rubab thesis

8/11/2019 Rubab Thesis

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GRAVITATIONAL WAVES

A thesis submitted for the degree of B.Sc. (Hons.)

RUBAB AMJAD0751-BH-PHY-10Session 2010-2014

DEPARTMENT OF PHYSICSGOVERNMENT COLLEGE UNIVERSITY

LAHORE


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RESEARCH COMPLETION CERTIFICATE

It is certified that research work contained in this thesis titledGravitationa Wav!"#has been carried out and co !leted b"R$%a% A&'a() #oll $o% 0751-BH-PHY-10&under " su!er'ision durin( his B*S+ ,Hon"*- studies in the sub)ect of !h"sics%

*ateS$.!rvi"or +Dr* M$%a"/ara Ha&!!(

*e!art ent of Ph"sics&, .ni'ersit"& /ahore%

S$%&itt!( t/ro$0/

Pro1* Dr* Ria2 A/&a( Contro !r o1 E3a&ination")C/air.!r"on *e!art ent of Ph"sics , .ni'ersit"& /ahore, .ni'ersit"& /ahore%


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DECLARATION

I&R$%a% A&'a( & #oll $o% 0751-BH-PHY-10& student of B%Sc Hons% in the sub)ectof Ph"sics session 2010-2014& hereb" declare that the atter !rinted in the thesistitledGravitationa Wav!"# is " own work and has not been sub itted in wholeor in !art for an" de(ree or di!lo a at this or an" other uni'ersit"%*ated+

Si(nature of the *e!onent 444444444444444444


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D!(i+at!( to

My Grandmother

Who

Love me from the depth

Of heart.


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Abstract ,ra'itational wa'es are 8ri!!les in s!ace-ti e%8 9ust like a boat sailin( throu(h theocean !roduces wa'es in the water& o'in( asses like stars or black holes !roduce(ra'itational wa'es in the fabric of s!ace-ti e% 6 ore assi'e o'in( ob)ect will !roduce ore !owerful wa'es& and ob)ects that o'e 'er" :uickl" will !roduce orewa'es o'er a certain ti e !eriod% ,ra'itational wa'es are usuall" !roduced in aninteraction between two or ore co !act asses% Such interactions include the binar" orbit of two black holes& a er(er of two (ala ies& or two neutron star orbitin(each other% 6s the black holes& stars& or (ala ies orbit each other& the" send out wa'esof 8(ra'itational radiation8 that reach the ;arth& Howe'er& once the wa'es do (et tothe ;arth& the" are e tre el" weak% his is because (ra'itational wa'es& like water wa'es& decrease in stren(th as the" o'e awa" fro the source% ,ra'itational wa'escarr" not onl" ener(" but also infor ation about how the" were !roduced and !ro'ideus with infor ation that li(ht cannot (i'e%

ha!ter1% INTRODUCTION


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9*9: Co"&o o07

os olo(" is the stud" of the ori(in& e'olution& and e'entual fate of the uni'erse alsothe stud" of the .ni'erse in its totalit"%

9*;: R! ativit7

he theor" of relati'it"& or si !l" relati'it" in !h"sics& usuall"enco !asses two theories b" 6lbert ;instein+

S!ecial relati'it" and(eneral relati'it"

S!ecial relati'it" is a theor" of the structure ofs!aceti e % S!ecial relati'it" is basedon two !ostulates+

1% he laws of !h"sics are the sa e for all obser'ers in unifor otion relati'eto one another !rinci!le of relati'it" %

2% hes!eed of li(ht in a 'acuu is the sa e for all obser'ers& re(ardless of their

relati'e otion or of the otion of the li(ht source%

,eneral relati'it" is a theor" of (ra'itation de'elo!ed b" ;instein in the "ears1


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echnicall"& (eneral relati'it" is a theor" of(ra'itation whose definin( feature isits use of the;instein field e:uations% he solutions of the field e:uationsare etric tensors which define theto!olo(" of the s!aceti e and how ob)ectso'e inertiall"%

9*


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?e know that li(ht alwa"s takes the shortest !ath between two !oints& which weusuall" think of as a strai(ht line% Howe'er& a strai(ht line is onl" the shortest distance between two !oints on a flat surface% n a cur'ed surface& the shortest distance between two !oints is actuall" a cur'e& technicall" known as a(eodesic%

i(ure 1%2+ 6 (eodesic in a cur'ed s!aceti e

9*>: S.a+!?ti&!

