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A N
EW U
ND
ERST
AN
DIN
G: C
UR
VED
SPA
CET
IME
In 1
916,
Ein
stei
n p
rese
nte
d t
he
wo
rld
a n
ew u
nd
erst
and
-in
g o
f th
e u
niv
erse
: his
Gen
eral
Th
eory
of R
elat
ivit
y. I
n t
his
th
eory
, sp
ace
is n
ot
an e
mp
ty v
oid
, bu
t an
invi
sib
le s
tru
c-tu
re
calle
d “
spac
etim
e”.
No
r is
sp
ace
sim
ply
a
thre
e-d
imen
sio
nal
gri
d t
hro
ug
h w
hic
h m
atte
r an
d l
igh
t an
d
ener
gy
mov
e. I
t is
a f
ou
r-d
imen
sio
nal
str
uct
ure
wh
ose
sh
ape
is c
urv
ed a
nd
tw
iste
d b
y th
e p
rese
nce
of m
atte
r.
Wh
at e
xact
ly is
th
is “s
pac
etim
e” t
hat
Ein
stei
n s
po
ke o
f?
For
man
y, “s
pac
etim
e” s
ou
nd
s lik
e a
scie
nce
fic
tio
n c
on
cep
t w
her
e ex
plo
rers
tra
vel t
hro
ug
h t
ime
on
the
way
to p
aral
lel u
niv
erse
s to
mee
t alie
n li
fe fo
rms.
In s
om
e w
ays,
it is
a s
imp
ler i
dea
than
that
. Th
e te
rm “s
pac
etim
e” s
imp
ly re
fers
to a
fram
e o
f ref
eren
ce in
wh
ich
the
thre
e d
imen
sio
ns
of s
pac
e (x
, y, z
) an
d t
he
dim
ensi
on
of t
ime
are
use
d to
des
crib
e m
oti
on
or a
ctio
n in
th
at fr
ame
of r
efer
ence
.
Wh
ere
spac
etim
e g
ets
inte
rest
ing
is w
hen
on
e co
nsi
der
s th
at it
is a
fram
e o
f ref
eren
ce t
hat
is n
ot
alw
ays
squ
are.
Wh
en a
mas
s is
pre
sen
t, su
ch a
s a
star
or
pla
net
, th
e sp
acet
ime
fram
e is
def
orm
ed
or
curv
ed b
y it
s p
rese
nce
. Th
is m
ean
s th
at t
he
fram
e o
f re
fere
nce
, by
wh
ich
we
mea
sure
th
e m
oti
on
of t
hin
gs,
is n
ot
a g
rid
of s
trai
gh
t lin
es a
nd
rig
ht
ang
les
bu
t a
gri
d t
hat
can
be
war
ped
an
d
curv
ed.
This
idea
co
ntr
asts
dir
ectl
y w
ith
New
ton’
s co
nce
pt
of
a fix
ed, i
mm
ovab
le, u
niv
ersa
l gri
d
up
on
wh
ich
all
mo
tio
n c
an b
e m
easu
red
.
Take
, fo
r in
stan
ce, t
he
orb
it o
f M
ars
aro
un
d t
he
Sun
. Fr
om
ou
r p
ersp
ecti
ve, i
t ap
pea
rs a
s th
ou
gh
Mar
s is
fo
llow
ing
a c
urv
ed p
ath
th
rou
gh
sp
ace.
We
thin
k it
is
curv
ed b
ecau
se w
e co
mp
are
it w
ith
an
invi
sib
le, f
ixed
, sq
uar
e b
ackg
rou
nd
.
Ho
wev
er, w
e ca
n a
lso
imag
ine
that
Mar
s is
follo
win
g a
str
aig
ht p
ath
thro
ug
h s
pac
e. I
t is
a “s
trai
gh
t”
pat
h b
ecau
se it
is fo
llow
ing
th
e sh
ape
of s
pac
etim
e. I
t ju
st s
o h
app
ens
that
sp
ace
itse
lf is
cu
rved
d
ue
to t
he
pre
sen
ce o
f th
e Su
n.
HO
W D
OES
CU
RV
ED S
PAC
ETIM
E EX
PLA
IN O
RB
ITS?
