On 2 + 2 dim Spacetimes and Stringy Black Holes

8/14/2019 On 2 + 2 dim Spacetimes and Stringy Black Holes

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1st ReadingMarch 9, 2007 10:44 WSPC/139-IJMPA 03619

International Journal of Modern Physics A1Vol. 22, No. 00 (2007) 124c World Scientific Publishing Company3

ON (2 + 2)-DIMENSIONAL SPACE TIMES,

STRINGS AND BLACK HOLES5

C. CASTRO

Center for Theoretical Studies of Physical Systems,7

Clark Atlanta University, Atlanta, GA 30314, USA

[email protected]

J. A. NIETO

Facultad de Ciencias Fsico-Matematicas de la Universidad Autonoma de Sinaloa,11

80010, Culiacan Sinaloa, Mexico

[email protected]

Received 9 November 2006Revised 5 January 200715

We study black hole-like solutions (spacetimes with singularities) of Einstein field equa-tions in 3 + 1 and 2 + 2 dimensions. We find three different cases associated with17hyperbolic homogeneous spaces. In particular, the hyperbolic version of Schwarzschilds

solution contains a conical singularity at r = 0 resulting from pinching to zero size r = 019the throat of the hyperboloid H2 and which is quite different from the static spherically

symmetric (3 + 1)-dimensional solution. Static circular symmetric solutions for metrics21in 2+2 are found that are singular at = 0 and whose asymptotic limit leads to

a flat (1 + 2)-dimensional boundary of topology S1R2. Finally we discuss the (1 + 1)-23dimensional BarsWitten stringy black hole solution and show how it can be embeddedinto our (3 +1)-dimensional solutions. Black holes in a (2 + 2)-dimensional spacetime25from the perspective of complex gravity in 1 + 1 complex dimensions and their quater-nionic and octonionic gravity extensions deserve furher investigation. An appendix is27included with the most general Schwarzschild-like solutions in D 4.

Keywords: Strings; black holes; 2 + 2 dimensions; general relativity.29

PACS numbers: 04.60.-m, 04.65.+e, 11.15.-q, 11.30.Ly

1. Introduction31

Through the years it has become evident that the 2 + 2 signature is not only

mathematically interesting1,2 (see also Refs. 35) but also physically. In fact, the33

2 + 2-signature emerges in several physical context, including self-dual gravity a la

Plebanski (see Ref. 6 and references therein), consistent N = 2 superstring theory35

as discussed by Ooguri and Vafa,7,8 N = (2, 1) heterotic string.912 Moreover, it has

been emphasized13,14 that MajoranaWeyl spinor exists in spacetime of 2+2 signa-37

ture. Even cosmologically there is a wisdom15 that the 2 + 2 signature is interesting.

1


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2 C. Castro & J. A. Nieto

In Refs. 2326 it was shown how a N = 2 supersymmetric WessZumino1NovikovWitten model valued in the area-preserving (super)diffeomorphisms group

is self-dual supergravity in 2 + 2 and 3 + 1 dimensions depending on the signatures3of the base manifold and target space. The interplay among W gravity, N = 2strings, self-dual membranes, SU() Toda lattices and SU() YangMills instan-5tons in 2 + 2 dimensions can be found also in Refs. 2326.

More recently, using the requirement of the SL(2, R) and Lorentz symmetries it7

has been proved16 that 2+2 target spacetime of a 0-brane is an exceptional signa-

ture. Moreover, following an alternative idea to the notion of worldsheets for world-9

sheets proposed by Green17 or the 0-branes condensation suggested by Townsend18

it was also proved in Ref. 16 that special kind of 0-brane called quatl 19,20 leads11

to the result that the 2 + 2-target spacetime can be understood either as 2 + 2-

worldvolume spacetime or as 1 + 1 matrix-brane.13

Another recent motivation for a physical interest in the 2 + 2 signature has

emerged via Duffs21 discovery of hidden symmetries of the NambuGoto action. In15fact, this author was able to rewrite the NambuGoto action in a 2+2 target space

time in terms of a hyperdeterminant, reveling apparently new hidden symmetries17

of such an action. More recently the Duffs observation has been linked with the

matrix-brane idea.2219

Considering seriously the possibility that the (2 + 2)-dimensional spacetime

is an exceptional signature one may wonder what is the connection between21

(2 + 2)-dimensional spacetime and other exceptional structures in physics such

as black-holes. In this respect it becomes convenient to discuss black-holes physics23

from modern perspective. In particular, it become convenient to clarify the many

subtleties behind the introduction of a true point-mass source at r = 039 and the25

admissible family of radial functions R(r) in the static spherically symmetric solu-

tions of Einstein field equations2938 (see also Refs. 4245).27

We begin by writing down the class of static spherically symmetric (SSS) vacuum

solutions of Einsteins equations46 studied by Refs. 2932, 5258, 4245 and 72,

among many others, given by a infinite family of solutions parametrized by a family

of admissible radial functions R(r) (in c = 1 units)

(ds)2 =

1 2GNMo

R

(dt)2

1 2GNMo

R

1

(dR)2 R2(r)(d)2 , (1.1)

where the solid angle infinitesimal element is

(d)2 = (d)2 + sin2()(d)2 . (1.2)

This expression of the metric in terms of the radial function R(r) (a radial gauge)

does not violate Birkoffs theorem since the metric (1.1), (1.2) expressed in terms29

of the radial function R(r) has exactly the same functional form as that required

by Birkoffs theorem and 0 r . Metrics of the form (1.1) were employed31by Ref. 94 based on the nonperturbative renormalization group flow and running

Newtonian coupling G = G(r) in quantum Einstein gravity.909333


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On (2 + 2)-Dimensional SpaceTimes, Strings and Black Holes 3

