Elliptic fibrations associated with the Einstein

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 39, NUMBER 10

OCTOBER 1998

Elliptic fibrations associated with the Einstein space–times Paweł Nurowskia) Dipartimento di Scienze Matematiche, Universita degli Studi di Trieste, Trieste, Italy, and Department of Mathematics, King’s College, London, United Kingdom

~Received 13 April 1998; accepted for publication 20 April 1998! ˜ over it. Given a conformally nonflat Einstein space–time we define a fibration P The fibers of this fibration are elliptic curves ~two-dimensional tori! or their degenerate counterparts. Their topology depends on the algebraic type of the Weyl tensor ˜ is a double branched cover of the bundle P of the Einstein metric. The fibration P of null direction over the space–time and is equipped with six linearly independent one-forms which satisfy a certain relatively simple system of equations. © 1998 American Institute of Physics. @S0022-2488~98!01210-9#

I. DEFINITIONS

In Ref. 1 we defined a certain differential system I on an open set U of R.6 We showed that a pair (U,I ) naturally defines a four-dimensional conformally nonflat Lorentzian space–time (M,g) which satisfies the Einstein equations R i j 5lg i j . In this paper we prove the converse statement. In particular, we give a construction which associates a certain six-dimensional elliptic ˜ with any conformally nonflat Lorentzian Einstein space–time. Moreover, we show fibration P ˜ how P may be equipped with a unique differential system which has all the properties of the system I on U. We briefly recall the definitions of the geometrical objects we need in the following. Let M be a four-dimensional-oriented and time-oriented manifold equipped with a Lorentzian metric g of ¯ ,k,l) on M with a coframe signature ~1,1,1,2!. It is convenient to introduce a null frame (m,m ¯ ,K,L) so that u i 5( u 1 , u 2 , u 3 , u 4 )5(M ,M ¯ 2KL. g5g i j u i u j 5M M

~1!

@Such expressions as u i u j mean the symmetrized tensor product, e.g., u i u j 5 21 ( u i ^ u j 1 u j ^ u i ). Also, we will denote by round ~respectively, square! brackets the symmetrization ~respectively, antisymmetrization! of indices, e.g., a (ik) 5 21 (a ik 1a ki ), a @ ik # 5 21 (a ik 2a ki ), etc.# The Lorentz group L consists of matrices l ij PGL(4,C) such that g jl 5g ik l i j l k l ,

l 3 2 5l 3 1 ,

l 2 2 5l 1 1 ,

l 2 1 5l 1 2 ,

l 2 3 5l 1 3 ,

l 2 4 5l 1 4 ,

l 3 3 5l 3 3 ,

l 3 4 5l 3 4 ,

l 4 2 5l 4 1 ,

l 4 3 5l 4 3 ,

~2! l 4 4 5l 4 4 .

We will denote the inverse of the Lorentz matrix l i j by ˜li j . The connected component of the identity element of L is the proper ortochroneous Lorentz group, which we denote by L↑1 . Given g and u i the connection one-forms G i j 5g ik G k j are uniquely defined by d u i 52G i j ∧ u j ,

G i j 1G ji 50.

~3!

a!

Permanent address: Instytut Fizyki Teoretycznej, Wydział Fizyki, Uniwersytet Warszawski, ul. Hoz˙a 69, Warszawa, Poland. Electronic mail: [email protected]

0022-2488/98/39(10)/5481/10/$15.00

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The connection coefficients G i jk are determined by G i j 5G i jk u k ~we lower and raise indices by means of the metric and its inverse!. Using them we define the curvature two-forms Rk i , the Riemann tensor R i jkl , the Ricci tensor R i j , and its scalar R by Rk i 5 12 R k im j u m ∧ u j 5dG k i 1G k j ∧G j i ,

R i j 5R k ik j ,

R5g i j R i j .