Ou
r so
lar
syst
em is
fille
d w
ith
ob
ject
s fo
llow
ing
orb
ital
pat
hs.
Pla
net
s o
rbit
th
e Su
n. M
oo
ns
orb
it
the
pla
net
s. S
atel
lites
an
d t
he
Inte
rnat
ion
al S
pac
e St
atio
n o
rbit
th
e Ea
rth
. W
hen
New
ton
loo
ked
u
p a
t th
e n
igh
t sk
y in
th
e 17
th c
entu
ry, h
e o
bse
rved
th
ese
orb
ital
mo
tio
ns
and
co
ncl
ud
ed t
hat
, in
ac
cord
ance
wit
h h
is la
ws
of m
oti
on
, th
ere
mu
st b
e a
forc
e ac
tin
g o
n th
ese
ob
ject
s to
kee
p th
em in
o
rbit
.
Wh
y m
ust
th
ere
be
a fo
rce?
Bec
ause
th
ese
ob
ject
s w
ere
clea
rly
mov
ing
in c
urv
ed p
ath
s. A
cco
rd-
ing
to N
ewto
n’s
law
of i
ner
tia,
th
e o
nly
way
th
at a
n o
bje
ct w
ou
ld c
han
ge
dir
ecti
on
fro
m a
str
aig
ht
pat
h w
as if
a fo
rce
was
act
ing
on
it. S
ince
the
pla
net
s an
d m
oo
ns
wer
e co
nst
antl
y ch
ang
ing
dir
ec-
tio
n (
an o
rbit
is
an i
nfin
ite
seri
es o
f tu
rns)
, th
ere
mu
st b
e a
forc
e co
nst
antl
y ac
tin
g o
n t
hem
. N
ewto
n’s
con
clu
sio
n w
as th
at th
e fo
rce
was
“gra
vity
” – a
n a
ttra
ctiv
e fo
rce
that
em
anat
ed fr
om
eac
h
mas
s an
d p
ulle
d o
ther
mas
ses
in to
war
d it
.
Ein
stei
n c
ame
to a
diff
eren
t co
ncl
usi
on
. Fr
om
his
per
spec
tive
, th
e p
lan
ets
and
mo
on
s in
orb
it in
o
ur s
ola
r sys
tem
are
no
t m
ovin
g in
cu
rved
pat
hs.
In fa
ct, t
hey
are
follo
win
g s
trai
gh
t lin
es t
hro
ug
h
spac
etim
e. I
t is
sp
acet
ime
that
is c
urv
ed, n
ot
the
pat
hs
of t
he
orb
itin
g b
od
ies.
Sin
ce t
hes
e b
od
ies
are
no
t cu
rvin
g o
r tu
rnin
g, t
her
e is
no
forc
e ac
tin
g o
n t
hem
. N
o fo
rce
is n
eed
ed t
o e
xpla
in t
hes
e o
rbit
s; t
her
efo
re, t
he
attr
acti
ve f
orc
e o
f g
ravi
ty d
oes
no
t ex
ist.
In
fac
t, th
e o
rbit
ing
mas
ses
are
sim
ply
reac
tin
g to
th
e sh
ape
of s
pac
etim
e.
Ho
w c
ou
ld th
e Ea
rth
, wh
ich
so
cle
arly
see
ms
to fo
llow
a c
urv
ed p
ath
aro
un
d th
e Su
n, b
e sa
id to
be
follo
win
g a
str
aig
ht p
ath
? F
irst
, lo
ok
clo
ser a
t wh
at “s
trai
gh
t” m
ean
s –
it d
escr
ibes
the
mo
tio
n o
f an
o
bje
ct t
hat
sta
ys p
aral
lel t
o a
lin
e in
its
fram
e o
f ref
eren
ce.
Seco
nd,
imag
ine
that
th
e g
rid
lin
es o
n
ou
r fr
ame
of r
efer
ence
are
cu
rved
. A
rou
nd
th
e Su
n, t
he
fram
e o
f ref
eren
ce is
cir
cula
r b
ecau
se t
he
mas
s o
f th
e Su
n h
as w
arp
ed s
pac
etim
e ar
ou
nd
it.