There are two interesting cases to study based on the boundary conditions1

obeyed by R(r): (i) the Hilbert textbook (black hole) solution4951 based on the

choice R(r) = r obeying R(r = 0) = 0, R(r ) r. And (ii) the controversial3(erroneous) AbramsSchwarzschild radial gauge based on choosing the cutoffR(r =

0) = 2GNM such that gtt(r = 0) = 0 which apparently seems to eliminate the5

horizon and R(r ) r. This was the original solution of 1916 found bySchwarzschild. However, the choice R(r = 0) = 2GM has a serious flawand is: how7

is it possible for a point-mass at r = 0 to have a nonzero area 4(2GNM)2 and a

zero volume simultaneously? so it seems that one is forced to choose the Hilbert9

gauge R(r = 0) = 0 and retain only those metrics that are diffeomorphic to the

Hilbert textbook black hole solution only.11

Nevertheless there is a very specific radial function (never studied before to our

knowledge) R(r) = r+2GNM(r)36 that yields a metric which is not diffeomorphic13

to the Hilbert textbook solution based on the Heaviside step functiona which is

defined (r) = 1 when r > 0, (r) = 1 when r < 0 and (r = 0) = 0 (the15arithmetic mean of the values at r > 0 and r < 0). The Heaviside step function

behavior at r = 0 given by (r = 0) = 0 will ensure us that now we can satisfy17

the required condition R(r = 0) = r = 0, consistent with our intuitive notion that

the spatial area and spatial volume of a point r = 0 has to be zero. Since r =19

x2 + y2 + z2, a negative r branch is mathematically possible and is compatible

with the double covering inherent in the FronsdalKruskalSzekeres6062 analytical21

continuation in terms of the u, v coordinates. Each point of spacetime inside

r < 2GNM is represented twice (black hole and white hole picture). However there23

is a fundamental difference (besides others) with the FronsdalKruskalSzekeres

extension into the interior of r = 2GM, their metric description is no longer static25

in r < 2GM, whereas in our case the metric is static for all values of r.

Thus the scalar curvature associated to the point mass delta function source27

2GNM(r)/R2(dR/dr) 39 does not always remain invariant of the radial gaugechosen. In the very special case chosen by Schwarzschild in 1916 given by R3 =29

r3 + (2GNM)3 the scalar curvature and measure remains the same as in the Hilbert

textbook choice R(r) = r due to the relation R2 dR = r2 dr. But this was a his-31

torical fluke. An unfortunate accident which has impeded the progress for 90 years

because many were misled into thinking that any radial gauge choice was always33

equivalent to a naive radial reparametrization r r of the Hilbert metric. It is notbecause having a family of nondiffeormorphic metrics, parametrized by a family of35

inequivalent radial gauges belonging to different gauge orbits, is not the same thing

as having a family of naive radial changes of coodinates r r associated to a fixed37and given fiduciary metric.

The reason why there are metrics which are not diffeomorphic to the Hilbert39

textbook solution is due to the fact that there are orbits obtained by exponentiation

aWe thank Michael Ibison for pointing out the importance of the Heaviside step function and theuse of the modulus |r| to account for point mass sources at r = 0.


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of generators of diffeomorphisms that yield diffeomorphisms which are not connected1

to the identity and which still may act trivially at infinity (Marsden theorem). The

identity element of the diffs group is in our case related to the Hilbert textbook3trivial radial gauge-function R(r) = r. Consequently, there are radial gauges which

are not obtained by a naive radial reparametrization r r of the Hilbert textbook5metric and correspond to metrics which are not physically equivalent to it. More-

over, Donaldson showed that in D = 4 one has an infinite number of inequivalent7

differential structures, i.e. manifolds that are homeomorphic (topologically equiva-

lent) but are not diffeomorphic. The presence of matter (singularity) at r = 0 and9

the different choices of inequivalent radial gauges should single out the particular

differential structure in D = 4.11

There is an essential technical subtlety required in order to generate (r) terms

in the right-hand side of Einsteins equations. One must replace everywhere r |r|13as required when point-mass sources are inserted. A rigorous mathematical treat-

ment of Colombeaus theory of nonlinear distributions can be found in Refs. 6366.15The Newtonian gravitational potential due to a point-mass source at r = 0 is given

by GNM/|r| and is consistent with Poissons law which states that the Laplacian17of the Newtonian potential GM/|r| is 4G where = (M/4r2)(r) in Newto-nian gravity. However, the Laplacian in spherical coordinates of (1/r) is zero. For19

this reason, there is a fundamental difference in dealing with expressions involving

absolute values |r| like 1/|r| from those which depend on r like 1/r.59 Therefore21the radial gauge must be chosen by R(|r|) = |r| + 2GNM(|r|). Had one not use|r| in the expression for the metric, one will not generate the desired (r) terms in23the right-hand side of Einsteins equations R 12gR = 8GNT = 0, andone would get an expression identically equal to zero (consistent with the vacuum25

solutions in the absence of matter) instead of the (r) terms.39

To sum up, by using R(|r|) = |r| + 2GNM(|r|), we safely have that R(|r|) =27|r| + 2GNM, when r > 0 and the horizon can the be displaced from r = 2GNMto a location as arbitrarily close to r = 0 as desired rHorizon 0. To be more29precise, the horizon actually never forms since at r = 0 one hits the singularity.