".a+!ti&! is an" athe atical odel that co bines s!ace and ti e into a sin(leinterwo'en continuu % he s!aceti e of our uni'erse is usuall" inter!reted froa ;uclidean s!ace !ers!ecti'e& which re(ards s!ace as consistin( ofthree di ensions&and ti e as consistin( of one di ension& theAfourth di ensionA%

i(ure 1%C+ fabric of s!aceti e
http://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Spacehttp://en.wikipedia.org/wiki/Time_in_physicshttp://en.wikipedia.org/wiki/Continuum_(theory)http://en.wikipedia.org/wiki/Universehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Three-dimensional_spacehttp://en.wikipedia.org/wiki/One_dimensionhttp://en.wikipedia.org/wiki/Fourth_dimensionhttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Spacehttp://en.wikipedia.org/wiki/Time_in_physicshttp://en.wikipedia.org/wiki/Continuum_(theory)http://en.wikipedia.org/wiki/Universehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Three-dimensional_spacehttp://en.wikipedia.org/wiki/One_dimensionhttp://en.wikipedia.org/wiki/Fourth_dimension


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ha!ter2%

GRAVITY AS A MENIFESTATION OF SPACETIMECURVATURE

6ccordin( to $ewtonian theor"& (ra'it" is relati'el" stron( when ob)ects are near each other& but weakens with distance and the bi((er the bodies& the ore their forceof utual attraction% his is known as Din'erse-s:uare lawE%

i(ure 2%1+ $ewtonFs law of uni'ersal (ra'itation

;*9: Fi! ( E8$ation o1 N!@tonian Gravit7

;lectro a(netis is de'elo!ed b" considerin( the electro a(netic C-force on achar(ed !article% So discussion of (ra'it" starts fro the descri!tion of (ra'itationalforce in classical& non relati'istic@ theor" of $ewton% In $ewtonian theor"& the(ra'itational force f on a test !article of (ra'itational ass m, at so e !osition is

f = mG g= mG 2%1

where g is (ra'itational field fro (ra'itational !otential at that !osition%

,ra'itational !otential is deter ined b" PoissonFs e:uation+

2

= 4 G 2%2


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is the (ra'itational atter densit" and G is $ewtonFs (ra'itational constant% hisis the field e:uation of $ewtonian (ra'it"%

;*;: In+o&.ati%i it7 o1 N!@tonian 0ravit7 @it/ S.!+ia r! ativit7

$ewton had assu ed that the force of(ra'it" acts instantaneousl"& and ;instein hadalread" shown that nothin( can tra'el at infinite s!eed& not e'en (ra'it" & bein( li ited b" the uni'ersal s!eed li it of the s!eed of li(ht% urther ore& $ewton had assu edthat the force of (ra'it" was !urel" (enerated b" ass& whereas;instein had shownthat all for s of ener(" had effecti'e ass and ust therefore also be sources

of (ra'it" % $ewtonian (ra'it" consistent with s!ecial relati'it" as there is no e !licit ti e

de!endence& which eans !otential res!onds instantaneousl" to a disturbance in

the atter densit" @ as si(nals cannot !ro!a(ate faster than c s!eed of li(ht % ?e

a" re ed" this !roble b" notin( that /a!lacian o!erator 2 in 1%1 is

e:ui'alent to inus the dF6le bertian o!erator in the li it c & and thus

odified field e:uation

= 4 G, 2%C

Howe'er& this does not "ield a consistent relati'istic theor"% his is still not /orent

co'ariant& the atter densit" does not transfor like /orent scalar%

Second funda ental difference between electro a(netic and (ra'itational forces& isthe e:uation of otion of !article of inertial assm1 in a (ra'itational field is (i'en b"

d2

xd t

2 = mG

m1 2%4

#atiomGm1 a!!earin( in the e:uation is sa e for all the !articles% In contrast& the

e:uation of otion of a char(ed !article is
http://www.physicsoftheuniverse.com/scientists_einstein.htmlhttp://www.physicsoftheuniverse.com/scientists_einstein.htmlhttp://www.physicsoftheuniverse.com/scientists_einstein.htmlhttp://www.physicsoftheuniverse.com/scientists_einstein.html


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d d

= qm

F . u 2%5

is the 4- o entu and u is the 4-'ilocit"% he ratio q/m in e:uation of otion of a char(ed !article in an electro a(netic field is not sa e for all !articles%

