Black Holes in Multi-Fractional and Lorentz-Violating Models

Eur. Phys. J. C (2017) 77:335 DOI 10.1140/epjc/s10052-017-4879-5

Regular Article - Theoretical Physics

Black holes in multi-fractional and Lorentz-violating models

Gianluca Calcagni1,a, David Rodríguez Fernández2,b, Michele Ronco3,4,c 1 Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain 2 Department of Physics, Universidad de Oviedo, Avda. Calvo Sotelo 18, 33007 Oviedo, Spain 3 Dipartimento di Fisica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy 4 INFN, Sez. Roma1, P.le A. Moro 2, 00185 Rome, Italy

Received: 29 March 2017 / Accepted: 28 April 2017 / Published online: 20 May 2017 © The Author(s) 2017. This article is an open access publication

Abstract We study static and radially symmetric black astrophysics, and even atomic physics can give complemen- holes in the multi-fractional theories of gravity with q- tary information on how matter and geometry behave when derivatives and with weighted derivatives, frameworks where the principles of general relativity and quantum mechanics the spacetime dimension varies with the probed scale and are unified or modified [1–7]. geometry is characterized by at least one fundamental length Among the most recent theories beyond Einstein grav- ∗.Intheq-derivatives scenario, one finds a tiny shift of ity, multi-fractional spacetimes [7–11] have received some the event horizon. Schwarzschild black holes can present an obstinate attention due to their potential in giving a physi- additional ring singularity, not present in general relativity, cal meaning to several concepts scattered in quantum grav- whose radius is proportional to ∗. In the multi-fractional the- ity. In particular, not only they allowed one to control the ory with weighted derivatives, there is no such deformation, change of spacetime dimensionality typical of all quantum but non-trivial geometric features generate a cosmological- gravities analytically, but they also recognized this feature constant term, leading to a de Sitter–Schwarzschild black as a treasure trove for phenomenology, since it leaves an hole. For both scenarios, we compute the Hawking tempera- imprint in observations at virtually all scales. The main ture and comment on the resulting black-hole thermodynam- idea is simple.√ Consider the usual D-dimensional action ics. In the case with q-derivatives, black holes can be hotter S = d D x −g L[φi ,∂] of some generic fields φi , where than usual and possess an additional ring singularity, while g is the determinant of the metric and ∂ indicates that the in the case with weighted derivatives they have a de Sitter Lagrangian density contains ordinary integer-order deriva- hair of purely geometric origin, which may lead to a solution tives. In order to describe a matter and gravitational field of the cosmological constant problem similar to that in uni- theory on a spacetime with geometric properties changing modular gravity. Finally, we compare our findings with other with the scale, one alters the integro-differential structure Lorentz-violating models. such that both the measure d D x → d Dq(x) and the derivatives ∂μ → Dμ acquire a scale dependence, i.e., they depend on a hierarchy of scales 1 ≡ ∗,2,.... Without any 1 Introduction loss of generality at the phenomenological level [7,11], it is sufficient to consider only one length scale ∗ (separat- Along the winding road to quantum gravity, it is easy to stop ing the infrared from the ultraviolet). The explicit functional by and get absorbed by any of the local views offered by form of the multi-scale measure depends on the symme- the scenery we find when classical general relativity (GR) is tries imposed but it is universal once this choice has been abandoned and the territory of pre-geometry, modified grav- made. For instance, theories of multi-scale geometry where D μ μ ity, discrete spacetimes, and all the rest, is entered. The ques- the measure d q(x) = μ dq (x ) is factorizable in the tion of how gravity is affected when it becomes quantum or coordinates are called multi-fractional theories and the D is changed by phenomenological reasons receives different profiles qμ(xμ) are determined uniquely (up to coefficients, answers according to the scale of observation; cosmology, as we will discuss below) only by assuming that the spacetime Hausdorff dimension changes “slowly” in the infrared a e-mail: [email protected] [7,11]. Below we will give an explicit expression. Quite sur- b e-mail: [email protected] prisingly, this result, known as second flow-equation theo- c e-mail: [email protected] 123 335 Page 2 of 17 Eur. Phys. J. C (2017) 77 :335 rem, yields exactly the same measure one would obtain by briefly the latter, we give a self-consistent discussion of demanding the integration measure to represent a determinis- the presentation issue in Sect. 2.1. In Sects. 2.2–2.4,we tic multi-fractal [8]. There is more arbitrariness in the choice study the properties of q-multi-fractional black holes, focus- of symmetries of the Lagrangian, which leads to different ing on the event horizon, the curvature singularity and the multi-scale derivatives Dμ defining physically inequivalent Hawking temperature. All these pivotal features of general- theories. Of the three extant multi-fractional theories (with, relativistic black holes are deformed by multi-scaling effects. respectively, weighted, q- and fractional derivatives) two of In Sect. 2.5, we notice that, for certain choices of measure and them (with q- and fractional derivatives) are very similar to presentation, quantum modifications of the ergosphere com- each other and especially interesting for the ultraviolet behav- bined with gravitational-wave data can efficiently constrain ior of their propagator. Although a power-counting argument the multi-scale length ∗. Black holes in multi-fractional fails to guarantee renormalizability, certain fractal properties gravity with weighted derivatives are analyzed in Sect. 3 and of the geometry can modify the poles of traditional particle we find that they are standard Schwarzschild black holes with propagators into some fashion yet to be completely under- a cosmological constant term. The specific form of the solu- stood [7]. The same fractal properties can affect also the big- tion depends on a parameter , related to the kinetic term for bang singularity, either removing it or altering its structure, the measure profile. In Sect. 3.1, we focus on the solution for as it might be the case for the theories with weighted and q- = 0. Then we analyze how the evaporation time is slightly derivatives [7]. Since the fate of singularities is an important modified by multi-scale effects in Sect. 3.1.1. The simplest element to take into consideration when assessing alternative case in which = 0 is studied in Sect. 3.2, showing in theories of gravity or particle physics, the next obvious step Sect. 3.2.1 that the consequences on the evaporation time are is to check what happens for black holes in multi-fractional also of the same order. In Sect. 4, we discuss similarities and spacetimes. differences between our results and black holes in Lorentz- This issue has not been tackled before and is the goal of violating theories of gravity, such as Hoˇrava–Lifshitz gravity, this paper. We study static1 and spherically symmetric black- non-commutative and non-local models. Finally, in Sect. 5 hole solutions in two different multi-fractional theories, with we summarize our results and outline possible future plans. q-derivatives and with weighted derivatives. In both cases, Throughout the paper, we work with kB = h¯ = c = 1, if the background-independent gravitational action has been not specified differently. known for some time but only cosmological solutions have been considered so far [9]. We find interesting departures 1.1 A note on terminology and fractional calculus from the Schwarzschild solution of GR. The size and topology of event horizons and singularities are indeed deformed Multi-fractional theories propose an extension of certain by multi-fractional effects. Conceptually, the interest of this aspects of fractional calculus to a multi-dimensional (D > 1 resides in the fact that it represents a top-down example where topological dimensions) multi-scale (scale-dependent scal- small-scale modifications of standard GR affect (even if in ing laws) setting. In this paper, we will not need the elegant a tiny way) the physics on large scales, namely the structure tools of fractional calculus [12–16] but some clarification of of the Schwarzschild horizon. Of the two solutions found in terminology may be useful. In all multi-fractional theories, the case with weighted derivatives, in one there is no defor- the ultraviolet part of the factorizable integration measure μ μ mation on the horizon radius, while in the other there is. The μ dq (x ) is, for each direction, the measure of fractional Hawking temperature (hence, in principle, also black-hole integrals, as discussed at length in [16]. The only (but impor- thermodynamics) is modified in both multi-fractional theo- tant) detail that changes with respect to the fractional inte- ries. grals defined in the literature [12–15] is the support of the In all the cases, we restrict ourselves to small deforma- measure, in this case the whole space [7,8,16] instead of the tions due to anomalous effects, consistently with observa- half line. The scaling property is the same as in fractional tional bounds on the scales of the geometry [7]. Because integrals and one can consider the multi-fractional measure of this, and as confirmed through computations, all the pre- as a proper multi-scale, multi-dimensional extension2 of the dictions we make (for instance, deviations in the evapora- latter. Hence the name of this class of theories. tion time of black holes) correspond to tiny deviations with On the other hand, only the theory dubbed Tγ [7] features respect to the standard framework. (the multi-scale extension [7,8] of) fractional derivatives and Our paper is organized as follows. In Sect. 2, we ana- we will not employ this naming for anything else. The models lyze static and spherically symmetric black-hole solutions in discussed in this paper have other types of operators, called multi-fractional gravity with q-derivatives. Having reviewed q-derivatives and weighted derivatives, neither of which is

