sign in sign up
<<
Home , Tetrad formalism

Ъ *<№ FEB 1« ^1>;

KFKI-1979-65

A. SEBESTYÉN

THE TETRADS OF OBSERVERS AND EINSTEIN'S EQUATIONS

4bagarim 9Umdenp of "Science* CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST КРК1-197Э-65

THE TETRADS OF OBSERVERS AND EINSTEIN'S EQUATIONS

A. Sebestyén Central Research Institute for Physics H-1525 Budapest, P.O.B.49. Hungary

Submitted to and Gravitation

BO ISSN 036« 5330 18ВЯ 989 S71 бвв 1 ABSTRACT In the paper it is shown that the tetrad formalism can be used to deduce the Lagrangian responsible for thd effects of gravitation. The fields of a tetrad specified by a certain type of observers can be considered as physical fields and then these fields can be rendered to follow the variational prin­ ciple. A Lagr&ngian can be built up along the classical routes» and the Buler-Lagrange equations turn out to be Einstein's equations. It is essential that local Lorents coveriance is used in this deduction.

АННОТАЦИЯ

В работе показано, что ответственная за эффект гравитации функция Лаграиха может быть выведена из формализма тетрада. Если указанные наблюда­ телями поля тетрада принять за физические поля, то эти поля подчиняются ва­ риационному принципу. Функция Лагранжа может быть получена классическим ме­ тодом, а уравнения Эйлера-Лагранжа предстааляпг собой уравнения Эйнштейна. При выведении важно предположение локальной Лореицовой ковариантности.

KIVONAT A dolgozatban megmutatjuk, hogy a gravitációs effektusokért felelSs Lagrang« függvény a tetrad formallzmusb61 levezethet6. A megfigyelők által kijelölt tetrad mezeit fizikaiaknak tekintve, e mezőkre variációs elvet le­ het kimondani. A Lagrange függvény a klasszikus eljárás szerint felépíthető és az Euler-Lagrange-egyenletek éppen az Einstein egyenletek lesznek. Lénye­ ges, hogy a levezetésben a lokális Lorentz kovarianciát fel kell tételezni. It has long been emphasised in the theory of general relati­ vity [1] that when one speaks about an observer then one is ac­ tually considering a tetrad of vectors U and U. /A-1,2,3/ defined at the points of the timelike world line of the observer. U is the tangent vector of this world line, the orthogonal triad of the vectors U. is specified by the measuring rods of the observ­ er in his own rest frame. U and U. , the components of these vectors, follow the orthonormality relations

и Ги 8 о о 9гв " *' 0»Л«'A WB *rs. " -*«'AB'' which can be written in the concise form

"ZV aß ; /1/ here and in the following, greek indices signifying the of the tetrad run through 0, 1, 2 and 3, and

10 0 0 0-100 Y a3* 0 0-10 0 0 0-1

In consequence of /1/ the (Uft ) is non-singular, thus it is easy to deduce the equivalent relation

pi k ü üp 9 г /2/ - 2 - where the summation convention is extended for greek indices

which are lowered and raised by Ye« and its•inverse. A measurement consists in assigning numbers calculated from data of measuring devices to the projections of on this orthonormal tetrad. For example, if the components of the energy- -momentum four vector of a particle i n a local chart are p then the observer measures the quantities

ü rp , о *r ' that is, assigns numerical values to these scalers at a point of his world line. Another example is the energy-momentum field with components T.j- The energy density, momentum density and stresses measured in the observer's rest frame are the func­ tions

Ги вт г в °« « ^в » Л'т,., , и. ип т О О ГВ О A rS А В Г8 /А, В - 1, 2, 3/

respectively. A tetrad of the fields U subject to the relatione /1/ and /2/ can be considered as tetrads specified by a swarm of observers starting off a epacelike hypersurface in an orientable space-time .

It is seen from /2/ that the fields Ua determine the of the space-time uniquely. However, the inverse is not

true. Indeed if Ue fulfil equation /2/ then there exist scalar

functions F e such that for the components Ö of the vector

fields Ua defined by

aap we have again equations /2/, that is

pi ü üpK . qik #

The necessary and sufficient conditions for this is that Feß satisfy the equations - 3 -