Fin
ally
, tra
ce t
he
pat
h o
f th
e Ea
rth
aro
un
d t
he
Sun
. N
oti
ce t
hat
its
pat
h, w
hile
ap
pea
rin
g c
urv
ed a
t fir
st, i
s ac
tual
ly m
ovin
g p
aral
lel t
o t
he
circ
ula
r sp
acet
ime
gri
d a
rou
nd
Ear
th. T
he
Eart
h is
follo
win
g a
str
aig
ht
pat
h a
rou
nd
th
e Su
n.
This
is h
ow
cu
rved
sp
acet
ime
exp
lain
s th
e o
rbit
al m
oti
on
of a
ll m
atte
r in
th
e u
niv
erse
, fro
m s
atel
-lit
es to
pla
net
s to
bin
ary
star
s to
sw
irlin
g g
alax
ies,
in a
dd
itio
n to
the
Spac
e Sh
utt
le a
nd
the
Inte
rna-
tio
nal
Sp
ace
Stat
ion
. Th
e p
rese
nce
of
a m
ass
has
an
ab
ility
to
cu
rve
loca
l sp
acet
ime.
A
s m
atte
r m
oves
nea
r th
e ce
ntr
al m
ass,
it fo
llow
s th
is c
urv
e o
f sp
acet
ime.
A S
ECO
ND
IMP
LIC
ATI
ON
: FR
AM
E-D
RA
GG
ING
A f
ew y
ears
aft
er E
inst
ein
su
bm
itte
d h
is t
heo
ry o
f cu
rved
sp
acet
ime,
Au
stri
an p
hys
icis
ts J
ose
ph
Len
se
and
Han
s Th
irri
ng
pre
dic
ted
that
a m
ass
cou
ld d
efo
rm
spac
etim
e in
a s
eco
nd
way
- th
rou
gh
“fra
me-
dra
gg
ing”
(1
918)
. Th
ey p
rop
ose
d th
at th
e ro
tati
on
of p
lan
ets
and
st
ars
or
any
rota
tin
g m
ass
twis
ts t
he
spac
etim
e fr
ame
nea
r th
at m
ass.
Fo
r ex
amp
le, n
ot
on
ly is
loca
l sp
acet
-im
e cu
rved
nea
r th
e Su
n, i
t is
twis
ted
by
the
Sun’
s ro
ta-
tio
n.
Len
se a
nd
Th
irri
ng
pre
dic
ted
th
at t
his
eff
ect
wo
uld
be
extr
emel
y sm
all,
and
bec
om
e sm
alle
r fa
rth
er fr
om
the
rota
tin
g m
ass,
bu
t it w
ou
ld o
ccu
r aro
un
d e
very
rota
tin
g m
ass,
be
it a
pla
net
, a s
tar,
a g
alax
y, o
r a
per
son
. A
s ye
t, it
is
un
clea
r h
ow
th
is p
hen
om
eno
n i
s su
pp
ose
d t
o o
ccu
r, b
ut
the
mat
hem
atic
al e
qu
atio
ns
con
clu
de
that
if m
asse
s cu
rve
spac
etim
e, t
hen
th
ey a
lso
mu
st t
wis
t sp
a-ce
tim
e.
DEM
ON
STR
ATI
ON
OF “
FR
AM
E-D
RA
GG
ING
” Th
e p
urp
ose
of t
his
act
ivit
y is
to d
emo
nst
rate
ho
w t
he
Eart
h t
wis
ts t
he
loca
l sp
acet
ime
fram
e, b
ut
do
es n
ot
affe
ct a
dis
tan
t sp
acet
ime
fram
e.
M
ATE
RIA
LS: P
aper
pla
te, h
on
ey, s
up
erb
all,
foo
d c
olo
rin
g, t
hu
mb
tack
, pu
shp
in, p
epp
erco
rn
P
RO
CED
UR
E:
1. P
oke
a t
ack
thro
ug
h t
he
cen
ter o
f th
e p
late
po
inti
ng
up
war
ds.
2. P
ou
r a la
yer o
f ho
ney
on
th
e p
late
. 2.
Pla
ce t
he
bal
l fir
mly
on
th
e ta
ck.
3. P
lace
a p
epp
erco
rn in
th
e h
on
ey n
ear t
he
edg
e o
f th
e p
late
. 4.