1 We take the Schwarzschild solution because it is the simplest one that 2 A multi-dimensional extension of fractional operators in classical describes a black hole. mechanics was first considered in [17]. 123 Eur. Phys. J. C (2017) 77 :335 Page 3 of 17 335 fractional. In particular, q-derivatives may be regarded as calculated. In the fractional frame, where the integration mea- an approximation of fractional derivatives [7, question 13] sure gets non-trivial contributions d D x v(x) = d D x(1+...) −1 μ but they are much simpler than them. In [8], the name “q- and derivatives are modified into operators vμ (x )∂μ, one derivative” was inspired by the coordinate labels in classical sees departures from GR. This point will be further explained mechanics and was consistently used in subsequent papers and discussed in Sect. 2.1. to indicate a specific operator we will discuss later. Unfor- In the light of Eqs. (1)–(3), it is not difficult to realize that tunately, this is the same name of another, quite different the solutions to Einstein equations are the same of GR when operator introduced by Jackson in 1909 [18] and utilized they are expressed in qμ coordinates, but non-linear modi- in Tsallis thermodynamics [19]. The difference in context fications appear when we rewrite the solution as a function will avoid confusion between our q-derivatives and Jackson of xμ by using the profiles qμ(xμ). In the first part of this derivatives. work, we shall show that these multi-fractional modifications affect not only the event horizon and the curvature singularity but also thermodynamic properties of black holes such as 2 Schwarzschild solution in the multi-fractional theory the Hawking temperature. with q-derivatives After having reviewed the issue of presentation in Sect. 2.1, we shall show in Sect. 2.2 that the position of the hori- In the multi-fractional theory with q-derivatives, gravity is zon of the Schwarzschild black hole is generally shifted in rather straightforward to work out [9]. In fact, one only has multi-fractional gravity with q-derivatives. Depending on the to make the substitution xμ → qμ(xμ) everywhere in the choice of the presentation and also on the way we interpret action. In other words, ordinary derivatives ∂μ are replaced the existence of a presentation ambiguity, the curvature sin- μ μ −1 μ by [1/vμ(x )]∂/∂x = vμ (x )∂μ (no index summation), gularity can remain unaffected or an additional singularity (or μ where vμ = ∂μq . As a result, the Riemann tensor in this even many additional singularities, if we take into account theory is logarithmic oscillations; see below) can appear, or there can be a sort of quantum uncertainty in the singularity position (Sect. 2.3). As we shall explain exhaustively in the follow- q ρ 1 q ρ 1 q ρ q τ q ρ q τ q ρ Rμσν = ∂σ μν − ∂ν μσ + μν στ− μσ ντ, vσ vν ing, the interpretation of the results depends on how we inter- (1) pret the presentation ambiguity in multi-fractional theories. A general feature is that those extra singularities we find are where the Christoffel symbol is non-local because they have a ring topology. Evaporation and the quantum ergosphere are considered in Sects. 2.4 and 2.5, respectively. q ρ 1 ρσ 1 1 1 μν = g ∂μgνσ + ∂ν gμσ − ∂σ gμν . (2) 2 vμ vν vσ 2.1 The choice of presentation Finally, the q-version of the Einstein–Hilbert action reads Before analyzing the Schwarzschild solution in the multi- 1 √ fractional formulation of gravity with q-derivatives, we q S = d D xv(x) −g(q R − 2 ) + S , (3) 2κ2 m review the so-called problem of presentation. We refer to [7,10,20] for further details. In [10], the presentation is μ where v(x) = μ vμ(x ) and Sm denotes the matter action. described as an ambiguity of the model we have to fix. In As the reader has certainly noticed, there is no difference [20], an interesting reinterpretation of the presentation issue between GR and multi-fractional gravity with q-derivatives is proposed: it would be not an ambiguity to fix but, rather, a when we write the latter in terms of qμ coordinates. In manifestation of a microscopic stochastic structure in multi- fact, the geometric coordinates qμ provide a useful way fractional spacetimes. Here we are going to allow for both of re-writing the theory in such a way that all non-trivial possibilities [7]. aspects are hidden. However, the operation we described as Geometric coordinates qμ(xμ) correspond to rulers which “xμ → qμ(xμ)” is only a convenient way of writing this the- adapt to the change of spacetime dimension taking place at ory from GR and it should not be confused with a standard different scales. However, our measuring devices have fixed coordinate change trivially mapping the physical dynamics extensions and are not as flexible. For this reason, physi- onto itself. The presence of a background scale dependence cal observables have to be computed in the fractional frame, (a structure independent of the metric and encoded fully in i.e., the one spanned by the scale-independent coordinates the profiles qμ(xμ), which will be given a priori) introduces a xμ. This poses the problem of fixing a fractional frame xμ, preferred frame (called fractional frame, labeled by the frac- where we can make physical predictions. To say it in other tional coordinates xμ) where physical observables must be words, while dynamics formulated in terms of geometric 123 335 Page 4 of 17 Eur. Phys. J. C (2017) 77 :335 coordinates is invariant under q-Poincaré transformations responding to, respectively, the so-called initial-point and μ μ μ ν ν μ q (x ) = ν q (x )+a , physical observables are highly final-point presentation), notably excluding x = 0. Then the x-frame dependent since, being them calculated in the frac- geometric distance q(x) =|q(xB) − q(xA)| is tional picture, they do not enjoy q-Poincaré (nor ordinary 3 α Poincaré) symmetries. As a consequence, multi-fractional ∗ x q(x) = x ± , (5) predictions for a given observable hold only in the selected α ∗ fractional frame. Thus, the choice of the presentation corresponds to fix a physical frame. We shall also comment on where the sign depends on the presentation choice (+, initial- another way of looking at the problem of the presentation, point presentation; −, final-point presentation). As men- which has been proposed recently [20]. According to this tioned, there is also a different way of interpreting the pre- new perspective, which we call “stochastic view” in contrast sentation issue, motivated in [20]. The multi-fractional cor- with the “deterministic view” where one must make a frame rection to spacetime intervals may be regarded as a sort of choice, the presentation ambiguity has the physical interpre- uncertainty on the measurement, so that the two presentation of an intrinsic limitation on the determination of dis- tations in (5) correspond to a positive or a negative fluc- tances due to stochastic properties of some multi-fractional tuation of the distance. From this point of view, we do theories.4 In this paper, we shall follow both possibilities not have to choose any presentation at all, since such an and underline how the interpretation of the results change ambiguity has the physical meaning of a stochastic uncer- according to the view we adopt. tainty. Interestingly, limitations on the measurability of dis- With the aim of defining fractional spherical coordinates tances can easily be obtained also by combining basic (and, in particular, the fractional radius), we are particularly GR and quantum-mechanics arguments, thereby confirming interested in analyzing the effect of different presentation that multi-fractional models encode semi-classical quantum- choices on the definition of the distance in multi-fractional gravity effects [20].5 Strikingly, this also provides an expla- theories. We will define a radial coordinate sensitive to the nation for the universality of dimensional flow in quantum- presentation. In the q-theory in the deterministic view, there gravity approaches: its origin is the combination of very basic are only two choices available, each representing a separate GR and quantum-mechanics features. model. On the other hand, according to the stochastic inter- In this section, we are interested in studying the pretation of the presentation ambiguity [20], there is no such Schwarzschild solution in the multi-fractional theory with q- proliferation of models because the two different available derivatives. To this aim, we first have to transform the multi- choices give the extremes of quantum uncertainty fluctua- fractional measure to spherical coordinates. This represents tions of the radius. a novel task since the majority of the literature focused on Webegin by recalling that, in the theory with q-derivatives, Minkowskian frames or on homogeneous backgrounds. Let there is a specific relation between the distance x = us start from the Cartesian intervals analyzed above. If we − > xB xA 0 measured in the fractional frame and the center our frame in spherical coordinates at xA, then we see > (unphysical) geometric distance q 0 (positive in order that x = r provided the angular coordinates are θA = θB to have a well-defined norm). To be more concrete, let us and φA = φB. Thus, we can rewrite Eq. (5)as consider the binomial measure α μ α ∗ r μ μ μ ∗ μ x q(r) = r ± . q (x ) = x + sgn(x ) . (4) α (6) α ∗ ∗