Also, R r for r 2GNM and one recovers the correct Newtonian limit in the31asymptotic regime. It is now, via the Heaviside step function, that we may maintain

the correct behavior R(|r|) = |r|, when r = 0, and such that we can satisfy the33required condition R(r = 0) = r = 0, consistent with our intuitive notion that

the spatial area and spatial volume of a point r = 0 has to be zero. The metric35

is smooth and differentiable for all r > 0 and one will have R = R = 0 (in theregion r > 0 empty of matter and radiation). The metric is discontinuous only at37

the location of the point mass singularity r = 0 whose worldline which may be

thought of as the boundary of spacetime. The scalar curvature is infinite at r = 039

due to the delta function point mass source at r = 0, it jumps from zero to infinity

at r = 0.41

And most importantly, a radial reparametrization r r(r) leaves invariantthe scalar curvature and the measures associated with a given choice of the radial


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function R1(r):

4R21(r)dR1(r)dt = 4R21 (r

)dR1(r)dt , (1.3a)

R1(r) = 2GNMR21(r)(dR1/dr)

(r) = R1(r)

= 2GNMR21 (r

)(dR1(r)/dr)

(r) . (1.4a)

Choosing a different radial function R2(r) gives under a radial reparametrization

r r(r):4R22(r)dR2(r)dt = 4R

22 (r

)dR2(r)dt , (1.3b)

R2(r) = 2GNMR22(r)(dR2/dr)

(r) = R2(r)

= 2GNM

R22 (r)(dR2(r

)/dr) (r

) . (1.4b)1

In the same manner that one must not confuse active and passive diffeomor-

phisms we have

R(r) = r(r) R(r) = 2GNMR2(dR/dr)

(r) = 2GNMR2(r)

(R(r))

= 2GNMr2(r)(dr/dr)

(r) = 2GNMr2(r)

(r(r)) . (1.5)

Because the scalar curvature is an explicit function of the radial function R(r)

given by this expression: 2GM(r)/R2(r)(dR/dr) = 2GM(R(r))/R2(r) we3can see that the scalar curvature does not remain invariant of the infinite number

of possible choices of the radial functions R(r), except in the anomalous case when5

R3 = r3 + (2GM)3 (the radial gauge chosen by Schwarzschild in 1916) that leads to

2GM(r)/r2, and which accidentally happens to agree with the scalar curvature7in the Hilbert gauge R(r) = r.

What remains invariant of the choices R(r) is the action

S = 116GN

2GNMo

R2(dR/dr)(r)

(4R2 dRdt)

= 116GN

2GNMo

r2(r)

(4r2 drdt) . (1.6)

The Euclideanized EinsteinHilbert action associated with the scalar curvature

delta function is obtained after a compactification of the temporal direction along a

circle S1 giving an Euclidean time coordinate interval of 2tE and which is defined

in terms of the Hawking temperature TH and Boltzman constant kB as 2tE =

(1/kBTH) = 8GNMo.

SE =4(GNMo)

2

L2Planck=

4(2GNMo)2

4L2Planck=

Area

4L2Planck. (1.7)


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It is interesting that the Euclidean action SE (in units) is precisely the same as1

the black hole entropy Sin Planck area units. This result holds in any dimensionsD 3. This is not a numerical coincidence. Furthermore, the action is invariant of3the choices of R(r), whether or not it is the Hilbert textbook choice R(r) = r or

another. The choice of the radial function R(r) amounts to a radial gauge that leaves5

the action invariant but it does not leave the scalar curvature, nor the measure of

integration, invariant. Only the action (integral of the scalar curvature) remains7

invariant.

The actionentropy connection has been obtained from a different argument,9

for example, by Padmanabhan40 by showing how it is the surface term added to

the action which is related to the entropy, interpreting the horizon as a boundary11

of spacetime. The surface term is given in terms of the trace of the extrinsic cur-

vature of the boundary. The surface term in the action is directly related to the13

observer-dependent-horizon entropy, such that its variation, when the horizon is

moved infinitesimally, is equivalent to the change of entropy dS due to the vir-15tual work. The variational principle is equivalent to the thermodynamic identity

T dS= dE+ p dV due to the variation of the matter terms in the right-hand side.17A bulk and boundary stress energy tensors are required to capture the Hawking

thermal radiation flux seen by an asymptotic observer at infinity as the black hole19

evaporates.

With these modern developments at hand one may proceed to find black-hole21

type solutions of the Einstein field equations for a (2 + 2)-dimensional space

time. In Sec. 2 we present static hyperbolic solutions in a (2 + 2)-dimensional23

spacetime and describe its differences with the corresponding solution in 3 + 1

dimensions. In Secs. 3 and 4, we present the straightforward computations of the25

static circular symmetric solutions of Einstein field equations in 2 + 2 dimensions.

Finally, in Sec. 5 we show how the 1 + 1 BarsWitten stringy black-hole solution27

can be embedded into the (3 + 1)-dimensional solution of the appendix and discuss

the stringy nature behind a point-mass. Black holes in a (2 + 2)-dimensional29

spacetime from the perspective of complex gravity in 1 + 1 complex dimensions

and its quaternionic and octonionic gravity extensions deserve furher investigation.31

In the appendix we construct Schwarzschild-like solutions in dimensions D 4.

2. Static Hyperbolic Symmetric Solution in 2 + 2 Dimensions33

Consider the vacuum static spherically symmetric solutions of Einstein field equa-

tions in a spacetime of (3 + 1)-signature

R = 0 (2.1)

of the form

ds2 = e(r)(dt1)2 + e(r) dr2 + R2(r)d2 , (2.2)where

d2 = d2 + sin2 d2 . (2.3)


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The solutions are

ds2 =1

R(dt1)2 + (dR/dr)

2

(1 /R)dr2 + R2(r)d2 , (2.4)

where is a parameter that has mass dimensions. Several remarks are now in order

pertaining whether or not a Wick rotation of the metric (2.4) furnishes solutions

to the vacuum field equations for the signature 2 + 2. A naive Wick rotation of the

angle coordinate i = in the above solutions (2.4) yieldssin2() sin2(i) = sinh2() , d2 d2 , (2.5)

and due to the two sign changes in (2.5) one would have a 1 + 3 signature instead1

of a split 2 + 2 signature.

