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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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Einstein's Special

Theory of Relativity


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!


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


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’


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’


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


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


Solving the problem:

ax = -g

x

Ground

x = 0

Ball Starting Point:

x = h

v = 0

t = 0


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’


Good vs. Bad Coordinate Transforms

bad

good

The 3D distance formula

y

s

x


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


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


Math Interlude

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

Mirrors

or of the observer

Michelson Morley Experiment

Laser

Detector


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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