Phy
sics
2D
Lec
ture
Slid
esO
ct 6
Viv
ek S
harm
aU
CSD
Phy
sics
New
Rul
es o
f Coo
rdin
ate
Tran
sfor
mat
ion
Nee
ded
•Th
e G
alile
an/N
ewto
nian
rule
s of t
rans
form
atio
n co
uld
not
hand
les f
ram
es o
f ref
s or o
bjec
ts tr
avel
ing
fast
–
V ≈
C(li
ke v
= 0
.1 c
or 0
.8c
or 1
.0c)
•Ei
nste
in’s
pos
tula
tes l
ed to
–
Des
truct
ion
of c
once
pt o
f sim
ulta
neity
( ∆
t ≠∆
t’)
–M
ovin
g cl
ocks
run
slow
er
–M
ovin
g ro
ds s
hrin
k
•Le
ts fo
rmal
ize
this
in te
rms o
f gen
eral
rule
s of c
oord
inat
e tra
nsfo
rmat
ion
: Lor
entz
Tran
sfor
mat
ion
–R
ecal
l the
Gal
ilean
tran
sfor
m
•X
’ = (x
-vt)
•T’
= T
–Th
ese
rule
s th
at w
ork
ok fo
r mul
e ca
rts n
ow m
ust b
e m
odifi
ed
for r
ocke
t shi
ps w
ith V
≈C
Dis
cove
ring
The
Cor
rect
Tra
nsfo
rmat
ion
Rul
e'
gu
ess
'
()
('
''
'
g
uess
)
xxv
Gx
xvt
xxvt
txvt
Gx
=−
→=
=+
+→
=−
Nee
d to
figu
re o
ut
func
tiona
l for
m o
f G !
G m
ust b
e di
men
sion
less
G
doe
s no
t dep
end
on x
,y,z
,tB
ut G
dep
ends
on
v/c
G is
sym
met
ricA
s v/
c→0
, G →
1R
ocke
t in
S’ (
x’,y
’,z’,t
’) fra
me
mov
ing
with
vel
ocity
v w
.r.t o
bser
ver o
n fra
me
S (x
,y,z
,t)Fl
ashb
ulb
mou
nted
on
rock
et e
mits
pul
se o
f lig
ht a
t the
inst
anto
rigin
s of
S,S
’ coi
ncid
eTh
at in
stan
t cor
resp
onds
to t
= t’
= 0
. Lig
ht tr
avel
s as
a s
pher
ical
wav
e, o
rigin
is a
t O,O
’
Do
a T
houg
ht E
xper
imen
t: R
ocke
t Mot
ion
alon
g x
axis
Spe
ed o
f lig
ht
is c
for b
oth
obse
rver
s
Exa
min
e a
poin
t P (a
t dis
tanc
e r f
rom
O a
nd r’
from
O’)
on
the
Sph
eric
al W
avef
ront
The
dist
ance
to p
oint
P fr
om O
: r =
ct
The
dist
ance
to p
oint
P fr
om O
: r’
= ct
’ C
lear
ly t
and
t’ m
ust b
e di
ffere
ntt ≠
t’
Dis
cove
ring
Lore
ntz
Tran
sfro
mat
ion
for (
x,y,
z,t)
Mot
ion
is a
long
x-x
’ axi
s, s
o y,
z u
ncha
nged
y’=y
,
z’ =
zE
xam
ine
poin
ts x
or x
’ whe
re s
pher
ical
wav
ecr
osse
s th
e ho
rizon
tal a
xes:
x =
r , x
’ =r’
22
22
22
2
'
'
( -
) ,
' (
-)
()
[]
1 o
r =
1
(/
)
('
')
(
'')
'
()
xct
Gxvt
Gt
xvt
c
vct
Gct
tc
c
xct
Gxvt
xct
G
Gc
vctvt
Gv
vtv
c
t
xxvt
γ
γ
==
⇒=
∴
⎡⎤
∴=
−+
−⎢
⎥⎣
⎦⇒
=−
==
+
== −
+
=
∴=
−
22
2
2
2
2
2
22
1si
nce
1
, (
'')
(
