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Einstein's Special Theory of Relativity

Einstein's Special Theory of Relativity. Changing Coordinates. A Simple (?) Problem Your instructor drops a ball starting at t 0 = 11:45 from rest (compared to the ground) from a height of h = 2.0 m above the floor. When does it hit the floor?. h = 2.0 m.

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Einstein's Special Theory of Relativity

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  1. Einstein's Special Theory of Relativity

  2. Changing Coordinates A Simple (?) Problem Your instructor drops a ball starting at t0 = 11:45 from rest (compared to the ground) from a height of h = 2.0 m above the floor. When does it hit the floor? h = 2.0 m Need to set up a coordinate system!

  3. A poor coordinate choice Acceleration: ax = -g cos ay= -g sin az = 0 y t = t0= 11:45:00 Ball Starting Point: x = (R + h) cos y = (R + h) sin z = 0 h = 2 m vx= 0 vy = 0 vz = V0 R = 6370 km Earth  = 36.1 x Ground Starting Point: x = R cos y = R sin z = 0 vx= 0 vy = 0 vz = V0 V0= 30 km/s z

  4. Rotating coordinates y x’ t = t0 = 11:45:00 y’ Coordinate change x’ = x cos + y sin y’ = y cos - x sin z’ = z Acceleration: a’x = -g a’y= 0 a’z = 0  x Ball Starting Point: x’ = R + h y’ = 0 z’ = 0 Ground Starting Point: x’ = R y’ = 0 z’ = 0 v’x= 0 v’y = 0 v’z = V0 v’x= 0 v’y = 0 v’z = V0 z z’

  5. Translating space coordinates t = t0 = 11:45:00 y y’ Coordinate change x’ = x - R y’ = y z’ = z Ball Starting Point: x’ = h y’ = 0 z’ = 0 v’x= 0 v’y = 0 v’z = V0 Acceleration: a’x = -g a’y= 0 a’z = 0 R x x’ Ground Starting Point: x’ = 0 y’ = 0 z’ = 0 v’x= 0 v’y = 0 v’z = V0 z z’

  6. Time translation Ball Starting Point: x = h y = 0 z = 0 y vx= 0 vy = 0 vz = V0 t = t0 = 11:45:00 Coordinate change t’ = t - t0 Acceleration: ax = -g ay= 0 az = 0 x t’ = 0 Ground Starting Point: x = 0 y = 0 z = 0 vx= 0 vy = 0 vz = V0 z

  7. Galilean Boost y’ x’ z’ y t = 0 Coordinate change x’ = x y’ = y z’ = z - V0t Ball Starting Point: x’ = h y’ = 0 z’ = 0 v’x= 0 v’y = 0 v’z = 0 Acceleration: a’x = -g a’y= 0 a’z = 0 x V0= 30 km/s Ground Starting Point: x’ = 0 y’ = 0 z’ = 0 v’x= 0 v’y = 0 v’z = 0 z

  8. Solving the problem: ax = -g x Ground x = 0 Ball Starting Point: x = h v = 0 t = 0

  9. Coordinate Changes: A summary Rotating Coordinates(around z-axis) x’ = x cos + y sin y’ = y cos - x sin z’ = z Space Translation(x-direction) x’ = x - a y’ = y z’ = z Galilean Boost (x-direction) x’ = x - vt y’ = y z’ = z Why these? Time Translation t’ = t - a y Rescaling Transformation x’ = fx y’ = fy z’ = fz y’ x x’

  10. Good vs. Bad Coordinate Transforms bad good The 3D distance formula y s x

  11. Good vs. Bad Coordinate Transforms If a coordinate transformation leaves the quantity s2 unchanged, then it must be good, and nature’s laws are the same in the original and final systems. Space Translation(x-direction) x’ = x - a y’ = y z’ = z Rotating Coordinates(around z-axis) x’ = x cos + y sin y’ = y cos - x sin z’ = z Rotations (any axis), Translations (any direction), and combinations of them

  12. Distance Invariance x’ = x cos + y sin y’ = y cos - x sin z’ = z The Distance between two points does not change when you perform a rotation or a space translation y Prove the following The distance between an arbitrary point P1 = (x,y,z) and the origin does not change when you perform a rotation around the z-axis y’ x’  x

  13. Math Interlude Hyperbolic Functions Trigonometric Functions

  14. The funny thing about light . . . What determines the speed of light in vacuum? Detector Double Star The speed of light is independent of the motion of the source Mirrors or of the observer Michelson Morley Experiment Laser Detector

  15. The funny thing about light . . . c = 2.998 108 m/s c’ = c + v y Galilean Boost (x-direction) x’ = x - vt y’ = y z’ = z v The speed of light should change as viewed by a moving observer x The speed of light is always c, independent of the motion of the source or of the observer

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