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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

Einstein's Special

Theory of Relativity


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!


Einstein s special theory of relativity

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


Einstein s special theory of relativity

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’


Einstein s special theory of relativity

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’


Einstein s special theory of relativity

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


Einstein s special theory of relativity

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


Einstein s special theory of relativity

Solving the problem:

ax = -g

x

Ground

x = 0

Ball Starting Point:

x = h

v = 0

t = 0


Einstein s special theory of relativity

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’


Einstein s special theory of relativity

Good vs. Bad Coordinate Transforms

bad

good

The 3D distance formula

y

s

x


Einstein s special theory of relativity

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


Einstein s special theory of relativity

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


Einstein s special theory of relativity

Math Interlude

Hyperbolic Functions

Trigonometric Functions


Einstein s special theory of relativity

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


Einstein s special theory of relativity

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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