Gravitational field equations on Fefferman space-times

Elisabetta Barletta1

Sorin Dragomir1

Howard Jacobowitz2

Abstract. The total space M H1 S1 of the canonical circlebundle over the 3-dimensional Heisenberg group H1 is a space-time with the Lorentzian metric F0 (Feffermans metric) associ-ated to the canonical Tanaka-Webster flat contact form 0 on H1.The matter and energy content of M is described by the energy-momentum tensor T (the trace-less Ricci tensor of F0) as aneffect of the non flat nature of Fefermans metric F0 . We study

the gravitational field equations R (1/2)Rg = T on M.We consider the first order perturbation g = F0 + h,

2 Gravitational field equations on Fefferman space-times

Let X C(T (C(M))) denote the horizontal lift of the tangent vec-tor field X C(T (M)) with respect to Grahams connection . IfS C(T (C(M))) and C(T (M)) are respectively the tangent tothe S1-action and the Reeb vector field associated to the contact form then S is a globally defined time-like vector field on C(M), thusgiving a time orientation of the Lorentzian manifold (C(M), F). Thesynthetic object (C(M), F ,

S) is then a space-time (cf. [11]).Our purpose through the present paper is to continue the investigation(cf. [2], [6], [10], [15]) of the relationship between CR and pseudoher-mitian geometry (and the underlying subelliptic theory, cf. [3]) andspace-time physics, as prompted by the occurrence of Feffermans met-ric associated to a pseudohermitian manifold (M, ) and, viceversa, bythe occurrence of strictly pseudoconvex CR structures associated toshear free null geodesic congruences on a Lorentzian manifold (cf. [24],[27], [30], [32]). By a result of J.M. Lee (cf. [28]) F is never an Ein-stein metric i.e. there is no R such that R = g where Ris the Ricci tensor field of the Lorentzian manifold (C(M), F). In thespirit of space-time physics, to produce a gravitation theory on C(M)one should look for solutions to Einstein field equations. In the presentpaper we formulate a number of fundamental questions, the first ofwhich is I) what is the convenient form of Einsteins equations,to be considered on C(M)? A moments thought shows that

(1) Ric(g) = 0

[Einsteins equations for empty space, where Ric(g) is the Ricci tensorof g Lor(C(M)), the dependent variable in equations (1)] is not anappropriate choice [e.g. by the aforementioned result in [28], Feffer-mans metric F is never a solution to (1)]. It is one of our purposes toanswer the question posed above, although confined to the case whereM = H1 = C R is the 3-dimensional Heisenberg group. This may beorganized as a strictly pseudoconvex CR manifold CR isomorphic tothe boundary of the Siegel domain = {(z, w) C2 : Im(w) > zz} (cf.e.g. [14], p. 14). Let M = C(H1) be the total space of the canonicalcircle bundle over H1 (cf. [14], p. 119). Let

0 = dt+ i (z dz z dz)

be the canonical contact form on H1. Then 0 is positively orientedand Tanaka-Webster flat. Let F0 Lor(M) be the Fefferman metricof the pseudohermitian manifold (H1, 0) (cf. [14], p. 128). The metricF0 is not flat and its curvature may be thought of as the effect of amatter distribution on M. The matter, or energy, content of M is thendescribed by the energy-momentum tensor T which is the traceless

E. Barletta, S. Dragomir and H. Jacobowitz 3

Ricci tensor of F0 . T is further discussed in 5, showing that Tdescribes an incoherent matter distribution (incoherent dust) on M. Mis a 4-dimensional manifold diffeomorphic to H1 S1 so that it maynot be covered by a globally defined coordinate system. However onemay adopt the globally defined nonholonomic frame

(2) X0 = 2S, X1 = L , X2 = L

, X3 =

0 ,

where S X(M) is the tangent to the S1 action on M, 0 X(M) isthe Reeb vector field of (H1 , 0), and L = /z i z /t is the Lewyoperator (cf. [14], p. 12). Components of tensor fields on M are thenintended with respect to {X : 0 3} and are globally definedsmooth functions on M, perhaps complex valued. Through the presentpaper we shall study Einsteins gravitational field equations

(3) R 1

2Rg = T

on M in the presence of the matter distribution described by T . HereR = Ric(g) and R are respectively the Ricci and scalar curvatureof the Lorentzian metric g Lor(M) [whose components g are theunknown functions in the PDEs system (3)]. Any physics discussionof gravity on M requires a solution to (3). While the search for exactsolutions to (3) is deferred to further work, we wish (in the spirit ofthe classical approach by A. Einstein, [17]) to linearize the field equa-tions (3) with the manifest purpose of producing at least a solutionto the linearized equations. Another fundamental question posed inthe present paper is then II) what is a convenient base pointg0 Lor(M) [about which one ought to linearize (3)] and what is anappropriate choice of perturbation matrix h? We consider firstorder perturbations

(4) g = F0 + h,

4 Gravitational field equations on Fefferman space-times

[where R11 is the pseudohermitian Ricci tensor associated to an ar-bitrary contact form on H1, cf. [14] and 2 of this paper] and onelinearized (5) by considering perturbations of the form 0 + (

E. Barletta, S. Dragomir and H. Jacobowitz 5

Acknowledgements Part of this paper has been written while H. Ja-

cobowitz visited (June and October 2014) at the Department of Mathemat-

ics, Informatics and Economy (DiMIE) of the University of Basilicata. He

expresses his gratitude for support from INdAM (Rome, Italy) and DiMIE

(Potenza, Italy). E. Barletta and S. Dragomir acknowledge support from

P.R.I.N. 2012.

Contents

1. Introduction 12. Fefferman space-times 53. Geodesic motion in the classical limit 114. Gravitational field equations 165. Conclusions and final comments 22Appendix A. Linearized Ricci curvature 23References 30

2. Fefferman space-times

Notations, conventions and fundamental results (in CR and pseu-dohermitian geometry) adopted through this paper are those in [14].Some of the basic material is recalled in this section, for the needs of themore physics oriented reader. Let H1 be the 3-dimensional Heisenberggroup i.e. the Lie group C R with the product(10) (z, t) (w, s) = (z + w, t+ s+ 2 Im (zw)) .H1 is a strictly pseudoconvex CR manifold, of CR dimension 1, withthe CR structure T0,1(H1) spanned by the Lewy operator

L = /z iz /t.