We also introduce the traceless Ricci tensor by S i j 5R i j 2 14 g i j R. Note that the vanishing of S i j is equivalent to the Einstein equations R i j 5lg i j for the metric g. We define the Weyl tensor C i jkl by C i jkl 5R i jkl 1 13 Rg i[k g l] j 1R j[k g l]i 1R i[l g k] j , and its spinorial coefficients C m by ¯ ∧K1C ~ L∧K2M ∧M ¯ ! 1 ~ C 1 1 R ! L∧M R235C 4 M 12 3 2 ¯ ! 1 1 S L∧M ¯, 1 21 S 33M ∧K1 21 S 32~ L∧K1M ∧M 2 22 ¯ ∧K2C ~ L∧K2M ∧M ¯ ! 2C L∧M R145 ~ 2C 2 2 121 R ! M 1 0 ¯ ! 2 1 S L∧M ¯, 2 21 S 11M ∧K2 21 S 41~ L∧K1M ∧M 2 44 1 2

¯ ∧K1 ~ C 2 1 R !~ L∧K2M ∧M ¯ ! 1C L∧M ~ R432R12! 5C 3 M 24 2 1 ¯ ! 1 1 S L∧M ¯. 1 12 S 31M ∧K1 41 ~ S 121S 34!~ L∧K1M ∧M 2 42

II. THE BUNDLE OF NULL COFRAMES

Let F ~M! denote the bundle of oriented and time oriented null coframes over M. This means that F ~M! is the set of all equally oriented null coframes u i at all points of M. The mapping p :F (M)→M, which maps a coframe u i at xPM onto x, gives the canonical projection. A fiber p 21 (x) in F ~M! consists of all the null coframes at point x which have the same orientation and time orientation. If u j is a null coframe at xPM, then any other equivalently oriented null coframe at x is given by u 8 i 5l i j u j , where l i j is a certain element of L↑1 . This defines an action of L↑1 on F ~M!. Thus, F ~M! is a ten-dimensional principal fiber bundle with L↑1 as its structural group. It is well known that the bundle F ~M! is equipped with a natural four-covector-valued one-form e i , i51, 2, 3, 4, the Cartan soldering form, which is defined as follows. Take any vector v c tangent to F ~M! at a point c. Let c be in the fiber p 21 (x) over a point xPM. This means that c may be identified with a certain null coframe u ic at x. Then, the formal definition of e i reads: e i ( v c )5 u ic ( p v c ). The first two components of e i are * complex and mutually conjugated. The remaining two are real. Altogether, they constitute a system of four well-defined linearly independent one-forms on F ~M!. In the following we will denote them by F5e 1 5e 2 ,

¯ 5e 3 , T5T

¯ 5e 4 . L5L

~4!

A theorem which we present below is a null coframe reformulation of the Elie Cartan theorem on affine connections. ¯ ,T,L) be the soldering form on F ~M!. Then Theorem 1: Let e i 5(F,F (i) the system of equations dei1vi j∧e j50,

g ki v i j 1g ji v i k 50

~5!

for a matrix of complex-valued one-forms v i j (i, j51,2,3,4) on F ~M! has a unique solution,

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S

¯ ,G,G ¯ ,V,V ¯ ) by (ii) v i j uniquely defines six complex-valued one-forms (E,E ¯ 2V ¯ 0 2E V 2G ¯ ¯ 0 2G V2V 2E i v j5 , ¯ ¯ 2G 0 2G V1V ¯ ¯ 2E 0 2V2V 2E

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~6!

¯ ,T,L,E,E ¯ ,G,G ¯ ,V,V ¯ ) are linearly independent at each point of F ~M!. (iii) the forms (F,F For completeness we sketch the proof. First, we show that if there is a solution to ~5! then it is unique. To do this we assume the existence of two solutions—v i j and vˆ i j . Subtracting de i 1 vˆ i j ∧e j 50 from de i 1 v i j ∧e j 50 we get ˆ i j ! ∧e j 50. ~ vi j2v

~7!