A Wick rotation of i = , (d)2 (d)2 yields a 2 + 2 signature butsince the range of the only remaining angle is [0, ], instead of [0, 2], and one

will no longer cover the space completely. Furthermore, since there is a signature

change (a sign change in one of the metric components g) the connection andcurvature expressions will be modified accordingly and there is no reason now why

the vacuum field equations should be satisfied. In the next section we will find

explicit solutions in the static circular symmetric case:

ds2 = e(R())(dt1)2 e(R())(dt2)2 + e(R())(dR())2 + (R())2d2 ,where the rho function R() is now a function of, the radius of a circle 2 = x2+y2.3

In order to construct solutions with topology H3 R where H3 is a three-dimensional pseudosphere (a hyperboloid) of radius R parametrized by the coordi-

nates , , as

x = R cosh cos , y = R cosh sin ,

t1 = R sinh cos , t2 = R sinh sin ,

(2.6)

where and 0 2; 0 2 such that the flat spacetimemetric in 2 + 2 dimensions is

ds2 = (dt1)2 (dt2)2 + (dx)2 + (dy)2

= (dR)2 + R2[cosh2 (d)2 sinh2 (d)2 (d)2] . (2.7a)From Eq. (2.6) we infer that the three-dimensional pseudosphere H3 is repre-

sented analytically by

(t1)2 (t2)2 + x2 + y2 = R2 . (2.7b)The curved spacetime metric we are interested involve the two functions =

(R) and f = f((R)) = f(R) such that

ds2 = ef()(d)2 + 2[cosh2 (d)2 sinh2 (d)2 (d)2]

= ef(R)

d

dR

2(dR)2 + 2(R)[cosh2 (d)2 sinh2 (d)2 (d)2]

= e(R)(dR)2 + 2(R)[cosh2 (d)2 sinh2 (d)2 (d)2] , (2.8)


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where we have defined e(R) ef(R)(d/dR)2. The flat spacetime metric (2.7) is1recovered from (2.8) in the limit R

such that (R)

0 and (R)

R.

Another interesting parametrization r 0, and ; 0 2 ist2 = r sinh , x = r cosh cos , y = r cosh sin , (2.9)

where r is the throat size of the two-dimensional hyperboloid H2 defined in termsof t2, x, y as

(t2)2 + x2 + y2 = r2 (2.10)and the flat spacetime metric (dt1)2 (dt2)2 + (dx)2 + (dy)2 can be recast as

ds2 = (dt1)2 + (dr)2 + r2[cosh2 (d)2 (d)2] . (2.11)Notice that we have a 2 + 2 signature in Eq. (2.11), as one should, and that there is3

a difference between the forms of the metric in Eqs. (2.7) and (2.11). The topology

corresponding to Eq. (2.7) is

H3

R where

H3 is a three-dimensional hyperboloid5

(a three-dimensional pseudosphere); whereas, instead, the topology corresponding

to Eq. (2.11) is R R H2.7R is the half-interval [0, ] representing the values of the radial coordinates.

In Eq. (2.7) the three-dimensional hyperboloid (pseudosphere) of fixed radius R9

is spanned by the three coordinates , , as indicated by Eq. (2.6). Whereas

in Eq. (2.11), one temporal variable t1 is characterized by the real line R and11whose values range from , +, and the other temporal variable t2 is one of thethree coordinates (t2, x , y) which parametrized the two-dimensional hyperboloid H213described by Eq. (2.10).

A curved spacetime version of Eq. (2.11) is

ds2 = e(r)(dt1)2 + e(r)(dr)2 + (R(r))2[cosh2 (d)2 (d)2] . (2.12a)The metric in Eq. (2.12a) whose signature is 2 + 2 is the hyperbolic version of15

the Schwarzschild metric. One can replace r R(r) since Einsteins equations donot determine the form of the radial function R(r) as explained in the appendix.17

The global topology of the solutions depends on the choices of R(r). We still must

determine what are the functional forms of (r) and (r). In order to go from19

the solid angle (d)2 = sin2()(d)2 + (d)2 to cosh2 (d)2 (d)2 one must firstperform the change of coordinates /2 + such that sin2 cos2() and21then Wick rotate = i so that cos2() cosh2 and (d)2 = (d)2.

In the appendix we find the solutions to Einsteins vacuum field equations in D

dimensions for metrics whose signature is (D 2) + 2 (two times) associated witha (D 2)-dimensional homogeneous space of constant positive (negative) scalarcurvature. In particular when D = 4 and the two-dimensional homogeneous space

H2 has a constant positive scalar curvature, like two-dimensional de Sitter space,the metric components, in natural units G = = c = 1, are given by

gt1t1 =

1 MR(r)

, grr =

(dR/dr)2

(1 M/R(r)) , = const (2.12b)23


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which are almost identical to the components appearing in the Schwarzchild solu-1

tions for signature 3 + 1. The two-dimensional hyperboloid defined by Eq. (2.10)

coincides with a two-dimensional de Sitter space of constant positive scalar curva-3ture. Anti-de Sitter space has a constant negative scalar curvature.