;*

6ssu e that in coordinate s"ste 1%5 etric is stationar"& eans all deri'ati'es0 g are ero%

he worldline of the !article freel" fallin( under (ra'it" is (i'en in (eneral b"(eodesic e:uation%

d2 x d

2 + d x

d d x d

= 0 2%7

?e assu e& howe'er the !article is o'in( slowl" so@

d xi

d d x

0

d

?e can i(nore C- 'elocit" ter s and (et

d2 x

d 2 + 00 c

2( dt d )2

= 0 2%K

?e ha'e etric stationar"& so we ha'e

000 = 0 00i =

1

2 ij j h00

Insertin( the 'alues of coefficients in 1%K (i'es


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d

2t

d 2 = 0 this shows

dt d Lconstant

d2

xd 2

= 12 c

2

(dt d )

2

h 00 (2.9)

$ow& b" co binin( the two e:uations we ha'e followin( e:uation of otion of the !article+

2%10

If we co !are this with usual $ewtonian e:uation of otion in 1%5 we see these two

are sa e b" identif"in( that h00 =2 c

2

Hence for slowl" o'in( !article the descri!tion of (ra'it" as s!aceti e cur'aturetends to $ewtonian theor" if the etric is such that& in the li it of weak (ra'itationalfield&

g00 =(1 + 2 c2 ) 2%11ro 1%11 & the obser'ant reader will ha'e noticed that the descri!tion of (ra'it" inter s of s!aceti e cur'ature has another i ediate conse:uence& na el" that theti e coordinate t does not& in (eneral& easure !ro!er ti e% If we consider a clock at

rest at so e !oint in our coordinate s"ste i%e%dxi

dt = 0 & the !ro!er ti e inter'al

d between two clicksF of the clock is (i'en b"+

c2d

2 = g dx dx= g00 c

2dt

2

ro which we find that

d =(1 + 2 c

2 )1

2

dt

d2 x

d 2 =

12

c2

h 00


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his (i'es the inter'al of !ro!er d corres!ondin( to an inter'aldt of coordinateti e for a stationar" obser'er near a assi'e ob)ect& in a re(ion where the

(ra'itational !otential is % Since is ne(ati'e& this !ro!er ti e inter'al is

shorter than the corres!ondin( inter'al for a stationar" obser'er at a lar(e distancefro the ob)ect& where M0 and so d = dt % hus& as a bonus& our anal"sis hasalso "ielded the for ula fortime dilation in a weak (ra'itational field%

;*=: E !+tro&a0n!ti"& in a +$rv!( ".a+!ti&!

;lectro a(netic field tensor F defined on a cur'ed s!aceti e (i'es rise to a 4-force f L :F Nu& which acts on a !article of char(e : with 4-'elocit"u% hus the e:uationof otion of a char(ed !article o'in( under the influence of an electro a(netic fieldin a cur'ed s!aceti e has the sa e for as that in 3inkowski s!aceti e& i%e.

d d

= qm

F . u 2%12

?here 0 the rest ass of the !article% In this case& howe'er& because of the cur'atureof s!aceti e the !article is o'in( under the influence of both electro a(netic forcesand (ra'it"% In so e arbitrar" coordinate s"ste & the !articleFs worldline is a(ain(i'en b"

2%1C

b'iousl"& in the absence of an electro a(netic field or for an unchar(ed !article &the ri(ht-hand side is ero and we reco'er the e:uation of a (eodesic%?e ust re e ber& howe'er& that the ener(" and o entu of the electro a(neticfield will itself induce a cur'ature of s!aceti e& so the etric in this case isdeter ined not onl" b" the atter distribution but also b" the radiation%

;*>: T/! C$rvat$r! T!n"or

d2

x

d 2 +

d x

d d x

d Lq

m0 F

d x

d


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?e can easure the cur'ature of a anifold at an" !oint b" considerin( chan(in( theorder of co'ariant differentiation% o'ariant differentiation is (enerali ation of !artialdifferentiation& it atters in which order co'ariant differentiation is !erfor ed%han(in( the order& chan(es the results%

i(ure 2%2+a 'ector o'ed fro !oint 6 back toitself alon( the cur'e indicated in the dia(ra & the