One sets as customary T1,0(H1) = T0,1(H1) (overbars denote complexconjugates). A relevant second order differential operator (occurringin the linearized field equations (7)-(9)) is the sublaplacian (cf. [14])

bu = (LLu+ LLu

), u C2(H1),

coinciding with the Hormander operator 12

(X2 + Y 2) (cf. [26]) asso-ciated to the Hormander system of vector fields {X, Y } on H1, whereL = 1

2(XiY ). The sublaplacian b is formally similar to the Laplace-

Beltrami operator of a Riemannian manifold, yet it isnt elliptic. Nev-ertheless b is a positive, formally self-adjoint, degenerate elliptic (inthe sense of [13]) operator which is subelliptic of order 1/2 (cf. [20])

6 Gravitational field equations on Fefferman space-times

and hence hypoelliptic (cf. [25], a feature that b shares with ellipticoperators). Also for every u C2(H1)

(11) bu =1

20u+ 2i

t

(zu

z z u

z

) 2|z|2

2u

t2

where 0 is the ordinary Laplacian (2/x2 + 2/y2) on R2 (withz = x+ iy).

A complex p-form is a (p, 0)-form if T0,1(H1) c = 0. A top degree(p, 0)-form is a (2, 0)-form. The canonical bundle C K(H1) H1is the bundle of all (2, 0)-forms (a complex line bundle). There is anatural action of R+ (the multiplicative positive reals) on K0(H1) =K(H1) \ {zero section} and the quotient space M = K0(H1)/R+ is thetotal space of a principal S1-bundle (the canonical circle bundle overH1). Let : M H1 be the projection. Let be a positively orientedcontact form on H1 i.e. a real 1-form such that

Ker() = H(H1) = Re {T1,0(H1) T0,1(H1)}

(the Levi, or maximally complex, distribution on H1), d is a volumeform on H1, and the Levi form associated to

G(Z,Z) = i(d)(Z,Z), Z T1,0(H1),

(i =1) is positive definite. If 1 = dz then each class [] mod R+

of K0(H1)x may be represented as

[] =[( 1

)x

]Mx , C , x H1 .

In particular the canonical circle bundle is trivial i.e.

(12) M S1 H1 , : [] 7(

||, x

),

is a C diffeomorphism. The Heisenberg group H1 also carries a Rie-mannian metric g [the Webster metric of (H1 , )] given by

g(V,W ) = G(V,W ), g(V, ) = 0, g(, ) = 1,

for any V,W H(H1). The pair (H(H1) , G) is a sub-Riemannianstructure on H1 (cf. [9]) and the Webster metric g is a contractionof G (cf. [31]). For each u C1(H1) the gradient of u is given byg(u, V ) = V (u) for any V X(H1). The horizontal gradient of u is

Hu = Hu

where H : T (H1) H(H1) is the projection associated to the decom-position T (H1) = H(H1) R.

E. Barletta, S. Dragomir and H. Jacobowitz 7

Given a smoothly bounded strictly pseudoconvex domain Cn(n 2) a Lorentzian metric F on S1, known nowadays as Fef-fermans metric (cf. [14]), was built by C. Fefferman (cf. [19]). A keyfeature of F was that its restricted conformal class

{eu F : u C(,R)}

is a biholomorphic invariant of . Here : S1 is the pro-jection. This prompted the question (also due to C. Fefferman, [19])whether F admits an intrinsic construction for each strictly pseudocon-vex real hypersurface M Cn+1, such that the restricted conformalclass is a CR invariant of M . The question was settled by J.M. Lee(cf. [28]) who built a Lorentzian metric F on the total space C(M) ofthe canonical circle bundle over a strictly pseudoconvex CR manifoldM , not necessarily embedded, in correspondence to any fixed positivelyoriented contact form on M . The metric F is computed in terms ofpseudohermitian invariants [of the pseudohermitian manifold (M, )]and the construction of F is such that, whenever M is the boundaryof a strictly pseudoconvex domain Cn, the Lorentzian manifold(C(), F) is conformally diffeomorphic to ( S1, F ). Throughthe present paper we only need J.M. Lees result for M = H1 i.e.

F = G + 2 (

) , G = 2g11 1 1 ,

=1

3

{d +

(i 1

1 i2g11 dg11

8

)},

where is a local fibre coordinate on M and 11 is the connection 1-

form of the Tanaka-Webster connection of (H1 , ) i.e. L = 11L.Let = X(H1) be the Reeb vector field of (H1, ) i.e. the uniquenowhere zero, globally defined tangent vector field on H1, transverse tothe Levi distribution H(H1), determined by

() = 1, c d = 0.

For simplicity we set 0 = 0 . Also if R is the curvature tensor field

of then we set T1 = L, T1 = L, T0 = and

RADBCTD = R

(TB , TC)TA , A,B,C, {1, 1, 0},

g11 = G(L, L), g11 = 1/g11 , R11 = R1

111 , = g

11R11 .

Here g11, R11 and are respectively the Levi invariant, the pseudo-hermitian Ricci tensor, and the pseudohermitian scalar curvature of(H1 , ). Let us set M = C(H1) for simplicity. By a result of R.C.

8 Gravitational field equations on Fefferman space-times

Graham (cf. [21]) is a connection 1-form on the principal bundleS1 M H1. In particular for = 0

G0 = 2 1 1 , = 1

3d,

so that is flat. Let S X(M) be the tangent to the S1 actionand choose the local fibre coordinate such that S = (3/2) / onU(0) = 1 [(0)H1], where (0) = {ei : | 0| < } with0 R. If V X(H1) then let V X(M) be the horizontal lift of Vwith respect to i.e.

V p Ker()p , (dp)V p = Vx , p M, x = (p) H1 .

