notas geometría y agujeros negros

8/8/2019 Notas Geometra y Agujeros Negros

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T O P I C S O F R I E M A N N I A N G E O M E T R Y

N i k o l a i V . M I T S K I E V I C H , G u a d a l a j a r a - N e w D e l h i

1 Algebra of Riemannian Geometry

PLEASE SEE THE PAGES 8 & 9 IN COLOURS DIRECTLY ONTHE SCREEN!!!

Quite a number of expressions and relations of Riemannian geometrywill be extensively used below. In existing literature [see for a traditional ap-proach, e.g., Eisenhart (1926, 1933, 1972); Schouten and Struik (1935, 1938);

as to a presentation of general relativity from the mathematical viewpoint,see Sachs and Wu (1977)], they are scattered in different parts of various pub-lications, and the notations differ substantially from one source to another.Many relations are naturally given without derivation. The reader may skipthis section but look into it merely for clarification of some formulae. It isworth emphasizing that we take everywhere the speed of light equal to unity(c = 1), the space-time signature being (+ ); the Greek indices arefour-dimensional, running from 0 to 3, the Einstein summation convention isadopted, including collective indices. Symmetrization and antisymmetriza-tion with respect to sets of indices are denoted by the standard Bach brackets,(

) and [

] respectively, with the indices in the brackets. Tetrad indices

are written in separate parentheses, e.g., F()(). For important details of ab-stract representation of the tensor calculus and its applications to field theory,see (Israel 1970, Ryan and Shepley 1975, Eguchi, Gilkey and Hanson 1980,Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick 1982, von Westenholz1986).

We admit that the tensor algebra in an arbitrary basis and in abstractnotations, is commonly known. Hence we begin with the Cartan forms. Tomake notations shorter, we shall sometimes employ collective indices, e.g. inthe basis of a p-form [cf. Mitskievich and Merkulov (1985)]:

dxa := dx1

dxp, thus a =

{1, 2, . . . , p

}(1.1)

(completely skew-symmetrized). Here exterior (wedge) product is supposedto be an antisymmetrization of the tensorial product,

dx1 dxp := dx[1 dxp]. (1.2)

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Similarly, we shall write tetrad basis of a p-form as (a) (then a collective

or individual index in the parentheses pertains to a tetrad field); this con-struction being analogous to dxa, and representing an exterior product of thecovector basis 1-forms, () = g()dx

. Here g() are the covariant tetradcomponents corresponding to the lower index of a set of four independentcovectors (enumerated by the tetrad index ()). We write the tetrad indexin its individual parentheses immediately after the root letter g, the same asthat of the metric tensor, only then followed by the (coordinated) componentnumber (here, ).1 Then a p-form can be written as

= adxa = (a)

(a), while = (1)pq (1.3)

where p = deg , q = deg (degrees or ranks of the forms and ).The axial tensor of Levi-Civita is defined with help of the correspondingsymbol,2

E = (g)1/2, E = (g)1/2. (1.4)We use the Levi-Civita symbol strictly as a symbol, that is, in the sense thatit represents a set of constant scalars, so that one cannot raise or lower itsindices; see however the footnote 2.

To find some important properties of, and to relate these propertiesto those of Kroneckers deltas, let us somewhat formally consider an artificial

1In the metric space, to which the pseudo-Riemannian space-time belongs, any tensor

(in particular, vector) can be taken in its covariant and contravariant form (with respect toany of its indices) due to existence of the metric tensor which in fact is a generalization of amere identity matrix; thus to the covariant basis () always corresponds the contravariantbasis X(): this is the alternative use of the term dual. However such a duality hasnothing in common with the duality involving the Levi-Civita axial tensor.

2This means that the Levi-Civita symbol itself can be simultaneously interpreted as acontravariant axial tensor density (of weight +1) and a covariant axial tensor density ofweight 1: = (g)1/2E = (g)1/2E . Remember that a tensor densityis the corresponding tensor multiplied by (g)w/2 where w is the weight of the tensordensity, so that, for example, the transformation law of a scalar density L (of weight w;

for w = 1, as the Lagrangean density usually is denoted), is L

(x) = |J|w L (x), |J|being absolute value of the Jacobian of the coordinate transformation. Thus the readermay find some discrepancies (from the physicists viewpoint) on p. 129 of the otherwisequite good book by B.F. Schutz (1980). One has to take into account that Schutz simplyused the terminology utilized by mathematicians, in a contrast to physicists!

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(however, well constructed) determinant

= D

(1.5)

(this D still represents merely a notation). Due to the well-known general

properties of determinants, we immediately see that D is skew symmet-

ric in both all upper and all lower indices (each time in four indices whichcoincides with the 4 4-structure of the very determinant!). From (1.5) onealready can understand the role of Levi-Civitas epsilon in dealing with 4

4-

determinants in general. It is easy to figure out that D D01230123,while D01230123 1. Moreover, an immediate opening of the determinant in theleft-hand side of (1.5) yields

D 4![ ]. (1.6)

Thus we have shown that

= 4![

], (1.7)

or for the Levi-Civita axial tensors,

EE = 4![ ]. (1.8)

(In the last two expressions, it is remarkable how the products of epsilons andof the Levi-Civita axial tensors without any contractions! are found tobe equal to products of the Kronecker deltas, also without contractions. Con-secutive contractions of these relations (let us restrict them to those for theLevi-Civita axial tensors) yield the general formula (we write it convenientlyusing collective indices, both free and dummy ones)

EagEbg = p!(4 p)!ba, (1.9)

so that in all particular cases we have as a continuation of (1.8) the followingrelations:

EE = 3![ ], EE = 2 2![],

EE = 3!, EE = 4!

, (1.10)

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or, more symmetrically,

EE = 0!4![ ], EE = 1!3![ ],

EE = 2!2![], EE = 3!1!, EE = 4!0!.

The second expression in (1.4) is, of course, the result of raising the indicesin the first expression: E= (g)1/2gggg= (g)1/2 det(g)= (g)1/2/g

= (g)1/2, Q.E.D.;3 det(g)= {det(g)}1.In our four-dimensional (essentially, even-dimensional) spacetime mani-

fold, the Levi-Civita axial tensor has the following properties:

Eag = (1)p(4p)Ega (1)pEga,EagE

bg

= p!(4 p)!b

a; #a = #b = p = 4 #g. . (1.11)While using here collective indices, we denote the number of individual in-dices contained in them, by #. The Kronecker symbol with collective indicesis totally skew by a definition,

ba 1p1p := 1[1 pp]

, b[a v

u] bvau, (1.12)

so that Aaba Ab (let Aa be skew in all its individual indices). Dual con-jugation of a form is denoted by the Hodge star [see Eguchi, Gilkey andHanson (1980)] put before the form. The star acts on the basis as

dxa := 1(4p)!Eagdxg, := a dxa,1 = 1

4!Edx

dx dx dx = g(dx)

, (1.13)

(dx) being the four-dimensional volume element corresponding to the fourcovectors dx0, dx1, dx2, and dx3. Thus = (1)p+1 (here even-dimensional nature of the manifold is essential); in particular, 1 = 1.It is worth stressing that the sequence of the (lower and upper) indices is ofgreat importance too, e.g. it was not the question of a chance when we wroteEag and not Eg

a [which in fact is equal to (1)pEag, cf. (1.11)].Scalar multiplication of Cartans forms is realized with help of consecutive

dual conjugations:

(dxk dxal) = (1)p+1 (p + q)!p!

[lk dx

a],

#a = p,

#k = q = #l

. (1.14)

3Quod erat demonstrandum (Latin).

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Here dxk, of course, does not mean lowering of the indices of coordinates

under the differential, this is, strictly speaking, the lowering of the collectiveindex only after differentiation. In (1.14) it is important that #k + 4 #(al)

40 The proof of (1.14) consists of the following steps: (I):

dxk dxal = Ealg(4 p q)! (dxk dx

g) ;

(II):

(dxk dxg) = Ekg

b

p!dxb;

(III): dxk dxal = EalgEkgb

p!(4 p q)!dxb;

and this easily reduces to (IV):

(1)pEalgEbkgp!(4 p q)! dx

b =(1)p+1(p + q)!

p!albkdx

b (1)p+1(p + q)!

p![lk dx

a].

In particular, when we take in (1.14) a scalar multiplication of two 1-forms, this gives

(dx

dx

) = (dx dx

)g

= g

. (1.15)

This last relation is equivalent to

(E dx) = EE = 3! g, (1.16)

as well as todx dx = g (1.17)

(in fact, this property should be considered as introduction of a symmetricmetric tensor g in the manifold under consideration). For the more generaltetrad basis () we have

() () = g()(). (1.18)

Contravariant objects are treated with the use of the corresponding coor-dinated vector basis or tetrad vector basis X() (where necessary, their

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tensor products). Here is usually considered as the partial differen-

tiation operator

x acting on (scalar) functions; see however commentsin the next footnotes. The scalar product operation is defined so thatdx dx = . The tetrad basis is related to its tetrad com-ponents (the covariant case was mentioned above; in the contravariant one,X() = g()

, hence X() dx = g(), while () = g()). Thusthere always exists a pair of conjugated tetrads: one is the set of linearlyindependent contravariant vectors, X(), and another, a set of linearly in-dependent covariant vectors, (), so that g()

g() = and, equivalently,

g()g() =

: remember the alternative concept of duality mentioned

above (not to be confused with the Hodge conjugation).The usual dual conjugation of a bivector will be also denoted by a star,

but this will be written over (or under) the pair of indices to which it isapplied, e.g.

