Quantum probes of timelike naked singularities in the weak field regime of global monopole spacetime
The formation of a naked singularity in global monopole spacetime is considered in view of quantum mechanics. Quantum test fields obeying the KleinGordon, Dirac and Maxwell equations are used to probe the classical timelike naked singularity developed at . We prove that the spatial derivative operator of the fields fails to be essentially self-adjoint. As a result, the classical timelike naked singularity formed in global monopole spacetime remains quantum mechanically singular when it is probed with quantum fields having different spin structures. Pitelli and Letelier (Phys. Rev. D 80, 104035, 2009) had shown that for quantum scalar ( ) probes the general relativistic global monopole singularity remains intact. For specific modes electromagnetic ( ) and Dirac field ( ) probes, however, we show that the global monopole spacetime behaves quantum mechanically regular. The admissibility of this singularity is also incorporated within the Gubser’s singularity conjecture.
Spacetime singularities are believed to be one of the inevitable consequences of the Einstein’s theory of relativity. It describes the ”end point” or incomplete geodesics for timelike or null trajectories followed by classical particles. The black hole and colliding plane wave spacetimes are the two important branches of this theory that the nature and characteristics of spacetime singularities are manifested. Another intriguing one is the Big-Bang-like cosmological singularities. According to the classical singularity classification devised by Ellis and Schmidt 1 (), curvature singularities can be grouped as scalar and nonscalar. The scalar curvature singularities are the strongest ones in the sense that the spacetime becomes inextendible and all the physical quantities, such as the gravitational field, energy density and tidal forces, diverge at the singular point. Singularities forming at the centre of black holes and in some colliding plane wave spacetimes are good examples for strong scalar curvature singularity. In black hole spacetimes singularities located at the centre () is hidden by horizon(s). In the cases where this singularity is not hidden, it is called the naked singularity. Whereas, the singularity occurring in the interaction region of Bell-Szekeres solution BS () which describes the nonlinear interaction of electromagnetic plane waves can be given as an example to nonscalar curvature singularity.
Naked singularity which is visible from outside needs further care as far as the weak cosmic censorship hypothesis is concerned. It is beleived that, naked singularity forms a threat to this hypothesis. Hence, understanding and the resolution of naked singularities seems to be extremely important for the deterministic nature of general relativity.
However, the scale where the singularities are forming is very small (smaller than the Planck scale), so that the classical general relativity methods in the resolution of the singularities are expected to be replaced by the quantum theory of gravity. Unfortunately, there is no consistent quantum theory of gravity yet. Since this theory is still ”under construction”, the alternative methods in healing the singularities are always attracted the attentions. String theory 2 (); 3 ()and loop quantum gravity 4 () constitutes two major study fields in resolving singularities. It is shown in string theory that some timelike singularities are resolved: the orbifold, the flop, and the conifold. The flop and the conifold occurs in the Calabi-Yau manifolds in which their resolution involves the use of light matters such as ”twisted sectors” and ”wrapped D-branes” 5 ()(and references therein).
A rather different approach is considered in amit () for resolving the timelike singularities in Reissner-Nordström and negative mass Schwarzschild solutions. In this approach, the spacetime is viewed as being made of two parts which are naturally connected across the singularity. In this study, it is shown that the Reissner-Nordström singularity allows for communication through the singularity and can be termed as ”beam splitter” since the transmission probability of a suitably prepared high energy wave packet is 25%.
Another alternative method; following the work of Wald 6 (), is proposed by Horowitz and Marolf (HM)7 (), which incorporates ”self-adjointness” of the spatial part of the wave operator. Hence, the classical notion of geodesics incompleteness with respect to point-particle probe will be replaced by the notion of quantum singularity with respect to wave probes.
The method of HM has been used successfully for other spacetimes to check whether the classically singular spacetimes are quantum mechanically regular or not. As an example; negative mass Schwarzschild spacetime, charged dilatonic black hole spacetime and fundamental string spacetimes are considered in 7 (). An alternative function space, namely the Sobolev space instead of the Hilbert space, has been introduced in 8 (), for analyzing the singularities within the framework of quantum mechanics. As a result, the occurrence of timelike naked singularity in the negative mass Schwarzschild solution is shown to be quantum mechanically regular. Helliwell and Konkowski have studied quasiregular 9 (), Gal’tsov-Letelier-Tod spacetime 10 (), Levi-Civita spacetimes 11 (); 12 (), and recently, they have also considered conformally static spacetimes 13 (); 14 (). Pitelli and Letelier have studied spherical and cylindrical topological defects 15 (), BanadosTeitelboimZanelli (BTZ) spacetimes 16 (), the global monopole spacetime 17 () and cosmological spacetimes 18 (). Quantum singularities in matter coupled dimensional black hole spacetimes are considered in 19 (). Quantum singularities are also considered in Lovelock theory 20 () and linear dilaton black hole spacetimes 21 (). The occurrence of naked singularities in a dimensional magnetically charged solution in Einstein-Power-Maxwell theory have also been considered 22 (). Recently, the formation of naked singularity in a model of gravity is considered in 23 ().
