l3 - Special relativity Velocities Doppler shift Hubble...

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Special relativity Velocities Doppler shift Hubble constant Einstein’s postulates Special relativity: two frames with velocity difference (general relativity deals with large acceleration differences). Postulates: 1 The laws of physics are the same in all inertial reference frames. 2 The speed of light in free space has the same value c = 1 / μ 0 ǫ 0 in all inertial reference frames. All we had to do was to apply these two postulates to obtain the Lorentz transformations.
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Special relativity Velocities Doppler shift Hubble constant Lorentz transformations Consider an event at ( x 1 , y 1 , z 1 , t 1 ) in inertial reference frame S 1 . For an observer in frame S 2 that moves at a velocity v relative to frame S 1 , the coordinates ( x 2 , y 2 , z 2 , t 2 ) at which this event is observed are x 2 = γ ( x 1 vt 1 ) (1) y 2 = y 1 (2) z 2 = z 1 (3) t 2 = γ p t 1 β c x 1 P (4) where β v / c and γ 1 / r 1 β 2 . 0.0 0.2 0.4 0.6 0.8 1.0 Velocity β 0 1 2 3 4 5 Lorentz factor γ
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Special relativity Velocities Doppler shift Hubble constant Relativisic velocity Take derivatives of Eqs. 1 through 4 to compare velocities as seen in the two frames. Since the velocity v between the frames does not change, γ = 1 / r 1 ( v / c ) 2 is constant so d γ = 0. dx 2 = γ ( dx 1 v dt 1 ) and dy 2 = dy 1 and dz 2 = dz 1 dt 2 = γ ( dt 1 β c dx 1 ) = γ ( dt 1 v c 2 dx 1 ) . The velocity in frame 2 is the change in position dx 2 divided by the change in time dt 2 . Divide numerator and denominator by dt 1 : v 2 , x = dx 2 dt 2 = γ ( dx 1 vdt 1 ) γ ( dt 1 ( v / c 2 ) dx 1 ) = ( dx 1 / dt 1 ) v ( dt 1 / dt 1 ) ( dt 1 / dt 1 ) ( v / c 2 )( dx 1 / dt 1 ) = v 1 , x v 1 v v 1 , x c 2 (5)
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Special relativity Velocities Doppler shift Hubble constant Relativistic velocity II Calculate v 2 , y and v 2 , z in the same manner: v 2 , y = dy 2 dt 2 = dy 1 γ b dt 1 ( v / c 2 ) dx 1 B = ( dy 1 / dt 1 ) γ b ( dt 1 / dt 1 ) ( v / c 2 )( dx 1 / dt 1 ) B = v 1 , y γ ± 1 v v 1 , x c 2 ² (6) v 2 , z = v 1 , z γ ± 1 v v 1 , x c 2 ² . (7)
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Special relativity Velocities Doppler shift Hubble constant Relativistic velocity: inverse equations The inverses of the relativistic velocity expressions are v 1 , x = v 2 , x + v 1 + vv 2 , x c 2 (8) v 1 , y = v 2 , y γ b 1 + vv 2 , x c 2 B (9) v 1 , z = v 2 , z γ b 1 + vv 2 , x c 2 B . (10)
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l3 - Special relativity Velocities Doppler shift Hubble...

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