F

:=1

2EF, (1.19)

thus F F. It is obvious that F is skew (we called it bivector for thisreason), so that the dual conjugation does not lead to any loss of information.Our choice of place for the star is justified here by the convenience in writingdown the so-called crafty identities where asterisks are applied to differentpairs of indices [see e.g. (Mitskievich and Merkulov 1985)]:

V

= V

V

+ 2

V

+ 2

V

. (1.20)

In fact, these different applications of asterisks simply point out non-coincidingbut equivalent ways to deal with the dual conjugation. For example, onemay write equally F as (1/2)F dx (1/2)F dx; and it is clearthat F = (1/2)Fdx

= (1/2)F dx. Let us consider now the tensorused in (1.20) without leaving out important details. This tensor is skew inits first pairs of indices as well as in their last pair ( V = V[][ ], but therelationship between the pairs of indices is not fixed. Thus the tensor canbe considered as a double 2-form, V = Vdx

dx . The left-hand sidein (1.20) can then be written as V dx (we consider this as anatural extension of the Hodge star concept, though with the basis one hasto deal with contravariant parts of objects; however, the rules introduced forthe usual Cartan forms, can be applied without contradictions in this casetoo). In order not to formally contradict to the general amenity regulationsin the geometry, one may, of course, write V

dx dx , as this was

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done in (1.14). Then this expression will be equal to

14

V

+

+

+

+

+

+

+

+

+

+

+

+

dx dx.

(1.21)It is not too convenient to write down all 24 terms appearing as the pre-liminary result of the multiplications in (1.21). Therefore let us look forsimplifications surely awaiting for our attention somewhere around the cor-

ner. The first of these simplifications are quite impatient, they fill the firsttwo adjacent places in the first and in the second lines, and they all fourcontribute equally, yielding the term Vdx dx (the denominator 4

just disappeared). Here you also see that the first two indices in V becameits last pair of indices, and vice versa, please take into account the crucialin this sense basis factor dx dx. Well, we now have to deal with theremaining 20 terms. Let it be a (not too bad) exercise since there already isenough room for the comparison, see (1.20) vis. (1.21).

In electrodynamics [cf. (Wheeler 1962, Israel 1970, Mitskievich andMerkulov 1985)] we consider here two particular cases to which the crafty

identities lead; these cases together make it possible (and easy) to explic-itly represent any invariant built of the electromagnetic field tensor F as afunction of only two invariants, I1 = FF

and I2 = F

F (the second

is, of course, only a pseudo-invariant, i.e., an axial scalar). First, we takeV = FF whose substitution into (1.20) with one additional contrac-tion yields

FF F F =

1

2FF

. (1.22)

Another choice of V = FF

, also with a contraction, leads to

F F =

1

4

F

F. (1.23)

An example of application of these identities is worth being given: taking the(pseudo-)invariant FF

FF and applying (1.23) to two interior factors,

one directly comes to F14

I2

F = 14I1I2. By the way, all invariants

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and pseudo-invariants containing odd total number of usual and dually con-

jugated Fs, identically vanish.

See the next two pages:

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CORRECTION!!!

THE CALCULATION OF V

4V = V

+

+

+

+

+

+

+

+

+

+

+

+

=

= 4V+2V ( )+4V 4V4V +4V(we have already used the symmetry properties V = V

[][] for mutual

reduction of similar terms). Taking now into account that, for example,V V, we combine some pairs of terms coming to 4V == 4V+2V

(

)+4V ( )4V ( ) .

Application of the notation = 2 with a division by (4) yields

V = VV2V + 2V.

Finally, let us rewrite the just obtained identity without using colours andinterchanging the next to last term and the last one:

V = V V + 2V 2V. (1.24)It is obvious that both left- and right-hand sides of this identity are equallyskew-symmetric in the free indices , as well as in , .

Contraction of the identity (1) in and yields

V = V V + 2V 2V . (1.25)

Though this is a special restricted case following from (1), we shall now seeits importance, in particular for electrodynamics. To this end we substitutein (2) V = F

F:

FF FF =1

2I1

(1.26)

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where I1 = FF is the electromagnetic first invariant. Another substitu-

tion, V

= F

F , leads toFF

= F

F =1

4I2

(1.27)

where the second (pseudo-) invariant is I2 = FF. An example of appli-

cation of the identities (3) and (4) is the straightforward determination ofany invariant being an arbitrary even power of F and/or F (in the case ofodd number of factors such a product invariant identically vanishes; in manycases, this is obvious without application of the above identities).

I gratefully acknowledge the detection of my error in three last coefficientsin formula (1.20) in my file on Topics of Riemannian Geometry by studentsof Licenciatura on September 11, 2008.

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For the Levi-Civita axial tensor one has

E E = 2, (1.28)

for the RiemannChristoffel curvature tensor,

R

= R + R 4R[[]], (1.29)

and for the Weyl conformal curvature tensor,

C C . (1.30)

By the way, a repeated application of the crafty identities to a construction

quadratic in the curvature tensor (with subsequent contractions, so that onlytwo indices remain free ones), leads to the well known Lanczos identities,

RR 14RRg 2RR + RRg

RR + R R 14R2g = 0(1.31)

[see our derivation of these identities (Mitskievich 1969) which differs radi-cally from that by Lanczos (1938) who used less straightforward integral re-lations]. Note that the invariant RR

is known as the BachLanczosinvariant, or the generalized GaussBonnet invariant;4 like the electromag-

netic invariant FF , it represents (in a four-dimensional world) a puredivergence, so that it does not contribute to the field equations in four di-mensions. Cf. also DeWitt (1965).

2 Differentiations in Riemannian Geometry

There are two closely related differentiation operators using the importantconcept ofconnectionin Riemannian geometry: the more modern nablaoper-ator with which we begin this section, and the comparatively older ;-operatorto be encountered in more traditional treatment of the Riemannian geometry.

Finally, the Lie derivative does not use the concept of connection (though itcan be written with the help of covariant derivatives). But the Lie derivativeserves rather for more general description of covariant derivatives than the

4The first of Lagrangians of the Lovelock (1971) series reduces to it.

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latter ones are needed for its conceptual introduction. From these differential

operators one easily can pass to many variants of the crucially important con-cept of Riemannian geometry, that of different kinds of transports in spacesendowed with curvature. This last way of treating curved manifolds is natu-rally related to one of the central object both of geometry and of physics incurved manifolds, the geodesic equation.

2.1 The Nabla Operator

The covariant differentiation axioms [see e.g. Ryan and Shepley (1975)] are:(1) vT has the same tensor rank and valence properties as T, v beingalways a vector.

(2) v is a linear operation:v(T + U) = vT + vU,au+bvT = auT + bvT.

(3) The Leibniz property5 holds (including the contraction operation in theRiemannian geometry).(4) Action of on a function6 is vf = vf, where v = v v()X().(5) Metricity property: vg = 0 (for a further generalization of the geometry,the non-metricity tensor appears on the right-hand side).(6) Zero torsion axiom:

uv vu = [u, v] + T, T = 0.(in a generalization of the geometry, the torsion tensor7 T becomes differentfrom zero).

These axioms8 are most simply realized when the connection coefficientsare introduced,

X()X() = ()()()X(); X()() = ()()()() (2.1)5Distributive property when product expressions are differentiated.6A scalar; the concept of scalar is somewhat different for the covariant differentiation

denoted by a semicolon; see more in the footnote 9.7

Like the curvature operator below, the torsion T (if= 0) here also describes a tensor,now of the rank three, by the rule: X()X()X()X()

X(),X()

= T()()()X()

[cf. (3.2)]. Then, of course, the connection coefficients following from (2.1), should bemodified.

8There could be also added the seventh (in some sense) axiom: [u,v] = [u,v] +R(u, v) [cf. the definition (3.1)], with R = 0, as an alternative to T = 0. Then, of course,

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(only one of these two relations is independent). When the structure coeffi-

cients of a basis are introduced with help of[X(), X()] = C

()()()X(), C

()()() =

()()() ()()() (2.2)

(for a holonomic, or coordinated basis, C = 0; as usually, we write inindividual parentheses only the tetrad and not coordinated basis indices),the general solution satisfying the whole set of axioms, from 1 to 6, is

()()() =

12g

()()

X()g()() + X()g()() X()g()()

+12 C

()()() + C

()()()g()()g

()() + C()()()g()()g

()()

.

