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8/14/2019 On 2 + 2 dim Spacetimes and Stringy Black Holes
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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
J. A. NIETO
Facultad de Ciencias Fsico-Matematicas de la Universidad Autonoma de Sinaloa,11
80010, Culiacan Sinaloa, Mexico
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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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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4 C. Castro & J. A. Nieto
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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8 C. Castro & J. A. Nieto
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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On (2 + 2)-Dimensional SpaceTimes, Strings and Black Holes 9
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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12 C. Castro & J. A. Nieto
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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On (2 + 2)-Dimensional SpaceTimes, Strings and Black Holes 13
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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14 C. Castro & J. A. Nieto
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)
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On (2 + 2)-Dimensional SpaceTimes, Strings and Black Holes 15
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
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16 C. Castro & J. A. Nieto
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
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18 C. Castro & J. A. Nieto
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
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On (2 + 2)-Dimensional SpaceTimes, Strings and Black Holes 19
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)
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20 C. Castro & J. A. Nieto
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
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On (2 + 2)-Dimensional SpaceTimes, Strings and Black Holes 21
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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