We show that Weyl-invariant dilaton gravity provides a description of black holes without classical space-time singularities. Singularities appear due to the ill behaviour of gauge fixing conditions, one example being the gauge in which theory is classically equivalent to standard General Relativity. The main conclusions of our analysis are as follows: (1) singularities signal a phase transition from broken to unbroken phase of Weyl symmetry; (2) instead of a singularity, there is a “baby universe” or a white hole inside a black hole; (3) in the baby universe scenario, there is a critical mass after which reducing mass makes the black hole larger as viewed by outside observers; (4) if a black hole could be connected with white hole through the “singularity,” this would require breakdown of (classical) geometric description; (5) the singularity of Schwarzschild BH solution is nongeneric and so it is dangerous to rely on it in deriving general results. Our results may have important consequences for resolving issues related to information loss puzzle. Though quantum effects are still crucial and may change the proposed classical picture, a position of building quantum theory around essentially regular classical solutions normally provides a much better starting point.

Black hole (BH) solutions in General Relativity (GR) typically contain space-time singularities, that is, hypersurfaces which particles or observers may hit in finite proper time, but on which curvature blows up to infinity. A textbook example is Schwarzschild black hole:

As there is a neighbourhood around singularity where space-time curvature radii become smaller than Planck length, the usual philosophy is to seek rescue in the quantum gravity. However, standard physical reasoning would prefer a situation where classical singularities are not present, even if quantum description is important. Our goal here is to argue that one can achieve this in a rather conservative way without introducing new degrees of freedom, higher derivative, or nonlocal terms in the action^{1}

The outline of the paper is as follows. In Section

Let us assume that the (physical) field content of a theory consists of the matter sector ^{2}

With Weyl invariance, an additional gauge symmetry is introduced. Though we have the additional field (dilaton

WIDG is not some exotic way of incorporating gravity into the theory. This is best seen in a so-called

Our philosophy here is to take

Before discussing general BHs, it is good to have some explicit examples for a demonstration. This section serves the purpose, and later on we will argue that the main results are generic in the WIDG description of BHs. Now, for practical purposes, we need an analytic solution, but with the nontrivial (i.e., not constant) scalar field because generic (dynamical) BH configurations are such. So, we can take one of the “hairy” spherically symmetric BH solutions from the literature, where the simplest contain just the metric and one physical scalar field as degrees of freedom. The simplest candidate appears to be BBM solution [^{3}

One such simple analytic example of “hairy” BH solution is Zloshchastiev solution [^{4}

This theory has two types of static spherically symmetric asymptotically flat BH solutions. Besides the standard Schwarzschild BH solution, for which space-time metric is as in (

The main properties and the thermodynamical behaviour of this BH are qualitatively similar to those of Schwarzschild BH. In ^{5}

We now proceed to show that, in WIDG picture,

However, classical trajectories of particles interacting with fields

^{6}

One important observable which must be addressed is the “physical proper time,” which obviously is not the proper time defined from the metric due to its Weyl noninvariance. The solution can be found in (^{7}

Which of the two possibilities is realised depends on the details of the physics near

^{8}

The figure shows horizon areas

The properties of BH solution in

On the example of BH studied above, we can understand why the new properties discovered here in

One can check that the analysis from the previous section can be repeated, with essentially the same conclusions, for

For example, let us take as our second example the string inspired theory containing the scalar field

However, our construction obviously does not work for BH solutions in which scalar field is trivial, that is, constant in the whole space-time, which include the very important case of Schwarzschild BHs. Moreover, for a large class of scalar potentials ^{9}

The full procedure of finding well-defined gauges in which solutions are regular cannot be performed simply because there are no analytic solutions for such “perturbed” Schwarzschild BHs. To get a taste of what is happening, let us consider a related problem of regulating generic spherically symmetric homogeneous solutions near a space-like singularity in GR. In GR (^{10}^{11}

There are gauges in which Schwarzschild BH is regular, obtained by using higher-derivative terms. An example, with the lowest order in derivatives, is a class of gauges obtained by applying Weyl-factor^{12}

The figure shows a nongeneric nature of Schwarzschild singularity and also a singular nature of

The emerging picture from our analysis is that

We have demonstrated that a two-derivative Weyl-invariant formulation of gravity provides a description of black holes essentially without classical singularities. Appearance of singularities in standard formulation of gravity (GR) is simply an artefact of the fact that particular choice of the gauge fixing of local scale symmetry, in which the theory is equivalent to GR, is not well defined at singularities. By studying classical black hole solutions in regular gauges, in which fundamental fields are all well defined in the whole space-time, we obtained the following main results: (1) singularity is a signal of a phase transition from broken to unbroken phase of Weyl symmetry; (2) instead of a singularity, there is a “baby universe” or white hole inside a black hole; (3) as black hole decreases its mass (e.g., by Hawking radiation), there is a critical mass after which reducing mass makes the black hole bigger as viewed by outside observers (in the baby universe scenario). We emphasise that these results are obtained in the theory which is classically completion of GR, without introducing new physical fields (normal or phantom) or higher-derivative terms in the action. In fact, our classical analysis is essentially equivalent to using particular field redefinitions, but allowing the possible natural extension of the parameter space (important in white hole scenario). The essential requirement is that there is at least one physical scalar field in the theory with the properties which Standard Model Higgs field fulfils.

These results may be of relevance for issues related to unitarity of the evolution of black holes, indicating that BHs do not evaporate completely through the process of Hawking radiation and that some information, of possibly unlimited amount, may simply go into remnant baby universe or white hole. Such scenarios may offer a resolution (see, e.g., [

It is interesting to compare our findings with those obtained in the context of cosmological singularities inside the same framework of Weyl-invariant formulation of gravity and presented in [^{13}

The analysis we presented is completely classical, and for several reasons it cannot be considered complete without mentioning quantum effects. First, one could worry about health of the Weyl invariance we used extensively, in view of the generally anomalous nature of scale symmetry. However, in theories with dilaton field, there are regularisation cases in which classical Weyl invariance is preserved in an effective action [

The author declares that there are no competing interests regarding the publication of this paper.

The author thanks L. Bonora, M. Cvitan, S. Pallua, and I. Smolić for stimulating discussions. The research was supported by the University of Rijeka under Research Support no. 13.12.1.4.05.

That some higher-derivative theories may resolve singularities is observed before (e.g., see [

In principle, one could add higher-curvature terms in the action; for example, at four derivatives, one allowed term is

As shown in [

We could instead use any of the proper “hairy” BH solutions from [

In [

Of course, not all scalars are expected to be regular on all physical configurations. For example, in General Relativity with minimally coupled scalar field

Regular Weyl invariants behave smoothly over

This is valid for all definitions of time coordinate

In the example from the previous section, potential (

We also see that the BH solution (

The

In these gauges, classical action contains higher-derivative terms typically leading to new degrees of freedom and/or breaking of unitarity. For this reason, results obtained in such gauges should be treated with care.

After the first version of this paper was submitted, a paper [