(2.3)

We give here the hint for deduction of (2.3): first, to look at the firstsquare brackets and represent them as X()g()() + X()g()() X()g()() =X()

X() X()

+ X()

X() X()

X() X() X() ; then to applynabla-differentiation and the definition of structure coefficients (2.2). Finally,

to do a recombination of the expression ()()()+

()()() into 2

()()()+

()()()

()()(), the last two terms forming the structure coefficient C

()()(). Then one

has to rewrite the result already into the form of (2.3), and all is done.In a coordinated basis (where the basis vectors and covectors are usually

written as and dx), the connection coefficients reduce to the Christoffel

symbols (which are written without parentheses before and after each index,

and which obviously are symmetric in the two lower indices):

=1

2g (g, + g, g,) . (2.4)

Orthonormal bases and those of NewmanPenrose [see Penrose and Rindler(1984a, 1984b)] are special cases of bases whose vectors have constant scalarproducts (i.e., the corresponding tetrad components of the metric are con-stant). For such bases the connection coefficients are skew in the upper andthe first lower indices; they are sometimes called Ricci rotation coefficients.Thus, in a coordinated basis, the differentiation of A = A yields

A = (A)dx

+ Adx

=

A, A

dx

=: A;dx

, (2.5)

the theory changes drastically, but in literature one can find an assertion that in this casethe former gravitational effects become those of the torsion. The author is not ready toswear that this is completely true, and the following exposition strictly belongs to theRiemannian geometry.

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and for V = V,

V =

V, + V

=: V

;, (2.6)

in accordance with the (old) traditional notations for the ;-covariant deriva-tive. From these definitions it is easy to see that the ;-covariant derivativesof vectors, A; and V

;, are components of rank 2 tensors. In general the

;-covariant derivatives of rank r tensors also are (in the sense of components)tensors of the rank r + 1 (cf. the axiom 1 of the -covariant derivative).

Let us revisit the definition of connection coefficients, using now the coor-dinated basis. Then dx = dx

; it is probably better to highlight more thesense of the index in this expression (that it is not a component number

of a vector, but the number of a covector in a coordinated basis), entering itinto a circle (the parentheses already have other meanings): dx = dx.

Then; =

, = ,

since the Kronecker delta is a set of constants, and the encircled index doesnot pertain to the coordinated components (now it has more in common witha tetrad component number, but for a non-orthonormal tetrad). Thus wecome to an equivalent of (2.1) in a coordinated basis, dx = dx.The subindex in =

can be treated similarly, only the sign will be

changed to the inverse one.

2.2 ;-covariant Differentiation

Thus, covariant differentiation can also be denoted by a semicolon (;) beforethe differentiation index (especially, in the elder literature9). Then, accordingto the covariant differentiation axioms (in particular, the axiom 1, howeverparadoxical this may appear), it leads to an increase by unity of the valenceof the corresponding tensor. As Trautman (1956, 1957) remarked, such acovariant derivative can be defined as

Ta; := Ta, + Ta

|

(2.7)

9 In the old-fashioned coordinated component notations, tensor components are usuallytreated as synonyms of the corresponding tensors, so it is then quite awkward to considerthem as a set of scalars, but in the modern abstract (coordinate-free) notations, tensor and connection components are dealt with just as a set of scalar functions. Here weuse a fairly innocuous mixture of both styles.

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where the coefficients Ta| determine the behavior of the quantity Ta (whichcould be not merely a tensor, but also a tensor density or some more generalobject) under infinitesimal transformations of coordinates, being a dimen-sionless infinitesimal scalar constant:

x = x + (x),

Ta := T

a(x) Ta(x) = Ta|,, (2.8)

i.e. Ta| = 1Ta/( ,). It is clear that the coefficients Ta| possess aproperty

(SaTb)| = Sa|Tb + SaTb| (2.9)which is closely related to the Leibniz property of differentiation. The co-

variant (in the sense of u) differentiation axioms are readily reformulatedin terms of ;-differentiation; in particular, the axiom 5 takes the formg; = 0 or, equivalently, E; = 0 (2.10)

which (in each of these two forms) immediately yields the definition (2.4) of.

We shall now show in general that after the ;-differentiation the resultingobject has property of a tensor of the same valences as those of the tensorbefore the differentiation, plus one additional covariant valence. To this endwe shall use collective indices, but in another sense than in the Cartan for-malism (now, without antisymmetrization). Thus the true analogue of the

oversimplified expressions (2.5) and (2.6) reads

Ta

bdxa b

= Tab; (dx

a b) . (2.11)Similarly, a chain of equations follows:

Tab(x) (dxa b) = Tab;(x) (dxa b) =x

xTa

b;(x) (dx

a b) .(2.12)

Now note that the successive contraction of the basis objects with collectiveindices yields

(dxa

b)

k

dxl = ak lb, (dxa b)

k

dxl = x

a

xk

xl

xb, (2.13)

so that finally

Tkl;(x

) =x

xxa

xkxl

xbTa

b;(x). (2.14)

This is precisely the transformation law we have spoken about above.

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2.3 The Lie Derivative

The definition of covariant derivative (2.7) may be deduced from the conceptof Lie derivative,

Ta := Ta, Ta| , Ta; Ta|;, (2.15)

which can be written for objects of the most general nature as

Ta := 1 (Ta(x) Ta(x)) (2.16)

[cf. Yano (1955)].10 Due to the property (2.9), the definition (2.15) of the Liederivative under an infinitesimal transformation yields the Leibniz property.

However, if we consider the finite-difference Lie derivative (2.16) when itis not subjected to an infinitesimal transport (the parameter is now finite,

so that we have to introduce instead of a new notation, say,

), then ageneralized Leibniz property has to be introduced,

(SaTb) =

Sa

Tb

+

Sa

Tb + Sa

Tb

.

Seemingly, it is quite different from the former one (see the first right-handside term). However this situation is typical for finite differences, automati-cally changing to the usual Leibniz property when one goes to the infinitesi-

mal case (in the lower order of), and only the last two right-hand side termssurvive.

10The quantity Ta(x) which will appear also in the proof of Noethers theorem, describesthe change of dependence of the components Ta due to the transformation of coordinatesfrom x to x. This may seem to be somewhat artificial, so we give here an explanationfor it in a form of parable. Consider a region of space-time filled with integral curvesof the vector field (x) and two points, P and P, belonging to the same curve, suchthat the coordinates ofP in the system {x} are numerically equal to those ofP in {x}:x|P = x|P. This corresponds to the situation when at both points there are indepen-dent observers having exactly the same mentality so that they ascribe to their respectivepositions in space-time the same numerical characteristics thus using these two coordi-nate systems, {x} and {x}, respectively. Let these observers have all necessary devicesfor measurement and reproduction of the field Ta at their respective points. Moreover,suppose that each of them can communicate with the other telepathically. Then the in-formation about measurement ofTa made by the observer at P

, sent instantaneously tothe observer at P and reproduced by him, is related to the independent measurement ofTa by this latter observer directly through the Lie derivative (2.16).

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It is important that both objects in the parentheses in (2.16), have the

same argument (non-primed one). This leads to a possibility to extend theoperation Ta; even to quantities of the type of a connection.Concerning the transformations of coordinates, we usually consider only

those which are reducible to infinitesimal ones (i.e., we speak about the con-nected component of the identity transformation). In this case it is alwayspossible to recover finite transformations through integration of the infinites-imal ones, but of course not such as inversions. Thus we may write

x = x + , = (x), (2.17)

being an infinitesimal constant, and an arbitrary differentiable vector field(note that a vector field is inextricably related to the general concept of con-

gruence which is central in the definition of the Lie derivative). The standardtransformation coefficients of tensor components by virtue of infinitesimalityread

x

x= +

,,

x

x= ,, (2.18)

(up to the first order of magnitude terms). In this sense

; = R(). (2.19)

The Lie derivative (2.15) reduces to the gradient projected on when it isapplied to a scalar function. Since the set of coordinates x is a collection offour scalar functions, this means that

x = (2.20)

for any vector field .Concerning the Lie derivative, it is worth also mentioning the relations

g = ; + ;, =

;; R (;;) R(). (2.21)

A vector field satisfying the equation

; + ; = 0, (2.22)

is called the Killing field. Its existence imposes certain limitations on thegeometry, existence of an isometry: g = 0 [see (Yano 1955, Eisenhart

1933, Ryan and Shepley 1975)]. The Lie derivative plays an important rolein the monad formalism, as we shall see below; this is why a version of thisformalism well adapted to description of the canonical structure of the fieldtheory, is called the Lie-monad formalism [see (Mitskievich and Nesterov1981, Mitskievich, Yefremov and Nesterov 1985)].

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3 The Curvature in Riemannian Geometry

It is convenient to define the concept of curvature in the Riemannian ge-ometry by introducing the curvature operator [see e.g. (Ryan and Shepley1975)],

R(u, v) := uv vu [u,v]. (3.1)This operator has the following five properties:1) It is skew in u and v.2) It annihilates the metric, R(u, v) g = 0.3) It annihilates any scalar function, R(u, v) f = 0 (e.g., R(u, v) = 0).4) This is a linear operator (cf. the axiom 2 of the covariant differentiation).5) The Leibniz property holds for it (this fact is remarkable since the curva-ture operator involves the second, not first order differentiation).