This is the simplest measure entailing a varying dimension Here we have defined q(r) ≡ q(r). In the deterministic with the probed scale and is a very effective first-order coarse- view, this formula states that the radius acquires a non-linear graining approximation of the most general multi-fractional modification whose sign depends on the presentation. In the = measure [7,11]. Let D 1 for the sake of the argument. stochastic view, we do not have any non-linear correction Changing the presentation corresponds to making a transla- of the radius but, rather, the latter is afflicted by an intrin- ( ) → ( ) = ( − ) tion: q x q x q x x . The second flow-equation sic stochastic uncertainty and it fluctuates randomly between theorem [11] fixes the possible values to x = xA, xB [7] (cor- ∗ α ∗ α r + α (r/∗) and r − α (r/∗) . In the first case, we just have a deformation of the radius, while in the second case we are 3 See Ref. [21] for a recent discussion of these symmetries. suggesting that a stochastic (most likely quantum [20]) fea- 4 Such a way of interpreting the presentation issue might hold rigor- ture comes out as a consequence of multi-fractional effects, ously only for the multi-fractional theory with fractional derivatives (but this point is still under study). On the other hand, the multi-fractional theory with q-derivatives is known to be an approximation of the theory 5 For this reason, and with a slight abuse of terminology, we will with fractional derivatives in the infrared [7], which guarantees that it interchangeably call these fluctuations of the geometry “stochastic” or also admits the stochastic view when not considered per se [20]. “quantum.” 123 Eur. Phys. J. C (2017) 77 :335 Page 5 of 17 335 namely the radius acquires a sort of fuzziness due to multi- both from a qualitative and a quantitative point of view. For fractional effects. instance, in the absence of log oscillations, according to the Including also one mode of log oscillations, which are initial-point presentation we shall find the horizon radius present in the most general multi-fractional measure [11], in rh r0 − δ, where r0 = 2MG is the usual Schwarzschild the spherical-coordinates approximation Eq. (6) is modified radius and δ>0 will be introduced later (i.e., a smaller hori- by a modulation term: zon with respect to GR), while rh r0 + δ (a bigger horizon with respect to GR) if we choose the final-point presenta- α ∗ r tion. On the other hand, in the stochastic view the results q(r) = r ± Fω(r) , α (7a) ∗ obtained with the initial-point and the final-point presenta- r r tions will be interpreted as the extremes of fluctuations of Fω(r) = 1 + A cos ω ln + B sin ω ln . (7b) ∞ ∞ relevant physical quantities. Then the horizon will be the one of GR but it will quantum-mechanically fluctuate around its Here A < 1 and B < 1 are arbitrary constants and ω is the classical value, r r0 ± δ. This shows how non-trivial local frequency of the log oscillations. The ultra-microscopic scale quantum-gravity features can modify macroscopic properties ∞ is no greater than ∗ and can be as small as the Planck such as the structure of black-hole horizons. length [7,20]. Notice that the plus sign is for the initial-point To summarize, we are going to analyze the multi- presentation, the minus for the final-point one, and both signs fractional Schwarzschild solution in six different cases: are retained in the interpretation of the multi-fractional modifications as stochastic uncertainties. The polynomial part of 1. in the deterministic view with the initial-point presenta- Eq. (7a) features the characteristic scale ∗ marking the tran- tion; sition between the ultraviolet and the infrared, regimes with 2. in the deterministic view with the final-point presenta- a different scaling of the dimensions. On the other hand, the tion; oscillatory part Fω(r) is a signal of discreteness at very short 3. in the stochastic view, where the presentation ambiguity distances, due to the fact that it enjoys the discrete scale corresponds to an intrinsic uncertainty on the length of invariance Fω(λωr) = Fω(r), where λω = exp(−2π/ω). the fractional radius, Averaging over log oscillations yields Fω =1 and Eq. (6) [8]. Indeed, in the stochastic view, the logarithmic oscillatory without and with log oscillations. part is regarded as the distribution probability of the measure that reflects a non-trivial microscopic structure of fractional 2.2 Horizons spaces [20]. We want to take expression (6) or the more general (7a)as Looking at Eqs. (1)–(3) and recalling the related discussion, it our definition of the radial geometric coordinates, while we is easy to realize that the Schwarzschild solution in geometric + leave the measure trivial along the remaining 2 1 directions coordinates qμ(xμ) (as well as all the other GR solutions) is a ( ,θ,φ) t . We will consider modifications in the radial and/or solution of the q-multi-fractional Einstein equations. Explic- time part of the measure for the theory with weighted deriva- itly, the Schwarzschild line element in the multi-fractional tives, while still leaving the angular directions undeformed. theory with q-derivatives is given by Note that q(r) = [q1(x1)]2 +[q2(x2)]2 +[q3(x3)]2 − (assuming the spherical system is centered at xμ = 0). In r r 1 q ds2 =− 1 − 0 dt2 + 1 − 0 dq2(r) fact, we derived Eq. (6) passing to spherical coordinates in the q(r) q(r) fractional frame and, of course, this is not equivalent to hav- +q2(r)(dθ 2 + sin2 θdφ2), (8) ing geometric spherical coordinates [10]asin(6). However, it is not difficult to convince oneself that the difference between where r0 := 2GM, M is the mass of the black hole and q(r) q(r) and [q1(x1)]2 +[q2(x2)]2 +[q3(x3)]2 is negligible is a non-linear function of the radial fractional coordinate r, with respect to the correction term in (6) at sufficiently large given by Eq. (6) in the case of the binomial measure without scales, which justifies the use of the spherical geometric coor- log oscillations and by Eq. (7a) in their presence. dinate q(r) as a useful approximation to the problem at hand. Our first task is to study the position of the event horizon. Notice, incidentally, that the geometric radius in the theory As anticipated, fixing the presentation we will find that the with fractional derivatives is q(r) exactly [7]. horizon is shifted with respect to the standard Schwarzschild In Sect. 2.2, we will analyze the effects of this multi- radius r0. In particular, choosing the initial-point presentation fractional radial measure on black-hole horizons and, con- the radius becomes smaller, while it is larger than the stan- sequently, on the Hawking temperature of evaporation. We dard value 2GM in the case of the final-point presentation. will see that, as announced, the initial-point presentation and The two shifted horizons obtained by fixing the presenta- the final-point presentation will produce different predictions tion can also be regarded as the extreme fluctuations of the 123 335 Page 6 of 17 Eur. Phys. J. C (2017) 77 :335

