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Wormhole Cosmic Censorship: An Analytical Proof

In this work we present an analytical proof of cosmic censorship in a Kerr-like phantom wormhole (WH) which contains a singularity that is not protected by an event horizon. We show that the naked singularity of this space-time is causally disconnected from the universe. To do so, we consider a slowly rotating limit and by means of the Hamilton-Jacobi theory separate the Hamiltonian of the geodesics into two polynomials. During this process we find a fourth conserved quantity. After examining the properties of these polynomials we conclude that the ring singularity is untouchable by any observer traveling in a geodesic of this space-time. We also derive the conditions on the four constants of motion that are necessary for a traveler to go back and forth both universes connected through the WH, and then compare its structure to that of a negative mass Kerr black hole.

In this work we present an analytical proof of cosmic censorship in a Kerr-like phantom wormhole (WH) which contains a singularity that is not protected by an event horizon. We show that this space-time singularity without event horizon is causally disconnected from the universe. To do so, we consider a slowly-rotating limit and by means of the Hamilton-Jacobi theory separate the Hamiltonian of the geodesics into two polynomials. During this process we find a fourth conserved quantity. After examining the properties of these polynomials we conclude that the ring singularity is untouchable by any observer traveling in a geodesic of this space-time. We also derive the conditions on the four constants of motion that are necessary for a traveler to go back and forth both universes connected through the WH.
Introduction.-In 1969, Roger Penrose studied gravitational collapse and suggested the existence of a "cosmic censor" who forbids the appearance of a naked singularity and hides it behind an event horizon [1]. Many years later (1999), he discussed the question of cosmic censorship and provided some arguments against and in favor of it [2], leading to the conclusion that the conjecture still did not posses a definite proof. In fact, the possibility that new mathematical techniques (twistor theory) were needed was also stated. Though there have been attempts to prove or disprove it, so far the question remains open. To this day, naked singularities remain one of the biggest mysteries of General Relativity and little is known about the geometrical nature of this region of the space-time. Not to mention the physical unknowns it hides, since Einstein's field equations are not valid at the singularity and there is no physical theory that can describe it.
In [3], and later on in [4], the metric of a Kerr-like wormhole (WH) was reported as an exact solution of Einstein's field equation with a phantom like source. It is described by three parameters: mass, angular momentum and a scalar field charge. It also contains a ring singularity without an horizon. The line element in Boyer-Lindquist coordinates is given by where With line element components ω and f where The parameters l 1 and l 0 have units of distance (l 2 0 > l 2 1 ) and are related to the size of the throat. While a and k 1 are parameters with units of angular momentum.
Analysis of metric (1) revealed that an observer could travel from one asymptotically flat universe to another through the WH and without facing strong tidal forces [5]. Studying its geodesics numerically it was found that, as along as the observer does not travel on the equatorial plane, he could reach the throat of the WH. Fig.(1) shows that none of the geodesic paths touch the ring singularity at l = l 1 and θ = π/2. In [5] it was conjectured that it could be possible to protect a singularity if it was surrounded with a WH's throat. However, this conclusion was obtained through the solution of the geodesic equations, which where solved by numerical methods. In this work we give an analytical proof, in the slowly rotating limit, that is not possible for a traveler to touch the ring singularity of this space-time, showing that the numerical results and the conjecture are valid. To do so, we perform a coordinate change in (1) and apply the slowly rotating limit to the line element components (2)-(5). This will allow the Hamiltonian of the geodesics to be separable in the new coordinates system, thus obtaining a fourth conserved quantity through the Hamilton-Jacobi formalism. Afterward, we analyze the separated equations of motion and their relationship with the four conversed quantities, giving us insight on whether the singularity is touched by the observer or not.
Of course, what we present here is not a definite proof of the cosmic censorship conjecture and lacks the generality to do so. However, the results obtained here are the first example of a space-time with a singularity without horizon but where the geodesics are unable to touch it. This result shows an example of a singularity without horizon that is causally disconnected, it can be a different and particular case of the cosmic censorship hypothesis and could be considered as an argument in favor of it.
Change of Coordinates and Slowly Rotating Limit.-Making Lx = l − l 1 and y = cos θ, with L 2 = l 2 0 − l 2 1 , changes the line element (1) and line element parameters (2)-(5) to Notice that the ring singularity now lies at: x = 0 and y = 0. Also it is important to mention that y = 1 is a coordinate singularity rather than an essential singularity.
For the slowly rotating limit we take a << L 2 to first order, and a ∼ k 1 . In this limit e λ ≈ 1 + (a 2 +k 2 1 )y 2L 2 k1(x 2 +y 2 ) , so (4) becomes The Hamiltonian.-From now on we work with the Hamiltonian of the geodesics. From (6) it follows: The Hamiltonian (9) is itself a constant of motion: 2H = κ, with κ = 0 for lightlike geodesics and κ = −1 for timelike geodesics.
In the metric (6) ∂ ∂t and ∂ ∂φ are Killing vectors, so p t = E and p φ = L are constants of motion. The conserved quantities E and L are associated with the energy and angular momentum of the particle falling into the WH, respectively.