The curvature tensor components (which are scalars from the point ofview of the -differentiation) are determined as

R()()()() := () R(X(), X())X(). (3.2)

These are the coefficients (components) of decomposition of the curvaturetensor with respect to the corresponding basis, and in the general case theyare

R()()()() = X()()()() X()()()() + ()()()()()()

()()()()()() C()()()()()().(3.3)

It is worth remembering that some authors use a definition of curvature withthe opposite sign, e.g. Penrose and Rindler (1984a,b) and Yano (1955) [seealso a table in (Misner, Thorne and Wheeler 1973)]. The structure of thecurvature operator yields a simple property,

Rw = w;; w;;. (3.4)

It is also easy to show that the standard algebraic identities hold,

R = R[][] = R,

R[] = 0, or equivalently, R = 0

(3.5)

(this last identity bears the name of Ricci, but sometimes it is called al-gebraic Bianchi identity). Important role is also played by the differential

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Bianchi identity,

R[;] = 0, equivalently, R ; = 0. (3.6)

The Ricci curvature tensor we define as R R := R, cf. Eisen-hart (1926, 1933), and Misner, Thorne and Wheeler (1973). There arealso introduced the scalar curvature, R := R, the Einstein tensor G =R (1/2)gR forming the left-hand side of Einsteins equations, and theWeyl conformal curvature tensor C,

C := R + g[R] + g[R] 13

Rg[g] (3.7)

(here written in the four-dimensional spacetime manifold). The latter (takenwith one contravariant index, and in a coordinated basis), does not feel amultiplication of the metric tensor by an arbitrary good function, so it isconformally invariant. Remember that the Weyl tensor works only for space-time dimensionality D 4. When D = 2, G 0, and for D = 3 theCotton tensor (Cotton 1899) is used when the conformal correspondence ofspace-times and the classification a la Petrov are considered, see (Garcia etal. 2004).

3.1 Cartans structure equations

In the formalism of Cartans exterior differential forms, see (Israel 1970, Ryanand Shepley 1975), the operation of exterior differentiation is introduced,

d := ()X() dx, (3.8)where is the discussed above wedge product (the operation of exteriormultiplication), so that dd 0. The operation d has its most simple repre-sentation in a coordinated basis (since the Christoffel symbols are symmetricin their two lower indices). It is however especially convenient to work in anorthonormalized or NewmanPenrose basis when the connection 1-forms

()() := ()

()()

() (3.9)

are skew-symmetric (()() = ()()). In the general case they are foundas a solution of the system of Cartans first structure equations,

d() = ()() (), (3.10)

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and relations for differential of the tetrad metric components,11

dg()() = ()() + ()(). (3.11)

Then one has just 24 + 40 equations for determining all the 64 com-ponents of the connection coefficients, so that the solution of this set ofequations should be unique. It is easy to accustom to find the specific so-lutions without performing tedious formal calculations; then the work takessurprisingly short time. Since reference frame characteristics (vectors of ac-celeration and rotation, and rate-of-strain tensor) below will turn out to bea kind of connection coefficients, this approach allows to calculate promptlyalso these physical characteristics, though for the rate-of-strain tensor the

computation will be rather specific.In their turn, the curvature tensor components are also computed easilyand promptly, if one applies Cartans second structure equations,

()() = d()() + ()

() ()(). (3.12)

Here the calculations are much more straightforward than for the connection1-forms, and they take less time. It remains only to read out expressions forthe curvature components according to the definition

()() :=1

2R()()()()

()

() (3.13)

being in conformity with the curvature operator properties via (3.12). TheRicci and Bianchi identities read in the language of Cartans forms as

()() () = 0 (3.14)

andd()() =

()() ()() ()() ()() (3.15)

respectively.The detailed deduction of the second Cartan structure equation (3.12)

from the definitions of curvature (3.3) and the curvature 2-form (3.13) canbe done as follows. When the (skew) symmetry properties are taken into

11For deduction of these relations consider the following calculations: dg()() = d(X() X()) =

()X()(X()X()) = ()()(X()X())+()()(X()X()) = ()()g()()+()()g()() = ()() + ()().

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account, as well as the definition of d (3.8), we immediately rewrite ()()

as

d()()()

()+()()()()+()()()()()() where we also used thedefinition of 1-form of connection (3.9) (and we shall use it below). Then the

next identical rewriting yields d()() + ()()()

()() () + ()() ()() +

()()()

() ()() where both terms with identically cancel, and we finallycome to (3.12).

3.2 Einsteins field equations

From the Bianchi identities (3.6), R; + R; + R; = 0, it followsafter two contractions (in and and then with multiplication by g) that

R 1

2R

;

= 0. (3.16)

The tensor G := R 12R whose covariant divergence thus identically

vanishes, is called Einsteins conservative tensor. The variational principleapplied to the first term (simply proportional to the scalar curvature R) inthe action integral for Einsteins field, directly yields this tensor as the left-hand side of equations; the same principle yields, via the term interpretedas Lagrangian density of the physical field(s) to be used as source(s) of thegravitational field, the energy-momentum tensor of the physical field(s) [the

other path to deduce these both tensors is to use the Noether theorem, see,e.g., Mitskievich (2006)]. All these variation results, naturally, appear withthe corresponding coefficients including, in particular,

g. Historically, theconservative tensor was first proposed by Einstein to be equated with theenergy-momentum tensor (multiplied by a certain coefficient) to intuitivelycome to the equations of gravitational field.

3.3 Parallel Transport and it Analogues

Let us give here a scheme of definition of transports. Denote first as Tv :=

v|Q v|PQ the change of a vector (or of some other object) under itstransport between points P and Q [see (Mitskievich, Yefremov and Nesterov1985)]. Then for the parallel transport

Pv

d= uv (3.17)

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and for the FermiWalker transport [cf. Synge (1960)]

FWvd

= uv + [(v uu)u (v u)uu] =: FWu v (3.18)

where u = dx/d is the tangent vector to the transport curve (for theFermiWalker transport, it is essential to choose canonical parameter alongthe transport curve in such a way that the norm uu = 1 would be constantand non-zero so the curve itself should be non-null for the FermiWalkertransport). Using the definitions (2.7), it is easy to generalize the transportrelations to arbitrary tensors or tensor densities; so for the FermiWalkertransport

FWTa

d= Ta;u

+ Ta|

(u;u

uu

;)u

. (3.19)

From (3.18), a definition of the FermiWalker connection seems to be di-rectly obtainable, together with the corresponding covariant FW differentia-tion operation which should lead also to the FW-curvature [cf. (Mitskievich,Yefremov and Nesterov 1985)]:

RFW(w, v) := FWu FWv FWv FWu FW[u,v]. (3.20)

The reader may find that in this generalization a difficulty arises: the deriva-tive FWu does not possess the linearity property. This trouble is howeverreadily removable if we begin with introduction of another type of derivative,

namely u (see below) which possesses all necessary properties of a covariant

derivative.

3.4 The Geodesic Equation

We give here the (equivalent to each other) forms of the geodesic equation(describing parallel transport of a vector along itself) for a canonical (affine)parameter (then absolute value of the vector does not suffer changes):

uu = 0,

u du = 0,d2x

d2 +

dx

ddx

d = 0,

dd

(gdx

d) = 1

2g,dx

ddx

d .

(3.21)

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A geodesic may be, of course, also null (lightlike), but if it is time-like, one has

d = ds. If a non-canonical parameter is used, an extra term proportionalto u should be added in the equations (3.21), with a coefficient being somefunction of the (non-canonical) parameter.

Below we shall make use of the just mentioned generalization of the FW-derivative (3.18) along a time-like congruence with a tangent vector field normalized by unity (the monad). It is convenient to denote such a derivativeas u,

uT = uT + Ta| (; ;) uBa, (3.22)where Ba is the (coordinated) basis of T: T = TaB

a. Then FWu uu. Thedefinition (3.22) can be also written as

T[]a; = Ta; + Ta|(; ;). (3.23)This -derivative acts as the usual u upon any scalar function f, and itannuls the vector , the metric tensor g, and the construction b = g (the three-metric or the projector in the monad formalism):

uf = uf, u = ug = ub = 0 (3.24)(action of a -derivative on a tensor is understood in the sense of (3.22); uis considered as a partial differentiation operator in uf). The corresponding-curvature operator is denoted as R(w, v) [see (Mitskievich, Yefremov andNesterov 1985)].