Now, let e m , m 51, 2, 3, 4, 5, 6, be a system of one-forms such that the ten one-forms (e i ,e m ) constitute a basis of one-forms on F ~M!. Let v i j 5 v i jk e k 1 v i j m e m and vˆ i j 5 vˆ i jk e k 1 vˆ i j m e m be the corresponding decompositions of the solutions. Then ~7! easily yields v i j m 5 vˆ i j m and v i @ jk # 5 vˆ i @ jk # . The defining properties of the solutions give also v (i j)k 505 vˆ (i j)k . Now, due to the identity A i jk 5A i @ jk # 2A j @ ik # 2A k @ i j # , which is true for any A i jk such that A (i j)k 50, we get v i jk 5 vˆ i jk . This shows that v i j 5 vˆ i j , hence the uniqueness. We proceed to the construction of a solution. Given a sufficiently small neighborhood O in M we identify p 21 (O ) with O 3L↑1 . Then, the soldering form may be written as e i 5l i j u j .

~8!

Taking de i and using Eq. ~3! one easily finds that

v i j 5l i k G k m˜lmj 2dl i k˜lk j

~9!

is a solution to ~5!. Given this solution one defines the forms ~E, G, V! by E52 v 1 3 ,

G52 v 2 4 ,

V5 21 ~ v 2 2 1 v 3 3 ! .

~10!

This is in accordance with ~6! due to v (i j) 50 and the reality properties of e i . ¯ ,T,L,E,E ¯ ,G,G ¯ ,V,V ¯ ) it is enough to To prove the linear independence of the system (F,F 1 ˜k 2 ˜k 1 ˜k 2 ˜k 1 ˜k observe that the six one-forms dl k l 3 , dl k l 3 , dl k l 4 , dl k l 4 , dl k l 1 , and dl 3 k˜lk 3 constitute a basis of right invariant forms on L↑1 . The theorem is proven.

III. THE STRUCTURE EQUATIONS

¯ ,T,L,E,E ¯ ,V,V ¯ ,G,G ¯ ) on F ~M! given by ~4! and Consider the ten well-defined forms (F,F ~10!. The differentials of the first four forms are given by ~5! and ~6!. In the basis ¯ ,T,L,E,E ¯ ,V,V ¯ ,G,G ¯ ) they assume the form (F,F ¯ ! ∧F1E∧T1G ¯ ∧L, dF5 ~ V2V

~11!

¯ ∧F ¯ 2 ~ V1V ¯ ! ∧T, dT5G∧F1G

~12!

¯ ∧F1E∧F ¯ 1 ~ V1V ¯ ! ∧L. dL5E

~13!

The differentials of the other forms may be easily calculated using the local representation ~9! and the well-known structure equations

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˜n l p d v i j 52 v i k ∧ v k j 1 21 R k smn l i k˜ls j ˜lm l l p e ∧e .

~14!

These differentials are by far much more complicated than the differentials of dF, dT, and dL. In particular, the decompositions of dF, dT, and dL onto the basis of two-forms associated with ¯ ,T,L,E,E ¯ ,V,V ¯ ,G,G ¯ ) have only constant coefficients. It turns out that in the differentials dE, (F,F dV, and dG coefficients which are functions appear. The zero sets of these functions have a well-defined geometrical meaning and define certain subsets of F ~M!. Now, the hope is that when we restrict ourselves to such subsets then the differentials of E, V, and G will have a much simpler form than their differentials on the whole F ~M!. Our aim now is to study this possibility.

IV. EXPLICIT EXPRESSIONS FOR THE BASIC FORMS ON F „M…

We concentrate on the analysis of dE52d v 1 3 . Let us introduce the matrices l i j (w,z,y) and l i j (w 8 ,z 8 ,y 8 ) such that

l i j ~ w,z,y ! 5

S

¯¯z ! u w u w 21 ~ 11y

u w u w 21¯y z

¯¯z ! z u w u w 21 ~ 11y

u w u w 21¯y

¯ w u w u 21 yz ¯¯z ! y u w u ~ 11y

w u w u 21 ~ 11yz ! u w u ~ 11yz !¯y

w u w u 21 ~ 11yz !¯z ¯¯z u 2 u w uu 11y

w u w u 21 y

¯z u w u 21

z u w u 21

u z u 2 u w u 21

u w u 21

u w uu y u 2

D

,

l i j ~ w 8 ,z 8 ,y 8 !