The main motivation in these studies is to understand whether these classically singular spacetimes turn out to be quantum mechanically regular if they are probed with quantum fields rather than classical particles.
Recently, a solution describing global monopole in the weak field regime has been presented in 24 (). This study showed that, the main contribution of the modified theory compared to the ordinary global monopole solution due to the Barriola and Vilenkin (BV) 25 () is that, in addition to admitting double and single horizons, it admits solution without horizon as well. And, the most important influence is seen on the nature of the singularity that occur at . In the case of BV, this singularity is spacelike, whereas in the case of theory, it has timelike nature.
Generally, solutions admitting black holes attracted more attention than the solutions admitting naked singularity. Recently, the influence of the modified theory on the thermodynamic quantities of an global monopole spacetime 24 () has been investigated and compared with BV spacetime in 26 (). The outcome of this investigation is that, theory modifies the thermodynamic quantities, but the shapes of curves for thermodynamic quantities with respect to the horizon are similar to the results within the frame of general relativity.
In this paper, we wish to investigate the occurrence of timelike naked singularities in global monopole spacetime within the context of quantum mechanics. The singularity at will be probed with three different types of quantum fields that obey Klein-Gordon, Maxwell and Dirac equations. The singularity for the BV spacetime will also be investigated with the spinor fields obeying Maxwell and Dirac equations.This will be the spinor field generalization of the study performed by Pitelli and Letelier 17 () for BV spacetime.
The appearance of naked singularities are also encountered in gauged supergravity theories. Gubser GUB () proposed a singularity conjecture to resolve singularities in these theories in the following way.
Conjecture: Large curvatures in scalar coupled gravity with four dimensional Poincare invariant solution are allowed only if the scalar potential is bounded above in the solution.
In this paper, the approach of Gubser will be incorporated to our analysis briefly to display its applicability in spacetimes which do not obey Poincare invariance.
The paper is organized as follows: In section II, we give the solution and the spacetime structure obtained in 24 (). The definition of quantum singularity is briefly reviewed in section III. Section IV is devoted for the quantum singularity analysis of the global monopole spacetime. Three different types of waves with different spins are used to probe the singularity. The spinor field generalization of the paper by Pitelli and Letelier 17 () is given in section V. In section VI, Gubser’s singularity conjecture is used to identify if the studied curvature singularity is bad or good. Finally, we give the concluding remarks of this study in section VII.
Ii The Metric for a Global Monopole in Theories and Spacetime Structure
ii.1 The Metric for a Global Monopole in Theories.
Recently, the metric describing the global monopole in theories for the static spherically symmetric systems has been presented in the weak field regime 24 (). The adopted action for such a gravitational field coupled to matter fields in theory is given by
in which is an analytic function of the Ricci scalar , here is the Newton constant and represents the action of the coupled matter fields given by
In the considered global monopole model, represents the Lagrangian density that gives the simplest global monopole model given by
in which and are constant parameters. The global monopole, that forms as a result of spontaneous symmetry breaking from global to during the phase transitions in the early universe is described by the self - coupling triplet of scalar fields given by the following ansatz,
with and is a constant parameter. The adopted metric for such a model is given by
where and are only function of . The field equation reads
In Eq. (6) represents minimally coupled energy momentum tensor of the matter field whose non-zero components are given by
Furthermore, the trace of the field equation (6) reads
with With reference to the paper 24 (), the solution to the field equations was obtained in the weak field regime which assumes the metric function in the form of and with the property that and smaller than unity. As a consequence of a weak field regime, the considered model of theory corresponds to a small correction on standard general relativity in such a way that, with Explicit form of is given in 24 () (Eq. 42 in 24 ()). Hence, corresponds to the standard general relativity. Employing these conditions in the field equations yields and resulting metric function with global monopole is found to be
where is the mass parameter and is a very small parameter ( since ) that measures the deviation from the standard general relativity. As stated in 24 (), for a typical Grand Unified Theory the parameter is in the order of GeV. Hence, Note that one can recover the result of BV if It is known that, the global monopole solution obtained by BV has one horizon only and the nature of the singularity at is spacelike.
ii.2 The Spacetime Structure
The structure of the solution obtained in 24 () and given in Eq. (12), has remarkable features that deserves to be investigated in detail. The obtained solution admit black holes with inner and outer horizons. To find the location of the horizon, we prefer to write the metric component in the following form
where and denote the outer and inner horizons respectively and given by
The Kretschmann scalar which indicates the formation of curvature singularity for the global monopole is given by
It is evident that is a typical central curvature singularity that is peculiar to the spherically symmetric systems. In order to find the nature or the character of the singularity at for the global monopole, we perform conformal compactification. The conformal radial or tortoise coordinate is given by
The retarded and advanced coordinates are defined as and respectively. Defining the Kruskal coordinates as
the metric can be written as
In order to bring infinity into a finite coordinate, we define
The corresponding Carter - Penrose diagrams for the following three possible cases are plotted and given in figures. The singularity located at is shown vertically on the Carter-Penrose diagram which indicates timelike character.