There is a physical singularity at r = 0, the location of the point mass source,

when the hyperboloid H2 degenerates to a cone since the throat size r has beenpinched to zero. When the radial function is chosen to be R3 = r3+(M)3 R(r =0) = M then grr(r = 0) = and gt1t1(r = 0) = 0. The proper circumference forthis choice R3 = r3 + (M)3 is

C(r, ) = 2R(r)cosh C(r = 0, ) = 2M cosh . (2.13)The proper area for a given value of r is

A(r) = 2R2(r)+

cosh d = 2R2(r)2 sinh (2.14)

and diverges as because the two-dimensional hyperboloid is not compact.5If one chooses R(r) = r, then R(r = 0) = 0, so the proper circumference is zero

(for finite ) and the proper area corresponding to r = 0 is 0 = since sinh 7approaches infinity faster than r2 approaches zero.

Another parametrization is

t2 = r cosh , x = r sinh cos , y = r sinh sin , (2.15)

where the thoat size r is defined in terms of t2, x, y as

(t2)2 + x2 + y2 = r2 (2.16)which can be obtained from Eq. (2.10) by r2 r2. Equation (2.16) representsanalytically the two disconnected branches of a two-dimensional hyperboloid:

ds2 = (dt1)2 (dt2)2 + (dx)2 + (dy)2

= (dt1)2 (dr)2 + r2[sinh2 (d)2 + (d)2] . (2.17)Notice the sign change dr2 in Eq. (2.15) as one must have if one persists in having9a 2 + 2 signature. In this case the coordinate r must be interpreted as a radial

time.11

The curved spacetime version of (2.17) would be

ds2 = e(r)(dt1)2 e(r)(dr)2 + (R(r))2[sinh2 (d)2 + (d)2] , (2.18)where (r) and (r) are two functions to be determined by solving Einsteins equa-

tions. The functional form of (r), (r) differs from the functions (r), (r) in13

Eqs. (2.12a) and (2.12b) due to a crucial sign change in the grr component of the

metric in Eq. (2.18).15

Concluding, we have 3 interesting cases described by the metrics of 2 + 2 signa-

ture given by Eqs. (2.8), (2.12) and (2.18). The 2 + 2 hyperbolic-symmetric version17

of Schwarzschilds 3 + 1 solution is given by Eqs. (2.12a) and (2.12b).


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From (3.7) we get

=

+

+

2R

R . (3.8)

Substituting (3.8) into (3.4) and (3.5) we obtain

=

R

R R

R

(3.9)

and

=

R

R R

R

, (3.10)

respectively. Equations (3.9) and (3.10) can be integrated to give

= aR

R

(3.11)

and

= bR

R, (3.12)

respectively, where a and b are constants. Substituting (3.11) and (3.12) into (3.8)

leads to

= aR

R+ b

R

R+

2R

R. (3.13)

The expressions (3.11)(3.13) can be solved. We get

= a ln R/c, (3.14)

= b ln R/d (3.15)

and

= a ln R/c + b ln R/d + 2ln R + f , (3.16)

where c, d and f are arbitrary constants. If we substitute (3.14)(3.16) into (3.6)

we find

12

a

R

R R

2

R2

1

4a2

R2

R2+

1

4

a

R

R

a

R

R+ b

R

R+

2R

R

12

b

R

R R

2

R2

1

4b2

R2

R2+

1

4

b

R

R

a

R

R+ b

R

R+

2R

R

+ 12

a R

R+ b R

R+ 2R

R

R

R R

R= 0 . (3.17)

This can be reduced to a +

1

2ab + b

R2

R2= 0 . (3.18)


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Excluding the solutions

R = const (3.19)

Eq. (3.18) gives

a +1

2ab + b = 0 . (3.20)

Therefore we have shown why the form of R = R() can be completely arbitrary

while one must have the following constraint among the constants:

b = 2a(a + 2)

, (3.21)

where we assumed that a + 2 = 0.1A trivial solution of Eq. (3.20) is a = b = 0 which leads to = = 0 and

= 2 ln(dR/d), when f = 0, yielding the metric

ds2 = (dt1)2 (dt2)2 + dR()2 + R2()d2 , (3.22)the flat spacetime metric is attained when R() = , and also for any function

R() with the asymptotic property such that for very large values of it behaves3

R .

4. An Explicit Nontrivial Solution5

We have seen that the trivial flat spacetime solutions (3.22) are obtained when

a = b = f = 0 and when R() = . In order to find interesting nontrivial solutions

we should have a nontrivial rho function R()

= . Let us consider two particular

cases of (3.21). In the first case taking a = 2 from Eq. (3.21) we get b = 1.Similarly, in the second case by setting a = 1 in Eq. (3.21) implies b = 2. Thus inthe first case (3.14)(3.16) become

= 2ln R/c, (4.1)

= ln R/d (4.2)and

= 2ln R/c ln R/d + 2ln R + f . (4.3)While in the second case we find

= ln R/c, (4.4) = 2ln R/d (4.5)

and

= ln R/c + 2ln R/d + 2ln R + f . (4.6)


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An interesting possibility arises by setting c = d = M and f = 0. In the first case we

get that the metric in 2 + 2 dimensions ends up being expressed in the R-variable as

ds2 =

R

M)2(dt1)

2

M

R

(dt2)

2 +

R

M

(dR)2 + R2(d)2 , (4.7)

while in the second case we obtain

ds2 =

M

R

(dt1)

2

R

M

2(dt2)

2 +

R

M

(dR)2 + R2(d)2 . (4.8)

Notice that in both solutions (4.7) and (4.8) there is a kind of duality in the two1

times t1 and t2 factors.

Equations (4.7) and (4.8) can be written as

ds2 =

R

M

(dt2)

2 +

R

M

(dR)2 + R2

(d)2 (dt1)2

M2

, (4.9a)

ds2 =

R

M

(dt1)

2 +

R

M

(dR)2 + R2

(d)2 (dt2)2

M2

. (4.9b)

As announced earlier, the form of the rho function R() is undetermined. Any3

arbitrary choice of R() solves Einsteins equations.