Then X = S is a globally defined time-like vector field on the

Lorentzian manifold (M, F) i.e. X is a time orientation, so that(M, F , X) is a space-time. If z = x+ iy and t are canonical coordi-nates on H1, we endow M with the local coordinates

(x) (x0 , xj) (, x , y , t ),

where 0 3, 1 j 3. With respect to (x) the Feffermanmetric F0 reads

F0 = 2[(dx1)2

+(dx2)2]

+2

3

[dx3 + 2

(x1 dx2 x2 dx1

)] dx0 .

We shall write the geodesics equations for the Lorentzian manifold(M, F0) as the Euler-Lagrange equations of the variational principle

(13)

F(x)x

x ds = 0

i.e.d

ds

(L

x

)=

L

x, L(x, x) F(x) x x .

Since

[F ] =

0 23y

2

3x

1

3

23y 2 0 0

2

3x 0 2 0

1

30 0 0

E. Barletta, S. Dragomir and H. Jacobowitz 9

the Euler-Lagrange equations of (13) are

(14)d2x0

ds2= 0,

(15)d2x1

ds2 2

3

dx0

ds

dx2

ds= 0,

(16)d2x2

ds2+

2

3

dx0

ds

dx1

ds= 0,

(17)d2x3

ds2 4

3x1dx0

ds

dx1

ds 4

3x2dx0

ds

dx2

ds= 0.

Straightforward integration of the ODE system (14)-(17) leads to thetwo families of geodesics of (M, F0)

(18) (s) = B0 , z(s) = x(s) + i y(s) = s+ , t(s) = a s+ b,

, C, a, b, B0 R,and

(19) (s) = A0 s+B0 , A0 , B0 R, A0 6= 0,

(20) (z(s), t(s)) =

(1i , k

)( ei s ,

(1

||2 + ||2

)s

),

, C, k R, = 23A0 .

The dot product in (20) is given by (10). A parallel of (14)-(17) to

dx

ds+

{

}x x = 0, x dx

ds,

furnishes the list of all nonzero Christoffel symbols of F0 i.e.

(21)

{102

}=

{120

}= 1

3,

{201

}=

{210

}=

1

3,

(22)

{301

}=

{310

}= 2

3x,

{302

}=

{320

}= 2

3y.

Consequently if G = det [F ] the Ricci curvature

r =

{

}|(

logG)||

+

+

{

}(logG)|{

}{

}

10 Gravitational field equations on Fefferman space-times

of (M, F0) is

(23) r00 =2

9, ri0 = 0, rij = 0, 1 i, j 3.

Indeed G = 4/9 hence

r =

{

}|{

}{

}.

Next [by (21)-(22)]

{00

}= 0 and for instance

r00 = {

0

}{0

}= 2

{120

}{201

}=

2

9.

The remaining components of r may be computed in a similar man-ner. The Lorentzian manifold (M, F0) has nonzero curvature, whichmay be thought of as the result of a distribution of matter on M.Therefore the matter, or energy, content of M is described by theenergy-momentum tensor

T = r 12r F , r = F F r , r = F

r .

Yet [by (23)] r = F 00r00 = 0 hence

(24) [T ] =

0 0 0 00 0 0 00 0 0 00 0 0 2

.It is our purpose, as explained in 1, to consider first order perturba-tions g = F0 + h (with

E. Barletta, S. Dragomir and H. Jacobowitz 11

function. Yet for this choice of h one has h00 = F(/ , /) = 0.Consequently, should one set x0 = = c and interpret as a timecoordinate, the geodesic motion equations on (M, F) will not reduce(for

12 Gravitational field equations on Fefferman space-times

not the proper time i.e. time as perceived by an observer attachedto C [and a change of parameter as given by (28) is needed]. Letvj = dCj/d , 1 j 3, and = v/c where v = (v1, v2, v3) and

v =(3

j=1(vj)2)1/2

. If

(28) s = () =

0

[gC(u)

(C(u) , C(u)

)] 12du

and x(s) = C(1(s)) then the geodesic equations are

(29)d2x

ds2+

{

}dx

ds

dx

ds= 0.

Here{

}= g {, } , {, } = 1

2

(g| + g| g|

),

f| = f/x , f C1(M).

We wish to derive the geodesic equations of motion (29) in the gravi-tational field g, in the weak field (

E. Barletta, S. Dragomir and H. Jacobowitz 13

where

H() =2

3

[2C2()

v1

c 2C1() v

2

c v

3

c

]= O().

On the other hand

h(C())dC

d

dC

d=

= c2[h00(C()) + 2h0j(C())

vj

c+ hjk(C())

vj

c

vk

c

]

(by dropping O(2))

c2[h00(C()) + 2h0j(C())

vj

c

]and (31) becomes

(32) gC()

(C(), C()

)= c2 [H() + h00(C())] .

As a consequence of (32)

() c [H() h00(C())]12

so that

(33)dx

ds(()) =

1

c[H() h00(C())]

12dC

d.

Differentiation with respect to in (33) together with

[H() h00(C())]1 =1

H()+

H()2h00(C()) +O(

2)

yields

(34) c2 [H() h00(C())]d2x

ds2=d2C

d 2

12

[H() h00(C())]1[

2

3cG() h00|j(C())

dCj

d

]dC

d

where

G() = 2C2()d2C1

d 2 2C1() d

2C2

d 2 d

2C3

d 2.

Yet

h00|j(C())vj

c= O()

and (34) reads

(35) c2 (H h00)d2x

ds2=d2C

d 2 1

3c

(1

H+

H2h00

)G()

dC

d.

14 Gravitational field equations on Fefferman space-times

Next (by (33)) {

}(C())

dx

ds(())

dx

ds(()) =

=1

c2 (H c h00)

{

}(C())

dC

d

dC

d

or (by dropping O(2))

(36)

{

}dx

ds

dx

ds=

1

H h00

{00

}.

Moreover {00

}= g {00, } = 1

2g g00|

so that

(37)

{00

}=

2g h00| .