'ector does not return to itself% his ha!!ens because thes!here is cur'ed%

or a scalar field& co'ariant deri'ati'e issi !l" the !artial deri'ati'e& but for so e arbitrar" 'ector field defined on a

anifold& with co'ariant co !onents a % he co'ariant deri'ati'e of this is (i'en

b"+a= a ad d

6 second differentiation then "ieldsc a = c ( a ) ac! ! c! ! a

c a (c ad

)d add

c d

ac! ( ! !d d) c! (! a a!d d)

?hich follows since a is itself a rank-2 tensor% Swa!!in( the indicesb andc to

obtain a corres!ondin( e !ression for c a and then subtractin( (i'es

c a c a= " a cd d

?here&

" a cd # ac

d c ad + ac! !d a! !cd 2%14


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" a cd

are co !onents of so e rank-4 tensor R. his tensor is called the ur'ature

tensor or #ie ann tensor% In flat re(ion coordinates ca

and its deri'ati'e are ero&

and hence+

" a cd

2%15

at e'er" !oint in the re(ion% ur'ature tensor easures the cur'ature in cur'ed anifold% he 'anishin( of

cur'ature tensor is necessar" and sufficient condition for a re(ion of a anifold to beflat%

;* : Pro.!rti!" o1 t/! +$rvat$r! t!n"or:

he cur'ature tensor 2%14 !ossesses a nu ber of s" etries and satisfies certainidentities& which we now discuss% he s" etries of the cur'ature tensor are osteasil" deri'ed in ter s of its co'ariant co !onents

" a cd = g a! " cd!

or co !leteness& we note that in an arbitrar" coordinate s"ste an e !licit for for

these co !onents is found& after considerable al(ebra& to be

" a cd =1

2 (d a g c d gac+c gad c a g d ) g!f ( !ac f d !ad f c )

ne can use this e !ression strai(htforwardl" to deri'e the s" etr" !ro!erties of thecur'ature tensor& but we take the o!!ortunit" here to illustrate a (eneral athe aticalde'ice that is often useful in reducin( the al(ebraic burden of tensor ani!ulations%

/et us choose so e arbitrar" !oint in the anifold and at !oint the connection

'anishes& $

ca L0& so& the cur'ature tensor takes the for @

" a cd =1

2 (d a g c d g ac + c gad c a g d ) P

ro this e !ression one a" i ediatel" establish the followin( s" etr" !ro!erties at +

" a cd = " acd


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" a cd = " a dc

" a cd = " cda

he first two !ro!erties show that the cur'ature tensor is antis" etric with res!ectto swa!!in( the order of either the first two indices or the second two indices% hethird !ro!ert" shows that it is s" etric with res!ect to swa!!in( the first !air of indices with the second !air of indices% 3oreo'er& we a" also easil" deduce thecyclic identity

" a cd + " acd + " ad c = 0

he cur'ature tensor also satisfies a differential identit"& which a" be deri'ed as

follows% /et us once a(ain ado!t a (eodesic coordinate s"ste about so e arbitrar" !oint P% In this coordinate s"ste & differentiatin( and then e'aluatin( the result at (i'es

( ! " a cd ) $=( ! " a cd ) $=( ! c a d ! d a c ) $

"clicall" !er utin( c, d ande to obtain two further analo(ous relations and addin(&one finds that at

! " a cd + c " a d! + d " a !c = 0

his is& howe'er& a tensor relation and thus holds in all coordinate s"ste s@ oreo'er&since P is arbitrar" the relationshi! holds e'er"where% his result is known as the

!ianchi identity %

;* : T/! Ri++i t!n"or an( +$rvat$r! "+a ar:

It follows fro the s" etr" !ro!erties of the cur'ature tensor that it !ossesses onl"two inde!endent contractions% ?e a" find these b" contractin( either on the first twoindices or on the first and last indices res!ecti'el"% #aisin( the inde a and thencontractin( on the first two indices (i'es

" acda = 0

ontractin( on the first and last indices (i'es in (eneral a non- ero result and thisleads to a new tensor& the "icci tensor % ?e denote its co !onents b"+

" a # " a cc


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6 further contraction (i'es thecurvature scalar or "icci scalar