Applying the slowly rotating limit to first order in a/L 2 to the Hamiltonian (9), yields By making use of the Hamilton-Jacobi theory, and after some algebraic work, it can be shown that (10) can be separated into two equations for variables x and y. As a result, a fourth conserved quantity K is found.
In the slowly rotating limit the momenta p x and p y are related to the velocitiesẋ = dx dτ andẏ = dy dτ , where τ is an affine parameter, through Then, using (12) we have where With the help of (14) and (15) a relationship between x and y can be found which no longer depends on the affine parameter τ .
This differential equation can be solved numerically in order to describe the motion of the freely falling test particle. However, we shall take a different approach and analyze the fourth degree polynomials (14) and (15). In particular, the nature of their roots, which will give us information about the accessible regions in the WH to our traveler.
The Polynomials X(x) and Y (y).-We now analyze the fourth degree polynomials (14) and (15). From (13) it can be seen that in order to obtain real velocities, we must have for a pair of coordinates x = x 0 and y = y 0 that X(x 0 ) > 0 and Y (y 0 ) > 0. Otherwise, this would result on a non-physical case and the particle could never reach said coordinates in the space-time. This is the motivation behind the study of the nature of the roots of (14) and (15).
In a fourth degree polynomial the nature of its roots is determined by the discriminants ∆ x , P 1 and P 2 , which of course depend on the coefficients of the polynomial [6]. In the case of (14), these discriminants are given by: with A = L 2 (E 2 + κ), B = K + L 2 , C = aLE/L and D = A + K.
As it will be later explained, we are interested in the scenario where X(x) = 0 has two pairs of complex conjugate roots. For this to be the case, the conditions on the discriminants (17) are as follows: ∆ x > 0 and either P 1 > 0 or P 2 > 0.
In order to cross the throat of the WH it is necessary that X(0) > 0, that is B > 0. Furthermore, if the traveler were to freely move through the upper universe (x > 0) and the lower universe (x < 0), it would be needed that X(x) > 0 for all x ∈ R. This is accomplished by demanding that B > 0 and that (14) has two pairs of complex conjugate roots.
So far the only condition for the quantities appearing in (17) is κ = 0, −1 and B > 0. Given this last condition notice that a term by term sign examination of ∆ x reveals that if A < 0 and D < 0, then ∆ x < 0. Performing a similar analysis, one can find that if P 2 > 0, it has to be that A > 0 and aditionally 4AB > D 2 . Also, with A > 0, then D > 0 so that P 1 > 0.
With this in mind, it can be stated that if X(x) > 0 for all x ∈ R, then the following conditions should be met: 1. A > 0 (Note that this is trivially satisfied for null geodesics).
Now, for the polynomial Y (y) its discriminants are somewhat more complicated than for X(x): It is important to mention that here we omitted a discriminant ∆ y , mainly due to it being a large expression but also because independently of its sign, (15) can have real roots, which is what we desire for this polynomial.
Just from the fact that Y (0) = −X(0) = −B it can be seen that the traveler would not be able to touch the ring singularity at x = y = 0. However, to travel through the WH in a plane y = y 0 , there should be some y 0 in which Y (y 0 ) > 0. For that matter, we search for cases where (15) has real roots.
The conditions on the discriminants (18) for four real roots in Y (y) = 0 are: P 1 < 0 and P 2 < 0. So, if X(x) > 0 for all x ∈ R then A > 0, and it is sufficient to impose that: D > 0, D 2 > 4AB and K > 0 in order to fulfill said conditions on (18). See Fig. (2). Thus we have proved that, in the slowly-rotating limit, a traveler following geodesic motion in this WH could never be able to touch the singularity of the space-time. Nevertheless, satisfying the conditions on the conserved quantities studied above, it would be possible for him to freely travel back and forth both asymptotically flat universes in a plane y = y 0 with Y (y 0 ) > 0.
Of course, the behavior of the curves (14) and (15) shown in Fig.(2) is not unique and will change depending on the four constants of motion through the discriminants (17) and (18). However, given the form of the coefficients of these polynomials is not possible to find curves where: X(x) > 0 for all x ∈ R and Y (y) > 0 for all y ∈ [−1, 1].
Conclusions.-Metric (1) describes the space-time of a Kerr-like phantom WH with a ring singularity which bounds the throat that is not protected by a horizon. We have shown that, through a coordinate change and applying a slowly-rotating limit, the geodesic equation (9) can be separated into two polynomials yielding a fourth constant of motion. Careful examination of the resulting polynomials, in particular the nature of their roots, led to the conclusion that the ring singularity is untouchable by any observer traveling in geodesic motion. This result shows that the WH contains another form of cosmic censorship, as the geodesics instead of entering into some horizon, here they are deviated to the space-time on the other side of the throat. Thus, in the same way as in a normal black hole, here the space-time singularity is causally disconnected from the rest of the universe, it is untouchable with geodesics. This confirms, at least for this space-time, a new concept of cosmic censorship first proposed by Roger Penrose. Despite this, it would be possible to cross back and forth both universes through the throat of the WH on a polar plane y = y 0 . In order to do so, the inequalities on the conserved quantities here reported need to be fulfilled.