There are three possible cases to be investigated.
ii.2.1 Case 1: When
The metric function, admits two positive roots and indicating the location of the outer and inner horizons of a black hole. The Penrose diagram for this case is shown in Fig.1.
ii.2.2 Case 2: When
The metric function, admits one horizon only. It can be interpreted as the extreme black hole. The Penrose diagram of this case is given in Fig. 2. Recently, the thermodynamic properties of the black hole solutions of global monopole is investigated and presented in 25 ().
ii.2.3 Case 3: When
In this case, the metric function, , does not admit real roots. Hence, the solution in this particular case is not a black hole solution and the singularity at becomes timelike naked singularity, as depicted in the Penrose diagram in Fig. 3. The choice of the parameters of the global monopole metric results with timelike naked singularity at or black hole solutions with one or two horizons. These results seem to show that the small correction to the standard general relativity produces significant changes on the spacetime structure of the BV metric obtained by Barriola and Vilenkin.
In this paper, we are aiming to investigate this singularity within the context of quantum mechanics. This classically singular spacetime will be probed with quantum waves obeying the Klein-Gordon, Maxwell and Dirac equations to check whether the timelike naked singularity is smoothed out or not.
ii.3 The Description of the Global Monopole Spacetime in a Newman-Penrose (NP) Formalism
The global monopole metric is investigated with the Newman-Penrose (NP) formalism, in order to clarify the contribution of the gravity. The set of proper null tetrads is given by
The non-zero spin coefficients in these tetrads are
As a result, we obtain the Weyl and the Ricci scalars as
so that the spacetime is Petrov type. The parameter representing the contribution of gravity is seen to effect only the Ricci components, leaving the mass term of an ordinary global monopole unchanged.
Iii Quantum Singularities
Horowitz and Marolf (HM) 7 (), by developing the pioneering work of Wald 6 (), have proposed a prescription which involves the use of quantum particles/waves to judge whether the classical timelike curvature singularities occurring in static spacetimes are smoothed out quantum mechanically or not. According to HM, the singular character of the spacetime is defined as the ambiguity in the evolution of the wave functions. That is to say, the singular character is determined in terms of the ambiguity when attempting to find a self-adjoint extension of the spatial part of the wave operator to the entire Hilbert space. If the extension is unique, it is said that the space is quantum mechanically regular. A brief review now follows:
Consider a static spacetime with a timelike Killing vector field . Let denote the Killing parameter and denote a static slice. The Klein-Gordon equation in this space is
This equation can be written in the form
in which and is the spatial covariant derivative on . The Hilbert space , is the space of square integrable functions on . The domain of an operator is taken in such a way that it does not enclose the spacetime singularities. An appropriate set is , the set of smooth functions with compact support on . The operator is real, positive and symmetric; therefore, its self-adjoint extensions always exist. If it has a unique extension then is called essentially self-adjoint 27 (); 28 (); 29 (). Accordingly, the Klein-Gordon equation for a free particle satisfies
with the solution
If is not essentially self-adjoint, the future time evolution of the wave function (35) is ambiguous. Then the HM criterion defines the spacetime as quantum mechanically singular. However, if there is only a single self-adjoint extension, the operator is said to be essentially self-adjoint and the quantum evolution described by Eq. (35) is uniquely determined by the initial conditions. According to the HM criterion, this spacetime is said to be quantum mechanically non-singular. In order to determine the number of self-adjoint extensions, the concept of deficiency indices is used. The deficiency subspaces are defined by (see Ref. 8 () for a detailed mathematical background)
The dimensions are the deficiency indices of the operator . The indices are completely independent of the choice of depending only on whether or not lies in the upper (lower) half complex plane. Generally one takes and , where is an arbitrary positive constant necessary for dimensional reasons. The determination of deficiency indices is then reduced to counting the number of solutions of ; (for ),
that belong to the Hilbert space . If there are no square integrable solutions ( i.e. , the operator possesses a unique self-adjoint extension and is essentially self-adjoint. Consequently, the way to find a sufficient condition for the operator to be essentially self-adjoint is to investigate the solutions satisfying Eq. (38) that do not belong to the Hilbert space.