Pa0pßp - PpaF% " *aß ' /4/ meaning that at any point of space-time the transformation /3/ is a Lorentz transformation from the reference frame of the tetrad {Vj to that of {ÖK Local Lorentz covariance implies that reference frames connected by transformations of type /3/ are all completely equivalent to each other in describing physical phenomena. Consequently» no equation expressing physical laws can yield restrictions for the functions P - other than equations /4/. At this point some interesting problems arise. The vectors U are specified by physical objects, that is by clocks and measuring rods of observers. This suggests that, in addition to other fields such as the , the U them­ selves should also be considered as physical fields. On the other hand, physical fields in the theory of general relativity are all subject to the variational principle, and the total Lagrangian must contain contributions of all the physical fields. In writing down the contributions L_ of the vectors и one may follow the classical argument taken over from special relati­ vity that it must be a linear combination of bilinear terms built up from the fields and their first partial derivatives. Naturally in the general theory covariant derivatives should be used in-, stead of partial derivatives. This, together with the global Lorentz invariance under the transformations of the reference frames /3/ with constant P «, suggests that L must be of the form

[üPr Ü AüPr *. - ;t PV ;r °р*,* + /51 + BüPr,t "рг^Чг^ > where the semicolon stands for covariant derivatives, А, В and с are constants, and as one coefficient in the total Lagrangian is arbitrary we choose the constant before the first term on the right hand side of /5/ to be equal to one. It is our proposition that if the considerations above con­

cerning the fields Ua are correct then the total Lagrangian L of 4 - general relativity is of the form

L - xbp • bg , /6/

where L- is made up of physical fields other than UQ, x is a con­ stant and L is given by /5/. Thus no separate term like v^gR must be added. It will also be shown that В is zero and A» -1 in /5/. In deducing the Buler-Lagrange equation» for (! we have to express the Metric tensor and the by the i ik Ü *-s and their derivatives as the g -s and their variations are e i determined by the ü -s and their variations respectively. How- tional problen and trto take account of the conditions /2/ by ever, it is easier to treat the gi k -s independently in the varia­ means of Lagrange multiplicators. Thus if the U -s are varied we have

+ Büpr,tV;t+DüP%rüpt,t + '7>

r p pr s rB 4 * Cüp ü r + Are(U Up -g )]d V - 0

as Lp in /6/ does not contain the U -s. In /7/ the constant

D - A + 1

was introduced for the sake of later convenience, Xik • Xki are the Lagrange multiplicators. The equations we get are the following

DU^ri-^aiAV^ri^ri^ri ) - ° >8> here Rjv is Ricci's tensor, that is, the contraction of Riemann's tensor *±klmt

Rik *R ikr ' - 5 -

If equation /8/ is written down for a tetrad Ua defined by /3/ and is used again we get

О |Г1 Р в ,Г p (i О ri p ;r'

P r + P Г - B(P„ ,Wftí 2F II 4 ) - О s I r pi a ,r pi) ' showing that unless В and D are both zero we get restrictions for the functions F - other than /4/ for a given tetrad of the vector fields U . Consequently local Lorentz covariance implies В • D • О reducing /8/ to

°er<"ti+xrl4C»ri) - °' or equivalently

»ik^ik^ik * ° /9/ and for the Lagrangian L we have

r pt pt L - v^[,l ü .U -U* u . + g p ;t ?r p r >t

• wVüPe-*re)+CüPrüpr] • The variation of the gi k -s gives the equation 0

"»ik" W^Vit^r + * ^VPV^VPV) * "J - /10/

- Aik-C9ik * ° '

pr where condition /2/ was used in evaluating U U and TiJc is the energy-momentum tensor we get from Lp. The first two terms in the square brackets in /10/ can be put in another form: - б -

pr pr fc pr fuKV uÜ * ) } +ifu üü -uü üü * } • P tt t№ ,t p ,r ,r p ,t'

P ftr ^ »tp jr |Г p ;t'

P t Pr t P t - J« 4 )tr^» ,t»p «' 4 ,t),r - - j^p'.rt - ^44't, - i» in view of the integrability conditions of Rienannian geometry and condition /2/. Using this result and equation /9/ we have for /10/1

^ik + Rik - 7*ikR " *ikA - 0 , which are Einstein's equations with the cosmological constant Л - 4 С. Thus we have proved our proposition and arrived at the in­ teresting result that if the fields of a tetrad specified by a certain type of observers are treated as physical fields, that is, if they are subject to the variational principle and local Lorentz covariance holds then Einstein's equations follow from a Lagrangian built up from the fields of the tetrad.

REFERENCE

[1] СИ. Misner, K.S. Thome and J.A. Wheeler: Gravitation, W.H. Freeman and Co., San Francisco, 1973 Kiadja a Központi Fizikai Kutató Intézet Felelős kiadós SzegS Károly Szakmai laktort Perjés Zoltin Hyelvi lektort Kóta József Példányszám« 260 Törzsszám 79-827 Készült a KFKI sokszorositö üzemében Budapest, 1979. november hé