Schwarzschild radius, if we interpret the presentation ambiguity as an intrinsic uncertainty on lengths coming from a stochastic structure at very short distances (or, equivalently, as a semi-classical quantum-gravity effect) according to [20]. From this perspective, the horizon remains r0 but now it is affected by small quantum fluctuations that become relevant for microscopic black holes with masses close to the multi- fractional characteristic energy E∗ ∝ 1/∗, i.e., when the Schwarzschild radius becomes comparable with the multi- fractional correction. From Eq. (8), the equation that determines the fractional event horizon rh is ( ) q(rh) = r0 , (9) Fig. 1 Behavior of the geometric radius q r as a function of the fractional radius r with α = 1/2and∗ = 1. The solid line is the rela- valid even for the most general multi-fractional measure tion for the initial-point representation; choosing instead the final-point (which we have not written here but can be found in [11]). presentation, we find the dotted line;thedashed line is the ordinary case q(r) = r. The interpretation of multi-fractional corrections as Looking at this implicit formula for rh in the case (7a), it is quantum/stochastic uncertainties would make the dashed line fuzzy by evident that the initial-point rh is inside the Schwarzschild adding random fluctuations between the two other curves in the limit horizon and, on the opposite, the final-point rh stays out- of large fractional radius r. Then, once we enter into the regime where side the Schwarzschild horizon. However, in order to make r ∼ δr (i.e., r < 1 in the plot), it is no longer allowed to talk about a radial distance r according to the stochastic view an explicit example and also to get quantitative results, let us restrict ourselves to the coarse-grained case without log oscillations. Then the above equation simplifies to fractional effects. The bottom line is that the singularity is still present but the causal structure of black holes generally 1−α ∗ α changes. In fact, novel features appear both for the final- r ± r = r . (10) h α h 0 point presentation and the case of a fuzzy radius. Consider first the measure without logarithmic oscillations. (i) In both If we also fix the exponent by choosing α = 1/2 (a value that the initial-point and the final-point presentations, there is no has a special role in the theory [7,20]), we can easily solve departure from the GR prediction on the curvature singularity the horizon equation analytically, obtaining at the center of the black hole, since q(0) = 0 (for the most general factorizable measure). (ii ) However, and contrary ip = + − 2 + < rh 2 ∗ r0 2 ∗ r0 ∗ r0 (11) to what one might have expected, if we choose the final- point presentation, a second essential singularity appears. for the initial-point presentation, while In fact, the geometric radius in the final-point presentation 1−α α q(r) = r −(∗ /α)r has two zeros where the line element fp = + 2 + − > = rh r0 2 ∗ r0 ∗ 2 ∗ r0 (12) (8) diverges, one at r 0 and one at the finite radius

− 1 for the final-point presentation. The superscripts distinguish r = α 1−α ∗ ⇒ r ∼ ∗ . (14) the two possibilities. On the other hand, following the interpretation of [20], we would have The second expression stems from the fact that the second flow-equation theorem [11] leaves freedom in picking the 2 /α α rh = r0 ± δ(r), δ(r) := 2 ∗ + r0∗ − 2∗ , (13) prefactor “ ∗ ”inEq.(4) and, making an -independent choice, the numerical factor in (14)isalwaysO(1).This where δ has the meaning of uncertainty on the position of r = ∗ locus corresponds to a ring singularity that is not the event horizon generated by the intrinsic stochasticity of present in the Schwarzschild solution of GR. (iii) Finally, in spacetime. In Fig. 1, we show the geometric radius q(r) as the stochastic view the reader might guess that the singularity a function of the fractional radius r for the two different is resolved due to multi-fractional (quantum) fluctuations of presentations we consider. the measure. Unfortunately, this is not the case. In fact, the origin r = 0 represents a special point because δ(0) = 0 2.3 Singularity and it does not quantum fluctuate. Therefore, in the origin multi-fractional effects disappear and the theory inherits the The next task is to study whether and how the curvature singularity problem of standard GR. Let us also mention that singularity of the Schwarzschild solution is affected by multi- stochastic fluctuations become constant in the limit α → 0 123 Eur. Phys. J. C (2017) 77 :335 Page 7 of 17 335

the presence of logarithmic oscillations of the measure, the Hawking temperature collapses to the standard behavior in the limit of large r0, while for small radii we encounter a series of poles in correspondence with the zeros of the geometric radius (see the previous subsection and, in particular, Fig. 2). The Hawking temperature can be deﬁned in the following manner: ip,fp := 1 d − r0 . Th 1 (15) 4π dr q(r) = ip,fp r rh

Imposing the same restrictions we made above for the hori- Fig. 2 Semi-log graph showing the behavior of the geometric radius zon, we can find the analytic expression for the multi- (7a) as a function of the fractional radius r,withA = B = ω = 1. fractional Hawking temperature: = = −2 α = / In this figure, ∗ 1and ∞ 10 .Here 1 2 and we chose the initial-point presentation. There are periodic zeros of q(r) which ∗ are additional singularities for x ∗, at ultra-short distances from the r0 1 + 2rip origin. The qualitative trend does not depend on the chosen presentation. h ip = , Cosmological observations constrain the amplitudes in the measure to Th 2 (16) values that avoid these singularities π ip + ip 4 rh 2 ∗rh − ∗ and the singularity might actually be avoided. However, α = r0 1 fp 2rh 0 is not a viable choice in the parameter space, unless log fp = , Th 2 (17) oscillations are turned on. We will do just that now. fp fp 4π r − 2∗r Considering the full measure (7a), we find that not only is h h the singularity not resolved, but in principle there may also ip,fp which, of course, reduce to lim → T = T := be other singularities for r = 0 due to discrete scale invari- ∗ 0 h H0 1/(4πr ) in the standard case. As expected, there are no ance of the modulation factor Fω(r). To see this in an analytic 0 appreciable effects at large distances r ∗ and the correct form, we first consider a slightly different version of the log- 0 α GR limit is naturally recovered. Given that, we can ask our- oscillating measure (7a), q(r) =[r + (∗/α)(r/∗) ]Fω(r), selves what happens to micro (primordial) black holes with where the modulation factor multiplies also the linear term. Schwarzschild radius close to or even smaller than ∗.Again This profile is shown in Fig. 2. The geometric radius van- we shall discuss all the three possibilities regarding the pre- ishes periodically at r = exp√(−nβ±)∞, n = 0, 1, 2,..., sentation. Let us start with the initial-point case and make an where β± = arccos[(A ± B A2 + B2 − 1)/(A2 + B2)]. expansion of Eq. (16) up to the first order in ∗ for r ∗: Since −1 ≤ A ≤ 1 and −1 ≤ B√ ≤ 1, the parameter β± 0 is well defined only when |B|≥ 1 − A2. In general, also ∗ 2∗ in the actual case (7a), these extra singularities appear only TH = TH0 > TH0 . (18) 2πr 2 r when one or both amplitudes A and B take the maximal 0 0 value |A|∼1 ∼|B|. Fortunately, observations of the cos- Thus, multi-fractional micro black holes are hotter than their mic microwave background constrain the amplitudes to be GR counterparts, which means that they should also evapo- smaller than about 0.5[22], which means that some protec- rate more rapidly. Such a result is somehow counter-intuitive tion mechanism avoiding large log oscillations is in action. since we found that, in the presence of putative quantum- This is also consistent with the fact that, in fractal geome- gravity effects (here consisting in a non-trivial measure), not try, these oscillations are always tiny ripples around the zero only is the information paradox [23–27] not solved, but it mode. even gets worse.6 This can be noticed immediately by comparing the solid line in Fig. 3 with the usual behavior repre- 2.4 Evaporation temperature sented by the dashed line. In the final-point presentation, the modification of the We continue the analysis of the Schwarzschild solution in Hawking temperature is given by Eq. (17), where the event multi-fractional gravity with q-derivatives by studying the fp horizon at which Th has to be evaluated is defined in Eq. thermodynamics of the black hole in the absence of log oscillations. In particular, we calculate the Hawking temperature 6 See Ref. [28] for a similar conclusion in the context of loop-quantum- for both presentations and compare it with the GR case. In gravity black holes. 123 335 Page 8 of 17 Eur. Phys. J. C (2017) 77 :335