S

u w 8 u w 8 21¯y 8 z 8

D

¯ 8¯z 8 ! u w 8 u w 8 21 ~ 11y

u w 8 u w 8 21¯y 8

¯ 8¯z 8 ! z 8 u w 8 u w 8 21 ~ 11y

w 8 u w 8 u 21 y 8¯z 8 ¯ 8¯z 8 ! u w 8 u ~ 11y

w 8 u w 8 u 21 y 8

w 8 u w 8 u 21 ~ 11y 8 z 8 !¯z 8 . ¯ 8¯z 8 u 2 y 8 u w 8 uu 11y

¯z 8 u w 8 u 21

u w 8 u 21

w 8 u w 8 u 21 ~ 11y 8 z 8 ! 5 u w 8 u ~ 11y 8 z 8 !¯y 8 z 8 u w 8 u 21

u w 8 uu y 8 u 2

u z 8 u 2 u w 8 u 21

Then, it is well known that L↑1 can be represented by L↑1 5UøU8 ,

~15!

U5 $ l i j ~ w,z,y ! such that ~ w,z,y ! PC3 ,wÞ0 % ,

~16!

U 8 5 $ l i j ~ w 8 ,z 8 ,y 8 ! such that ~ w 8 ,z 8 ,y 8 ! PC3 ,w 8 Þ0 % .

~17!

where

On the intersection UùU8 , the coordinates (w,z,y) and (w 8 ,z 8 ,y 8 ) shall be related by w52

w8 , z 82

w 8 52

w , z2

z5

1 , z8

1 z 85 , z

y52z 8 ~ 11y 8 z 8 ! ,

~18!

y 8 52z ~ 11yz ! .

~19!

Thus, we can cover any p 21 (O )>(O 3L↑1 ) by the two charts O 3U and O 3U 8 . Now, on O consider the coframe u i of ~1!. Inserting l i j 5l i j (w,z,y) or l i j (w 8 ,z 8 ,y 8 ) to the formulas ~8!, ~9!, ~14! and using the definitions ~1!, ~4!, ~10! we easily obtain the following two lemmas. Lemma 1: On O 3U the forms F, T, L, E, V and G read F5

uwu ¯ 1y ¯¯z !~ M 1zK ! 1y ¯ zM ¯ L#, @~ 11y w

¯ 1y¯y L # , ¯¯z u 2 K1y ~ 11y ¯¯z ! M 1y ¯ ~ 11yz ! M T5 u w u @ u 11y

~20! ~21!

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1 ¯ 1z ¯ K1zM ¯M #, @ L1zz uwu

~22!

1 @ dz1G 321z ~ G 211G 43! 1z 2 G 14# , w

~23!

L5

E5

Paweł Nurowski

1 1 dw 2ydz2yG 322 ~ 112yz !~ G 211G 43! 2z ~ 11yz ! G 14 , 2 w 2

G5w @ dy2y 2 dz2y 2 G 322y ~ 11yz !~ G 432G 12! 2 ~ 11yz ! 2 G 14# .

~24! ~25!

Lemma 2: On O 3U8 the forms F, T, L, E, V and G read F5

u w 8u ¯ 1z 8 L ! 1y ¯ 8¯z 8 !~ M ¯ 8 z 8 M 1y ¯ 8K # , @~ 11y w8

¯ 1y ¯ 8¯z 8 u 2 L1y 8 ~ 11y ¯ 8¯z 8 ! M ¯ 8 ~ 11y 8 z 8 ! M 1y 8¯y 8 K # , T5 u w 8 u @ u 11y

V52

~27!

1 ¯ #, ¯8M @ K1z 8¯z 8 L1z 8 M 1z u w 8u

~28!

1 @ dz 8 1G 411z 8 ~ G 121G 34! 1z 8 2 G 23# , w8

~29!

L5

E5

~26!

1 1 dw 8 2y 8 dz 8 2y 8 G 412 ~ 112y 8 z 8 !~ G 121G 34! 2z 8 ~ 11y 8 z 8 ! G 23 , 2 w8 2

G5w 8 @ dy 8 2y 8 2 dz 8 2y 8 2 G 412y 8 ~ 11y 8 z 8 !~ G 342G 21! 2 ~ 11y 8 z 8 ! 2 G 23# .