" # ga " a = " aa 2%1>

his is a scalar :uantit" defined at each !oint of the anifold%

;* : T/! Ein"t!in t!n"or:

he co'ariant deri'ati'es of the #icci tensor and the cur'ature scalar obe" a !articularl" i !ortant relation& which will be central to our de'elo! ent of the fielde:uations of (eneral relati'it"% #aisin(a in the Bianchi identit" and contractin( withd (i'es

gad ( ! " a cd + c " a d! + d " a !c )= 0

! " c + c " a!a + d " !c

a = 0

.sin( the antis" etr" !ro!ert" in the second ter & (i'es

! " c + c " ! + d " !ca = 0

$ow b" raisin( b and contractin( withe& we find ! " c + c " + d " c

a = 0 2%17

$ow a(ain usin( the !ro!erties we a" write the third ter as

a " ca = a " ca = a " ca= " c

So fro 1%17 we obtain

2 " c c " = (2 " c c " )= 0

inall" raisin( the inde c& we (et

( " c 12 g c " )= 0he ter in !arentheses is called the ;instein tensor

Ga # " a

1

2ga "


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2%1K

It is clearl" s" etric and thus !ossesses onl" one inde!endent di'er(ence

Ga

&which 'anishes and it is this tensor that describes the cur'ature of s!aceti e in the

field e:uations of (eneral relati'it"%


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;* : T/! Gravitationa 1i! ( E8$ation"

;insteinFs su((estion that (ra'it" is a anifestation of s!aceti e cur'ature wasinduced b" the !resence of atter% ?e ust therefore obtain a set of e:uations thatdescribe :uantitati'el" how the cur'ature of s!aceti e at an" e'ent is related to theatter distribution at that e'ent% hese will be the gravitational field equations & or

#instein equations & in the sa e wa" that the 3a well e:uations are the field e:uationsof electro a(netis % 3a wellFs e:uations relate the electro a(netic fieldF at an"e'ent to its source& the 4-current densit" at that e'ent% Si ilarl"& ;insteinFs e:uationsrelate s!aceti e cur'ature to its source& the ener("= o entu of atter%

;*9 : T/! En!r07 Mo&!nt$& T!n"or

o construct the (ra'itational field e:uations& we ust first find a !ro!erl" relati'isticor covariant wa" of e !ressin( the source term % In other words& we ust identif" a

tensor that describes the atter distribution at each e'ent in s!aceti e% /et usconsider so e (eneral ti e-de!endent distribution of electricall" neutralnon$interacting !articles& each of rest ass 0% his is co onl" called dust in theliterature% 6t each e'ent P in s!aceti e we can characteri e the distribution

co !letel" b" (i'in( the atter densit" and C-'elocit"u as easured in so einertial fra e%or si !licit"& let us consider the fluid in itsinstantaneous rest frame S at P& in which

u L % In this fra e& the !ro!er densit" is (i'en b" 0 L 0n0& where 0 is the rest

ass of each !article and n0 is the nu ber of !articles in a unit 'olu e% In so e other fra e O& o'in( with s!eed v relati'e to S& the 'olu e containin( a fi ed nu ber of !articles is /orent contracted alon( the direction of otion% Hence& in O the nu ber

densit" of !articles is L % &' 0 % ?e now ha'e an additional effect& howe'er& since

the ass of each !article in O is Q L % &m0 % hus& the atter densit" in SQ is

(= % &2

0 2%1

?here " is the #icci tensor& " is the cur'ature scalar anda, b, c are constants%

/et us now consider the constantsa, b, c % irst& if we re:uire that e'er" ter in . is linear in the second deri'ati'es of g then we see i ediatel" that c L

0% So we ha'e+

= a" + " g

o find the constants a and b we recall that the ener("= o entu tensor satisfies

) = 0 @ thus&

= (a " + g )= 0

6lso we ha'e@

( " 12 g )= 06nd so& re e berin( that

g = 0

?e obtain&

=(12 a+ )g " = 0

hus we find that = a

& and so the (ra'itational field e:uations take the for


26/42

a( " 12 g ")= / ) for consistenc" with the $ewtonian theor"& we re:uirea 1 and so&

( " 12 g " )= / ) 2%27

?here / =8 G

c4 % ;:uation constitutes ;insteinFs (ra'itational field e:uations&

which for the athe atical basis of the theor" of (eneral relati'it"%

?e can obtain an alternati'e for of ;insteinFs e:uations b" writin( 2%< in ter s of

i ed co !onents&

g0( " 12 g " )= / g 0 )