()
')
'
1
'1
,
'(
)
'
'[1
xxvt
xx
vtvt
xxvt
xxvt
vt
xx
tx
xt
tv
vvv
xt
tv x
v
v
v
c
v
tv
ct
γ γγ
γγ
γ
γγ
γγ
γγ
γ
γ
γ
γγγ
=+
⇒=
−+
∴
⎡⎤
⎡⎤
∴=
−+
=−
+⎢
⎥⎢
⎥⎣
⎦⎣
⎛⎞
⎛⎞
−=
−⎜
⎟⎜
⎟⎝
⎠
⎦
⎡⎤
⎛⎞
∴=
+⎝
−⎢
=−
−+
=
⎥⎜
⎟⎝
⎠⎣
⎦
⎛⎞
⇒=
−⎜ ⎝
⎠
+⎟ ⎠
21
vxt
cγ
⎡⎡
⎤⎛
⎞−
⎜⎟
⎢⎥
⎝
⎤−
=⎢
⎠⎥
⎣⎣
⎢⎦
⎦⎥
Lore
ntz
Tran
sfor
mat
ion
Bet
wee
n R
ef F
ram
es
2
' ' ''(
)y
yz
zv
ttx
t c
xv x
γ γ=
⎛⎞
=−
⎜⎟
⎝
=−
⎠
=Lore
ntz
Tran
sfor
mat
ion
2
' '
'') '
'
(
vtx
xvt
tc
y zx
y z
γ γ=+
⎛⎞
=+
⎜⎟
⎝
=
⎠
=
Inve
rse
Lore
ntz
Tran
sfor
mat
ion
As
v→0
, Gal
ilean
Tra
nsfo
rmat
ion
is re
cove
red,
as
per r
equi
rem
ent
Not
ice
: SPA
CE
and
TIM
E C
oord
inat
es m
ixed
up
!!!
Lore
ntz
Tran
sfor
m fo
r Pai
r of E
vent
s
Can
und
erst
and
Sim
ulta
neity
, Len
gth
cont
ract
ion
& T
ime
dila
tion
form
ulae
from
this
Tim
e di
latio
n: B
ulb
in S
fram
e tu
rned
on
at t
1&
off
at t 2
: W
hat ∆
t’di
d S
’ mea
sure
?tw
o ev
ents
occ
ur a
t sam
e pl
ace
in S
fram
e =>
∆x
= 0
∆t’
= γ
∆t
(∆t=
pro
per t
ime)
S
x
S’
X’
Leng
th C
ontra
ctio
n: R
uler
mea
sure
d in
S b
etw
een
x 1&
x2
: W
hat ∆
x’di
d S
’ mea
sure
?tw
o en
ds m
easu
red
at s
ame
time
in S
’ fra
me
=> ∆
t’ =
0 ∆
x=
γ(∆
x’ +
0 )
=> ∆
x’ =
∆x
/ γ(∆
x=
prop
er le
ngth
)
x 1x 2
rule
r
Lore
ntz
Vel
ocity
Tra
nsfo
rmat
ion
Rul
e '
''
21
x''
''
21
x'
2
x'
2
x'
2'
In S
' fra
me,
u
,
u,
u1
For v
<<
c, u
(G
ali
divi
de b
y dt
'
'
lean
Tra
ns. R
esto
r
()
(
ed)
)
x
x
xxx
dxtt
dt
dxvdt
vdt
dxc
vdt
dtdx
dxv
dxc
uu
u
d
vv
t
cvγ
γ
−=
=−
−=
− −==
=
=−
−
−
−
SS
’v
u
S an
d S’
are
mea
surin
g
ant’s
spe
ed u
alo
ng x
, y, z
ax
es
2
'
2
'
2
divi
de b
y dt
on
(1)
Ther
e is
a c
hang
e in
vel
ocity
in th
e di
rect
ion
to S
-S' m
otio
n
',
' '(
H
)
'
S
!
R
()
x
yyy
uudy
dy dy
vdt
dtdx
dyc
u
uv
dydt
dxc
v cγ
γ
γ= =
= ⊥
−
−
=
=
−
Vel
ocity
Tra
nsfo
rmat
ion
Per
pend
icul
ar to
S-S
’ mot
ion
'
2
Sim
ilarly
Z
com
pone
nt o
fA
nt' s
vel
ocity
tra
nsfo
rms
(1)
as
zz
x
uu
v cu
γ=
−
Inve
rse
Lore
ntz
Vel
ocity
Tra
nsfo
rmat
ion
'x
'
' '
2 ' 2 ' 2
Inve
rse
Vel
ocity
Tra
nsfo
rm:
(1
u
)
11 ()
yy
z
x
z
x
x
x
uv
vu uu
v c u vc
u
c
u u
γ γ
=+
=++
=+
As
usua
l, re
plac
e v
⇒-v
Doe
s Lo
rent
zTr
ansf
orm
“wor
k” ?