3.5 The Laplacian and De Rhamian Operators

Alongside with exterior differentiation d which increases the degree of a formby unity, in Cartans formalism another differential operation is also usedwhich diminishes this degree by unity,

:= d . (3.25)In fact this is the divergence operation,

(hdxh

) = ph;

dxh

, p = #h + 1, (3.26)which does not however fulfil the Leibniz property: when applied to a prod-uct, e.g., of a function and a 1-form, f a, it results in

(f a) = f a (df a),

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while it acts on a function (0-form) simply annihilating it, f 0. Onemight call this a generalized Leibniz property. Of course, similarly to theidentity dd 0, there holds also the identity 0. A combination of bothoperations gives the de Rham operator (de-Rhamian),12

:= d + d (d + )2 (3.27)

(remember divergence of a gradient in Euclidean geometry). A form anni-hilated by the de-Rhamian, is called harmonic form (remember harmonicfunctions and the Laplace operator). In contrast to , the de-Rhamian actsnon-trivially on 0-forms, and in contrast to the exterior differential d, it actsnon-trivially on 4-forms too (also in the four-dimensional world). The follow-

ing classification elements of forms are also used: an exact form is that whichis an exterior derivative of another form; a closed form is that whose exteriordifferential vanishes (an exact form is always closed, though the converse isin general not true); a co-exact form is a result of action by on some formof a higher (by unity) degree; a co-closed form is itself identically annihilatedby . In (topologically) simplest cases it is possible to split an arbitrary forminto sum of a harmonic, exact, and co-exact forms [see Eguchi, Gilkey andHanson (1980), Mitskievich, Yefremov and Nesterov (1985)].

For an arbitrary form ,

= a;

;dxa

pR

h

|dx

h, #a = p. (3.28)

The number of individual indices entering the collective index a of the com-ponents of the form , may be any from 0 to 4, while the quantity a| isdetermined by (2.8). The expression (3.28) can be reduced identically to

= (p!)1[h;; + pR h +p(p 1)

2R h]dx

h (3.29)

where a = h, #a = 2 + #h = p, and p may be equal also to zero.We compare here explicitly the results of action of de-Rhamiam on forms

of all degrees in a four-dimensional world:

f : f = f,;, (3.30)

A = Adx : A = (A;; + RA)dx, (3.31)

12The same symbol, , also denotes the usual Laplacian.

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F =1

2Fdx

:

F =

1

2(F;

; + 2RF + R

F)dx, (3.32)

T = 13!

Tdx : T = 1

3!(T;

; + 3RT + 3R

T )dx

= (A;; + RA) dx,

(3.33)

V =1

4!Vdx

: V = 14!

V;;dx = f;; 1. (3.34)

From the last two expressions it is clear that there exists a symmetry withrespect to an exchange of a basis form to its dual conjugate with a simultane-ous dual conjugation of components of the form [compare (3.30) and (3.34),(3.31) and (3.33)]. Here we made use of obvious definitions

A = 13!

TE, f = 1

4!VE

,

alongside with the easily checkable identity,

(2RE + 3R

E)E 0.

References

Choquet-Bruhat, Y., DeWitt-Morette, C., and Dillard-Bleick, M. (1982)

Analysis, Manifolds, and Physics (Amsterdam: North-Holland).Cotton, E. (1899) Ann. Fac. Sci. Toulouse II 1, 385.

DeWitt, B.S. (1965) Dynamical Theory of Groups and Fields (NewYork: Gordon and Breach).

Eguchi, T., Gilkey, P.B., and Hanson, A.J. (1980) Phys. Reports 66,213.

Eisenhart, L.P. (1926) Riemannian Geometry(Princeton, N.J.: Prince-ton University Press).

Eisenhart, L.P. (1933) Continuous Groups of Transformations (Prince-ton, N.J.: Princeton University Press).

Eisenhart, L.P. (1972) Non-Riemannian Geometry (Providence, RI:American Mathematical Society).

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Garca, A.A., Hehl, F.W., Heinicke, Ch., and Macas, A. (2004) Class.

Quantum Grav. 21, 1099.Islam, J.N. (1985) Rotating fields in general relativity (Cambridge,Cambridge University Press).

Israel, W. (1970) Commun. Dublin Inst. Adv. Studies A 19, 1.

Lanczos, C. (1938) Ann. Math. 39, 842.

Lanczos, C. (1962) Revs. Mod. Phys. 34, 379.

Lovelock, D. (1971) J. Math. Phys. 12, 498.

Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973) Gravitation(SanFrancisco: W.H. Freeman).

Mitskievich, N.V. (1969) Physical Fields in General Relativity(Moscow:Nauka). In Russian.

Mitskievich, N. (2006) Relativistic Physics in Arbitrary Reference Frames(New York: Nova Science Publishers, Inc.).

Mitskievich, N.V., and Merkulov, S.A. (1985) Tensor Calculus in FieldTheory (Moscow: Peoples Friendship University Press). In Russian.

Mitskievich, N.V., and Nesterov, A.I. ((1981) Exper. Techn. der Physik29, 333.

Mitskievich, N.V., Yefremov, A.P., and Nesterov, A.I. (1985). Dy-namics of Fields in General Relativity (Moscow: Energoatomizdat) InRussian.

Penrose, R., and Rindler, W. (1984a) Spinors and Space-Time. Vol. 1.Two-Spinor Calculus and Relativistic Fields (Cambridge: CambridgeUniversity Press).

Penrose, R., and Rindler, W. (1984b) Spinors and Space-Time. Vol.2. Spinor and Twistor Methods in Space-Time Geometry. (Cambridge:Cambridge University Press).

Ryan, M.P., and Shepley, L.C. (1975) Homogeneous Relativistic Cos-mologies (Princeton, N.J.: Princeton University Press).

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Sachs, R.K., and Wu, H. (1977) General Relativity for Mathematicians

(New York, Heidelberg, Berlin: Springer).Schmutzer, E. (1989) Grundlagen der Theoretischen Physik(Mannheim,Wien, Zurich: BI-Wissenschaftsverlag), Teile 1 & 2.

Schmutzer, E. (2004) Projektive Einheitliche Feldtheorie mit Anwen-dungen in Kosmologie und Astrophysik (Frankfurt am Main: Wis-senschaftlicher Verlag Harry Deutsch GmbH). Mit einem Anhang vonA.K. Gorbatsievich.

Schouten, I.A., and Struik, D.J.(1935) Einfuhrung in die neueren Meth-oden der Differentialgeometrie. Vol. 1 (Gronongen: Noordhoff).

Schouten, I.A., and Struik, D.J. (1938) Einfuhrung in die neuerenMethoden der Differentialgeometrie. Vol. 2 (Gronongen: Noordhoff).

Schutz, B.F. (1980) Geometrical Methods of Mathematical Physics (Cam-bridge: Cambridge University Press).

Synge, J.L. (1960) Relativity: The General Theory(Amsterdam: North-Holland).

Synge, J.L. (1965) Relativity: The Special Theory(Amsterdam: North-Holland).

Trautman, A.(1956) Bulletin de lAcademie Polonaise des Sciences,Serie de sciences math., astr. et phys. IV, 665 & 671.

Trautman, A. (1957) Bulletin de lAcademie Polonaise des Sciences,Serie de sciences math., astr. et phys. V, 721.

Vladimirov, Yu.S., Mitskievich, N.V., and Horsky, J. (1987) Space,Time, Gravitation (Moscow: Mir). Revised English translation of aRussian edition of 1984 (Moscow: Nauka).

Westenholz, C. von (1986) Differential Forms in Mathematical Physics

(Amsterdam: North-Holland).Wheeler, J.A. (1962) Geometrodynamics (New York: Academic Press).

Yano, K. (1955) The Theory of Lie Derivatives and Its Applications(Amsterdam: North-Holland).

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4 Deduction of the Schwarzschild solution

First to use the ideas of symmetry to determine the general form of metricto describe the spherically symmetric gravitational field (spacetime) withoutinitially excluding dependence of metric on the t variable. The variable r isintroduced from the spherical symmetry (in the angular sense) like this wasdone by Synge.ds2 = e2(r,t)dt2 e2(r,t)dr2 r2 d2 + sin2 d2 ;(0) = edt, (1) = edr, (2) = rd, (3) = r sin d;

d(0) = e(1) (0), d(1) = e(0) (1), d(2) = 1r

e(1) (2),d(3) = 1r e(1) (3) + cot r (2) (3);(0)(1) =

e(0) + e(1), (2)(1) = 1r e(2), (3)(1) = 1r e

(3),(3)(2) =

cot r

(3);

(0)(1) =

e

e(1) (0) +

e.

e(0) (1),(0)(2) =

(0)(1) (1)(2) = e2r (0) (2),

(0)(3) = e2r (0) (3),(2)(1) = e2r (1) (2) e

r(0) (2),

(3)(1) =

e

2

r (1)

(3)

e

r (0)

(3),

(3)(2) = 1r2

1 e2 (3) (2).Please check the following properties: (0)(i) = +

(i)(0),

(i)(j) = (j)(i), and

the same for s. These properties for s in fact follow from the relations(3.11) given in the Topics of Riemannian Geometry, and they then applyonly in the case of constant components of g()(), while for s they comefrom the general properties of the curvature tensor and do apply without anyrestrictions.

ThusR(0)(1)(1)(0) =

e e

.

e,

R(0)(2)(0)(2) = R(0)(3)(0)(3) = e

2r ,

R(2)(1)(2)(1) = R(3)

(1)(3)(1) = e2

r ,

R(2)(1)(2)(0) = R(3)

(1)(3)(0) = e

r,

R(3)(2)(3)(2) =1

r2

1 e2 ;

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R(0)(0) = e e.

e

2e2

r ,R(1)(1) =

e

e. e 2 e2r ,R(0)(1) = 2er ,R(2)(2) = R(3)(3) = (

) e2r

1r2

1 e2 .

