1. Special Relativity, 4 · Postulates of Special Relativity 1. All laws of physics – mechanical...
Transcript of 1. Special Relativity, 4 · Postulates of Special Relativity 1. All laws of physics – mechanical...
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1. Special Relativity, 4lecture 5, September 8, 2017
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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
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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.
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“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.”
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G
TIME DILATION
LENGTH CONTRACTION
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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
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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
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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
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EIII.EE?IItoniE#IeIEno£2 - EE = 1grieve; ;)see : ← r=#%.
so we end up win ,
⇐ '
¥2=
-Y⇐uyg
⇐;.Y¥Iy⇒
torrentz.EE?BeBnshosBo#*.dgshaBE
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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
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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
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back to the airport
H would see: E+B!
A would see: B
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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
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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
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remember:
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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!
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and the coil?yup. right observation all along.
Electric and magnetic fields, depending on the relative frames
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the punch line.
Maxwell’s Equations were good all along.
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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
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@ 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 @ ?
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TRANSFORMATION OF VELOCITIES
the sidewalk evolves
@ SO
|¥¥oes '← a so
-2← wvt @
3 velocities going on here
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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
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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
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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
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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
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y@
→ yi SO•→
"
•
in @ calculate @ quantities vyasti 's. ,,"
-
itfptu'
+ vffv×
7
.
* T ×'
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
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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.
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* 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 .