1. special relativity, 4 · postulates of special relativity 1. all laws of physics – mechanical...

1. Special Relativity, 4lecture 5, September 8, 2017

housekeepingremember to check the course page:

chipbrock.org

and sign up for the feedburner reminders

testing, testing:

someone please go to the syllabus and create a fake pdf according to the instructions and then try to upload it to the dropbox in the D2L site. Let me know friday. Okay?

someone did and it worked for us

Homework workshop is Tuesday

your homework “attempts” due midnight Monday night into D2L

2

Postulates of Special Relativity

1. All laws of physics – mechanical and electromagnetic – are identical in co-moving inertial frames.

taking Galileo seriously, and then adding Maxwellcalled “The Principle of Relativity”

2. The speed of light is the same for all inertial observers.taking Maxwell seriously…that “c” in M.E. is a constant.

“The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an

“absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty

space in which electromagnetic processes take place.”

G

TIME DILATION

LENGTH CONTRACTION

Galilean Transformations →"

Lorentz Transformations"

E 's derivation

@ @ @ SO @ SO

e-he1¥" @

j¥§E#@

← Epaneeiagu.

IN @ :

@Y

× '2+y' 2- ( ct

')2=0

÷¥i×::# a .

after time tdt2=×2+y2 How to connect them ?

x7y2 . cE=o

Try a G. T.

:×

' = x . ut

t'

= t

×'

2+42 . Kt'5=0

( × . utT+y2 - ktT=O

x2 + Cutt .zxut + up - (at

spoils invariance

E : search fn a linear transformation

that preserves postulate # 2

×'

= xx + Utdetermine 471€18

t'

= ext 8T

x '=o×'

= xxtut when

I'Ife ¥t

'

= Exist

÷ at

so : × '=0=xu #←yet

y=- xu

x '= xx - xut = x( x. at )

from ×

'2

+ y's

- ( ct' )2=o a)

: '= Exist

£4x-ut5 . EC Ex

-18+5=0×2×2+ Nutt

'

. zsixut - c2e2x2

.is#.zc2exVt=0x2(x2_c2E)-t2(-x2u2+c282)-zx2xut.2c2ex8t=O

EIII.EE?IItoniE#IeIEno£2 - EE = 1grieve; ;)see : ← r=#%.

so we end up win ,

⇐ '

¥2=

-Y⇐uyg

⇐;.Y¥Iy⇒

torrentz.EE?BeBnshosBo#*.dgshaBE

G1

Why ? Because Lorentz worked backwards from Maxwell 's Equations :

Q.

What transformations on × & t world leave

Maxwell 's Equations Invariant ?

A .

×'

= 8 ( × - nt )1%95 waned by Poincare 1904

t

'= r ( t -

ny )

→ how about the other way?

"

inverse transform"

?

U → u

'

1- not

1

remember?

Weird alert #2: Two identical physical outcomes... from entirely different physical causes for situations which

Weird alert #1: Two different physical outcomes... for situations which differ only by the frame of reference

G1

back to the airport

H would see: E+B!

A would see: B

so the original problems are solved by:

the Lorentz transformations in x and t actually mix electric and magnetic fields

so

A magnetic field in one frame

is a mixture of magnetic and electric fields in another frame

Anelectric fieldinoneframeis a mixture of electric and magnetic fields in another frame

so the original problems are solved by:

the Lorentz transformations in x and t mix electric and magnetic fields

A magnetic field in one frame is a mixture of magnetic and electric fields in another frame

Anelectricfieldinoneframeis a mixture of electric and magnetic fields in another frame

EandBaretwomanifesta0onsofonething:theElectromagne0cField

remember:

remember?

+Q

––– –– ––––q

+B

Situa4on#2

++++++++ ++qv I

novelocity,noforce

+Q

––– –– ––––q

+B

Situa4on#2,fixedbyRela4vity

+ +++++++ ++qv I

novelocity,nowthere’saforce!

F = QELorentz-contracted…nowanelectrosta4cforce

but there should have been a force!

and the coil?yup. right observation all along.

Electric and magnetic fields, depending on the relative frames

the punch line.

Maxwell’s Equations were good all along.

G2

Length contraction redux

@ @ In @Lo

= Xz'

- X ! fixed

→U

Lo

Tamm|#=x.s what about @ ?

tricky . . think about times fn length measurement

1) SO stick sitting stin ⇒×

' xi

⇐OFtape T T ⇒ same length .

tomorrow today× 's . ×

,

'

= Lo

2) @ watching @ go by . ..