~30! ~31!

¯ , K↔L, Note that this Lemma follows from the previous one by applying transformations M ↔M 1↔2, and 3↔4, where the last two transformations refer to the tetrad indices. Now one can easily find the differential of dE. ¯ ,T,L,E,E ¯ ,V,V ¯ ,G,G ¯ ) of one-forms on Lemma 3: The decomposition of dE onto the basis (F,F 21 p (O ) reads ¯ ! 1 b F∧L1 c T∧F ¯ 1a ~ L∧T2F∧F ¯ ! 1 a L∧F, dE52V∧E1 f T∧F1b ~ L∧T1F∧F where f ,b, b , c ,a, a are well-defined functions on p 21 (O ). The functions f ,b, b , c ,a, a are given by

f5

c5

1 F, uwu2

1 C, w2

¯ ~ 21 f¯z 1y ¯ f !, b5w

a5w ~ 41 c z 1y c ! ,

¯ 2 ~ 21 f¯z¯z 1y ¯ f¯z 1y ¯ 2f !, b 5w

a 52w 2 ~ 121 c zz 1y 2 c 1 21 y c z ! 2 121 R,

¯ S 232zS 131zz ¯ ~ S 121S 34! 1 21¯z 2 S 221 21 z 2 S 112z ¯ 2 zS 242z 2¯z S 141 21 z 2¯z 2 S 44 , F5 21 S 332z and C5C 4 24C 3 z16C 2 z 2 24C 1 z 3 1C 0 z 4 on O 3U and by

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f5 c5

1 F 8, u w 8u 2

1 C 8, w 82

¯ 8 ~ 21 f¯z 8 1y ¯ 8f !, b5w

a5w 8 ~ 41 c z 8 1y 8 c ! ,

Paweł Nurowski

¯ 8 2 ~ 21 f¯z 8¯z 8 1y ¯ 8 f¯z 8 1y ¯ 82f !, b 5w

a 52w 8 2 ~ 121 c z 8 z 8 1y 8 2 c 1 21 y 8 c z 8 ! 2 121 R,

¯ 8 S 142z 8 S 241z 8¯z 8 ~ S 121S 34! 1 21¯z 8 2 S 11 F 8 5 21 S 442z 1

1 2

¯ 8 2 z 8 S 132z 8 2¯z 8 S 232 21¯z 8 2¯z 8 2 S 33 z 8 2 S 222z

and C 8 5C 0 24C 1 z 8 16C 2 z 8 2 24C 3 z 8 3 1C 4 z 8 4 on O 3U8 . The following three cases are of particular interest. ~a! ~b! ~c!

The metric g of the four-manifold M satisfies the Einstein equations R i j 5lg i j and is not conformally flat. This case is characterized by F[0 and CÓ0. The metric g is conformally flat but not Einstein. This case corresponds to C[0, FÓ0. The metric g is of constant curvature. This means that C[F[0.

In the first two cases there is a canonical choice of certain six-dimensional subsets in F ~M!. This is defined by the demand that on such sets certain components of dE should identically vanish. This approach is impossible in case ~c! since this implies an immediate reduction of dE to the form dE52V∧E1 121 RL∧F.

~32!

V. DISTINGUISHED SUBSET OF F „M…

From now on we consider case ~a!. This is the most interesting generic Einstein case. Imposing the restrictions ~a! on dE we immediately see that ¯ 1a ~ L∧T2F∧F ¯ ! 1 a L∧F, dE52V∧E1 c T∧F where c ,a, a are the same as in the Lemma 1. Since we are in the not conformally flat case ~a! we have c Þ0. This makes possible the restriction to such a set W , p 21 (O ) in which a identically vanish. Thus we consider W 5W 1 øW 2 ,

~33!

where W 1 5 $ ~ x;w,z,y ! P ~ O 3U ! such that c y1 41 c z 50 % , W 2 5 $ ~ x;w 8 ,z 8 ,y 8 ! P ~ O 3U8 ! such that c y 8 1 41 c z 8 50 % , or ~what is the same due to the nonvanishing of w and w 8 ) W 1 5 $ ~ x;w,z,y ! P ~ O 3U ! such that Cy1 41 C z 50 % ,

~34!