In a vacuum, the scalar curvature R = 0, thus Einsteins equations re-duce to R()() = 0. When we take ()() (0)(1), we see that = (r).Einsteins equations now take the formR(0)(0) =

e

e 2 e2r = 0,

R(1)(1) = e e 2e2

r

= 0,

R(2)(2) = R(3)(3) = ( ) e2r 1r2

1 e2 = 0.

From R(0)(0) + R(1)(1) = 0 we see that + = f(t) (we already knowthat (r), i.e. does not depend on t. Now, from ds2 we see that the stillnot determined function f(t) can be simply eliminated by merger with thevariable t of which now depends nothing. This is equivalent to put f(t) = 0in our final results. Thus we simply put + = 0. Substituting = into the equation R(2)(2) = 0, we come to the simple equation only for :

(e2) = 1e2

r , which yields e2 = 1 Cr , C being the only survived integra-

tion constant. It is now easy to show that the full set of Einsteins equations

in this case is completely solved with this , and the solution in the form ofds2 isds2 = (1 2M

r)dt2 (1 2M

r)1dr2 r2 d2 + sin2 d2. M easily can be

shown to be the central mass (in the units involving the Newtonian gravita-tional constant) creating this gravitational field. The very solution bears thename Schwarzschild solution having been obtained by Karl Schwarzschildafter few days since Einsteins paper containing the final form of his gravita-tion theory appeared in December of 1915.

5 Interpretation of the Schwarzschild metric:

the first steps

The further steps: 1. In the region where r M, consider the geodesicequation in non-relativistic limit and weak field approximation. A compari-

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son with the Newtonian theory then gives the interpretation of M.

2. Beginning with r > 2M, consider exact application of the geodesic equa-tion to the radial free fall of a test particle on the Schwarzschild centre. Usings as the proper time parameter, calculate duration of the fall till r = 2Mand till r = 0 (the field singularity). Then change to the variable t as atime parameter and show that the fall duration from r > 2M to r = 2Mis infinite. A discussion of these conclusions.

Let us take the geodesic equation in the form dd (gdx

d ) =12g,

dx

ddx

d .Obviously, two conservation laws follow from it for = 0 and = 3 (there isone more independent conservation law with a mixture of and completingthe picture of absence of angular momentum as one of the cases of conserva-tion, but we use the angular laws only as a justification of the radial fall

treatment). The law corresponding to = 0 has its exact expression as

gttdt

ds= E, (5.1)

being the first integral of motion named kinetic energy of the test particleper unit of its rest mass (please dont confuse Ewith !).

The first step in our consideration is here to obtain the physical interpre-tation of the constant M in ds2. To this end we take the radial componentof the geodesic equation, that is

dds

grr dr

ds

= 1

2

gtt

dtds

2+ grr

drds

2(in the case of a radial fall) which identically reads

d2r

ds2= 1

2gtt. (5.2)

All this is done exactly, without approximations.It is easy to see that in the non-relativistic limit ds dt; at the same

time, we explicitly differentiate gtt with respect to r. This finally givesd2r

dt2 = M

r2 , being precisely the 2nd Newtonian law when the mass M isattracting a test mass falling onto it (in fact, M = Gm where G is theNewtonian gravitational constant, and m is value of the central mass in itsusual units). You see: the approximative calculation was used at the finalstep, and only once.

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The second step of our consideration of the Schwarzschild metric is to

reveal the main property of its horizon, r = 2M. We here use for anothertime the ( = 0)-conservation law, g00dtds

= E. Writing down ds2, dividing itby ds2, and taking into account the energy conservation law, we come exactly

to

drds

2= E2 gtt, so that for a particle falling downward in the sense of r,

ds = drE2 gtt , (5.3)which can be considered as another first integral of motion, tohether with(5.1). To simplify the further calculation, let us take (only in this secondstep) E= 1 which means that the falling particle starts falling from spatialinfinity with zero velocity (the state of rest), though we, of course, may followits fall beginning with any final point outside r0 > 2M where this particlealready had non-zero velocity towards the origin. The interval of the propertime s between the particle was passing the point r0 and then came to someminor value of r, is s =

rr0

r1/2dr2M

. The result of integration is obvious, andit is finite both for r = 2M and for r = 0. But this is true only for the propertime of the falling particle. If we would try to describe the time of fall interms of the variable t, the result will be infinite already for r = 2M. Thenwhat occurs with this r = 2M?!

Recall that the physical sense of coordinates is determined by their sig-nature. Due to its definition used in these notes, the sign + marks thetimelike dimension and the sign , the other three spatial dimensions. Butin Schwarzschilds ds2 t is timelike and r, spacelike, only outside the hori-zon r0 = 2M, while inside the horizon t becomes spacelike and r, timelike;moreover, on the horizon these two coordinated completely lose their math-ematical meaning, and there is nothing in the Schwarzschild metric whatcould indicate in which direction is the future and in which is the past of thistimelike r inside the horizon, even we would somehow define what mean pastand future for the variable t outside it. However there is a proper way todo this introducing Novikovs coordinates simultaneously in whole spacetimewhile beginning with the usual Schwarzschildean coordinates. We shall not

realize this program here in its complete form, but we shall now come toa sufficiently comprehensive hint how to do this, beginning with somewhatsimpler synchronous coordinates.

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6 The synchronous coordinates

Definition: Synchronous coordinates are such in which the time coordinateT lines represent a non-rotating timelike geodesic congruence parameterizedby proper time along the lines, while the spatial coordinates lines belong tospacelike hypersurfaces perpendicular to the T-congruence. (Such congru-ences are also called normal congruences since absence of rotation is thenecessary and sufficient condition of existence of global hypersurfaces orthog-onal normal to these congruences.) Synchronous coordinates do notneed any special kind of geometry of the manifolds in which they are built, sothat one can construct synchronous coordinates in every pseudo-Riemannianspacetime.

Let us first make it clear what mean the concepts of coordinates and ofcoordinates lines. The first corresponds to independent variables existingin the manifold under consideration. If we choose one of these variables(one coordinate) and consider it running through all values it can take, whilefixing values of all other variables, we describe thus a line of this coordinatemarked by all constant values which take the other coordinates on this line.If this line goes through the origin (where all coordinates, including that towhich belongs this line, are equal to zero; this concept of the origin is in factnot so simple as it could seem initially), it is marked (labeled) by n 1 zeros(n is the number of all independent variables on this manifold, thus it isn-dimensional), and this special line is then called the axis of the coordinatewhich we did not fix in this paragraph. It is interesting that when we considera coordinates transformation, the lines of coordinates become automaticallytransformed in a way which deserves to be named complementary to theway of transformation of the very coordinates. An example seems to me hereto be appropriate. Let us consider a two-dimensional spacetime {t, x} andapply to these coordinates the usual Galilean transformation (first written,by the way, by Newton): t = t, x = x vt. The question is: how will thenchange the axes of coordinates? By the definition of the axes, we have to takefor the axis ofx the condition t = 0, but since t = t, this is at the same timethe condition by which the axis of x is determined, consequently, the axes

of x and of x coincide. As to the axis of t, we have to put x = 0; clearly,this gives the following equation to find this axis: x = vt, thus the axis of t

differs from that of t, and we did really come to a certain complementarity(about which I never heard or read before). It remains to be added onlythat the t-lines here exactly coincide with the new time coordinates lines

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appearing after the usual Lorentz transformation with the same velocity v

as in the Galilean transformation several lines above.We now turn to of synchronous coordinates in the Schwarzschild space-time. We already know that the timelike (outside the horizon, though thereis a theorem asserting that the timelike, spacelike or null behaviour of ageodesic cannot change) radial geodesic of a particle freely falling towardsthe origin, is determined by the first integrals (5.1) and (5.3). They can becombined (to exclude ds) as

dt +Edr

gtt

E2 gtt= 0, (6.1)

a relation which equivalently describes radial geodesics of the free fall, likewe considered them in the beginning of section 2, though in another manner.But we need these geodesics to the end of introducing the coordinate lines ofT in the synchronous system which now is under construction. How we cometo these coordinate lines? Since the relations between coordinates T, R, onthe one hand, and t, r, on the other, have to be of the type of T = T(t, r)and R = R(t, r), and recalling the general definition of coordinate lines inthe second paragraph of this section (that coordinate lines of the coordinateT are determined by fixation of the alternative coordinate, here R: dR = 0),we have to consider the equation dR = Rt dt +

Rr dr = 0 as an analogue of

(6.1). Therefore let the definition of the new coordinate R be13

dR = Edt + E2dr

gtt

E2 gtt, (6.2)

an equation having as its left- and right-hand sides only total differentials(which is of great importance), and fixation of R, i.e. dR = 0, then reallyleads to determination of timelike geodesic lines which we need as coordinatelines of T. Why not construct dT(t, r) in a similar manner, that is dT =dt + F(r)dr (though we now are not concerned in geodesic properties ofcoordinate lines ofR), and then determine the unknown function F(r)? Theway to determine it is, moreover, clear: the mutual orthogonality of dT and

dR and the unit property of the T T-component of the new synchronousmetric. Look: dR dT = GRT = 0, and then dT dT = GT T = 1 (under G we

13We introduced here and in (6.3) an extra constant factor E in dR and dT which isnecessary for obtaining GTT = 1 and for achieving more symmetry in these expressions,as well as for having a simple form of (6.4).