Po
ke a
pu
shp
in in
th
e to
p o
f th
e su
per
bal
l. 5.
Sq
uee
ze a
dro
p o
f fo
od
co
lori
ng
in t
he
ho
ney
aro
un
d t
he
bal
l. 5.
Tw
ist
the
sup
erb
all a
s fa
st a
s yo
u c
an.
6. O
bse
rve
the
effe
cts
of t
he
twis
tin
g b
all o
n t
he
ho
ney
.
QU
ESTI
ON
S: 1.
How
doe
s th
e ho
ney
reac
t diff
eren
tly
near
the
ball
than
far f
rom
the
ball?
2.
Wha
t do
the
part
s of
the
mod
el re
pres
ent?
The
bal
l? T
he h
oney
? T
he p
eppe
rcor
ns?
The
food
col
orin
g?
3. W
hat c
ause
s th
e “d
ragg
ing”
in th
e m
odel
?
SUM
MA
RY:
The
bal
l rep
rese
nts
the
Eart
h, t
he
ho
ney
rep
rese
nts
sp
acet
ime,
an
d th
e p
epp
erco
rn re
pre
sen
ts a
dis
tan
t mas
s
(sta
rs, p
lan
ets,
etc.
) in
sp
acet
ime.
Th
e fo
od
co
lori
ng
is u
sed
to
hig
hlig
ht
the
ho
ney
’s m
oti
on
, an
d d
oes
no
t re
pre
sen
t an
ast
ron
om
ical
ob
ject
. Th
e ro
tati
on
of t
he
Eart
h d
oes
twis
t th
e sp
acet
ime
fram
e lik
e th
e b
all t
wis
ts
the
ho
ney
, alt
ho
ug
h it
is n
ot
cau
sed
by
“fri
ctio
n” b
etw
een
th
e Ea
rth
an
d lo
cal s
pac
etim
e. T
he
theo
ry o
f gen
-er
al re
lati
vity
sta
tes
that
sp
acet
ime
and
mas
ses
hav
e a
mys
teri
ou
s m
utu
al “g
rip”
on
eac
h o
ther
.
EA
RTH
WH
ITE
DW
ARF
(avg
.)
SIZ
E
1
2,75
6 km
~
13,
000
km
MA
SS
6
x 10
24 k
g
~ 1
, 100
,000
x 1
024 k
g
MO
DEL
S O
F SP
AC
ETIM
E U
sin
g a
pla
stic
fram
e an
d s
pan
dex
sh
eet,
you
can
cre
ate
a si
mp
le m
od
el o
f cu
rved
sp
acet
ime.
On
th
is m
od
el, y
ou
can
dem
on
stra
te s
ever
al p
hen
om
ena
that
ar
e ca
use
d b
y th
e cu
rvat
ure
of s
pac
etim
e.
MAT
ERIA
LS
* Sp
and
ex s
hee
t, 6’
x 6’
--
(ord
er fr
om fa
bric
sto
re o
r web
site
) *
PVC
pip
es a
nd
join
ts -
- (c
ut to
bui
ld a
5’x
5’x
3’ f
ram
e; c
ompl
ete
inst
ruct
ions
at h
ttp:
//ei
nste
in.s
tanf
ord.
edu/
) *
Larg
e w
eig
hts
--
(bag
s of
san
d or
wat
er, i
ron/
bras
s w
eigh
t)
* Sm
all a
nd
larg
e b
alls
--
(sup
erba
lls, s
teel
/bra
ss b
all b
eari
ngs,
ping
pon
g ba
lls, g
olf b
alls
, bea
ch b
alls
, etc
.) A
& B
-- S
ATE
LLIT
E A
ND
PLA
NET
AR
Y O
RB
ITS
1. P
lace
a h
eavy
mas
s in
th
e ce
nte
r of t
he
shee
t to
cre
ate
a si
zab
le d
epre
ssio
n.
2. P
ass
ou
t a
smal
l bal
l to
eac
h p
air o
f stu
den
ts. T
ake
turn
s to
try
ou
t ch
alle
ng
es.