2.5 Effects on the quantum ergosphere

In Ref. [29], it was shown that the recent discovery of gravitational waves can provide, at least in principle, a tool to place observational constraints on non-classical geometries. In particular, a way to obtain an upper bound on the multi-fractional length ∗ consists in comparing the mass shift M, due to quantum ﬂuctuations of the horizon, with the experimental uncertainty δMBH on the mass of the ﬁnal black hole in the GW150914 merger. Such a mass shift M can be related to the appearance of a quantum ergosphere (see Refs. [29,30]). Here we want to reconsider this analysis in the framework of

Fig. 3 Behavior of the Hawking temperature TH as a function of r0, the multi-fractional theory with q-derivatives. In other words, for α = 1/2and∗ = 1. The solid line is the black-hole solution in we are going to study the formation of the quantum ergo- the multi-fractional theory with q-derivatives in the initial-point pre- sphere in the multi-fractional Schwarzschild black hole (8) dotted line dashed sentation; the is for the final-point presentation; the with the objective to see if it is possible to find constraints line represents TH for the GR Schwarzschild solution. If we regard the multi-fractional part as a quantum uncertainty on the radius and the pre- on ∗. In this section only, we ignore log oscillations. sentation ambiguity as the two possible signs for the fluctuations, then The mass shift M is related to a corresponding change TH would quantum fluctuate around the standard Hawking temperature of the radial hypersurfaces q(r) by q(r) = 2MG.In order to find the width r of the ergosphere, we have to plug Eq. (6) into the above expression, thereby obtaining

(12). As the reader can easily understand by looking at Fig. r = MG =: , −α MG (19) 3, the behavior is even worse with respect to the initial-point 2 1 ± (∗/r)1 fp case. In fact, the dotted line (which represents Th as a function of r0) increases more rapidly than the other two curves where the plus sign holds for the initial-point presentation as the black-hole mass decreases. Therefore, again we find and the minus sign for the final-point presentation. Accord- 2 that multi-fractional effects do not cure the GR information ing to Ref. [29], noting that M ∼ E∗/MBH in the absence paradox but make it even more prominent. However, it is of multi-fractional effects [30] (here E∗ is some quantum- fp <δ δ = interesting to look at the behavior of Th for very small gravity scale) and imposing M MBH (with MBH ( black holes. We can see that there is a value of r0 where O M) if we are considering the GW150914 merger), one the Hawking temperature vanishes. Thus, in multi-fractional obtains a very high bound, E∗ < 1058 GeV. In the case q-gravity in the final-point√ presentation, (micro) black holes of the multi-fractional theory with q-derivatives, if we use with r0 = (5 − 2 5)∗ ≈ 0.5∗ do not emit Hawking radia- the initial-point presentation then we have a plus sign in the tion. Even so, however, they are unstable since, as clear from denominator of Eq. (19) and the energy bound is even higher, the figure, any increase +δM or decrease −δM of their mass since M <M. Things completely change with the final- would make them emitting rather efficiently. point presentation, where the upper bound on E∗ is The third possibility is to regard multi-fractional modifications as an uncertainty on relevant physical quantities. 1−α < δ − ∗ = ± δ E∗ MBH MBH 1 In that case, we have TH TH0 T , i.e., the Hawking r temperature fluctuates around the GR value. As for the other 1−α quantities we analyzed, the magnitude of such random fluc- ∗ = 1 − 1058 GeV . (20) tuations depends on how large ∗ is and it decreases as M r (or, equivalently, rh) increases. To summarize, the theory with q-derivative does not solve For α 1, the upper bound remains E∗ < 1058 GeV for any the information paradox of GR, a datum consistent with the sensible value of ∗. However, in the limit α → 1 the upper problems one has when quantizing gravity perturbatively bound dramatically lowers, regardless how small is the ratio here [7]. On the other hand, approximating the theory to the ∗/r. This shows that the correction to the quantum ergo- stochastic view the information paradox is not worsened and sphere, combined with gravitational waves measurements, the role of the random fluctuations in this respect is not yet can be used to severely constrain the multi-fractional theory clear. This may indicate that the theory with fractional deriva- with q-derivatives in the final-point presentation for big val- tives is better behaved than its approximation the q-theory, ues (i.e., close to 1) of α. Note, however, that values α ∼ 1 again consistent with previous findings [7]. do not have any theoretical justification. 123 Eur. Phys. J. C (2017) 77 :335 Page 9 of 17 335

Adopting the stochastic view instead, the correction term choose but not U(v). However, keeping black holes and in the denominator of Eq. (19) would result from the quantum cosmology as separate entities this restriction is lifted. uncertainty on the radius, i.e., q(r) = r ± δr. Given that, the The metric gμν in the Jordan frame is not covariantly con- only constraint coming from the quantum-ergosphere calcu- served, just like in a Weyl-integrable spacetime. For conve- −α lation is (∗/r)1 < 1. However, this inequality is always nience, we will move to the Einstein frame, which is obtained satisfied as far as we consider solar-mass or super-massive after performing the Weyl mapping black holes for which the radius r of the ergosphere exceeds − the multi-fractional length ∗ by several orders of magnitude. gμν = e gμν, (23) In this case, multi-fractional effects on the ergosphere might be relevant only for primordial (microscopic) black holes so that the action (21)inD = 4 reads with r <∗. On the other hand, according to the stochastic view, it is meaningless to contemplate distances smaller 1 4 μ − δ Sg = d x −g R − ∂μ∂ − e U . (24) than the multi-fractional uncertainty r. Consequently, we 2κ conclude that this argument cannot be used to constrain the scale ∗. In this frame, although the metricity condition ∇σ gμν = 0 is satisfied, the dependence in the measure profile cannot be 3 Schwarzschild solution in the multi-fractional theory completely absorbed. As we will see, black-hole solutions are with weighted derivatives highly sensitive to the choice of ω, which may even hinder their formation. For illustrative purposes, we will examine The gravitational action in the theory with weighted deriva- the cases = 0(ω fixed) and =−3/2(ω = 0). At this tives is similar to the one of scalar–tensor models, with point, it is important to recall a key feature of these theories. In the crucial difference that the role of the scalar field is standard GR, at the classical level one has the freedom to pick v( ) = played by the non-dynamical measure weight x either the Jordan or the Einstein frame, leading to equivalent v ( 0)...v ( D−1) v ( μ) = ∂ μ( μ) 0 x D−1 x , where μ x μq x . Since predictions; at the quantum level, these frames are inequiv- this is a fixed profile in the coordinates, one does not vary alent and one must make a choice based on some physical the action with respect to it and the dynamical equations of principle. In the multi-fractional case, the existence of the motion are therefore different from the scalar–tensor case. non-trivial measure profile v(x) that modifies the dynamics However, even if it is not dynamical, the measure profile renders both frames physically inequivalent already at the affects the dynamics of the metric so much that the resulting classical level. A natural question is which one is “preferred” cosmologies depart from the scalar–tensor case [9]. for observations. The answer is the following. Measurements As for scalar–tensor models, we can identify a “Jordan involve both an observable and an observer. Given the nature frame” (or fractional picture) and an “Einstein frame” (or of multi-scale spacetimes, both feel the anomalous geome- integer picture) related to each other by a measure-dependent try in the same way if they are characterized by the same conformal transformation of the metric. In the Jordan frame, scale, while they are differently affected by the geometry the action for multi-fractional gravity with weighted deriva- otherwise. This is due to the fact that measurement appara- tives in the absence of matter is given by [9] tus have a fixed scale and do not adapt with the changing √ geometry. In the multi-fractional field theory with weighted 1 D /β μ Sg = d xe −g R − ∂μ∂ − U(v) , derivatives and in the absence of gravity, this occurs in the 2κ2 fractional picture, while in the integer picture the dynamics (21) reduces to that of an ordinary field theory. In the presence of with gravity, the integer picture (Einstein frame) is no longer trivial (see Eq. (24)), but the interpretation of the frames remains 9ω 2 1 1 := e β + (D − 1) − , (22) the same. Therefore, the Jordan frame is the physical one [9]. 4β2 2β∗ β Physical black holes as those found in astrophysical observations can be formally described within the Einstein frame, ( ) = v( ) ω where x ln x is not a Lorentz scalar field and is while to extract observables one has to move to the Jordan an arbitrary constant (not to be confused with the frequency frame. of log oscillations). In D = 4 topological dimensions, β = β∗ = 1 is fixed by the theory. In [9], one demanded that U = 0 in order to support consistent solutions with cosmo- 3.1 Black-hole solution with = 0 logical constant. Since this quantity is measure-dependent but background independent, if we want to describe both In this section, we will examine the spherically symmetric black holes and consistent cosmologies, we have freedom to solution when the “kinetic term” of the measure vanishes: 123 335 Page 10 of 17 Eur. Phys. J. C (2017) 77 :335

