Einsteins Special Theory of Relativity Coordinate Changes A
- Slides: 15
Einstein’s Special Theory of Relativity Coordinate Changes
A Simple (? ) Problem Your instructor drops a ball starting at t 0 = 1: 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!
A Poor Coordinate Choice t = t 0 = 1: 45: 00 2 m Earth y h= Acceleration: ax = -g cos ay = -g sin az = 0 7 R 3 6 = m k 0 = 36. 1 V 0= 30 km/s z Ball Starting Point: x = (R + h) cos y = (R + h) sin z=0 vx = 0 vy = 0 vz = V 0 x Ground Starting Point: x = R cos vx = 0 y = R sin vy = 0 z=0 vz = V 0
Rotating Coordinates y t = t 0 = 1: 45: 00 y' Acceleration: a'x = -g a'y = 0 a'z = 0 z z' x' Coordinate change x' = x cos + y sin y' = y cos - x sin z' = z x Ball Starting Point: x'= R + h v'x = 0 y' = 0 v'y = 0 z' = 0 v'z = V 0 Ground Starting Point: v'x = 0 x' = R v'y = 0 y' = 0 v'z = V 0 z' = 0
Space Translation t = t 0 = 1: 45: 00 y y' Ball Starting Point: x' = h v'x = 0 y' = 0 v'y = 0 z' = 0 v'z = V 0 Acceleration: a'x = -g a'y = 0 a'z = 0 z R z' Coordinate change x' = x - R y' = y z' = z x x' Ground Starting Point: v'x = 0 x' = 0 v'y = 0 y' = 0 v'z = V 0 z' = 0
Time Translation y t = t 0 = 1: 45: 00 Coordinate change t' = t - t 0 Ball Starting Point: x=h vx = 0 y=0 vy = 0 z=0 vz = V 0 Acceleration: ax = -g ay = 0 az = 0 x t' = 0 z Ground Starting Point: vx = 0 x=0 vy = 0 y=0 vz = V 0 z=0
Galilean Boost y' t=0 Ball Starting Point: x' = h v'x = 0 y' = 0 v'y = 0 z' = 0 v'z = 0 Acceleration: a'x = -g a'y = 0 a'z = 0 Ground Starting Point: x' = 0 v'x = 0 y' = 0 v'y = 0 z' = 0 v'z = 0 V 0= 30 km/s z' z y Coordinate change x' = x y' = y z' = z - V 0 t x x'
Solving the Problem ax = -g Ground x=0 Ball Starting Point: x=h v=0 x t=0
Coordinate Changes: A Summary Space Translation (x-direction) x' = x - a y' = y z' = z Time Translation t' = t - a Rescaling Transformation x' = fx y' = fy z' = fz Rotating Coordinates (around z-axis) x' = x cos + y sin y' = y cos - x sin z' = z Galilean Boost (x-direction) x' = x - vt y' = y z' = z Why these? y y' x' x
Good Vs. Bad Coordinate Changes bad The 3 D distance formula good y s x
Distance Formula & Good Coordinate Changes If a coordinate transformation leaves the quantity s 2 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
Sample Problem 1. 1 The Distance between two points does not change when you perform a rotation or a space translation Prove the following The distance between an arbitrary point P 1 = (x, y, z) and (0, 0, 0) does not change when you perform a rotation around the z-axis y y' x' x x' = x cos + y sin y' = y cos – x sin z' = z
Math Interelude: Hyperbolic Functions Trigonometric Functions
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 or of the observer Michelson Morley Experiment Mirrors Laser Detector
The Funny Thing About Light (2) c = 2. 998 108 m/s c' = c + v y Galilean Boost (x-direction) x' = x - vt y' = y z' = z v x The speed of light should change as viewed by a moving observer The speed of light is always c, independent of the motion of the source or of the observer
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