" 0 1

2

0 = / ) 0

6nd contractin( b" settin( 0 = & we thus find " = /) & where ) = ) 0 0

% Hence

we can write ;insteinFs e:uations 1%22 as+

" = / () 1

2) g ) 2%2K

In four-di ensional s!aceti e g has 10 inde!endent co !onents and so in

(eneral relati'it" we ha'e 10 inde!endent field e:uations% ?e a" co !are this with $ewtonian (ra'it"& in which there is onl" one (ra'itational field e:uation%

urther ore& the ;instein field e:uations are non-linear in the g whereas

$ewtonian (ra'it" is linear in thefield %

6 re(ion of s!aceti e in which ) = 0 is called e !t"& and such a re(ion istherefore not onl" de'oid of atter but also of radiati'e ener(" and o entu % It can be seen that the (ra'itational field e:uations for e !t" s!ace are+

" = 0 2%2


27/42

onsider the nu ber of field e:uations as a function of the nu ber of s!aceti edi ensions@ then& for two& three and four di ensions& the nu bers of field e:uations

and inde!endent co !onents of are " as shown in the table%

$o% of s!aceti e di ensions 2 C 4 $o% of field e:uations C > 10 $o% of inde!endent co !onents 1 > 20

hus we see that in two or three di ensions the field e:uations in e !t" s!ace guarantee that the full curvature tensor must vanish % In four di ensions& howe'er&there are 10 field e:uations but 20 inde!endent co !onents of the cur'ature tensor% Itis therefore !ossible to satisf" the field e:uations in e !t" s!ace with a non-'anishin(cur'ature tensor% #e e berin( that a non-'anishin( cur'ature tensor re!resents a

non-'anishin( (ra'itational field& we conclude thatit is only in four dimensions or more that gravitational fields can e0ist in empty space .

;*9


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" 00 1 1

2 ij jhh 0

Substitutin( our a!!ro i ate e !ressions for g and R "" into 1%25 & in the weak-

fieldF li it we thus ha'e

1

2 ij j h00 1 / ( 2 00

1

2) ) 2%C1

or si !licit" we consider a !erfect fluid% 3ost classical atter distributions ha'e -c

2 and so we a" in fact take the ener("= o entu tensor to be that of

dust& i%e%

) = u u

his (i'es+

) = c 2

/et us also assu e that the !articles akin( u! the fluid ha'e s!eeds u in our coordinate s"ste s that are s all co !ared with c% ?e thus ake the a!!ro i ation

% 1 1 and hence u0 1 c2

% herefore e:uation 2%1C reduces to

1

2 ij i j h00 1

1

2/ c

2

Here we a" write@

ij i j=2

6nd we ha'e+

h00 = 2 /c2 and / = 8 G /c4

Here is (ra'itational !otential% $ow we finall" (et the for +

2

1 4 G

his is PoissonFs e:uation in $ewtonian (ra'it"% his identification 'erifies our earlier assertion thataLR1 in the deri'ation of ;insteinFs e:uations%


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Con+ $(in0 r!&ar6":

here is now the co !letion of the task of for ulatin( a consistent relati'istic theor"of (ra'it"% his has led us to the inter!retation of (ra'it" as a anifestation of s!aceti e cur'ature induced b" the !resence of atter and other fields % his !rinci!le is e bodied athe aticall" in the ;instein field e:uations 1%1K % In there ainder of this& we can e !lore the !h"sical conse:uences of these e:uations in awide 'ariet" of astro!h"sical and cos olo(ical a!!lications%

C/a.t!r


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6s discussed earlier& a weak (ra'itational field corres!onds to a re(ion of s!aceti ethat is onl" sli(htl"F cur'ed% hus& throu(hout such a re(ion& there e ist coordinate

s"ste s x

in which the s!aceti e etric takes the for

g = +h Here& |h | JJ 1 C%1

and the first and hi(her !artial deri'ati'es of h are also s all%

Such coordinates are often ter ed :uasi-3inkowskian coordinates& since the" allow

the etric to be written in a close-to-3inkowski for % learl"&h ust be

s" etric with res!ect to the swa!!in( of its indices% ?e also note that& when !re'iousl" considerin( the weak-field li it& we further assu ed that the etric was

stationar"& so that0 g L 0 h L 0 where 0 is the ti elike coordinate% In our

discussion& howe'er& we wish to retain the !ossibilit" of describin( ti e-'ar"in( weak

(ra'itational fields& and so we shall not ake this additional assu !tion here%It is stressed an" ti es& coordinates are arbitrar" and& in !rinci!le& one couldde'elo! the descri!tion of weak (ra'itational fields in an" coordinate s"ste % If onecoordinate s"ste e ists in which C%1 holds& howe'er& then there ust be an" suchcoordinate s"ste s% Indeed& two different t"!es of coordinate transfor ation connect:uasi-3inkowskian s"ste s to each other+ (lobal /orent transfor ations andinfinitesi al (eneral coordinate transfor ations& both of which are discussed here%