CH
ALL
ENG
E A
– Y
ou
are
sen
din
g a
sat
ellit
e to
exp
lore
a d
ista
nt
pla
net
b
ehin
d a
nea
rby
pla
net
. St
and
on
op
po
site
sid
es o
f th
e ce
nte
r mas
s
(th
e “n
ear p
lan
et”)
. R
oll
the
bal
l (yo
ur s
atel
lite)
so
th
at it
reac
hes
yo
ur p
artn
er (t
he
“dis
tan
t p
lan
et”)
wit
ho
ut
hit
tin
g t
he
cen
ter m
ass.
CH
ALL
ENG
E B
– Y
ou
wan
t to
pu
t a
sate
llite
into
orb
it a
rou
nd
th
e “n
ear p
lan
et”.
Wo
rk to
get
her
an
d fi
gu
re o
ut
ho
w b
est
to
pu
t yo
ur “
sate
llite
” in
to o
rbit
aro
un
d t
he
cen
ter m
ass.
C &
D --
BEN
DIN
G S
TAR
LIG
HT
& G
RA
VIT
ATI
ON
AL
LEN
SEs
1. P
lace
a h
eavy
mas
s in
th
e ce
nte
r o
f th
e sh
eet
to c
reat
e a
siza
ble
dep
ress
ion
. R
emin
d s
tud
ents
th
at e
lec-
tro
mag
net
ic w
aves
(su
ch a
s lig
ht)
follo
w t
he
curv
e o
f sp
acet
ime
just
like
mas
ses
do.
2.
Pu
t st
ud
ents
in g
rou
ps
of
thre
e. O
ne
per
son
is a
“sta
r” e
mit
tin
g li
gh
t in
th
e fo
rm o
f a
ph
oto
n-b
all.
On
e p
erso
n is
an
ob
serv
er o
n E
arth
. T
he
thir
d p
erso
n is
th
e “m
arke
r”.
CH
ALL
ENG
E C
- D
emo
nst
rate
ho
w s
tarl
igh
t b
end
s ar
ou
nd
larg
e m
asse
s. R
oll
a b
all f
rom
th
e “s
tar”
on
on
e si
de
of t
he
shee
t to
the
“Ear
th” o
n th
e o
ther
sid
e o
f th
e sh
eet.
Th
e st
ar a
nd
Ear
th s
ho
uld
be
dir
ectl
y o
pp
osi
te
each
oth
er.
The
thir
d p
erso
n s
po
ts t
he
po
int
wh
ere
the
“ph
oto
n-b
all”
turn
s to
war
ds
the
Eart
h.
On
ce,
she
spo
ts t
hat
po
int,
she
stan
ds
at t
he
end
of l
ine
com
ing
fro
m t
he
Eart
h t
hro
ug
h t
he
“tu
rnin
g p
oin
t”.
She
is n
ow
th
e “a
pp
aren
t st
ar.”
CH
ALL
ENG
E D
– D
emo
nst
rate
a g
ravi
tati
on
al l
ens.
Hav
e th
e “s
tar”
stu
den
t ro
ll tw
o “p
ho
ton
-bal
ls”
aro
un
d
op
po
site
sid
es o
f th
e ce
ntr
al m
ass.
Kee
p tr
yin
g u
nti
l bo
th b
alls
reac
h th
e “E
arth
” at
the
sam
e ti
me.
Tw
o
stu
den
ts m
ark
the
“tu
rnin
g p
oin
ts” a
nd
sta
nd
in li
ne
wit
h e
ach
po
int a
nd
the
Eart
h. T
hes
e st
ud
ents
are
tw
o “a
pp
aren
t st
ars”
cre
ated
fro
m t
he
ligh
t o
f a s
ing
le s
tar.
* S
tars
em
it li
gh
t in
all
dir
ecti
on
s, so
sta
rlig
ht d
oes
no
t ju
st g
o a
rou
nd
on
e si
de
of a
cen
tral
mas
s. If
the
star
an
d th
e ce
ntr
al m
ass
are
alig
ned
pro
per
ly, t
he
star
will
ap
pea
r on
mu
ltip
le s
ides
of t
he
cen
tral
mas
s.