W 2 5 $ ~ x;w 8 ,z 8 ,y 8 ! P ~ O 3U8 ! such that C 8 y 8 1 41 C z88 50 % .

~35!

On W we have ¯ 1 a L∧F. dE52V∧E1 c T∧F

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˜ of W in which c 521. One can still simplify this relation by restricting oneself to a subset P 0 21 ˜ is a subset of p (O ), which is given by Then, P 0 ˜ 5P ˜ øP ˜ , P 0 01 02

~36!

˜ 5 $ ~ x;w,z,y ! P ~ O 3U ! such that Cy1 1 C 50, C1w 2 50 % P 4 01 z

~37!

˜ 5 $ ~ x;w 8 ,z 8 ,y 8 ! P ~ O 3U8 ! such that C 8 y 8 1 1 C 8 50, C 8 1w 8 2 50 % . P 4 02 z8

~38!

where

and

˜ we have It follows from the construction that on P 0 ¯ ∧T1 a F∧L. dE52V∧E1F VI. ELLIPTIC FIBRATION

˜ . We study the geometry and topology of the set P 0 ˜ The equations defining P 0 may be written as y52

1 Cz , 4 C

C1w 2 50,

y 8 52

1 C z88 , 4 C8

C 8 1w 8 2 50.

~39! ~40!

We see that the first pair of equations uniquely subordinates y to z and y 8 to z 8 . The second pair gives a relation between w and z and w 8 and z 8 . Thus, locally among the parameters (x;w,z,y) in O 3U @respectively, (x;w 8 ,z 8 ,y 8 ) in O 3U8 ], only x and z ~respectively, x and z 8 ) are free. This ˜ is six dimensional. Moreover, P ˜ is fibered over O with two-dimensional fibers. shows that P 0 0 These are locally parametrized by z or z 8 . To discuss the topology of fibers one observes that the relation between w and z ~respectively, w 8 and z 8 ) is purely polynomial. Over every point of O it has the form w 2 52C 4 14C 3 z26C 2 z 2 14C 1 z 3 2C 0 z 4

~41!

w 8 2 52C 0 14C 1 z 8 26C 2 z 8 2 14C 3 z 8 3 2C 4 z 8 4 .

~42!

or

Assume for a moment ~a! that w50 and w 8 50 are the allowed values of the parameters and ~b! that equations 2C 4 14C 3 z26C 2 z 2 14C 1 z 3 2C 0 z 4 50,

~43!

2C 0 14C 1 z 8 26C 2 z 8 2 14C 3 z 8 3 2C 4 z 8 4 50

~44!

for z and z 8 have only distinct roots. Then, the relations ~41!–~42!, as being fourth order in the parameters z and z 8 , describe a two-dimensional torus. ~This is a well-known fact of classical algebraic geometry, see, e.g., Ref. 2. I am very grateful to Roger Penrose for clarifying this for me!.3 Let us comment on ~a! and ~b!. ~a! We know that w and w 8 cannot be zero by their definitions. So to have a torus fibration over the Einstein space–time we need to accept that w and w 8 may vanish. A price paid for this ¯ ,T,L,E,E ¯ ,V,V ¯ ,G,G ¯ ) will be singular on the resulting fibration at is that some of the forms (F,F

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these values of w and w 8 . With these remarks, from now on, we accept that w and w 8 may vanish. ˜ over M which, over the neighborhood O ,M is This enables us to introduce the fibration P given by ˜ 5P ˜ øP ˜ , P 1 2

~45!