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he abo'e !ro!ert" su((ests a con'enient alternati'e 'iew!oint when describin(weak (ra'itational fields% Instead of considerin( a sli(htl" cur'ed s!aceti e

re!resentin( the (eneral-relati'istic weak field& we can considerh si !l" as a

s" etric rank-2 tensor field defined on the flat 3inkowski back(round s!aceti e in

artesian inertial coordinates% Howe'er& thath does not transfor as a tensor

under a (eneral coordinate transfor ation but onl" under the restricted class of (lobal/orent transfor ations%


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;:uation C%5 is 'iewed as a (au(e transfor ation rather than a coordinatetransfor ation% $ow that we ha'e considered the coordinate transfor ations that

!reser'e the for of the etric g in C%1 & it is useful to obtain the corres!ondin(

for for the contra'ariant etric coefficients g

% It is strai(htforward to 'erif"

that& to first order in s all :uantities& we ust ha'e

g = +h


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h = h

2

3

If we choose the functions3 ( x) so that the" )ustif"

2

3

= h

hen we ha'e h

L0

he i !ortance of this result is& in this new (au(e& each of last three ter s on lefthand side of C%> 'anishes and field e:uations in new (au(e beco e

2 h = 2 4 ) C%7

Pro'ided h 5ati5f6 th! guag! c7'diti7'5

h = 0 C%K


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his is a constant tensor on each h"!ersurface of constant ti e% $e t& we will use thisfor ula to deter ine the (ra'itational wa'es (enerated b" a ti e-'ar"in( atter source%

C/a.t!r=* Gravitationa @av!"

,ra'itational wa'es are ri!!les inthecur'ature of s!aceti e that !ro!a(ate as a wa'e& tra'ellin( outward fro thesource%

ro !re'ious to!ic the linearised field e:uations of (eneral relati'it" could bewritten in the for of a wa'e e:uation

2h V 5v L R2% 5v 4%1

Pro'ided that theh V 5v satisf" the /oren (au(e condition

h V 5v L 0. 4%2

his su((ests the e istence of (ra'itational wa'es in an analo(ous anner to that inwhich 3a wellFs e:uations !redict electro a(netic wa'es% Here is discussion of the !ro!a(ation& (eneration and detection of such (ra'itational radiation% 6s inthe !re'ious cha!ter& there was ado!tion of the 'iew!oint thath 5v is si !l" as" etric tensor field under (lobal /orent transfor ations defined on a flat3inkowski back(round s!aceti e%
http://en.wikipedia.org/wiki/Curvaturehttp://en.wikipedia.org/wiki/Spacetimehttp://en.wikipedia.org/wiki/Wavehttp://en.wikipedia.org/wiki/Curvaturehttp://en.wikipedia.org/wiki/Spacetimehttp://en.wikipedia.org/wiki/Wave


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i(ure 4%1+ wo su!er assi'e black holes s!iral to(ether after their (ala ies ha'e er(ed& sendin( out(ra'itational wa'es%

=*9: Ana o07 %!t@!!n Gravitationa an( E !+tro&a0n!ti+ Wav!"

Before (oin( on to discuss (ra'itational wa'es in ore detail& it is instructi'e toillustrate the close analo(" with electro a(netic wa'es% B" ado!tin( the /oren(au(e condition

A 5 L 0& the electro a(netic field e:uations in free s!ace take the for 2 A 5

L 0% hese ad it !lane-wa'e solutions of the for

A 5 L 67 5 e5! 2i 4 x

38,

where the 7 5 are the constant co !onents of the a !litude 'ector% he fielde:uations a(ain i !l" that the 4-wa'e'ector # is null and the /oren (au(e

condition re:uires that7 5

% 5 L 0& thereb" reducin( the nu ber of inde!endentco !onents in the a !litude 'ector to three% In !articular& if we a(ain consider awa'e !ro!a(atin( in the 0C-direction then6% 5 8 L 2%,0 , 0 , %3 and the /oren(au(e condition i !lies that 7 0 L 7 C& so that

67 5 8L 270 , 7 1 , 7 2 , 7 0 3.

he /oren (au(e condition is !reser'ed b" an" further (au(e transfor ation of

the for A 5 M A 5 W : * !ro'ided that 2 : L 0% 6n a!!ro!riate (au(e

transfor ation that satisfies this condition is


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: L e5! 2 i4 x *

where is a constant% 4his "ields7 5 L 7 5 Wi% 5& and so

7 0 L 7 0 Wi

%, 7 1 L 7 1 , 7 2 L 7 2.

