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Published byKieran Cleveland Modified about 1 year ago

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Transformation of space and time t = (t' + x' v/c²)t' = (t – x v/c²) x = (x' + vt')x' = (x – vt) y = y'y' = y z = z'z' = z ct x y z v/c 00 v/c 00 00 10 00 01 ct' x ' y ' z ' X = = = X ' Spacetime interval: For two events e 0 = (t 0, x 0, y 0, z 0 ) = (t 0 ', x 0 ', y 0 ', z 0 ') and e 1 = (t 1, x 1, y 1, z 1 ) = (t 1 ', x 1 ', y 1 ', z 1 ') we find that the spacetime interval s 2 = (c t) 2 – ( x 2 + y 2 + z 2 ) = (c t') 2 – ( x' 2 + y' 2 + z' 2 ) is Lorentz-invariant ( t = t 1 – t 0 ; x = x 1 – x 0 ; …) Lv Lv X = L v X ' Lorentz transformation Four-vector x ct ct' x' Light 1 Unit hyperbola: Relativistic spacetime diagram

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t = (t' + x' v/c²)t' = (t – x v/c²) x = (x' + vt')x' = (x – vt) y = y'y' = y z = z'z' = z ct x y z v/c 00 v/c 00 00 10 00 01 ct' x' y' z' X = = = X ' Velocity transformation: u x = dx/dt dx = (dx' + vdt') dt = (dt' + dx' v/c²) dx/dt = (dx'+ vdt')/ (dt'+ dx' v/c²) = (dx'/dt'+ v)/ (1 + dx'/dt' v/c²) dy/dt = dy' / (dt'+ dx' v/c²) = dy'/dt' / ( 1 + dx'/dt' v/c²) u x = (u x '+ v)/(1 + u x 'v/c²)reverse transformation: u x ' = (u x – v)/(1 – u x v/c²) u y = u y '/ (1 + u x 'v/c²)replace v by –v:u y ' = u y / (1 – u x v/c²) u z = u z '/ (1 + u x 'v/c²)u z ' = u z / (1 – u x v/c²) special case u = u x : (that is: u is parallel to the relative velocity v of the reference frames) u = (u'+ v)/(1 + uv/c²) u' = (u – v)/(1 – uv/c²) dt' Lv Lv Velocity transformation

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Velocity transformation: u x = (u x '+ v)/(1 + u x 'v/c²)reverse transformation: u x ' = (u x – v)/(1 – u x v/c²) u y = u y '/ (1 + u x 'v/c²)replace v by –v:u y ' = u y / (1 – u x v/c²) u z = u z '/ (1 + u x 'v/c²)u z ' = u z / (1 – u x v/c²) Four-Velocity U U ' U (τ is proper time of the object) X X X Coordinate transformation: X = L v X' d X = L v d X' τ is Lorentz-invariant U = L v U'

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x t y Structure of spacetime Elsewhere x Past Future ct y Elsewhere ct' x'x' s² = 0 s² < 0 t > 0 t' < 0 3e8 m x ct Past Future s² > 0 ct' x'x' Light (c = Δx/ Δt) 1 s 1 m

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The Twin Paradox A and B are twins of age 20. B decides to join an expedition to a nearby planet. The distance between Earth and the planet is d = 20 light years and the space ship used for the expedition travels at a speed of v = 4/5 c. -What time is it on Earth and on the spaceship respectively when B arrives at the planet? After a short stop at the planet the spaceship returns with the same speed. -How old are A and B at B’s return? t = (t' + x' v/c²)t' = (t – x v/c²) x = (x' + vt' )x' = (x – vt) t = 0 t = t 1 t = 2t 1 t'= t 1 '=t 2 ' t' = 0 t" = t 2 ' t" = 2t 2 ' Earth planet spaceship Time passing on Earth while spaceship turns around t 1 : x 1 = x planet ;x 1 ' = 0 t 1 = 25 y t 1 ' :x 1 = x planet ;x 1 ' = 0 t 1 ' = 15 y t 2 ' : x 2 = x planet ;x 2 ' = 0 t 2 ' = 15 y t 2 : x 2 = 0 ; t 2 ' = 15 y t 2 = 9 y t = t 2

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The Twin Paradox and Doppler Shift Earth planet

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Relativistic Doppler Effect f R = f E [ (1 – v/c) / (1 + v/c) ] ½ for source moving away from receiver (or vice versa) f R = f E [ (1 + v/c) / (1 – v/c) ] ½ for source moving towards receiver (or vice versa) (f R : frequency of receiver in receiver’s frame; f E frequency of emitter in emitter’s frame) “longitudinal Doppler shift” Transversal effect due to time dilation: f R = f E 1 ct' x' Plain wave emitted source at rest in S' Plain wave emitted source at rest in S x 1 ct 0

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0 K4 K 2.721 K2.729 K T = 0.02 mK (Dipole subtracted) Cosmic Microwave Background (Nobel prize in physics 2006: George Smoot, John Mather) COBE (COsmic Background Explorer) satellite Planck (2009) WMAP (2001) COBE (1989)

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