˜ 5 $ ~ x;w,z,y ! P ~ O 3C3 ! such that Cy1 1 C 50, C1w 2 50 % , P 4 1 z

~46!

where

H

˜ 5 ~ x;w 8 ,z 8 ,y 8 ! P ~ O 3C3 ! such that C 8 y 8 1 1 C 8 50, C 8 1w 8 2 50 P 2 4 z8

J

~47!

and the transition functions between (w,z,y) and (w 8 ,z 8 ,y 8 ) coordinates are given by ~18!. ˜ is a torus fibration over O provided that Eqs. ~43! and ~b! The discussion in ~a! means that P ~44! have distinct roots. It is well known that the number of distinct roots in ~43! and ~44! is directly related to the algebraic ~Cartan–Petrov–Penrose4–6! classification of space–times. Thus, ˜ is a torus fibration over O . In the if the space–time M is algebraically general in O , then P ˜ algebraically special cases the fibers of P are degenerate tori. These topologically are: ~II! a torus with one vanishing cycle in the Cartan–Petrov–Penrose type II, ~D! two spheres touching each other in two different points in the Cartan–Petrov–Penrose type D, ~III! a sphere with one singular point in the Cartan–Petrov–Penrose type III, ~N! two spheres touching each other in one point in the Cartan–Petrov–Penrose type N. The pure situations considered so far may be a bit more complicated when the Cartan– Petrov–Penrose type of the Einstein space–time varies from point to point. Imagine, for example, that along a continuous path from x to x 8 in O the Cartan–Petrov–Penrose type of the Einstein ˜ over x 8 is only a torus with one vanishing space–time changes from I to II. Then the fiber of P cycle although the fiber over the starting point x was a torus. It is clear that more complicated ˜ over different points of M can have topologies II, situations may occur, and that the fibers of P D, III, and N. Fibrations of this type are widely used in algebraic geometry. They are called elliptic, since their fibers can be any kind ~even degenerate! of an elliptic curve. VII. THE MAIN THEOREM

In Secs. IV–VI, for the clarity of presentation, we restricted ourselves to the neighborhood O ˜ over O . This, however, can be easily prolonged of M. We ended up with an elliptic fibration P to an elliptic fibration over the whole M. To see this it is enough to observe that a fiber over any point in O is essentially defined by the Weyl tensor. Since the Weyl tensor is uniquely defined on the whole M then we can use equations like ~41! to uniquely define the elliptic fibers over all M. Summing up the information from Secs. IV–VI we have the following theorem. Theorem 2: Given a four-dimensional conformally nonflat space–time M satisfying the ˜ →M with the following propEinstein equations R i j 5lg i j one naturally defines a fibration P:P erties. (1) A fiber P 21 (x) over a point xPM is a (possibly degenerate) elliptic curve C given by C 5C 1 øC 2 , C 1 5 $ ~ w,z ! PC2 such that w 2 52C 4 14C 3 z26C 2 z 2 14C 1 z 3 2C 0 z 4 % , C 2 5 $ ~ w 8 ,z 8 ! PC2 such that w 8 2 52C 0 14C 1 z 8 26C 2 z 8 2 14C 3 z 8 3 2C 4 z 8 4 % , where the transition functions between (w,z) and (w 8 ,z 8 ) coordinates are given by w52

w8 , z 82

z5

1 . z8

(2) The degeneracy of a fiber depends on the algebraic type of the space–time metric and may change from point to point.

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˜ of dimension six immersed in the (3) There is a unique construction of a certain surface P 0 ˜ is fibered over M and P ˜ may be viewed as an extension of bundle of null coframes F ~M!. P 0 ˜ achieved by adding to each fiber of P ˜ at most four points. P 0 0 ¯ ¯ ,V,V ¯ ,G,G ¯ ) on P ˜ with the following properties: (4) There are ten one-forms (F,F ,T,L,E,E (a) Forms T,L are real, all the other are complex valued. (b) The forms are defined in two steps. First, by restricting the soldering form components e i ˜ and second, by exand the Levi–Civita connection components v ij from F ~M! to P 0 ˜ tending the restrictions to P . ¯ ,T,L,E,E ¯ ) constitute the basis of one-forms on P ˜. (c) (F,F ˜: (d) The forms satisfy the following equations on P ¯ ! ∧F1E∧T1G ¯ ∧L, dF5 ~ V2V ¯ ∧F ¯ 2 ~ V1V ¯ ! ∧T, dT5G∧F1G ¯ ∧F1ELF ¯ 1 ~ V1V ¯ ! ∧L, dL5E

~48!