It m
ay a
pp
ear a
s if
ther
e ar
e se
vera
l sta
rs a
rou
nd
the
cen
tral
mas
s. In
fact
, th
ese
star
s ar
e al
l par
t of t
he
ligh
t co
min
g fr
om
a s
ing
le s
tar.
E --
BIG
GER
OB
JEC
TS =
GR
EATE
R G
RA
VIT
Y?
Wh
ich
cau
ses
mo
re c
urv
atu
re o
f sp
acet
ime
(str
on
ger
gra
vity
) -- t
he
Eart
h o
r a w
hit
e d
war
f sta
r?
Even
th
ou
gh
th
ey a
re t
he
sam
e si
ze, t
he
wh
ite
dw
arf i
s m
uch
m
ore
mas
sive
an
d c
reat
es a
gre
ater
cu
rvat
ure
of
spac
etim
e.
Dem
on
stra
te th
e d
isti
nct
ion
wit
h a
wh
iffle
bal
l an
d a
bas
ebal
l o
n th
e sp
and
ex s
hee
t. T
he
wh
iffle
bal
l cre
ates
a m
uch
sm
alle
r d
epre
ssio
n o
r cu
rve
of s
pac
etim
e th
an t
he
bas
ebal
l cre
ates
.
Wh
y? B
ecau
se t
he
bas
ebal
l is
mo
re m
assi
ve.
In t
his
cas
e, m
ass
mat
ters
, no
t si
ze.
EART
H
STA
R
MO
DEL
S OF S
PACE
TIM
E (CO
NT.
) Q
uest
ions
A
& B
1.
Wha
t are
the
thre
e ke
y va
riabl
es y
ou m
ust c
ontr
ol w
hen
dire
ctin
g a
sate
llite
to w
here
you
wan
t it t
o go
? 2.
Why
is it
impo
ssib
le fo
r you
to p
ut y
our “
sate
llite
” int
o pe
rpet
ual o
rbit
in th
is m
odel
, eve
n th
ough
we
do it
in s
pace
all
the
time?
C
& D
3.
Whe
re d
oes t
he st
ar a
ppea
r to
be to
the
Eart
h-ba
sed
obse
rver
?
4. W
hy is
one
star
calle
d an
“app
aren
t sta
r”?
5. T
his p
heno
men
on is
calle
d a
“gra
vita
tiona
l len
s”. W
hat d
oes i
t hav
e to
do
with
gra
vity
?
6. H
ow is
this
phe
nom
enon
sim
ilar t
o w
hat a
lens
doe
s to
light
? E
7. W
hat c
hara
cter
istic
det
erm
ines
the
amou
nt o
f spa
cetim
e cu
rvat
ure
or g
ravi
tatio
nal s
tren
gth
-- v
olum
e or
mas
s?
8. W
hat a
bout
the
Sun
and
a ne
utro
n st
ar?
The
Sun
is a
bout
100
,000
tim
es la
rger
, but
the
neut
ron
star
has
a li
ttle
mor
e m
ass.
Th
e Su
n co
vers
muc
h m
ore
spac
e; d
oes t
his c
ause
mor
e cu
rvat
ure?
9.
Do
mor
e de
nse
obje
cts a
lway
s cre
ate
mor
e cu
rvat
ure
than
less
den
se o
bjec
ts?
Answ
ers -- #1 -- Initial speed, initial direction, distance of orbit from planet; #2 -- Friction betw
een ball and sheet reduces speed causing ball to lose m
omentum
; no friction in space so satellite’s mom
entum rem
ains constant; #3 -- At the turning point, to the side of the central m
ass; #4 -- The star appears to be to the side of the central m
ass, but is actually behind the central mass; #5 -- The curvature of spacetim
e, which bends the starlight, is
referred to as gravity; #6 -- Just like a lens, the curvature of spacetime around the central m
ass bends incoming light and focuses it to a com
mon
point; #7 -- Mass determ
ines curvature or gravitational strength; #8 -- Even though the Sun covers more space, the m
ore massive object alw
ays creates m
ore curvature or gravity; #9 -- No. A
denser object may not have as m
uch mass as a less dense object. C
ompare a steel ship and a quarter.
The ship is less dense (it floats), but has more overall m
ass.
James Overduin, Pancho and Evelyn Eekels, and Kate Stephenson