2 − ≈ 8 = 0 ⇒ ω = e 2 , (25) where Tvac 34 K [33]. However, in this scenario, the 3 Stefan–Boltzmann law receives a sub-leading contribution 1 4 − as a consequence of integrating out in the presence of some Sg = d x −g R − e U . (26) 2κ2 ∞ ν → ∞ ν w(ν), w(ν) = measure proﬁle, i.e., 0 d 0 d with 7 +δw(ν) ρ = σ 4 +δρ Taking the variation with respect to gμν, we get 1 , so that T (for more details, see [22]). Nevertheless, the correction is small. Taking for instance the 1 − binomial measure Rμν − gμν R − e U = 0. (27) 2 1−α 3 We restrict to an isotropic, static and radially symmetric ν ω(ν) = 1 + , (34) geometry. Thus, our Ansatz is ν∗ μ ν 2 2 gμνdx dx =−γ (r)dt + γ (r)dr 1 2 − with α = 1/2,ν∗ 3 × 109 cm 1 [22], and integrating +γ ( ) 2 θ 2 + 2 θ φ2 . 3 r r d sin d (28) out over all frequencies, ∞ 3 After some manipulations (γ3 can be consistently set to 1), ν ρ = 240σ dνw(ν) ∼ σ T 4 + δρ , (35) the Einstein equations read (primes denote derivatives with 2πν 0 e T − 1 respect to r and the r dependence is implicit in all functions) 4725 T 9 T = (γ γ ), δρ = σ T 4 √ ζ + O , (36) 0 1 2 (29) π 4 ν∗ 2 ν∗ 16 2 γ 1 γ γ1 γ 0 = γ − γ 2 + − 1 − r 2 − 2γ + 2 , − 1 1 2 2 we get δρ(T )/(σ T 4 ) ≈ 10 4, which becomes even 2γ2 r 2γ1 r γ2 vac vac (30) smaller for lower temperatures. Since we are interested only in the order of magnitude of χ, we can just adopt the standard plus a master equation for U: power law

2 γ 1 − γ ρ ∼ 4, U =−v 1 − 2 . (31) T (37) γ2r γ1 r and set the value of χ as in (33), ignoring any other anomalous Restoring coordinate dependence, a consistent solution is contribution. Moreover, according to (33), one sees that even given by for a black hole of mass 1010 M, it is safe to assume that (GM)2 1/χ. χ r0 2 1 Let us pause for a moment and discuss one of the main γ1(r) = 1 − ± r ,γ2(r) = , U =∓v(x)χ, r 6 γ1(r) results of this paper. Just assuming a non-trivial dimensional (32) flow in the Hausdorff dimension of spacetime (i.e., a non- trivial multi-fractional measure), we have just shown that the which is a two-parameter family with a cosmological poten- simplest black-hole solution is the Schwarzschild–de Sitter tial. Several caveats are in order. First, although the functions solution, where the cosmological constant term is caused by γ1 and γ2 depend only on the radius, the “potential” term U is the multi-scaling nature of the geometry. This offers a possi- factorized in the coordinates, since it depends on the measure ble reinterpretation of the cosmological constant [34,35]asa weight v(x) (which we did not approximate by a radial profile purely geometric term arising from the scaling properties of as done in the theory with q-derivatives). Second, the exis- the integration measure. Since, in this case, there is no rea- tence of the “hair” χ = const was foreseeable since we have son to expect a huge value of χ due to quantum fluctuations considered a non-zero “potential” coupled to gravity. Third, of the vacuum energy (as it would be the case in quantum the sign in front of the r 2 term is arbitrary but, in order to get a field theory), then we do not have the problem of fine tuning Schwarzschild–de Sitter solution, we pick the minus sign. In large quantum corrections. This step towards the solution of [31,32], the cosmological constant χ was expressed in terms the cosmological constant problem is somehow analogous of a temperature Tvac by means of the Stefan–Boltzmann law, to what happens in unimodular gravity, as noted in [9]. In so that unimodular gravity, as a consequence of fixing the determinant of the metric gμν, the source of the gravitational field π 3 4 χ = 4 Tvac ≈ −66 2, is given only by the traceless part of the stress-energy ten- 2 10 eV (33) 15 mPl sor and, thus, all potential energy is decoupled from gravity

7 Since is a not dynamical measure proﬁle, we do not vary the action 8 We have employed the conversion factor 1 K = 8.6217 × 10−5 eV 28 (21) with respect to it. and mPl = 1.22 × 10 eV. 123 Eur. Phys. J. C (2017) 77 :335 Page 11 of 17 335

(see, e.g., Ref. [36]). In this way, χ appears as an integration with constant rather than a parameter of the Lagrangian [37,38]. (0) (0) (0) (0) However, unimodular gravity also has the feature of break- T = T − T ,δT(= ) =−T lim δv (43) H BH vac 0 H → ing time diffeomorphisms as recognized for the first time r rh in Ref. [39], whose consequences are still to be completely and understood. The multi-fractional scenario has the advantage of formally preserving full diffeomorphism invariance [21], χ (0) MG (0) rh MG although in this case the “diffeomorphism” transformations T = TH0, T = χ ∼ , (44) BH 2πr 2 vac 12π 3π are deformed with respect to those of general relativity. h At this point, it is interesting to discuss the causal structure ( ) where we have approximated r ∼ r 3 . Since 1 >δv(x)> of our manifold. Imposing γ1(rh) = 0, we distinguish three h h 0, one expects to get a redshift. Two comments are in order. horizon radii (r0 = 2MG) The first is that the temperature now depends on the spacetime coordinates through the non-trivial measure profile v(x), and, ( , ) 6 ( ) 2 r 1 2 ∼− MG ± , r 3 ∼ r + M2G2χ . as stated before, this implies that one can have a spacetime- h χ h 0 1 3 dependent redshift. The second is that the temperature has (0) (38) two sources: one is the standard black-hole temperature TBH ( ) and the other, T 0 , comes from the de Sitter background, (1) (2) vac rh is unphysical since it√ is negative. In order for rh to can be related to the effective temperature scale of the cos- /χ > be physical, it should be 6 MG, which means that mological vacuum energy. The equilibrium point is achieved χ (2) (3) in the small- limit rh is the cosmological horizon. rh when is the apparent inner horizon which reduces to the standard Schwarzschild radius when χ → 0. Hereafter, we shall con- (0) (0) 1 3 sider only this horizon. T = T → M = MC . (45) BH vac 2G 2χ Undoing the Weyl mapping, the solution in the Jordan frame is This condition would set a critical mass scale MC above 1 which accretion takes place at a higher rate than evapo- gμν = g . v( ) μν (39) χ ∝ / 2 x ration. Plugging in the estimate (33)(G 1 MPl), 52 23 MC ≈ 10 kg ≈ 10 M. Even for the largest monster Moreover, since in the Jordan frame the Hawking temper- black hole ever discovered so far, with M ≈ 1010 M [40], γ ( ) = ature is given by (recall that 1 rh 0), accretion cannot compete with evaporation. γ ( ) 1 1 r TH(x) = lim 4π r→rh v(x) 3.1.1 Consequences on the evaporation time of black holes γ (r)v(x) − γ (r)v(x) = 1 1 1 lim 2 It is interesting to ask oneself whether the anomalous geom- 4π r→rh v(x) etry can lead to significant differences on the evaporation γ ( ) χ 1 1 r 1 r0 1 time of black holes, such extremely massive objects, with = lim = − rh lim π r→r v( ) π 2 r→r v( ) 4 h x 4 rh 3 h x masses at least comparable with the solar mass, will have (0) 1 small Hawking temperatures. In particular, for this case, = , − TH lim (40) δρ( )/(σ 4) ≈ 8 r→rh v(x) TH TH 10 , so that the approximation (37)is well justified also here. According to the standard Stefan– (0) =| / 2 − χ / |/( π) with TH r0 rh rh 3 4 , the Hawking tempera- Boltzmann law, the power emitted by a perfect black body ture in the Einstein frame, it is immediate to notice a shift due in repose (E = M)is to the anomalous geometry. From previous work [7], we can safely infer that the contribution from the anomalous geom- dE P = σ A T 4 =− =−M˙ , (46) etry to observables is rather tiny at large scales. Hence, we h H dt write σ being the Stefan–Boltzmann constant. Note that, in this v( ) + δv( ) + O(δv2), x 1 x (41) theory, the horizon area Ah remains unchanged √ 2 so that Ah = dθdφv(x) gθθgφφ = dθdφ r sin θ r=rh r=rh ∼ (0) + δ = π 2. TH(=0) TH T(=0) (42) 4 rh (47) 123 335 Page 12 of 17 Eur. Phys. J. C (2017) 77 :335