¯ ∧T1 a L∧F, dE52V∧E1F ˜. with a certain function a on P ˜ ~respectively, on P ˜ ) may be obtained from the The explicit formulas for the forms on P 1 2 expressions of Lemma 1 ~respectively, Lemma 2! by inserting the relations y52C z /(4C) and w 2 1C50 @respectively, y 8 52C z88 /(4C 8 ) and w 8 2 1C 8 50].

VIII. RELATION BETWEEN THE ELLIPTIC FIBRATION AND THE BUNDLE OF NULL DIRECTIONS

˜ constitutes a double branched cover of the Penrose bundle Finally we note that the bundle P P of null directions over the space–time. To see this consider the map f given by f

˜ { ~ x;w,z,y ! → ~ x,z ! PO 3C, P 1 f

˜ { ~ x;w 8 ,z 8 ,y 8 ! → ~ x,z 8 ! PO 3C. P 2 ˜ ùP ˜ the coordinates z and z 8 are related by z 8 51/z, then the two Since on the intersection P 1 2 copies of C, which appear in the above relations may be considered as two coordinate charts ~say around the North and the South pole, respectively! on the two-dimensional sphere. This sphere is the sphere of null directions at a given point of O which can be seen as follows. ¯ 1z ¯ K1zM ¯ M at xPO for all the values Consider directions of all the one-forms L(z)5L1zz of the complex parameter z. These directions are in one to one correspondence with null directions ¯ via L(z)52g(k(z)). These null directions do not form a sphere yet, k(z)5k1zz¯ l2zm2z¯ m since the direction corresponding to the vector l is missing. But the missing direction is included in the family of null directions corresponding to the directions of one-forms K(z 8 )5K1z 8¯z 8 L ¯ 1z ¯ 8 M at x. Thus the space parametrized by z and z 8 subject to the relation z 8 51/z is in 1z 8 M one to one correspondence with the sphere of null directions at x. This proves the assertion that f is a cover of P . The map f is a double cover since if z ~or z 8 ) is not a root of ~43! @respectively, ~44!# then f 21 (z) is a two-point set. In at most four cases when z is a root f 21 (z) is a one-point set. This proves that f is a singular cover. As we know the singular points are branch points, which proves the statement from the beginning of this section.

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J. Math. Phys., Vol. 39, No. 10, October 1998

Paweł Nurowski

ACKNOWLEDGMENTS

I am deeply indebted to Andrzej Trautman who focused my attention on the problem of encoding Einstein equations on natural bundles associated with a space–time. Without his influence this work would never have been completed. I would also like to express my gratitude to Paolo Budinich, Mike Crampin, Ludwik Da¸browski, Jerzy Lewandowski, Roger Penrose, David Robinson, and Jacek Tafel for helping me at various stages of the present work. This research was partially supported by: Komitet Badan´ Naukowych ~Grant No. 2 P03B 017 12!, Consorzio per lo Sviluppo Internazionale dell Universita degli Studi di Trieste and the Erwin Schro¨dinger International Institute for Mathematical Physics. P. Nurowski, ‘‘On a certain formulation of the Einstein equations,’’ J. Math. Phys. 39, 5477 ~1998!. D. Mumford, ‘‘Tata lectures on Theta II,’’ Prog. Math. 43, 3.12–3.13 ~1984!. 3 R. Penrose ~private communications!. 4 E. Cartan, ‘‘Sur les espaces conformes generalises at l’universe optique,’’ Compt. Rendus Acad. Sci. Paris 174, 857–859 ~1922!. 5 R. Penrose, ‘‘A spinor approach to general relativity,’’ Ann. Phys. ~N.Y.! 10, 171–201 ~1960!. 6 A. Z. Petrov, ‘‘Classification of spaces defining gravitational fields,’’ Sci. Not. Kazan State Univ. 114, 55–69 ~1954!. 1 2