B" choosin( L Ri7 0 /% & on dro!!in( !ri es we ha'e 7 0 L 0% In the new(au(e& the a !litude 'ector has )ust two inde!endent co !onents& 7 1 and 7 2&and the electro a(netic fields are trans'erse to the direction of !ro!a(ation% B"introducin( the two linear !olariHation 'ectors

e 51 L 0& 1& 0& 0 and e 52 L 0& 0&1& 0 &

he (eneral a !litude 'ector can be written as+

7 5 L a ! 1 + ! 2 ,

wherea andb are arbitrar" in (eneral& co !le constants%If b L 0 then as the electro a(netic wa'e !asses a free !ositi'e test char(e this willoscillate in the 01-direction with a a(nitude that 'aries sinusoidal with ti e%Si ilarl"& if a L 0 then the test char(e will oscillate in the 02-direction% he

!articular co binations of linear !olari ations (i'en b" b L Xia (i'e circularl" !olari ed wa'es& in which the utuall" ortho(onal linear oscillations co bine insuch a wa" that the test char(e o'es in a circle%

=*;: T/! 0!n!ration o1 0ravitationa @av!"

/et us su!!ose that we ha'e a atter distribution the source locali ed near the ori(in

of our coordinate s"ste that we and take our field !oint x

to be a distance r fro that is lar(e co !ared with the s!atial e tent of the source% ?e a" thereforeuse the compact$source appro0imation discussed in last cha!ter% ?ithout loss of (eneralit"& we a" take our s!atial coordinates i to corres!ond to the centre-of o entu F fra e of the source !articles& in which case fro17%CK we ha'e

h00 =

4 G9 c

2

8 & hi 0 = h0 i= 0 4%C

he re ainin( s!atial co !onents of the (ra'itational field are (i'en b" the


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inte(rated stress within the source& which a" be written in ter s of thequadrupole formula 17%44 as

hij (ct * x)= 2 Gc

68 [ d

2 2 ij (ct )d t

2 ]r 4%4

he :uadru!ole- o ent tensor of the source is

1 i+2ct3L U

002ct, 6

3 yi y + d 4 6

4%5

hus& we see that& in the co !act-source a!!ro i ation& the far field of the sourcefalls into two !arts+ a stead" field 4%C fro the total constant assF 3 of the source

and a !ossibl" 'ar"in( field 4%4 arisin( fro the inte(rated internal stresses of thesource% It is clearl" the latter that will be res!onsible for an" e itted (ra'itationalradiation%or slowl" o'in( source !articles we ha'e 00 c2& where is the !ro!er densit" of the source& and so the inte(ral 4%5 a" be written as

1 i+ 2ct3L c2 U

2ct, x

0i 0 +d C x

4%>

hus& the (ra'itational wa'e !roduced b" an isolated non-relati'istic source is

!ro!ortional to the second deri'ati'e of thequadrupole moment of the atter densit"distribution% B" contrast& the leadin( contribution to electro a(netic radiation is thefirst deri'ati'e of the dipole o ent of the char(e densit" distribution%his funda ental difference between the two theories a" be easil" understood fro

ele entar" considerations% .sin( to denote either the !ro!er ass densit" or the

!ro!er char(e densit"& the 'olu e inte(ral ; d9 o'er the source is constant inti e for both electro a(netis and linearised (ra'itation and so (enerates noradiation% $ow consider the ne t o ent

: 0i d9 & i%e% the di!ole o ent% or electro a(netis & this (i'es the !osition of the centre of char(e of the source& which can o'e with ti e and hence ha'e a non-ero ti e deri'ati'e@ this !ro'ides the do inant contribution in the (eneration of electro a(netic radiation%or (ra'itation& howe'er

:

0i d9 (i'es the centre of ass of the source and& for an isolated s"ste &conser'ation of o entu eans that it cannot chan(e with ti e and so cannot

contribute to the (eneration of (ra'itational wa'es% hus& it is the (enerall" uchs aller :uadru!ole o ent& which easures the sha!e of the source that is do inant


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rubab thesis

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