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now understand the new metric in T, R, , and coordinates; the angular

coordinates remain without any change in the synchronous coordinates sincethey were orthogonal to t and r, hence also to T and to R). The calculationsare extremely simple:(1) dR dT = dt dt + EF(r)

gttE2gtt

dr dr where dt dt = gtt = 1/gtt, dr dr =grr = gtt, while dt dr = 0. Therefore in order to have dR dT = 0, we take

dT = Edt +

E2 gttgtt

dr. (6.3)

(2) dT dT = 1.(3) A non-trivial fact is that it is easy to express the old radial variable r in

terms of the combination of new coordinates T and R,

d(R T) = drE2 gtt . (6.4)This relation is immediately integrable and gives the concrete dependencer(R T), at least implicitly. As to the last two variables and whichdid not change, the metric coefficients corresponding to them now involver2(R T). Finally, the Schwarzschild spacetime is described in synchronouscoordinates by the metric

ds2 = dT2 E2

dR

2

E2 1 + 2Mr(RT) r2(R T) d2 + sin2 d2 . (6.5)

From these expressions we see that in the synchronous coordinates there re-main no traces of the singularity earlier detected on the horizon in Schwarz-schildean coordinates. This singularity only marked the inadmissible charac-ter of the usual Schwarzschildean coordinates on the horizon, without beingin any sense related to the physical properties of the Schwarzschild solu-tion; in the latter, there exists only the singularity at r = 0. Naturally,one might choose the constant E in such a way that there would appearzero in the denominator at some r, but this would simply be an inadequate

restricted choice of the T-congruence in constructing the synchronous coor-dinates, nothing more. As an exercise, the case ofE= 1 is proposed in whichyou have to perform all calculations up to the final results which will thenbe quite simple; then to draw a spacetime diagram in the T R-plane givingthere some lines of constant r, including r = 0 (singularity) and r = 2M

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(horizon). Look to which types pertain these two lines (timelike, spacelike,

or null). Using as a definition of null (lightlike) lines ds2

= 0, draw in thisdiagram (approximately) from some convenient point on the horizon takenas a vertex, both generatrices of the light cone (to the future as well as tothe past). Try to imagine how can go a timelike world line of a test parti-cle from in T to the future, and determine the region from where thisparticle (say, a spaceship) can escape hitting the singularity, and from whereit cannot. How could you relate this behaviour of a particle to the gravita-tional collapse phenomenon? Finally, answer the question: if we would liketo compare the point-like object at the origin (r = 0) of the Schwarzschildspacetime with some type of a particle, which type would be acceptable that of a usual massive particle, of a photon, or of a tachyon? Defini-

tively, the source of the Schwarzschild field is a tachyon with themass M, moving with infinite 3-velocity, while always remainingat r = 0 (especially clear when we use a Penrose diagram).14

7 The Kerr solution deduced

Here we shall introduce the famous Kerr black hole solution using our gener-alization of the so-called falling box method first published by Sommerfeld(with a reference to Lenz) where the Schwarzschild solution was deduced.From 1949 till 1984 that was the only metric deduced by this extrav-

agant method. Then I have invented similar, though more sophisticatedapproaches to semi-intuitively come first to the Kerr solution, then to thegravitational-electric ReissnerNordstrom, and finally to the most compli-cated three-parameter KerrNewman solution [see Vladimirov, Mitskievichand Horsky (1987)]. Now, with the example of the Kerr spacetime, we shallmake acquaintance with this SommerfeldLenz method severely criticized byquite serious physicists, but after the deduction with its help of all fourblack hole metrics, safely survived at least for being used in the popularliterature on gravitation.

We begin with the Minkowski spacetime supposedly applicable far from

14An addition of as small as you like charge (or angular momentum, or both) toSchwarzschilds centre immediately results in the change of the spacelike (r = 0)-line(of the point-like singularity) to timelike and, consequently, of the tachyonic particle toa usual massive one, with the corresponding complete instantaneous restructuring of thePenrose or synchronous coordinates diagram.

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localized sources of the gravitational field. Thus, taking for that far the

mark of infinity , we may write (in fact, using the spherical coordinates)ds2 = dt

2 dr2 r2 d2 + sin2 d2 = (0) 2 (1) 2 (2) 2 (3) 2 (7.1)from where the structure of all four basis covectors

() becomes obvious.

A transition to a non-uniformly rotating reference frame is done by locallyapplying Lorentz transformations so that every point has its own speed ofmotion directed towards an increasing angle . The absolute value of thisvelocity is a function V which depends, generally speaking, on both coor-dinates r and . Such a Lorentz transformation is not equivalent to thetransformation of coordinates in the domain being studied (in practice, this

domain is the whole of space), but is limited only to the transformation ofthe basis at each point. Thus, we have

(0) =

(0) V (3)

/

1 V2, (1) = (1) , (2) = (2) ,(3) =

(3) V (0)

/

1 V2.

(7.2)

Now let the box with the observer be released from infinity. In this casewe can write a new basis in which time has slowed down, and the lengthsin radial direction have shortened. This is equivalent to substitution of

()

first in (7.2) and then the resulting expressions into

(0) = (0)

1

v2, (1) = (1)/

1

v2, (2) = (2), (3) = (3). (7.3)

We have thus assumed that the observer makes his measurements in therotating frame and notices the relativistic changes in his observations. Nowlet us do the reverse local Lorentz transformations with respect to (7.2) thistime applied to the basis (7.3):

(0) =

(0) + V (3)

/

1 V2, (1) = (1), (2) = (2),(3) =

(3) + V (0)

/

1 V2.

(7.4)

We now insert into(7.4) the () basis which is expressed in terms of the() from (7.3), and then write this expression in terms of () from (7.2).

We postulate that the resulting basis (7.4) remains orthonormalized. A fewmanipulations yield

ds2 =

1 v21V2

(0)

2 (1 v2)1(1) 2 (2) 2

1 + v2V2

1V2

(3)

2+ 2V v

2

1V2 (0)

(3) .

(7.5)

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The principle of correspondence with Newtons theory presupposes, as

this was assumed by Sommerfeld in his falling-box deduction of the Schwarz-schild solution, that

gtt = 1 2Mr

1 + 2N, (7.6)N being the Newtonian gravitational potential considered there as a spher-ically symmetric solution of the Laplace equation. Here this should be alsoa solution of the Laplace equation, but now under the rotational symmetryand not spherical. Therefore it is now worth considering oblique spheroidalcoordinates in flat space. These coordinates, r, , and , are defined as

x + iy = (r + ia)ei sin , z = r cos ,x2 + y2

r2 + a2+

z2

r2= 1. (7.7)

We know that in the spherical coordinates (1/r) = 0 when r = 0, and thisequality holds under any translation of coordinates. Let this translation nowbe purely imaginary and directed along the z axis, i.e. x x, y y, andz zia. Then it is easy to find that r =

x2 + y2 + z2 (r2a2 cos2

2iar cos )1/2 = r ia cos . From here the expression for Newtons potentialin oblique spheroidal coordinates follows,

N =M

r N = Re M

r ia cos =Mr

r2 + a2 cos2 , (7.8)

since the Laplace equation is satisfied simultaneously by both the real andthe imaginary parts of the translated potential. Hence we can get with thehelp of (7.6) (the abbreviation 2 = r2 + a2 cos2 is henceforth used)

v2

1 V2 =2Mr

2. (7.9)

We determine the velocity V using a model of rotating ring of some ra-dius r0 for the source of the Kerr field, this ring being stationary relative to

the rotating reference frame (7.3). On the one hand, V = (x2 + y2)1/2

=

(r2 + a2)1/2

sin , corresponds to the well-known relationship between an-gular and linear velocities in spherical coordinates. On the other hand, it isclear that the reference frame cannot rotate as a rigid body in all the world,otherwise the frame wouldnt be extendible beyond the light cylinder aswe dropped our box from infinity. Therefore the angular velocity has also

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to be a function of position. The ring lies naturally in the equatorial plane,

so that the angular momentum along the z axis is

L = mV

r20 + a2, (r = r0, = /2). (7.10)

We now introduce an important hypothesis which establishes a connectionbetween the angular momentum and the Kerr parameter a, which is alsocharacteristic for spheroidal coordinates (7.7), namely we put a = L/m Theselast three statements yield (r = r0, = /2) = a (r

20 + a

2). If we now add asecond hypothesis, that the field is independent of the choice of the ring radius(depending only on its angular momentum), then we get = a/(r2 + a2),and finally

V = a(r2 + a2)1/2 sin . (7.11)

It only remains for us to choose the expression for a basis () which would

correspond to the assumed rotational symmetry (i.e., to the oblique spheroidalcoordinates). The reader may substitute differentials of the coordinates x, y,and z from (7.7) him/herself into the Minkowski squared interval, hence get-ting a quadratic form with a non-diagonal term. This term, which containsthe product drd, can be excluded by a simple change of the azimuth angle:d d + a(r2 + a2)1dr (a total differential), thus leading to a diagonalquadratic form. If now the square roots of the separate summands are taken,we get the final form of the initial basis

()

:

(0) = dt,

(1) =

2

r2+a2dr,

(2) =

2d,

(3) =

r2 + a2 sin d.