Two
rock
ets
trav
el in
oppo
site
dire
ctio
ns
An
obse
rver
on
eart
h (S
) m
easu
res
spee
ds =
0.7
5cA
nd 0
.85c
for A
& B
re
spec
tivel
y
Wha
t doe
s A
mea
sure
as
B’s
spe
ed?
Plac
e an
imag
inar
y S’
fram
e on
Roc
ket A
⇒v
= 0.
75c
rela
tive
to E
arth
Obs
erve
rS
Con
sist
ent w
ith S
peci
al T
heor
y of
Rel
ativ
ity
Exa
mpl
e of
Inve
rse
velo
city
Tra
nsfo
rm
Bik
er m
oves
with
spe
ed =
0.8
cpa
st s
tatio
nary
obs
erve
r
Thro
ws
a ba
ll fo
rwar
d w
ith
spee
d =
0.7c
Wha
t doe
s st
atio
nary
ob
serv
er s
ee a
s ve
loci
ty
of b
all ?
Pla
ce S
’ fra
me
on b
iker
Bik
er s
ees
ball
spee
d
u X’=
0.7c
Spee
d of
bal
l rel
ativ
e to
st
atio
nary
obs
erve
r u X
?
Can
you
be
seen
to b
e bo
rn b
efor
e yo
ur m
othe
r?A
fram
e of
Ref
whe
re s
eque
nce
of e
vent
s is
RE
VE
RS
ED
?!!
SS’
11
''
11
(,
)
(,
)
xt
xt
u
22
''
22
(,
)
(,
)
xt
xt
I take off f
rom SD
I arrive in SF
''
21
2 '
For w
hat v
alue
of v
can
'
0
vx
ttt
tct
γ⎡
∆⎤
⎛⎞
∆=
−=
∆−
⎜⎟
⎢⎥
⎝⎠
⎣⎦
∆<
I Can
t ‘be
see
n to
arr
ive
in S
F be
fore
I ta
ke o
ff fr
om S
D
SS’
11
''
11
(,
)
(,
)
xt
xt
u
22
''
22
(,
)
(,
)
xt
xt
'
22
2
' 21
2
'
'
For w
hat v
alue
of v
0
v
can
: N
ot a
l low
e
u1
< '0
dc
vx
tt
cc
vc
u
vx
vct
c
vx
ttt
tct
γ⎡
∆⎤
⎛⎞
∆=
−=
∆−
⎜⎟
⎢⎥
⎝⎠
⎣⎦
∆<
∆∆
∆=
∆⇒
<
⇒>
⇒
<⇒
∆
>
Rel
ativ
istic
Mom
entu
m a
nd R
evis
ed N
ewto
n’s
Law
s N
eed
to g
ener
aliz
e th
e la
ws
of M
echa
nics
& N
ewto
n to
con
firm
to L
oren
tzTr
ansf
orm
and
the
Spe
cial
The
ory
of R
elat
ivity
: Exa
mpl
e : pmu
=
12
Bef
ore v 1
’=0
v 2’
21
Afte
r V
’
S’
S
12
Bef
ore
vv
21
Afte
r V
=0
P =
mv
–mv
= 0
P =
0
''
'1
2
''
12
12
21
12
22
2 2
'
2
''
befo
reaf
ter2
0,
, '
2
11
1
1
1
2
2
',
p
pafter
before
mv
pmv
m
vv
vv
vV
vv
vV
vvv
vvV
vv c
p
vv
cc
mV
mv
cc
−−
−−
==
==
=
−=
+
=−
−−
−+
=
≠
+=
=−
()
()
()
2
3/2
22
2
22
2
2
3/2
2
2
2 3/2
1(
/)
12
()(
1
(/
) : R
elat
ivis
t
)2
1(
/)
1(
/)
1
ic
1(
)
( /
/)
mu
pmu
uc
ddud
usedt
dtdu
mmu
udu
Fc
dtuc
uc
mc
mu
mu
duF
dtc
u
dpd
mu
Fdt
dtu
duc
cdt
mF
c
u
γ=
=−
=
⎡⎤
−−
⎢⎥
=+
×⎢
⎥−
−⎣
⎦⎡
⎤−
+⎢
⎥=
⎢⎥
−
⎛⎞
⎜⎟
==
⎜⎟
−⎝
⎠
⎡⎣⎦
⎤⎢
⎥=
⎢⎥
−⎣
⎦ 3/2
2
Forc
e
Sinc
e A
ccel
erat
ion
a =
Not
e: A
s /
1, a
0 !!!