A study reveals that a rho function R() given by

1

R=

1

+

1

M, (4.10)

in units of G = = c = 1 is an appropriate choice. When = 0, R = 0 and when

= we have R( = ) = M, so we do recover an asymptotically flat spacetimemetric at spatial = given by

ds2 = (dt1)2 (dt2)2 + (dR)2 + R2(d)2 = (dt1)2 (dt2)2 + M2(d)2 . (4.11)Asymptotic infinity is defined by the condition R( = ) = M. It is the three-5

dimensional asymptotic boundary of the (2+2)-spacetime. It is a three-dimensional

manifold of topology S1 R2. The radius of S1 is R = M. When = 0 we have in7Eq. (4.7) that R( = 0) = 0, so the metric component g22( = 0) = and there isa metric singularity at = 0 as expected. Conversely, in Eq. (4.8) the singularity9

occurs in the component g11( = 0) = , instead.

5. Stringy 1 + 1 Black Holes Embedded in 3 + 1 and11

2 + 2 Dimensions

One of the main topics of the present work has been to link the 2+2 signature with13

the black hole concept, i.e. spacetimes with singularities. We have shown that there

are many different interesting ways to do this. In Sec. 2 we presented three very15

diferent cases associated with hyperboloids. In particular, in the static hyperbolic-

symmetric version of the Schwarschild case given by Eqs. (2.12a) and (2.12b), there17

is singularity at r = 0 which is associated with the conical geometry resulting from


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having pinched to zero size r = 0 the throat of the hyperboloid H2 and which1is quite different from the spherically symmetric case in 3 + 1 dimensions. In the

static circular symmetric case developed in Secs. 3 and 4 we obtained solutions with3singularties at = 0 and whose asymptotic limit leads to a flat (1 + 2)-dimensional boundary of topology S1R2 where the radius ofS1 is R( = ) = M.5

One further interesting possibility may arise if we split the 2 + 2 metric as the

diagonal sum of two 1 + 1 metrics in the form

ds2 = gab(x)dxa dxb + gmn(y)dy

m dyn , a, b = 1, 2 , m, n = 3, 4 . (5.1)

In this case one may look for solutions like

ds2 =dudv

1 uv +dwdz

1 wz , (5.2)where we have set the value of the mass parameter 2M = 1. Such mass parameter is

required on physical grounds and also because the denominators in Eq. (5.2) must7

be dimensionless.The metric of Eq. (5.2) can be understood as the diagonal sum of two 1 + 1

black holes solutions9597 and whose singularities are located at uv = 1 and wz = 1

respectively. There are two horizons. The region outside the first horizon is indicated

by u 0 v and v 0 u; and the region inside the first horizon is indicatedby 1 uv 0 and u, v 0. Similar considerations apply to the second horizon byexchanging u w and v z. The lightcone coordinates are defined by

u =1

2exp[x + t1 + log(1 e2x)] = X+ T1 ,

v = 12

exp[x t1 + log(1 e2x)] = X T1 ,(5.3a)

w = 12

exp[y + t2 + log(1 e2y)] = Y + T2 ,

z = 12

exp[y t2 + log(1 e2y)] = Y T2 .(5.3b)

Conformally flat solutions of the form

ds2 = e(x,y,t1,t2)[(dx)2 (dt1)2 + (dy)2 (dt2)2] , (5.4)where (x, y, t1, t2) has a similar singularity structure as the metric in Eq. (5.2)9

are worth exploring also.

The BarsWitten black hole (1 + 1)-dimensional metric (setting 2 M = 1) is

ds2 = (dr)2 tanh2(r)(dt)2 = dudv1

uv

(5.5)

with

u =1

2exp[r + t + log(1 e2r)] ,

v = 12

exp[r t + log(1 e2r)] .(5.6)


15/24



The Euclidean analytical continuation of the metric in Eq. (5.5) is obtained by1

setting = it, such that the metric is ds2 = dr2 + tanh2 r d2 and its Euclidean

geometry has the shape of a semiinfinite cigar that asymptotically approaches R13S1 for r . We should notice that the Lorentzian metric of Eq. (5.5) has asingularity at a complex value r = 0 + i/2 (setting 2M = 1) since tanh2(i/2) =5

tan2(/2) = which is consistent with the singularities at the location whereuv = 14e2r(1e2r)2 = 1, when r = 0+i/2, and a horizon at r = 0, since uv = 07when r = 0.

However this is not the end of the story. The BarsWitten black hole in (1 + 1)-

dimensional is obtained from a gauged Sl(2, R)/U(1) WZNW model with central

charge c = 2 + 6/k and is a consistent bosonic string background solution in a 1 + 1

target background given by the two-dimensional coset Sl(2, R)/U(1). Namely, the

CFT corresponding to the gauged Sl(2, R)/U(1) WZNW model with central charge

c = 2 + 6/k is a solution of equations derived from the vanishing beta functions

required by conformal invariance of the nonlinear sigma model. For example, therelevant massless bosonic closed-string fields in a (D = 26)-dimensional target back-

ground (a different CFT) are the antisymmetric tensor B(X(a)); the dilaton

(X(a)) and the gravitational field g(X(a))); where a = 1, 2 are the

worldsheet variables. The conditions for the vanishing of the one loop beta func-

tions, required by Weyl invariance of the nonlinear sigma model, to leading order

in the string tension turn out to be99

R + 14

H H 2DD = 0 , (5.7a)

DH 2(D)H = 0 , (5.7b)