T

proper

ma ran

melwefweame }different times ? wrong legtn .

SO must make L @ tz=tyfsimple

@ SO

massL=×z - X

,

@ t, =tz= O|¥E L=r(xkutD - rlx

,

'

- uti )-

L=8[ xz'

- x,

'

. at

'z+ut,

'

]

but also : tz=8( th -

uzzxz'

) = O ⇒ tz'

= aaxz'

t,

= 8Ct,

'

.

uax ,

' )=o tie uax ,

'

L=8[ xz

'

. x,

'

- n÷×i+

u÷× ! ]

L=8[ ( xz

'

.

4) ( i. Ea ) ]=y,÷=%.[

1 - ⇒ ( x 'z*,

'

)

L = FYE Lo

L =

Lg⇒ L measures stick to be less than Lo

what does @ see if sfiohm @ ?

TRANSFORMATION OF VELOCITIES

the sidewalk evolves

@ SO

|¥¥oes '← a so

-2← wvt @

3 velocities going on here

In G. T.

' . what you expect u

'

= v - u

T ^

sosmpeetenedqot , @ (£

speed of @ relative to @

speed of somethingin @

But now we do LT .

×'

= 8 ( × - at )

EI=r( daxz - ug¥ - daatt ) = 8k€ - u )q

but t '=8( A -

pzx ) v p=uz

d¥t=r( I -

zv )← ! Ku

- u )II. - - v'

=

DEE ,

=

-

Ki -Rev)1 V - U✓

=-

Relativistic Velocity TransformationI - ear

But wait...

there's more v 's transverse to u ?

they transform also → t & t'

U×=VI + n →

- ← along u

It B- Vx'

C

vy= I

84 + revs ]

Vz= VI

r[ it Eux'

]

1 v - u & u =N'+ n

V=

--

1- a v 1 +

Iv'

C C

viuextremes of " V =

-

I

+Bzw1) n very small B=% n pro ⇒ v = via G. T.

1 1 1 1

2) v very small

Bzv =

uzzv no

g. → o ⇒ u = vtu GT.

3)

n'= C a light beam in @

u = # = Ctu.

(= C ✓ItB- C

C tu zhdC

(>

,+p=the

Mass is complicated .

→ not willing topart

with momentum conservation

^

}

@

#x identical

Bftp.OB

A• each throws

theirA. B ban at

T one another

¥@•gB¥gj

A

• they collide elastically & recoil

@

y@ µ→y

' SO•→ •

vycB) tits ...,,t

"- itfprti

.

×p T ×

'

+ vftv Y

"¥@ v.a)

0• a y

after lots of practice : IVI = /v' I frame ±p=±IC

y@

→ yi SO•→

"

•

in @ calculate @ quantities vyasti 's. ,,"

-

itfptu'

+ vffv×

7

.

* T ×'

yII@v.caBe a s

Vy (B)= ~I(B)_ vy=

I

8( It

uvIe↳) )4 'tEu×

'

]⇒B) =0

V 'y( B) = V'

Momentum CONSERVATION ! ⇒ along yaxis

VYCB ) = Y'

Epy ( before ) = Epycactev )o -

in @ in @

MAV - MBVYCB ) = - Mart MBVYCB )

MAV - MB of = - MAV + mB¥IUKIV '/=v

ZMAV = Zmczef

mB= 8mA 8

y@

→ yi SO•→

"

•

vy( B) tits ...,,t"

- itfptv'

MOMENTUM CONSERVATION ! ⇒ along y + offer×

.

y'"jIR@T ×

'

o• a TY '

Epy ( before ) = Epy ( after )-

in @ in @

MAV - MBVYCB ) = - Mart MBVYCB )

MAV - MB of = - MAV + MBY'

lU/=1v'/=vZMAV = ZMB if

mB= 8mA 8

it n is vanishingly small ...

Miz = Ma → ="

mo" "

rest mass

"

but as a increases MB > Ma ,MB > Mo

Mo called" relativistic mass

"

M ( u ) =

=✓ I - n%znot popular . .

but consistent.

* ahem *

continuing I = ME

p→

= more relativistic momentum

Transform a

Force- →

F = d- P still

dt

- →

= mdu +

udm← dt

= ME + I dmmen ) =

mo=-

✓ I - n4e

dt ~ differentiate

→ d

E = nor [ die +udf in It

at II. ]

ugh . worse .

. if T is not along

F ? really ugly .