Iv Quantum Singularities in Global Monopole Spacetime
iv.1 Klein-Gordon Fields
The massive Klein-Gordon equation for a scalar particle with mass can be written as
For the metric (5), the Klein-Gordon equation can be splitted into a time and spatial part and written as
In analogy with Eq. (33), the spatial operator for the massless case is
and the equation to be solved is Using separation of variables, , we get the radial part of Eq. (38) as
whose solutions represents spin bosonic waves and a prime denotes the derivative with respect to . The spatial operator is essentially self adjoint if neither of two solutions of Eq. (42) is square integrable over all space . Because of the complexity in finding exact analytic solution to Eq. (42), we study the behavior of near and
iv.1.1 The case of r
The case is topologically different compared to the analysis for ordinary global monopole solutions reported in 17 (). The asymptotic behavior of the global monopole metric when is not conical and given by
For the above metric, the radial equation (42), for becomes,
whose solution is
where and are arbitrary integration constants, and are the first and second kind Bessel functions. The square integrability of the above solution for each sign is checked by calculating the squared norm of the above solution in which the function space on each constant hypersurface is defined as The squared norm for the metric (43) is given by,
Our calculation has revealed that the obtained solution at infinity fails to satisfy square integrability condition i.e. . Hence, the solution at infinity does not belong to the Hilbert space.
iv.1.2 The case of r
The approximate metric near the origin is Schwarzschild like and given by
The radial equation (42), for the above metric reduces to
in which , and the solution is obtained in terms of first and second kind of Bessel’s functions and given by
where and are arbitrary integration constants. The square integrability of the above solution is checked by calculating the squared norm for the metric (47) which is given by,
which is always square integrable near Consequently, the spatial operator is not square integrable over all space and therefore, it is not essentially self-adjoint. Hence, the classical singularity at remains quantum mechanically singular when probed with fields obeying the Klein-Gordon equation.
In the next subsections, the singularity will be probed with spinorial fields obeying Maxwell and Dirac equations. We prefer to use same method and terminology reported in 23 ().
iv.2 Maxwell Fields
The Newman-Penrose formalism will be used to find the source-free Maxwell fields propagating in the space of global monopole spacetime. The four coupled source-free Maxwell equations for electromagnetic fields in the Newman-Penrose formalism is given by
where and are the Maxwell spinors, and are the spin coefficients to be found and the bar denotes complex conjugation. The null tetrad vectors for the metric (5) are defined by
The directional derivatives in the Maxwell’s equations are defined by and We define operators in the following way by assuming ()
The non-zero spin coefficients are given in Eq.s (27-28). The Maxwell spinors are defined by 30 ()
where and are the components of the Maxwell tensor in the tetrad and tensor bases, respectively. Substituting Eq.s (59-62) into the Maxwell’s equations together with non-zero spin coefficients, the Maxwell equations become
The equations above will become more tractable if the variables are changed to
Then, we have
The commutativity of the operators and enables us to eliminate each from above equations, and hence we have
The variables and can be separated by assuming a separable solution in the form of
The separation constants for Eq. (75) and Eq. (76) are the same, because or, in other words, the operator acting on is the same as the operator acting on if we replace by . However, for Eq. (77) we will assume another separation constant. Furthermore, by defining , and , the radial equations can be written as
where and are the separability constants and denotes the frequency of the photon wave.
The definition of the quantum singularity for Maxwell fields will be the same as for the KleinGordon fields. Here, since we have three equations governing the dynamics of the photon waves, the unique self-adjoint extension condition on the spatial part of the Maxwell operator should be examined for each of the three equations for all space.
iv.2.1 For the case
The corresponding metric is given in Eq. (43). Hence, the radial parts of the Maxwell equations, (81) , (82) and (83), become
Thus, the solutions in the asymptotic case are
in which are integration constants, and The square integrability condition at infinity is checked by calculating the squared norm of each solution
Calculations has revealed that the obtained solutions do not belong to the Hilbert space because
iv.2.2 The case r
The metric near is given in Eq. (47). Hence, the radial parts of the Maxwell equations (81), (82) and (83) for this case are given by
in which and solutions are obtained as,
where are constants, and are Bessel and Neumann functions. The above solutions is checked for square integrability. Calculations have revealed that
which indicates that the obtained solutions are square integrable. As a result, the spatial part of the Maxwell operator is not essentially self-adjoint and therefore, the occurrence of the timelike naked singularity in gravity is quantum mechanically singular, if it is probed with photon waves.
iv.3 Dirac Fields
The Newman-Penrose formalism will also be used here to find the massless Dirac fields (fermions) propagating in the space of global monopole spacetime. The Chandrasekhar-Dirac (CD) equations in the Newman-Penrose formalism are given by
where and are the components of the wave function, and are the spin coefficients. The non-zero spin coefficients are given in Eq.s (27,28). The directional derivatives in the CD equations are the same as in the Maxwell equations. Substituting non-zero spin coefficients and the definitions of the operators given in Eq.s (59-62) into the CD equations leads to