Next one may observe that

(38) g = F h +O(2)where (cf. e.g. [4])

(39) [F ] =

0 0 0 3

01

20 y

0 01

2x

3 y x 2|z|2

[the inverse of F = F0(/x

, /x)] and h = FF h. Onemay conclude (by (37)-(38))

(40)

{00

}=

2F h00| .

Equations (35)-(36) and (40) imply geodesic motion (29) is governedby

(41)d2C

d 2 1

3c

(1 +

Hh00

) GH

dC

d=c2

2F k h00|k

i.e. for = j (respectively for = 0)

(42)d2Cj

d 2 1

3c

G

Hvj =

c2

2F jk h00|k ,

E. Barletta, S. Dragomir and H. Jacobowitz 15

(43) 13

(1 +

Hh00

) GH

=c2

2F 0k h00|k .

By (39) and [1 + (/H)h00]1 1 (/H)h00 equation (43) simplifies

to

(44) 13

G

H=

3c2

2

h00t

and substitution from (44) into (42) leads to

(45)d2Cj

d 2=c2

2F jk h00|k .

Next (by taking into account (39))

F 1k h00|k =1

2X(h00) , F

2k h00|k =1

2Y (h00), F

3k h00|k = V (h00),

where

X =

x+ 2y

t, Y =

y 2x

t,

V = y

x x

y+ 2|z|2

t= y X xY,

(so that L = 12(X iY )). Summing up, equations (44)-(45) are

(46) G() = 9c2

2H()

h00t

(C()),

(47)d2C1

d 2=c2

4X(h00)C() ,

d2C2

d 2=c2

4Y (h00)C() ,

(48)d2C3

d 2=c2

2V (h00)C() .

An inspection reveals (46) as a linear combination of (47)-(48). Finallyif

d2r

d 2=d2Cj

d 2

(

xj

)r()

then (47)-(48) become

(49)d2r

d 2=c2

4{X(h00)X + Y (h00)Y }r() .

For each u C1(H1) let Hu be the horizontal gradient of u withrespect to the canonical contact form 0. If E1 = (1/

2)X and E2 =

(1/

2)Y then Hu =2

a=1Ea(u)Ea and (49) becomes

(50)d2r

d 2=

(H

)r()

16 Gravitational field equations on Fefferman space-times

where

(51) = c2

2h00 .

Viceversa, given the classical potential , the motion of a particle willfollow a geodesic of (M, g) if the g00 term of the metric has the formg00 = (2/c2). The other components of g enter our asymptoticscheme only through the assumptions that they are time independentand nearly Feferman (i.e. close to the components of F0). As shownin the successive 4 if (26) is a solution to Einsteins field equations [ina nonempty space, whose matter content is described by the energy-momentum tensor (24)] to order O() then b = 0.

4. Gravitational field equations

By recent work of C.D. Hill & J. Lewandowski & P. Nurowski (cf.[24]) local embeddability of 3-dimensional CR manifolds M is closelytied to the existence of Lorentzian metrics on M R satisfying Ein-steins equations R = g . The knowledge that CR structures maybe associated, in a natural manner, to certain classes of Lorentzian met-rics is certainly older and goes back to the work by I. Robinson & A.Trautman (cf. [30]) and A. Trautman (cf. [32]). Cf. also [27] and [10].Fefferman metrics belong to this class, and they may actually be char-acterized (cf. [21]) as the Lorentzian metrics g on the 4-dimensionalmanifold M admitting a null Killing vector field K X(M) such thatK cW = K cC = 0 and Ric(K,K) > 0, where W , C and Ric are theWeyl, Cotton and Ricci tensors of g. Given such a Lorentzian metricg, the leaf space M/K may be organized, at least locally, as a C

manifold M carrying a strictly pseudoconvex CR structure and a pos-itively oriented contact form such that (M, g) be locally isometric to(C(M), F). To (locally) embed a 3-dimensional CR manifold M oneneeds (cf. e.g. [12]) to determine two functionally independent (local)CR functions fa, a {1, 2}, on M , so that (f1, f2) : M C2 is (lo-cally) a CR immersion. The subtle approach to the problem by C.D.Hill et al. (cf. op. cit.) is to write the Cartan structure equations (forthe Einstein metric g at hand)

d + =

1

2R

with respect to a special (local) frame { : 0 3} [with a, a {1, 2}, complex 1-forms and b, b {3, 4} real forms such that g admitsa particularly simple local representation i.e. g12 = g21 = 1, g03 = g30 =1, and g = 0 otherwise] and find indices (0 , 0) such that 00 =0 is (as a consequence of Einsteins equations) an involutive complex

E. Barletta, S. Dragomir and H. Jacobowitz 17

Pfaffian system on (an open subset of) M. Then, by a combination ofthe classical Frobenius theorem (for real Pfaffian systems) and existenceof isothermal coordinates (on a Riemann surface) one may represent00 as 00 = h d for some complex function with d d 6= 0,whose projection on M gives the desired (local) CR function. Theexistence of a solution g to the field equations (3) [rather than R =g ] written on an open set of M R is expected [in view of Hill-Lewandowski-Nurowskis scheme, producing CR functions] to requirepeculiar properties of the base CR structure. These properties are so farunknown and will be addressed in further work. It should be mentionedthat we neither continue nor explain the results in [24] but merelyconduct our calculations with respect to a special (globally defined)coframe chosen as in [24] i.e.

1 = 1 , 2 = 1 , 3 = 2F0(S , ), 0 = ,

so that 3 = 0. Then the Fefferman metric F0 admits the simplerepresentation

F0 = 2{

1 2 + 3 0}.

Let {X : 0 3} be the dual frame i.e. (X) = . Then

(52) X1 = L , X2 = L

, X3 =

0 , X0 = 2S,

where horizontal lifting is meant with respect to Grahams connection1-form 0 = (1/3) d. If F = F0(X , X) and [F

] = [F ]1 then

F12 = F21 = 1, F03 = F30 = 1,

F 12 = F 21 = 1, F 03 = F 30 = 1,

() 6 {(12), (21), (03), (30)} = F = 0, F = 0.Let D be the Levi-Civita connection of (M, g) and RD its curvaturetensor field. We adopt the conventions

DXX = X ,

RX = RD(X , X)X , R = R

.

