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