!Its
har
der t
o ac
cele
rate
whe
n yo
,
F
u ge
t clo
se to
spee
d of
l
a=1
(/
h
)
ig
m
tuc
du dt
uc
⎡⎤
−⎣
⎦⇒
→→
Rel
ativ
istic
M
omen
tum
Forc
e A
nd
Acc
eler
atio
n
Rea
son
why
you
can
t qui
tege
t up
to th
e sp
eed
of li
ght !
PEP
PEP --
II a
ccel
erat
or sc
hem
atic
and
tunn
el v
iew
II a
ccel
erat
or sc
hem
atic
and
tunn
el v
iew
A L
inea
r Par
ticle
Acc
eler
ator
3/2
2eE
a=1
(/
)m
uc
⎡⎤
−⎣
⎦
Acc
eler
atin
g E
lect
rons
Thr
u R
F C
aviti
es
Fitti
ng a
5m
pol
e in
a 4
m b
arnh
ouse
2
farm
boy
sees
pol
e c
ontra
ctio
n fa
ctor
1(3
Stud
ent w
ith p
ole
runs
/5)
4/5
says
pol
e ju
st fi
ts i
with
v=(
3/5)
n th
e ba
rn fu
lly!
c
cc
−=
2D S
tude
ntfa
rmbo
y
2
Stud
ent s
ees b
arn
con
tract
ion
fact
or
1(3
/5)
4/5
says
bar
n is
onl
y 3.
2m lo
ng
Stud
, to
ent w
ith p
ole
runs
o sh
ort
to c
onta
in
ent
ire 5
m p
ole
!
with
v=(
3/5)
cc
c
−=
Farm
boy
says
“You
can
do
it”
Stu
dent
say
s “D
ude,
you
are
nut
s”
V =
(3/5
)c
Is th
ere
a co
ntra
dict
ion
? Is
Rel
ativ
ity w
rong
?
Hom
ewor
k: Y
ou fi
gure
out
who
is ri
ght,
if an
y an
d w
hy.
Hin
t: Th
ink
in te
rms
of o
bser
ving
thre
e e
vent
s
Fitti
ng a
5m
pol
e in
a 4
m b
arnh
ouse
?
'
'2
0
' 0
0
0L =
pro
per l
engt
h of
pol
e in
S'
Even
t A :
arriv
al o
f rig
ht e
nd o
f pol
eat
= le
ngth
of b
arn
in S
< L
left
end
of b
arn:
(t =
0, t'
=0) i
s ref
eren
ce
In
fram
e
L
=L1
S: le
ng(
/th
of p
ol)
The
te
imes
in
l
vc
−
' 2
20
B
''
0C
22
't
1(
/)
1(
/)
'1
t1
(/
)
two
fram
es a
re re
late
d:
Tim
e ga
p in
S' b
y w
hich
eve
nts B
and
C
fail
to b
e si
mul
t
1(
aneo
u
/
s
)
BC
BC
ll
vc
tvc
vv
Lt
lv
vvc
vc
==
−=
−
==
⇒
=−
−
2D S
tude
ntfa
rmbo
y
V =
(3/5
)c
Ans
wer
: S
imul
tane
ity!
Farm
boy
sees
two
even
ts a
s si
mul
tane
ous
2D s
tude
nt c
an n
ot a
gree
Fitti
ng o
f the
pol
e in
bar
n is
rela
tive
!
A: A
rriv
al o
f rig
ht e
nd o
f pol
e at
left
end
of b
arn
B: A
rriv
al o
f lef
t end
of p
ole
at le
ft en
d of
bar
n C
: Arr
ival
of r
ight
end
of p
ole
at ri
ght e
nd o
f bar
nS
= Ba
rn fr
ame,
S' =
stud
ent f
Let
ram
e
Farm
boy
Vs
2D S
tude
nt
Pol
e an
d ba
rn a
re in
rela
tive
mot
ion
u su
ch th
at
lore
ntz
cont
ract
ed le
ngth
of p
ole
= P
rope
r len
gth
of b
arn
In re
st fr
ame
of p
ole,
E
vent
B p
rece
des
C