As D is torsion-free [X , X ] =(

)X. Hence

R = X()X

()

+ +

and the Ricci curvature is

(53) R = X()X

()

+ .

18 Gravitational field equations on Fefferman space-times

From now on we represent the perturbation matrix as

[h ] =

a v v uv b v b u

, h = h (X , X) ,with a, b, u, C(H1 , R) and v, , C(H1 , C). Indeed ifb = h12 = h21 then b = h(X1, X2) = h(X2, X1) = b so b is real valued.

Lemma 1. The Ricci tensor of (M, g) is given by

(54) R00 = 2 +

2ba+ 4(u b) 2i

[L(v) L(v)

],

(55) R10 = R01 =

2

[L2(v) LL(v) + 0L(a)

]+

+i [L(2u b) 0(v)] + 2 ,

(56) R20 = R02 =

2

{L2(v) LL(v) + 0L(a)

}

i [L(2u b) 0(v)

]+ 2 ,

(57) R30 = R03 =

2

{bu+ L0(v) + L0(v) +

20(a)

}+

i [L() L()

]+ 2 ,

(58) R11 = [0L(v) L2(u)

]+ i 0(),

(59) R22 = [0L(v) L

2(u)] i 0(),

(60) R33 =

2b +

[20(b) + L0() + L0()

],

(61) R21 =

2

[2LL (u+ b) + 0L(v) + 0L(v) + L2() + L

2()]

+

+2 i [L() L()

] i 0(u+ b) 2,

(62) R12 =

2

[2LL(u+ b) + 0L(v) + 0L(v) + L2() + L

2()]

+

+2 i [L() L()

]+ i 0(u+ b) 2 ,

(63) R31 = R13 = i [L() 0()

]+

+

2

[L0 (u+ b) + 20(v) + L2() LL() + L0()

],

(64) R32 = R23 = i [0() L()

]+

E. Barletta, S. Dragomir and H. Jacobowitz 19

+

2

[L0(u+ b) + 20(v) + L

2() LL() + 0L()

],

to order O(). One has

R20 = R10 , R22 = R11 , R21 = R12 , R32 = R31 .

In particular the scalar curvature of (M, g) is given by

(65) R = [b (2u+ b) +

20(a)

]+ 2

[L0(v) + L0(v)

]+

+[L2() + L

2()]

+ 2 i [L() L()

] 2

to order O().

The computational details leading to Lemma 1 are relegated to Ap-pendix A. Let

T /x /x = 2 /x3 /x3 = T X Xbe the energy-momentum tensor as considered in 2. Its componentswith respect to the frame (52) are T = 2 3

3 hence T = ggT

is given by

[T

]=

2(1 + 2u) 2 2 2

2 0 0 02 0 0 02 0 0 0

+O(2).Also the trace-less Ricci tensor of (M, g) is[

R 1

2Rg

]=

=

R00 R01 R02 R03 12 RR10 R11 R12 12 R R13R20 R21 12 R R22 R23

R30 12 R R31 R32 R33

+O(2)hence (by Lemma 1 and

[L,L

]= 2i0) Einsteins equations R

12Rg = T are [to order O()]

(66) ba 4{

2b+ i[L(v) L(v)

]}= 0,

(67) 0(La) + 2i L(2u b) + L2(v) LL(v) = 0,

(68) b (u+ b) + L0(v) + L0(v) + L2() + L

2()+

+4i[L() L()

] 2 = 0

(69) L2(u) 0L(v) i 0() = 0,

20 Gravitational field equations on Fefferman space-times

(70) b(2u+ b) + 20(a) + 2LL(u+ b) + L0(v) + L0(v)

2i[L() L()

]+ 2i 0(u+ b) + 2 = 0

(71) L0 (u+ b) + 20(v) + L2() LL() + L0()++2i

[L() 0()

]= 0,

(72) b 2 20(b) + 2[L0() + L0()

]= 0,

(merely conjugate equations were omitted). Equations (66)-(72) areEinsteins equations on M linearized about F0 . To provide a peudoher-mitian analog to Einsteins results (producing a spherically symmetricsolution to the gravitational field equations, cf. [17]) one should look forsolutions with Heisenberg spherical symmetry i.e. h(x) = f(r) withr = |x| for some functions f to be determined from (66)-(72). Here|x| = (|z|4 + t2)1/4 is the Heisenberg norm of x = (z, t) H1. Whilethis is left as an open problem, a nontrivial solution to (66)-(72) fur-nishing a diagonal perturbation matrix [h ] = diag (a, , , ) maybe determined as follows. If b = u = 0 and = v = 0 then the system(66)-(72) becomes

(73) ba = 0,

(74) 0(La) = 0,

(75) L2() + L2() 2 = 0,

(76) 0() = 0,

(77) 20(a) + 2 = 0,

(78) L0() 2i L() = 0,

(79) b = 0.

We need the following

Lemma 2. Any real valued CR function on H1 is a constant.

Proof. Let f be a R-valued CR function i.e. f is a C1 solutionto the tangential Cauchy-Riemann equations Lf = 0. By complexconjugation Lf = 0 hence

0 =[L , L

]f = 2i 0(f)

implying that f is a constant. Q.e.d.

Let (a, , ) be a solution to (73)-(79). As [L , 0] = 0 equation (74)shows that 0(a) is a real valued CR function on H1 hence (by Lemma

E. Barletta, S. Dragomir and H. Jacobowitz 21

2) 0(a) = C for some C R. Then [by (77)] = 0 and the system(73)-(79) reduces to

(80) ba = 0,

(81) L2() + L2() = 0,

(82) 0() = 0.

Equation (82) shows that (z, t) = f(z) + i g(z) for some real valuedfunctions f , g of one complex variable. Then (80)-(82) reads [cf. also(11)]

(83) axx + ayy = 0,

(84) fxx fyy + 2 gxy = 0,with z = x + iy. Moreover 0(a) = C yields a(z, t) = Ct + F (z) with0F = 0 [by (83)]. A solution with spherical symmetry F (z) = (|z|)is () = 1

2log (the fundamental solution to Laplace equation in the

plane) hence

(85) a(z, t) = C t+1

2log |z|.