4(D)2

4DD

+ R +1

12 HH

= 0 , (5.7c)

where

H = B + B + B , (5.7d)

is the third rank antisymmetric tensor field strength that is invariant under the9

transformations B = . For details of quantum nonlinear sigmamodels, conformal field theory, supersymmetry, black holes and strings we refer to11

the monograph by Ketov.98

The only consistent (2+2)-dimensional gravitational backgrounds on whichN=132 strings7,8 (strings with worldsheet supersymmetry) can propagate are those that

are self-dual and which solve the Plebanski heavenly equations in 2 + 2 dimensions.15

Self dual gravitational backgrounds in four dimensions are Ricci flat whose metric

is given in terms of a Kahler potential. However, the metric in Eq. (5.2) is not17

Ricci flat since the (1 + 1)-dimensional black hole metric is not Ricci flat. Such

metric in Eq. (5.5) is not a solution of the vacuum Einstein field equations, it is19

a solution of Eqs. (5.7) (without KalbRamond fields B) where the role of the

dilaton = ln(1 uv) is essential.21


16/24



Nevertheless, we will show how the BarsWitten (1 + 1)-dimensional black hole

metric can be embedded into the (3 + 1)-dimensional solutions of the appendix, up

to a conformal factor e, since the latter metrics were Ricci flat by construction.The embedding of the (1+1)-dimensional metric (5.5) into the conformally rescaled

(3 + 1)-dimensional solutions of the appendix are obtained by introducing the mass

parameter 2M (in units of G = c = 1) in the appropriate places in order to have

consistent units, and by writing

e(r)

1 2MR(r)

= tanh2

r

2M

, e(r)

(dR/dr)2

1 2M/R(r) = 1 , (5.8)

leading to the solutions for (r) and R(r) respectively

e =1

1 2M/R(r) tanh2

r

2M

, (5.9a)

where dR

1 2M/R = R + 2Mln

R 2M2M

=

dr

tanh r/2M= 2Mln

sinh

r

2M

. (5.9b)

This last equation (5.9b) yields the functional form R(r) (tortoise radial variable)

in implicit form for the radial function R(r). From Eq. (5.9b) one can infer that

R(r = 0) = 2M , R(r ) R r . (5.10)The radial function R has a lower (ultraviolet cutoff) bound given by 2M. The fact1

that the point r = 0 can have a nonzero proper area but zero volume seems to

indicate a stringy nature underlying the very notion of a point-mass itself. The3

string worldsheet has area but no volume. Aspinwall27,28 has studied how a string

(an extended object) can probe spacetime points.5

Notice that if we allow for complex values of r, like r = 0 + i2M(/2), that

furnish singularities in the metric (5.5), one must include a constant of integration

R0 = 2M(1 + i/2) to the solution in Eq. (5.9b):

R 2M

1 +i

2

+ 2Mln

R 2M

2M

= 2Mln

sinh

r

2M

(5.11)

such that when one plugs in the value r = 0 + i2M(/2) in the right-hand side of

Eq. (5.11), it coincides with the left-hand side of (5.11) when the value of the radial7

function R(r = 0 + i2M /2) = 2M(1 + i/2), after an analytical continuation

into the complex plane is performed. This is just a consequence of the relation9

ln[sinh(i/2)] = ln[i sin(/2)] = ln i = i/2.

This complex analytical continuation into regions where r, R are complex-valued11

roughly speaking amounts to looking into the interior of the point-mass. Having

complex coordinates to probe into the interior of a point-mass is not so farfetched.13

This suggests that quantum spacetime might be intrinsically fractal, meaning that


17/24


18/24



in 1 + 1 quaternionic dimensions, and gravity in 8 + 8 real dimensional can be seen1

as octonionic gravity in 1 + 1 octonionic dimensions.102,103

To illustrate this, let us write the following complex line element in four complex-dimensions:

ds2 =dz1 dz1 + dz1 dz11 z1z1 z1z1 +

dz2 dz2 + dz2 dz21 z2z2 z2z2 . (5.15)

Complex gravity requires that g = g() + ig[] so that now one has g =

(g),102104 which implies that the diagonal components of the metric gz1z1 =

gz2z2 = gz1z1 = gz2z2 must be real, and which in turn implies that a real slice of

the 4-complex dimensional space spanned by the four complex variables z1, z2, z1,

z2 may be taken by imposing the following two constraints:

z1 = z

1 , z2 = z

2 (5.16)

and upon doing so one ends up with a four real-dimensional space of signature 2 + 2

whose real line element is

ds2 =dz1 dz1 + dz1 dz

1

1 z1z1 z1z1+

dz2 dz2 + dz2 dz

2

1 z2z2 z2z2, (5.17)

where z1, z2 are the complex coordinates of the 1 + 1 complex dimensional space

time (2+ 2 real dimensional) while z1 , z

2 are their complex conjugates, respectively.

After defining

z1 =1

2(X+ iT1) , z

1 =1

2(X iT1) ,

z2 =1

2(Y + iT2) , z2 =

12

(Y iT2) ,(5.18)

the metric in Eq. (5.14) coincides preciselywith the metric in Eq. (5.2) comprised of3

the diagonal sum of two black hole solutions in 1 + 1 real dimensions. The quater-

nionic and octonionic versions of Eq. (5.16), in conjunction with the generalized5