For a process involving some energy (mass) loss, we compute characteristic time derived from αQED measurements [41], −36 −27 the time needed to jump from an initial energy Ei to a ﬁnal t∗ ≈ 10 s, ∗ ≈ 10 m, we get energy Ef . Inserting (40)into(46), ( ) − t 0 t − ≈ 10 16, (54) v(x)4 t − dE = σdt. (48) (0) A T 4 → h H r rh where we have employed Eq. (33) and ( t)0 refers to the evaporation time predicted by the standard lore. Such devia- At this point, we will consider a toy-model geometry where tion is independent of the presentation adopted. As it stands, only the time and radial directions are anomalous, v(x) = multi-fractional effects entail slight changes on the evapora- v0(t)v1(r), so that tion time on black holes, therefore coinciding with the usual 4 model in the large-scale regime. Ef v ( )4 tf 1 rh 1 − dE = 4πσ dt. (49) ( ) 4 v (t) Ei 2 0 ti 0 3.2 Black-hole solution with =−3/2 rh TH

Under the approximation (41), the right-hand side of (49) The simplest version of multi-fractional gravity with weighted can be rewritten as derivatives is in the absence of the fake “kinetic” term in the Jordan frame action, ω = 0( =−3/2). In this case, the v tf dependence cannot be eliminated in the equations of motion πσ − δv ( ) . 4 1 4 0 t dt (50) as we did before. The metric components now receive a direct ti contribution from the anomalous geometry, so that, in order Adopting the deterministic view with the initial-point pre- to preserve staticity and radial symmetry, we have to consider sentation in this last part of the analysis, we set the binomial a radial measure weight independent of angular coordinates, measure without log oscillations for each anomalous direc- v(x) = v(r). This must be regarded as an approximation of tion, the full theory because we do not have the liberty to change −α coordinates via a Lorentz transformation, which is not a sym- 1 0 t∗ metry of the theory.9 As in the case with q-derivatives, the v0(t) = 1 + δv0(t), δv0(t) = , t difference with respect to the exact case will be in sub-leading 1−α terms that do not change the qualitative features of the solu- ∗ v1(r) = 1 + δv1(r), δv1(r) = . (51) tion. Two other assumptions we will have to enforce in order r to get an idea of the solution will be that of small geometric (3) corrections and α = 1/2. Having thus cautioned the reader, Taking rh ∼ r , from (49) we get h we can proceed. 256 3 2 3 3 5/2 5/2 Considering a large-scale regime where multi-scale effects π 5G E − E + 12G 2G∗ E − E f i f i v ( ) + δv ( ) 15 are small, 1 r 1 1 r , for the black-hole metric (28) + 3χ 5− 5 + 9/2 − 9/2 we have γ = 1/γ and G 28G Ef Ei 60 2G ∗ E E 2 1 f i α 0 t∗ t γ1 ˜γ1 + δγ1,γ3 1 + δγ3, (55) = 4πσ t − 4 , (52) α0 t∗ where γ˜ = 1 − r /r − χr 2/6. At zeroth order in the M2χ = − 1 0 with t tf ti. Considering a process where we jump from expansion, the linearized Einstein equations are an initial state to a final state with zero energy, for example = = 3r δv2 + 2δγ + δγ , the evaporation of a black hole, we have Ei M0 (the initial 0 1 3 3 2(r0 − r) r mass) and Ef = Mf = 0. Then 2 r0 2 2 3 2 256 7168 0 = − 1 δγ − δγ3 + − δv π 3 2 3 + π 3 4 5χ r r 3 r 2 r 2 2 G M0 G M0 3 15 2 − δγ + δγ . (56) +256π 3 2 + 2 2χ r 2 1 1 GM0 2GM0 ∗ 12 60G M0 15 α In the deterministic-view initial-point presentation, 0 t∗ t described by means of the binomial profile in the r com- 4πσ t − 4 . (53) α0 t∗ 9 ≈ On the other hand, the Fourier transform is well defined even when Given some test black hole of mass M0 M, for the natural the measure weight is v(r), as is clear from an inspection of the plane choice α0 = α = 1/2[7] and taking the most stringent waves [7,42]. 123 Eur. Phys. J. C (2017) 77 :335 Page 13 of 17 335 ponent (51), one can easily find the non-trivial analytic 3.2.1 Consequences on the evaporation time of black holes solution We can repeat the same procedure to derive the evaporation 2 time of black holes for this specific theory. Starting from (48), δγ = ∗ r − 3 + 1 − r0 1 3 ln 1 2 8r0 4r0 r r 1 M0 dM = σt, (63) ∗ r0 ∗ 1 r 1 π 2 4 − + + − 4 0 rh TH ln 2 8r r 16 2r0 r r 0 where 3 r0∗ r − ln − 1 2 M0 2χ 16 r r0 1 dM 512 3 5r0 π M0r0 r0∗ + 1 ∗ π 2 4 3 1 r0 1 r 4 0 rh TH 5 4 δγ3 = 1 + ln 1 − − ln 1 − , 8 r0 r r r0 448 64 + π 3 M r 4χ + π 3 M r 2, (64) (57) 15 0 0 3 0 0 we can immediately derive the evaporation time and com- wherein we have imposed the standard solution (classical pare it with the one from the standard framework. For a test black hole in the presence of a cosmological constant) in the black hole with M0 ≈ M, one obtains again Eq. (54), the limit ∗ → 0. The potential U is obtained from the equations only difference being in decimals. Thus, although black-hole of motion and is non-zero for consistency: solutions and predictions for the Hawking temperature are inequivalent for the two values of considered here, devia- ∗ tions with respect to standard GR are found to be of the same U χ 1 + + O(χ 2). (58) r order.