As to equation (84) one looks for solutions of the form f(z) = A()and g(z) = B() with = x/|y| so that(

1 2)A() 2 A() = [ B() +B()]

according to whether y > 0. In particular if(1 2

)A() = k, B() = m,

for some constants k,m R then

A() = k log

1 1 + + `, B() = m log ||+ p,

with k, `, m, p R. Finally

(86) = k log

x |y|x+ |y|+ `+ i [m log xy

+ p] .Summing up [by (85)-(86) and = 0]

(87) g =

[t+

1

2log |z|

] (0)2

+

+

[log

x |y|x+ |y| i log xy

] (1)2 +

22 Gravitational field equations on Fefferman space-times

+

[log

x |y|x+ |y|+ i log xy

] (2)2 ++2(0 3 + 1 2

)is a solution to Einsteins equations (3) to order O().

5. Conclusions and final comments

The total space M of the principal circle bundle over the 3-dimen-sional Heisenberg group H1, endowed with the Fefferman metric F0and the time orientation 0 S, is a space-time. The metric F0 isnot flat and one may identify, as foundational for general relativitytheory, geometry to matter content of space described by the energy-momentum tensor T = 0 u

u where 0 = 2 and the four-velocityflow is u = 3 .

The formal similarity between T (the trace-less Ricci tensor of F0)

and (9.7) in [1], p. 263, allows for the physical interpretation of T asa field of non-interacting incoherent matter described by a four-vectorfield of flow u and a scalar proper density field 0, moving at low [by

also comparing T with (9.84) in [1], p. 277] velocity.

Linearization of Einsteins equations R 12 Rg = T about F0is expected to produce linear PDEs whose principal part is the wave op-erator (the Laplace-Beltrami operator of the Lorentzian metric F0).Indeed the linearized field equations (73)-(78) involve the sublaplacianb of (H1 , 0), which is the same as on functions not depending onthe time coordinate [for, by a result of J.M. Lee, [28], = b].

Geodesic motion on (M, F0 + h) in the classical limit of veloc-ities v/c

E. Barletta, S. Dragomir and H. Jacobowitz 23

subelliptic theory and its applications to CR geometry (cf. [16] for thestate-of-the-art).

Appendix A. Linearized Ricci curvature

Our purpose in Appendix A is to give a proof of Lemma 1. Themethods consist of a mix of tensor calculus (within pseudohermitiangeometry on H1) and principal bundle techniques (cf. [23]).

Let D be the Levi-Civita connection of (M, F0) and let us set

DXX = X. Using

[X , X ] =(

)X ,

2 g(DXY , Z) = X(g(Y, Z)) + Y (g(X,Z)) Z(g(X, Y ))++g([X, Y ], Z) + g([Z,X], Y ) g([Y, Z], X),

for any X, Y, Z X(M), one derives(88) = g

+B

where

B =1

2

{ +

+(

)gg

+(

)gg

},

=1

2{X(g) +X(g)X(g)} .

We shall use (88) to compute to order O(). To this end we deter-

mine from (cf. [2] for n = 1 and = 0)

(89) DXY = (XY ) (d0)(X, Y ) + 0

([X , Y

])S,

(90) DX = 0,

(91) DX = (X) ,

(92) DXS = DSX =

1

2(JX) ,

(93) DSS = DS = DS = D

= 0,

for any X, Y H(H1). Here is the Tanaka-Webster connectionof (H1 , 0). Also is short for 0. Using (89)-(93) one profits fromthe progress in [2], relating the Lorentzian geometry on (M, F0) topseudohermitian geometry on (H1 , 0), and bearing a strong analogyto the techniques of B. ONeill, [29]. However the results in [29] arederived for Riemannian submersions while : (M, F0) (H1 , g0)

24 Gravitational field equations on Fefferman space-times

isnt even semi-Riemannian (the fibres of are degenerate). Formula(89) is useful together with

(94)[X , Y

]= [X, Y ]

(cf. (35) in [2] for n = 1) showing that 0([X, Y ]) = 0. Formulas

(89)-(93) yield

(95) 01 = 10 = i

1 ,

02 =

20 = i 2 , 12 = 21 = i 3 ,

and the remaining connection coefficients are zero. Similar to (38)

g = F h +O(2)

with the new meaning of components of tensors involved, as related tothe nonholonomic frame {X : 0 3} rather than {/x : 0 3}. Hence

(96) g =

2F {X(h) +X(h)X(h)}

with the corresponding modification of (88). Next (95) implies

(97)[B

]0,3 =

=

0 i g03g

2 i g03g1 0i g03g

2 2i g13g2 B12 i g33g

2

i g03g1 B21 2i g23g1 i g33g10 i g33g

2 i g33g1 0

where

B12 = i(3 + g23g2 g13g1

), B21 = i

(3 + g23g

2 g13g1).

Note that h = F F h are

[h ] =

u b v b vu v v a

hence B simplifies to

(98)[B0

]0,3 =

0 i i 0i 0 0 0i 0 0 0

0 0 0 0

+O(2),(99)

[B1

]0,3 =

E. Barletta, S. Dragomir and H. Jacobowitz 25

=

0 i [1 + (u b)] i 0

i [1 + (u b)] 2i i i i i 0 0

0 i 0 0

+O(2),(100)

[B2

]0,3 =

=

0 i i [1 + (u b)] 0i 0 i 0

i [1 + (u b)] i 2i i 0 0 i 0

+O(2),(101)

[B3

]0,3 =

=

0 i v i v 0i v 0 i 0i v i 0 00 0 0 0

+O(2).Then (88) yields

(102) 000 =

2(a), 100 =

2L(a), 200 =

2L(a), 300 = 0,

(103) 010 = 001 = i +

2[L(u) (v)],

110 = 101 = i [1 + (u b)] +

2

[L(v) L(v)

],

210 = 201 = i , 310 = 301 = i v +

2L(a),

(104) 020 = 002 = i +

2

[L(u) (v)

], 120 =

102 = i ,

220 = 202 = i[1 + (u b)] +

2

[L(v) L(v)

],

320 = 302 = i v +

2L(a),

(105) 030 = 003 = 0,

130 =

103 =

2

[(v) L(u)

],

230 = 203 =

2[(v) L(u)], 330 = 303 =

2(a),

and in particular

(106) 0 = 0.