(7.12)

A mere substitution of these expressions into (7.5) yields the standardform of the Kerr metric in terms of the BoyerLindquist coordinates,

ds2 =

1 2Mr2

dt2 2 dr2 2d2

r2 + a2 + 2Mra2 sin2

2 sin2 d2 + 2 2M ra sin

2 2

dtd.(7.13)

We remind that2 = r2 + a2 cos2 ; (7.14)

the notations = r2 2Mr + a2 (7.15)

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and

2

= (r2

+ a2

)2

a2

sin2

(7.16)are also frequently used (in the KerrNewman case = r2 2Mr + a2 + Q2where Q is electric charge of the KerrNewman centre given in the units oflength, like this is done with M as we have seen in the deduction of theSchwarzschild metric).

We give now a transparent hint to three anagrams of the Kerr metricwith different combinations of three terms containing dt and/or d. First,these terms can be combined in two constructions giving a part (for two di-mensions) of the complete four-dimensional orthonormal basis with rotationin the Kerr spacetime,

1 2Mr2

(dt + d)2

r2 + a2 +

2Ma2r sin2

2 2Mr

sin2 d2 (7.17)

with

=2Mar sin2

2 2Mr . (7.18)

Second, there is another combination of the same terms into a part of thepseudo-rotational orthonormal basis,

2

2dt2

2 sin2

2(d

wdt)2 (7.19)

where15

w =2Mar

. (7.20)

And finally, dt-d terms can be combined into a very simple construction

2

dt a sin2 d2 sin2 2

r2 + a2

d adt2 (7.21)

including both rotation and pseudo-rotation [see Mitskievich and VargasRodrguez (2005) about them]. Similar situation occurs in the general (three-

parametric) KerrNewman case too.15Please dont confuse two very different letters from very different alphabets, and w!

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8 Checking the Kerr solution

Now we have to verify if the metric (7.13) really satisfies Einsteins equationsin a vacuum. To this end we have first to find the connection 1-forms usingthe 1st Cartan structure equations. As the simplest one, we choose theorthonormal basis from (7.13) and (7.21), so that

(0) =

(dt a sin2 d), (1) = dr,(2) = d, (3) = sin [(r

2 + a2)d adt]; (8.1)

consequently,

d = a

(0) + 1 sin

(3), dt = r

2

+ a

2

(0) + a sin

(3). (8.2)

We now write down the direct d-differentiation of the basis covectors,and, in parallel, the respective 1st Cartan equations:

d(0) = M(r2a2 cos2 )a2r sin2

3

(1) (0)a2

3 sin cos (2) (0) 2a

3 cos (2) (3),

d(0) = (0)(1) (1) (0)(2) (2) (0)(3) (3);(8.3)

d(1) =

a2

3 sin cos (2)

(1),

d(1) = (1)(0) (0) (1)(2) (2) (1)(3) (3); (8.4)d(2) = r

3(1) (2),

d(2) = (2)(0) (0) (2)(1) (1) (2)(3) (3);(8.5)

d(3) = r

3 (1) (3) + r2+a23 cot (2) (3) + 2 ar sin3 (1) (0),

d(3) = (3)(0) (0) (3)(1) (1) (3)(2) (2).(8.6)

We see that here the problem of algebraically solving the system of equationsis somewhat more complicated than in the deduction of the Schwarzschildsolution. Of course, there are present similar obvious terms of the type

d(2)

=r

3 (1)

(2)

where it is clear that (2)

can pertain only to theconnection 1-form and not to the basis covector following it in the exte-rior product since otherwise we should encounter (2)(2) which is identicallyequal to zero. Hence there should be (2)(1) (multiplied by

(1) which has noother choice), and this (2)(1) should be proportional to

(2), the coefficient

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of proportionality being quite obvious. However even here we run into an

additional term in (2)

(1), that containing (1)

and giving no contributionto the equation (8.5); this term being important in (8.4) where the termwhich was the only one being present in (8.5), loses its significance (simplydisappears). To this collection of terms in our problem here two new triosof terms now are added: one trio being marked with the indices 0, 2, 3 andanother, by 0, 1, 3 (in all mutually non-reducible successions). This couldprobably be not quite obvious, but such a situation (with each single trio) istypical in calculations of the connection 1-forms, and checking this rule viathe practical use easily shows that we deal with them correctly. We now findthat in all but one ((1)(2)) connection 1-forms there now appear new terms(for the first trio, in the connection 1-forms where two of the indices 0, 2,

and 3 are involved; similarly for the second trio with two indices of 0, 1, and3). Let us see how they have to be treated. In the first trio, we write inthe corresponding terms still not determined constant coefficients A, B, C(and we work similarly with the second trio introducing other three constantcoefficients, say, P, Q, R):

(0)(1) =M(r2a2 cos2 )a2r sin2

3

(0) + Par sin3 (3),

(0)(2) = a23 sin cos (0) + A a

3cos (3),

(0)(3) = Ba

3 cos (2) + Q ar sin3

(1),

(1)(2) =

a2

3sin cos (1)

r

3(2),

(1)(3) = r3 (3) + R ar sin3 (0),(2)(3) = r2+a23 cot (3) + Ca

3cos (0)

(8.7)

(we gave here the complete set of the connection 1-forms, though with thealready mentioned terms containing still not determined coefficients). Theseconstant coefficients are now determined by substitution of these connection1-forms into the 1st structure equations written in (8.3)-(8.6). A comparisonwith d() found there in parallel lines yields two sets of algebraic equations(one for each trio) whose solutions are A = B = C = 1, P = Q = R = 1.

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Thus the final form of s reads

(0)(1) = M(r2a2 cos2 )a2r sin2

3

(0) ar sin3 (3),(0)(2) = a23 sin cos (0) + a

3cos (3),

(0)(3) = a

3 cos (2) ar sin

3 (1),

(1)(2) = a23 sin cos (1) r

3(2),

(1)(3) = r

3 (3) ar sin 3 (0),

(2)(3) = r2+a23 cot (3) + a

3cos (0).

(8.8)

Further we have to calculate the components of the Riemann curvaturetensor using the second Cartan structure equations where most time and

persistence are needed while performing differentiation in the first right-handside term, but all these calculations are by no means problematic. Of course,in this first term it is advisable first to express ()() in terms of the coordi-nated basis [although the very ()() will continue to pertain to the tetradbasis (8.1)]. Then

(0)(1) =M(r2 a2 cos2 )

4(dt a sin2 d) + ar sin

2

2d, (8.9)

(0)(2) = 2a2

4sin cos (dt a sin2 d) + a

2sin cos d, (8.10)

(0)(3) = a2

cos d ar2

dr, (8.11)

(1)(2) = a2

2

sin cos dr r

2d, (8.12)

(1)(3) = r

2sin d, (8.13)

(2)(3) = r2 + a2

4cos [(r2 +a2)dadt] + a

4cos (dta sin2 d) (8.14)

[sometimes written with fragments of basis covectors for convenience, sincefinally we have to bring results to their standard form with respect to thebasis (8.1)]. Some of these expressions seem to be far from elegance, butwhat could be demanded from such a mixture of bases we used in theirrepresentation?!

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References

Hawking, S.W., and Ellis, G.F.R. (1973) The Large Scale Structure ofSpace-Time (Cambridge: CUP).

Hobson, M.P., Estathiou, G.P., and Lasenby, A.N. (2006) General Rel-ativity. An Introduction to Physicists. (Cambridge: CUP).

Islam, J.N. (1985) Rotating Fields in General Relativity (Cambridge:CUP).

Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973) Gravitation(SanFrancisco: W.H. Freeman).

Mitskievich, N.V. (2006) Relativistic Physics in Arbitrary PeferenceFrames (New York: Nova Science Publishers).

Mitskievich, N.V., and Vargas Rodrguez, H. (2005) Gen. Relat. andGrav. 37, 781.

Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., and Herlt,E. (2003) Exact Solutions of Einsteins Field Equations, 2nd edition(Cambridge: CUP).

Synge, J.L. (1960) Relativity: The General Theory(Amsterdam: North-

Holland).

Teukolsky, S.A. (1973) Astrophys. J. 185, 635.

Vladimirov, Yu.S., Mitskievich, N.V., and Horsky, J. (1987) Space,Time, Gravitation (Moscow: Mir).

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