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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
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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
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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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