Moreover

(107) 011 =

2

[2L() ()

], 111 = 2i +

2

[2L(b) L()

],

26 Gravitational field equations on Fefferman space-times

211 =

2L(), 311 = L(v),

(108) 021 =

2

[L() + L() (b)

],

121 = i +

2L(), 221 = i +

2L(),

321 = i+

2

[L(v) + L(v)

],

(109) 012 =

2

[L() + L() (b)

],

112 = i +

2L(), 212 = i +

2L(),

312 = i+

2

[L(v) + L(v)

],

(110) 031 = 013 =

2L(),

131 = 113 = i +

2

[(b) + L() L()

],

231 = 213 =

2(), 331 =

313 =

2[(v) + L(u)],

and in particular

(111) 1 = L(u+ b).

Moreover

(112) 022 =

2

[2L() ()

], 122 =

2L(),

222 = 2i +

2

[2L(b) L()

], 322 = L(v),

(113) 032 = 023 =

2L(), 132 =

123 =

2(),

232 = 223 = i +

2

[(b) + L() L()

],

332 = 323 =

2

[(v) + L(u)

],

and then

(114) 2 = L(u+ b).

Similarly

(115) 033 =

2(),

133 =

2

[2 () L()

], 233 =

2

[2 () L()

],

333 = (u),

E. Barletta, S. Dragomir and H. Jacobowitz 27

and then

(116) 3 = (u+ b).

Next (by (53) and X0(0) = 0)

(117) R00 = X (00) +

00

00

where (by (102) and (106), (111), (114), respectively by (102)-(103))

00 =

j00

j =

= L(u+ b) 100 + L(u+ b) 200 + (u+ b)

300 = O(

2),

00 = 2[1 + 2(u b)] + 2i

[L(v) L(v)

]+O(2),

X(00)

=

2ba.

Consequently (117) becomes

R00 =

2ba+ 2[1 + 2(u b)] 2i

[L(v) L(v)

]+O(2)

and (54) is proved. Next (by X0(1) = 0)

R10 = X (01) +

01

01 ,

X (01) =

2

[L2(v) LL(v) + L(a)

]+

+i L(u b) i L() i (v),01

= i L(u+ b) +O(

2),

01 = 2 + i

[L(b) L()

]+O(2),

hence (55) is proved. Collecting the information in (106), (111), (114)and (116) the contracted Christoffel symbols are given by

(118) =

0, = 0,

L(u+ b), = 1,

L(u+ b), = 2,

(u+ b), = 3,

henceR20 = X (

02) +

02

02 ,

X (02) =

2

[L2(v) LL(v) + L(a)

]+ i

[L() L(u b) + (v)

],

02 = i L(u+ b) +O(2),

02 = i

[L() L(b)

] 2 +O(2),

28 Gravitational field equations on Fefferman space-times

and (56) is shown to hold as well. Similarly

R30 = X(03)

+ 03 03 ,

X(03)

=

2

{bu+ L(v) + L(v) +

2(a)},

03 = O(

2),

03 = i

[L() L()

] 2 +O(2),

R11 = X(11)X1

(1)

+ 11 11 ,

X(11)

= [L2(b) + L(v)

]+ i

[2L() + ()

],

X1(1)

= L2(u+ b),

11 = O(

2), 11 = 2 i L() +O(

2),

thus leading to (57)-(58). The calculation of R21 is a bit trickier and afew computational details are provided below. By (109) and (118)

(119) 12 = i (u+ b) +O(2).

Moreover

12 =

01

02 +

11

12 +

21

22 +

31

32 =

[by (102), (107), (109) and (110)]

={i [1 + (u b)] +

2

[L(v) L(v)

]}012+

+

2

[L() + L() (b)

]202 +

{i+

2

[L(v) + L(v)

]}232+

+{i +

2

[(b) + L() L()

]}312 +O(

2)

or

(120) 12 = i

[L() L()

]+ 2 +O(2).

Next [by (109) and (118)]

(121) X(12)

=

2

[L(v) + L(v) + L2() + L

2()]

+

+i [L() L()

],

(122) X1(2)

= LL(u+ b).

Finally, substitution from (119)-(122) into

R21 = X(12)X1

(2)

+ 12 12

leads to

R21 =

2

[2LL (u+ b) + L(v) + L(v) + L2() + L2()

]+

E. Barletta, S. Dragomir and H. Jacobowitz 29

+2 i [L() L()

] i (u+ b) 2

which is (61) in Lemma 1. Similar calculations

R12 = X(21)X2

(1)

+ 21 21 ,

X(21)

= X(12), X2

(1)

= LL(u+ b),

21 = i

3 = i (u+ b) +O(

2),

21 =

2

1 = i

[L() L()

]+ 2 +O(2),

R31 = X(13)X1

(3)

+ 13 13 ,

X(13)

=

2

[L(u+ b) + 2(v) + L2() LL() + L()

]+ i L(),

X1(3)

= L(u+ b),

13 = O(

2), 13 = i () +O(

2),

lead to (62)-(63). Indeed

31 = 13 , X3

(1)

= L(u+ b),

hence R13 = R31. Finally

R22 = X(22)X2

(2)

+ 22 22 ,

X(22)

= [L2(b) + L(v)

] i

[2 + ()

],

X2(2)

= L2

(u+ b) ,

22 = O(

2), 22 = 2 i L() +O(2),

R32 = X(23)X2

(3)

+ 23 23 ,

X(23)

=

2

[L (u+ b) + 2(v) +

+ L2() LL() + L()

] i L(),

X2(3)

= L(u+ b),

23 = O(

2), 23 = i () +O(2),

R33 = X(33)X3

(3)

+ 33 33 ,

X(33)

=

2b +

[2(u) + L() + L()

],

X3(3)

= 2(u+ b),

33 =

3

3 = O(

2),

lead to (59)-(60) and (64) in Lemma 1. Q.e.d.

