Relativity Hawking Mlodinow p 19-p37 Successive Paradigms Kuhn Relativity Newton: positions are relative Special relativity: positions and time are relative Next time: General Relativity 1
What is a scientific theory? Self consistent framework describing reality 3 philosophical stands We can know reality because we are part of it, created at the image of the Creator, world is mathematical, evolution etc.. (e.g. Einstein) We can only describe our interactions with reality (Kant) We construct reality: there is nothing out there! (Berkeley) Paradigm (Thomas Kuhn). Facts, methods, theory are intimately linked + Social/cultural context (not accepted by all scientists) Clear impact on Vision of the world Can only be falsified (Karl Popper) Paradigms are incomplete Usually cannot fit everything Experimental errors (how do we know) Tension with some facts => Scientific revolutions Is there a theory of everything? Hope shared by most scientists At least successive mappings Gödel theorem: in any complex enough logical system, some propositions are undecidable Relationship with free will? 2
Successive Paradigms Greeks: Geocentric Aristotle: preferred frame of reference: earth Copernicus /Kepler: Heliocentric Galileo/Newton Classical mechanics relative space, absolute time addition of velocities Doppler shift Special relativity Velocity of light is absolute time is no more absolute: mixed with space mass and energy are equivalent General relativity Gravity is due to curvature of space Quantum Mechanics Waves and particles Modern cosmology: Homogeneous and Isotropic 3
Train stories v x Pig Pong in train Pig Pong seen in earth frame Newton Physics is the same in any frame in uniform motion Positions are relative, not absolute: Time is absolute =>Velocities add x i,lab = x i,train + x train dx i,lab dt = dx i,train dt + dx train dt 4
Galileo Group How do we think about this? Coordinates z { x i + Δx, y i + Δy, z i + Δz} { } x i, y i, z i y lenght = Δx 2 + Δy 2 + Δz 2 What are the transformations which keep the length constant? Translations Rotations e.g., x i = x i + x 0i for x 0i = a + bt no change in Newton's law F = m d 2 x dt 2 x' i = x i cosθ + ysinθ y' i = x i sinθ + ycosθ 5
Doppler Shift An experience that we all have: Change of pitch of a train passing by A way to think about it Regular signal in rest frame of moving object t Δt lab Δt lab > Δt rest if object is going away Δt rest x Same thing happens with light If the velocity of light is not infinite Roehmer and the satellites of Jupiter Different periods depending of whether Jupiter is going towards us or away! 1676! 6
Special Relativity Not only velocity of light appears finite but constant 2 reasons: Maxwell equations describing electromagnetism Michelson and Morley 1887 Einstein What transformations will keep velocity of light c constant? Minkowski: transformations that keep constant t Δt Δτ 2 = Δt 2 1 ( Δx 2 + Δy 2 + Δy 2 ) = ( Proper time) 2 c 2 For light propagating at speed of light c, Δτ 2 = 0 Δx x 7
Poincaré Group Translations Rotations Lorentz transformations: Boosts For a system with velocity v along x axis with respect to the lab x lab = γ ( x rest + vt rest ) with γ = y lab = y rest 1 1 v2 c 2 2 Δt lab 1 c 2 ΔxΔ 2 lab + Δy 2 2 2 ( lab + Δz lab ) = Δt lrest 1 c 2 Δx 2 rest + Δy 2 2 ( rest + Δz rest ) z lab = z rest t lab = γ t rest + v c x 2 rest Mixes space and time, velocities do not add anymore! 8
Consequences Clocks are not running at same speed in different frames v x A clock in train light bouncing between two mirrors The same clock seen in earth frame Appears to slow down Longitudinal dimension (at given t lab ) in lab also appears to shrink! transforming in such a way that E 2 pc ( ) 2 ( ) 2 = mc 2 1 Δt lab = γδt rest γ = Energy and momentum are also mixed <= velocities do not add 9 1 v2 c 2 In particular at rest E = mc 2 Equivalence of mass and energy For small p : E mc 2 + p2 2m = mc2 + mv2 2 for m 0 p = γ mv when velocity c E Impossible to reach c unless m = 0
Speed larger than speed of light? www.nature.com/news/2011/110922/full/news.2011.554.html http://arxiv.org/pdf/1109.4897 Lots of discussion! Too early to give up relativity 730km too small by 60ns= 18m 10
Next Time Wednesday October 28 Stephen Hawking & Leonard Mlodinow A briefer History of Time p. 38-67 Method: Read and write down a few of your own questions (distinguishing between facts, methods, theoretical framework and meaning) If possible email by the previous Monday night 8:00 p.m. sadoulet at berkeley dot edu 11