Einsteins field equations, will be the subject of future investigations. The quater-

nionic analog of two-dimensional conformal field theory in four dimensions has been7

studied by S. Vongehr.105 It is interesting to see (if possible) how one can construct

four-dimensional quantum nonlinear sigma models within the context of quantum 3-9

branes (conformal field theories in the four-dimensional worldvolume of the 3-brane)

and find the analog of the coupled equations (5.7) associated with the vanishing of11

the beta functions in two-dimensional CFT; namely from the perspective of a four-

dimensional quaternionic conformally invariant field theory formulated on Kulkarni13

four-folds (the four-dimensional analog of Riemann surfaces) corresponding to 3-

branes moving in curved target spacetime backgrounds. The cancellation of the15

four-dimensional conformal anomaly should constrain the type of backgrounds on

which 3-branes can propagate.17

It is worth mentioning that black hole solutions in a two times context have

been considered by some authors. In particular Kocinski and Wierzbicki107 con-19

sidered Schwarzschild type solution in a KaluzaKlein theory with two times. In


19/24



fact, using noncompactified KaluzaKlein theory with internal signature of the1

form 2 + 3 these authors determine a spherical symmetric solution. Vongehr108

also considered examples of black holes within the context of the two-times physics3formulation of Bars (see Ref. 106 and references therein). Their basic examples

coreponds essentially to a solutions associated with the signatures 1 + 1 and 2 + 3.5

Finally, the four-dimensional KaluzaKlein approach to general relativity in

2 + 2 as a local product of a (1 + 1)-dimensional base manifold and a (1 + 1)-7

dimensional fiber space109,110 warrants further investigation in so far that 2 + 2

gravity can be described by a (1 + 1)-dimensional YangMills gauge theory of dif-9

feormorphims of the two-dimensional fiber space coupled to a (1 + 1)-dimensional

nonlinear sigma model and a scalar field; i.e. this formulation of 2 + 2 gravity11

by109,110 is more closely related to the stringy picture of the BarsWitten black

hole in 1 + 1-dimensions. Thus, it seems interesting to pursue further research to13

see the possible connection between the present work and these other approaches.

For example, to study black holes solutions in noncommutative geometry,73 in par-15ticular Finsler spaces,6771,79,80 phase spaces7478,81,82 and the implications of the

minimal Planck scale41 stringy uncertainty relations83,84 in black holes physics.868917

Appendix A. Schwarzschild-like Solutions in Any Dimension D > 3

Let us start with the line element

ds2 = e(r)(dt1)2 + e(r)(dr)2 + R2(r)gij di d . (A.1)Here, the metric gij corresponds to a homogeneous space and i, j = 3, 4, . . . , D 2.The only nonvanishing Christoffel symbols are

121 = 12 , 222 = 12

, 211 = 12e ,

2ij = eRRgij , i2j =R

Rij ,

ijk =

ijk ,

(A.2)

and the only nonvanishing Riemann tensor are

R1212 = 1

2 1

42 +

1

4 ,

R1i1j = 1

2eRRgij ,

R2121 = e

1

2

+1

4

2

1

4

,

R2i2j = e

1

2 RR RR

gij ,

Rijkl = Rijkl R2e

ikgjl il gjk

.

(A.3)


20/24



The field equations are

R11 = e1

2

+

1

4 2

1

4

+

(D

2)

2

R

R

= 0 , (A.4)

R22 = 12

14

2 +1

4 + (D 2)

1

2

R

R R

R

= 0 , (A.5)

and

Rij = e

R2

1

2( )RR RR (D 3)R2

gij

+k

R2(D 3)gij = 0 , (A.6)

where k = 1, depending if gij refers to positive or negative curvature. From thecombination e+R11 + R22 = 0 we get

+ =2R

R. (A.7)

The solution of this equation is

+ = ln R2 + a , (A.8)

where a is a constant.1

Substituting (A.7) into Eq. (A.6) we find

e(RR 2RR (D 3)R2 = k(D 3) (A.9)or

RR + 2RR + (D

3)R2 = k(D

3) , (A.10)

where

= e . (A.11)

The solution of (A.10) for an ordinary D-dimensional spacetime (one temporal

dimension) corresponding to a (D 2)-dimensional sphere for the homogeneousspace can be written as

=

1 16GDM

(D 2)D2RD3

dR

dr

2

grr = e =

1 16GDM(D

2)D2RD3

1

dR

dr

2, (A.12)

where D2 is the appropriate solid angle in (D 2)-dimensional and GD is theD-dimensional gravitational constant whose units are (length)D2. Thus GDM3

has units of (length)D3 as it should. When D = 4 as a result that the two-

dimensional solid angle is 2 = 4 one recovers from Eq. (A.12) the four-5

dimensional Schwarzchild solution. The solution in Eq. (A.12) is consistent with


21/24



Gauss law and Poissons equation in D 1 spatial dimensions obtained in the1Newtonian limit.

For the most general case of the (D 2)-dimensional homogeneous space weshould write

= ln(k DGDMRD3

) 2 ln R , (A.13)

where D is a constant. Thus, according to (A.8) we get

= ln

k DGDM

RD3

+ const (A.14)

we can set the constant to zero, and this means the line element (A.1) can be

written as

ds2 =

k

DGDM

RD3 (dt1)

2

+(dR/dr)2

k DGDMRD3

(dr)2 + R2(r)gij di d . (A.15)

One can verify, taking for instance (A.5), that Eqs. (A.4)(A.6) do not determine3

the form R(r). It is also interesting to observe that the only effect of the homo-

geneous metric gij is reflected in the k = 1 parameter, associated with a positive5(negative) constant scalar curvature of the homogeneous (D2)-dimensional space.

Acknowledgments7

We wish to thank the referee for his numerous and critical suggestions to improve

this work. J. A. Nieto thanks L. Ruiz, J. Silvas and C. M. Yee for helpful comments.9

This work was partially supported by grants PIFI 3.2. C. Castro thanks M. Bowers

for hospitality and Sergiu Vacaru for many discussions about Finsler geometry and11

related topics.

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On 2 + 2 dim Spacetimes and Stringy Black Holes

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