absoluteness of Light Speed Reference Frames Ball moves at 10 km/hr in reference Light moves at exactly frame of plane speed c Ball moves at 910 km/hr Light moves at exactly speed in reference frame of c (not c + 900 km/hr) someone on ground • Motion can be defined with respect to a particular • Einstein claimed that light should move at exactly c frame of reference in all reference frames (now experimentally verified) What’s surprising about the The Michaelson Morley Experiment Originally thought that the absoluteness of the speed of light? speed of light was relative to some unknown medium, the Light moves at exactly aether. speed c This would be an absolute reference frame. Michelson and Morely tried to measure the speed of the Earth through the aether. This was done at Case-Western Reserve University in 1817.. The could find no evidence for the aether. Light moves at exactly speed c (not c + 900 km/hr) http://www.upscale.utoronto.ca/PVB/Harrison/SpecRel/Flash/MichelsonMorley/MichelsonMorley.html 3 4/1/10 Relativity of Motion at Low Speeds Thought (Gedanken) Experiments • Einstein explored the consequences of the absoluteness of light speed using “thought experiments” • The consequences will be easiest for us to visualize with thought experiments involving spaceships in freely floating reference frames (no gravity or acceleration) Relativity of Motion at Low Speeds Relativity of Motion at High Speeds 4 4/1/10 Light Speed is Absolute Relativity of Motion c + 0.9c = c !?! Trying to Catch up to Light How does relativity affect our • Suppose you tried to view of time and space? catch up to your own headlight beams • You’d always see them moving away at speed c • Anyone else would also see the light moving ahead of you 5 4/1/10 Path of Ball in a Stationary Train Path of Ball in a Moving Train • Someone outside the train would see the ball travel a longer path in one up-down cycle • The faster the train is moving, the longer that path would be • Thinking about the motion of a ball on a train will prepare us for the next thought experiment Time Dilation The Time Dilation Formula Light path in • We can perform a c 2t ′ 2 + v 2t 2 = c 2t 2 thought experiment your Light path in with a light beam reference frame of frame replacing the ball other 2 spaceship 2 2 v 2 t ′ = t − 2 t • The light beam, c moving at c, travels a longer path in a moving object v 2 t ′ = t 1− 2 c • Time must be passing more slowly there € 6 4/1/10 The Time Dilation Formula Simultaneous Events? • Time will appear to pass more slowly in a moving object by an amount depending on its speed • Time almost halts for objects nearing the speed of light • In your reference frame, red and green lights on other spaceship appear to flash simultaneously Simultaneous Events? Length Contraction • But someone on the other spaceship sees the green • Similar thought experiments tell us that an object’s light flash first—simultaneity is relative! length becomes shorter in its direction of motion 7 4/1/10 Length Contraction or Rotation? http://www.tempolimit-lichtgeschwindigkeit.de/tompkins/node1.html Mass Increase No relativity Length Contraction Length Contraction+ Finite speed of light • A force applied to a rapidly moving object produces less acceleration than if the object were motionless • This effect can be attributed to a mass increase in the moving object Due to finite speed of light, object will be rotated. http://faraday.physics.utoronto.ca/PVB/Harrison/SpecRel/Flash/ContractInvisible.html Velocity Addition Formulas of Special Relativity Velocity of first 2 ship in your frame = v v 1 Time Dilation : t ′ = t 1− c 2 Velocity of second 2 ship in frame of 1st = v2 v Length Contraction : l ′ = l 1− c 2 Velocity of second ship in your frame : v + v m 1 2 v v Mass Increase : m ′ = 1+ 1 × 2 2 c c v 1− 2 c € € 8 4/1/10 Deriving E = mc2 Why does this seem so strange? 2 In our world, v is much smaller than c m0 1 v m = ≈ m01 + 2 for small v 2 2 2 c v v Time Dilation : t ′ = t 1− c 2 1− 2 v 2 c Length Contraction : l ′ = l 1− c 2 m Mass Increase : m ′ = v 2 1− 2 1 c Total energy = mc 2 ≈ m c 2 + m v 2 0 0 Since v/c is very small: 2 € Mass-Energy of t’ = t, l’ = l, and m’ = m Kinetic Energy From Gamov’s Mr. Thompson’s Journey object at rest In our everyday experience (what if the speed of light was 30 km/hr) € A Drive through Tuebingen Every Reference Frame is Equal • According to you, time slows down in a moving spaceship • According to someone on that spaceship, your time slows down • Who is right? • You both are, because time is not absolute but depends on your perspective 9 4/1/10 Spacetime Diagram of a Car Worldlines • A worldline shows an object’s path through spacetime in a spacetime diagram – Vertical worldline: no motion – Diagonal worldline: constant-velocity motion – Curved wordline: • A spacetime diagram plots an object’s position in accelerating motion space at different moments in time Worldlines for Light Worldlines and Relativity • Worldlines for light go at 45° angles in diagrams with light-seconds on one axis and seconds on the other • Worldlines look different in different reference frames 10 4/1/10 Different Spacetime Coordinate Systems for Different Inertial Worldlines and Relativity Reference Frames Δx2 - cΔt2 = Δx’2 - cΔt’2 (in this case Δx2 = Δx’2 - cΔt’2 t’ t A B x’ x • Two observers move past each other. Imagine two surveyors mapping a property. They both a different • Both have their own spacetime coordinate system coordinate grid. They want to measure the distance between two trees. • They measure the same event. • Observer A thinks the events are simultaneous 2 2 2 2 What is constant: the distance x +y = x’ +y’ • Observe B does not. What are not constant - x, y • They can agree on one thing “interval” between two different events in spacetime: x2 + y2 + z2 – (ct)2 Light Cones Solution t t’ t t’ t t Δx = cΔt Flash of light x’ x’ x x x x Andrew Hamilton http://casa.colorado.edu/~ajsh/sr/sr.shtml (excellent web site!!) 11 4/1/10 Do the effects predicted by Tests of Relativity relativity really occur? • First evidence for absoluteness of speed of light came from the Michaelson-Morley Experiment performed in 1887 • Time dilation happens routinely to subatomic particles the approach the speed of light in accelerators • Time dilation has also been verified through precision measurements in airplanes moving at much slower speeds • Prediction that E=mc2 is verified daily in nuclear reactors and in the core of the Sun A Paradox of Non-Relativistic Test Relativity for Yourself Thinking • If speed of light were not absolute, you would see the car coming toward you reach the collision point before the car it struck • If speed of light were not absolute, binary stars would not look like two distinct points of light • No paradox if light • You can verify relativity by simply looking through a speed is same for telescope at a binary star system everyone 12 4/1/10 A Journey to Vega A Journey to Vega • The distance to Vega is • However, your twin on about 25 light-years Earth would have aged 50 years while you aged • But if you could travel to only 2 Vega at 0.999c, the round trip would seem to take only • There’s a seeming two years! contradiction to this • At that speed, the distance to conclusion:What does Vega contracts to only 1 your twin see in his light-year in your reference reference frame as he frame watches the Earth recede? • Going even faster would make the trip seem even shorter! The Twin Paradox The Twin Paradox The problem is solved when we realize that the twin does not stay in a single inertial reference frame.
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