Moreover, we can compute the new horizon radius rh. Restricting ourselves to a small deformation, 4 Comparison with other exotic black holes

The literature on black-hole solutions in quantum-gravity =ˆ + δ , ˆ = (3), rh rh r rh rh (59) approaches and alternative or modified theories of gravity is rapidly increasing [43–55]. The majority of these stud- ies involve some departure from standard Lorentz symme- we have tries, which are the local symmetries of GR. As discussed above, Lorentz transformations are not symmetries of space- 0 = γ (r ) γ (rˆ ) + δrγ (r)| =ˆ 1 h 1 h 1 r rh time when a short-distance multi-scale (in some cases, multi- ˜γ (ˆ ) + δγ (ˆ ) + δ γ˜ ( )| 1 rh 1 rh r 1 r r=ˆrh fractal) behavior is considered. Despite deep conceptual and = δγ (ˆ ) + δ γ˜ ( )| , formal differences between multi-fractional gravity and other 1 rh r 1 r r=ˆrh (60) scenarios, it may be interesting to discuss, at least qualita- so that tively, differences and similarities of black-hole solutions in some of the presently available Lorentz-violating models of ∗ gravity. δr =− 1 + r 2χ . (61) 32 0 A common feature in the wide landscape of quantum gravities is a deformation of event horizons. Departures from GR Once the horizon position is known, computing the Hawk- at very high energy (generally at the Planck scale) affect the ing temperature is straightforward: position of the event horizon of GR solutions. These modifications of the causal structure of black-hole spacetimes γ ( ) 1 1 r are usually tiny, being them suppressed by an ultraviolet TH(=0) = lim 4π r→rh v(r) energy scale, and become relevant only in the short-distance limit. As we shown and explained in this work, in multi- 1 ∗ 1 − 3r − r 3χ , (62) fractional gravity such effects are governed by ∗ (or, more π 2 0 h 12 rh rh generally, by a series of scales n, in the most general multi- fractional geometry [7]). For instance, a modification of the from which it is immediate to note that, when ∗ → 0, causal structure of black-hole solutions appears also in the (3) (0) rh rh and TH(=0) TH . Hoˇrava–Lifshitz scenario [56]. In that framework, Lorentz symmetries are broken by the choice of a preferred foliation 123 335 Page 14 of 17 Eur. Phys. J. C (2017) 77 :335 of the manifold. As a direct consequence, the theory allows for such microscopic black holes. Also in multi-fractional for signals of arbitrarily large speed. Nonetheless, in order to gravity does standard thermodynamics change, but the evap- preserve a notion of causality, a Killing horizon is invoked oration process can either get faster or slower, depending on [57–59], called universal horizon. The universal horizon is the presentation. Even if the behavior of TH improves for a a defined by the condition n ξa = 0, where n is the unit microscopic black holes, evaporation can never be avoided time-like normal vector to the space-like hypersurfaces and in the GUP scenario. On the other hand, here we found that ξa is a Killing vector field. Constant-time hypersurfaces can TH vanishes for special values of the black-hole mass, in the never cross it. In Hoˇrava–Lifshitz gravity, black holes can case of the multi-fractional theory with q-derivatives in the have either the universal horizon alone or both a universal final-point presentation. horizon and the standard Killing horizons of GR. In all the Finally, it is interesting to compare our findings in the studied solutions, the main characteristic is that such a uni- theory with weighted derivatives with the results in non- versal horizon lies always inside the Killing horizon. We have commutative models of gravity. Extending the idea of non- shown here that the horizon shrinks also in multi-fractional commutativity to a general covariant theory is still an theories with q-derivatives in the initial-point presentation, open challenge, mainly due to the clash between non- while the opposite effect takes place in the final-point presen- commutativity (which makes explicit reference to spacetime tation. Moreover, regardless of the presentation choice, we coordinates) and invariance under diffeomorphisms. Viable have shown that multiple singularities (and horizons) come ways of implementing a theory of gravitational interactions up if we take into account logarithmic oscillations of the mea- on non-commutative spaces have been proposed in the last sure. On the contrary, there are no effect on the singularity ten years and, using different assumptions, they lead to dif- in Hoˇrava–Lifshitz gravity. Such differences do not come as ferent results. A common feature is that singularities can- a surprise, since the two theories have a very similar scaling not be avoided, as we found here for multi-fractional black of the spectral dimension [60] but they do not have the same holes. However, from our perspective, the most remarkable symmetries. thing is that, in all non-commutative black-hole solutions, One of the main motivations for studying alternatives to it is possible to relate non-commutative correction terms (or modifications of) GR is the desire to solve the space- to the charges of classical black holes [70]. In particular, time singularities of GR. We have shown that, in general, the cosmological constant is generated by non-commutative singularity avoidance cannot be realized in the black holes effects. This is intriguingly similar to our result for the multi- of the multi-fractional theories with q- and weighted deriva- fractional theory with weighted derivatives [71,72]. Both a tives. This disappointing result should be compared with the collection of small no-go theorems [21] and the fact that non- achievements of both discrete-geometry theories such as loop commutative gravity is less developed prevent us from estab- quantum gravity and recently proposed non-local extensions lishing any rigorous connection between non-commutativity of GR, where singularity avoidance is a real possibility. A and multi-fractional theories. However, this analogy gives class of self-consistent theories of non-local gravity, con- further support to the possibility of interpreting charges as a sidered as a direct extension of higher-derivative models, result of non-trivial geometries [35,73]. Similar results are has been recently developed in [61,62]. In these theories, also available in modified f (R) theories of gravity involv- black-hole solutions are generally free of singularities [44] ing non-Riemannian spacetime properties as non-metricity and also cosmological solutions seem to enjoy finiteness at [74,75]. In that case, the electric charges of black holes are early times, in the form of a bounce [63]. The same outcome given a geometric interpretation [74–76]. has also been related to the possibility of having conformal invariance in the ultraviolet [64,65]. Among our findings there is the modification of the ther- 5 Conclusions modynamical properties of black holes. Departures from the Hawking temperature can be encountered in the most diverse In this paper, we studied static and spherically symmetric scenarios, for instance those with a generalized uncertainty black-hole solution in two different multi-fractional theories: principle (GUP). The GUP paradigm is an effective model with q-derivatives and with weighted derivatives. In both that can be obtained by naively combining the quantum cases, we found departures from the Schwarzschild solu- Heisenberg uncertainty principle with the limit on localiza- tion of GR. In multi-fractional gravity with q-derivatives, tion represented by the Schwarzschild radius. This combina- we considered two different views, one where the presen- tion gives rise to a generalized uncertainty principle. Black tation of the measure must be fixed and another where it holes in the GUP framework have been studied mainly in [66– reflects a stochastic uncertainty. In general, the position of 69], where it was found that sub-Planckian black holes are the event horizon changes and the Hawking temperature is non-singular and obey a modified thermodynamics. In partic- modified as described in the text. In multi-fractional gravity ular, the Hawking temperature scales with M instead of M−1 with weighted derivatives, static and spherically symmetric 123 Eur. Phys. J. C (2017) 77 :335 Page 15 of 17 335 black-hole solutions have a cosmological-constant term, i.e., might eventually screen Tγ from singularities. Our results can they are Schwarzschild–de Sitter black holes. The cosmolog- serve as a guiding line in anticipation of a thorough study of ical constant arises from non-trivial geometry and it is not this alternative multi-fractional scenario. related to quantum fluctuations of the vacuum (we focused on classical spacetimes), in analogy with what found also in Acknowledgements G.C. is under a Ramón y Cajal contract and is unimodular gravity. supported by the I+D Grant FIS2014-54800-C2-2-P. D.R.F. is supported by the FC-15-GRUPIN-14-108 Research Grant from Principado The outlook for future research may span different direc- de Asturias and partially supported by the Spanish Grant MINECO-16- tions. One is to explore in greater detail the differences and FPA2015-63667-P. M.R. thanks Instituto de Estructura de la Materia similarities with other quantum gravities discussed in Sect. 4, (CSIC) for the hospitality during an early elaboration of this work. The in particular the behavior of microscopic black holes and contribution of G.C. and M.R. is based upon work from COST Action CA15117, supported by COST (European Cooperation in Science and the possibility to give black-hole charges a purely geometric Technology). pedigree. Although the information paradox problem is not resolved according to the models examined in the present Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm work, possible non-trivial predictions related to anomalous ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, effects may arise in quantum entanglement entropy calcu- and reproduction in any medium, provided you give appropriate credit lations, since at smaller scales it is expected to get larger to the original author(s) and the source, provide a link to the Creative modifications. Another option could be to limit the attention Commons license, and indicate if changes were made. Funded by SCOAP3. to multi-fractional theories and study rotating (Kerr) black holes with the hope of finding novel phenomenology. A third possibility, however, is the following. 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