30 Gravitational field equations on Fefferman space-times

References

[1] R. Adler & M. Bazin & M. Schiffer, Introduction to general relativity,McGraw-Hill Book Co., New York, 1965.

[2] A. Aribi & S. Dragomir & A. El Soufi, A lower bound on the spectrum ofthe sublaplacian, Journal of Geometric Analysis, (3)25(2015), 1492-1519.

[3] A. Aribi & S. Dragomir, Dirichlet and Neumann eigenvalue problems onCR manifolds, to appear in Ricerche di Matematica, 2016.

[4] E. Barletta, Hormander systems and harmonic morphisms, Ann. Sc. Norm.Sup. Pisa, 2(2003), 379-394.

[5] E. Barletta & S. Dragomir & K.L. Duggal, Foliations in Cauchy-Riemanngeometry, Mathematical Surveys and Monographs, Volume 140, AmericanMathematical Society, 2007; 256 pp. ISBN: 978-0-8218-4304-8

[6] E. Barletta & S. Dragomir & H. Jacobowitz & M. Soret, b-Completionof pseudo-Hermitian manifolds, Class. Quantum Grav., 29(2012), 095007(27pp) doi:10.1088/0264-9381/29/9/095007

[7] E. Barletta & S. Dragomir & H. Jacobowitz, Linearized pseudo-Einsteinequations on the Heisenberg group, submitted to Journal of Geometry andPhysics, 2016.

[8] E. Barletta & S. Dragomir & H. Urakawa, Pseudoharmonic maps from non-degenerate CR manifolds to Riemannian manifolds, Indiana Univ. Math.J., 50(2001), 719-746.

[9] E. Barletta & S. Dragomir, Jacobi fields of the Tanaka-Webster connectionon Sasakian manifolds, Kodai Math. J., 29(2006), 406-454.

[10] E. Barletta & S. Dragomir & M. Magliaro, Wave maps fromGodels universe, Class. Quantum Grav., (19)31(2014), 195001doi:10.1088/0264-9381/31/19/195001

[11] J.K. Beem & P. Ehrlich & K. Easley, Global Lorentzian Geometry, MarcelDekker, Inc., New York-Basel-Hong Kong, 1996.

[12] A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex,CRC Press, Boca Raton-Ann Arbor-Boston-London, 1991.

[13] J.M. Bony, Principe du maximum, inegalite de Harnack, et unicite duprobleme de Cauchy pour les operateurs elliptiques degenere, Ann. Inst.Fourier, Grenoble, (1)19(1969), 277-304.

[14] S. Dragomir & G. Tomassini, Differential Geometry and Analysis on CRManifolds, Progress in Mathematics, Vo. 246, Birkhauser, Boston-Basel-Berlin, 2006.

[15] S. Dragomir & A. Minor, CR immersions and Lorentzian geometry. PartI: Pseudohermitian rigidity of CR immersions, Ricerche di Matematica,(2)62(2013), 229-263; CR immersions and Lorentzian geometry. Part II: ATakahashi type theorem, ibidem, (1)63(2014), 15-39.

[16] S. Dragomir, Cauchy-Riemann geometry and subelliptic theory, LectureNotes of Seminario Interdisciplinare di Matematica, 7(2008), 121-162.

[17] A. Einstein, The foundation of the general theory of relativity, The collectedpapers of Albert Einstein, Vol. 6, pp. 146-200, The Berlin years: writings1914-1917, Ed. by A.J. Kox & M.J. Klein & R. Schulman, Princeton Uni-versity Press, 1997.

[18] C. Fefferman, The Bergman kernel and biholomorphic equivalence of pseu-doconvex domains, Invent. Math., 26(1974), 1-65.

E. Barletta, S. Dragomir and H. Jacobowitz 31

[19] C. Fefferman, Monge-Ampere equations, the Bergman kernel, and geometryof pseudoconvex domains, Annals of Mathematics, (3)103(1976), 395-416.

[20] G.B. Folland, A fundamental solution for a subelliptic operator, Bull.A.M.S., (2)79(1973), 373-376.

[21] R.C. Graham, On Sparlings characterization of Fefferman metrics, Amer.J. Math., (5)109(1987), 853-874.

[22] D. Jerison & J.M. Lee,[23] S. Kobayashi & K. Nomizu, Foundations of Differential Geometry, Vol. I,

Interscience Publishers, New-York, 1963.[24] C.D. Hill & J. Lewandowski & P. Nurowski, Einsteins equations

and the embedding of 3-dimensional CR manifolds, arXiv:0709.3660v2[math.DG] 22 May 2008

[25] L. Hormander, Hypoelliptic second order differential equations, Acta Math.,119(1967), 147-171.

[26] J. Jost & C-J. Xu, Subelliptic harmonic maps, Trans. A.M.S.,(11)350(1998), 4633-4649.

[27] L. Koch, Chains on CR manifolds and Lorentz geometry, Trans. Amer.Math. Soc., 307(1988), 827-841.

[28] J.M. Lee, The Fefferman metric and pseudohermitian invariants, Trans.Amer. Math. Soc., (1)296(1986), 411-429.

[29] B. ONeill, The fundamental equations of a submersion, Michigan Math.J., (4)3(1968), 459-469.

[30] I. Robinson & A. Trautman, Cauchy-Riemann structures in optical geome-try, Proc. of the Fourth Marcel Grossmann Meeting on General Relativity,Ed. by R. Ruffini, Elsevier Science Publ., 1986, pp. 317-324.

[31] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geometry,24(1986), 221-263.

[32] A. Trautman, Deformations of the Hodge map and optical geometry, Jour-nal of Geometry and Physics, 1(1984), 85-95.