Sp

ecial Relativity: B

ackgrou

nd

…E

instein

was b

oth

ered b

y a differen

ce in N

ewto

nian

dyn

amics an

d M

axwell’s electro

dyn

amics

It loo

ked as th

ou

gh

New

ton

ian d

ynam

ics were

ind

epen

den

t of referen

ce frame

bu

t that M

axwell’s

equ

ation

s did

dep

end

on

the ch

oice o

f reference

frame

A referen

ce frame

is wh

ere you

make yo

ur m

easurem

ent

Fo

r examp

le, wh

en m

easurin

g th

e speed

of a car, w

ep

retend

the ro

ad is statio

nary, even

tho

ug

h th

e earth is

rotatin

g, revo

lving

arou

nd

the su

n, an

d th

e sun

isrevo

lving

arou

nd

the cen

ter of th

e galaxy

Th

e ether

Mich

elson

& M

orley set o

ut to

find

the eth

er

Sin

ce sou

nd

travels thro

ug

h air,th

ere mu

st be so

me m

ediu

mth

ou

gh

wh

ich lig

ht travels

Th

e earth is m

ovin

g aro

un

d th

e sun

at 30,000 m/s; co

mp

areth

e speed

of lig

ht p

arallel to th

at mo

tion

with

the sp

eed o

flig

ht p

erpen

dicu

lar to th

at mo

tion

usin

g an

interfero

meter

Th

ey learned

the sp

eed o

f ligh

t did

n’t ch

ang

e at all

Som

e postu

latesE

instein

mad

e several po

stulates to

form

ulate

special relativity

1.A

ll laws o

f natu

re are the sam

e in all u

nifo

rmly m

ovin

gfram

es of referen

ceT

his m

eans if yo

u th

row

a ball straig

ht u

p w

hile stan

din

g o

n th

eearth

, or o

n a m

ovin

g train

, it will co

me straig

ht b

ack do

wn

Yo

u can

no

t ind

epen

den

tly measu

re con

stant m

otio

n– o

nly

con

stant m

otio

n relative to

som

ethin

g else

2.T

he sp

eed o

f ligh

t in free sp

ace has th

e same m

easured

value fo

r all ob

servers, regard

less of th

e mo

tion

of th

eso

urce o

r the m

otio

n o

f the o

bserver; th

at is, the sp

eedo

f ligh

t is a con

stant

Th

e speed

of lig

ht is th

e speed

of lig

ht is th

e speed

of lig

ht is th

esp

eed o

f ligh

t

2

Sim

ultan

eityS

imu

ltaneity ju

st mean

s “thin

gs h

app

enin

g at th

esam

e time

”W

hat d

oes th

at mean

for th

ing

s mo

ving

relative toeach

oth

er?T

wo

events th

at are simu

ltaneo

us in

on

e frame o

freferen

ce need

no

t be sim

ultan

eou

s in a fram

em

ovin

g relative to

the first fram

e

2

Defin

e a ligh

t clock

to b

e a pu

lse of lig

ht

bo

un

cing

betw

een tw

o m

irrors

Co

mp

are the p

ath o

f the lig

ht in

two

differen

t reference fram

es:

Sin

ce the sp

eed o

f ligh

t is con

stant, it ap

pears th

elig

ht clo

ck run

s slow

er to th

e station

ary ob

server

Tim

e dilation

Tim

e Dilation

IIIf th

e time in

the referen

ce frame m

ovin

g w

ith th

eclo

ck is t0 , then

the tim

e t in a referen

ce frame

mo

ving

with

velocity v

relative to th

e clock is

We d

efine th

e Lo

rentz facto

r g to b

e

So

we g

etN

ote th

at g > 1 fo

r all v > 0…

. An

d th

at for sm

all v, itis alm

ost 1. T

hat is g

oo

d, sin

ce we d

on

’t no

ticech

ang

es in tim

e for th

ing

s mo

ving

arou

nd

the earth

†

t=t0

1-

v2

c2

†

g=

11-v

2

c2

†

t=gt0

Th

e Tw

in P

aradox

We’ve ju

st said th

at mo

ving

clocks ru

n slo

w

Bu

t that d

epen

ds o

n w

ho

’s mo

ving

!

If we see a sp

ace ship

zoo

m b

y, we th

ink it’s

mo

ving

Ho

wever, fro

m th

e persp

ective of th

e peo

ple o

n th

esp

ace ship

, we’re m

ovin

g an

d th

ey are station

ary!

To

qu

ote th

e text, “relativity effects are always

attribu

ted to

the o

ther g

uy

”

Th

e Tw

in P

aradox II

Su

pp

ose w

e have tw

o tw

ins

On

e is an astro

nau

t

On

e is an acco

un

tant

Th

e astron

aut leaves o

n a sp

aceship

wh

ich travels

at 99.5% th

e speed

of lig

ht

Th

is causes a tim

e dilatio

n o

f

or 1 year o

n th

e space sh

ip =

10 years on

Earth

†

t=t0

1-

v2

c2

=t0

1-

(0.995c) 2

c2

=t0

1-

0.99=10t0

Th

e Tw

in P

aradox III

Su

pp

ose th

e astron

aut leaves fo

r a 1 year trip: 6

mo

nth

s ou

t and

6 mo

nth

s back.

Fro

m h

er persp

ective, she’s ag

ed o

ne year

Ho

wever, h

er twin

has ag

ed 10 years!

Wait a seco

nd

… D

idn

’t we say th

at “relativityeffects are alw

ays attribu

ted to

the o

ther g

uy?

”

Th

at mean

s from

her p

erspective, th

e Earth

wen

taw

ay at 99.5% th

e speed

of lig

ht, so

sho

uld

n’t sh

eh

ave aged

mo

re than

her tw

in?

Th

e Tw

in P

aradox IV

Th

e Tw

in P

aradox V