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Weinberg [15, 16]. Asymptotic safety can be de- asymptotic darkness for large black holes, as we scribed in the following heuristic way. Consider now discuss. a truncated (for technical reasons, see [19]) ef- Obviously, since any theory of quantum grav- fective action for 4-d metric gravity with cosmo- ity must reduce to general relativity, a black hole logical constant coupled to matter must form for an experiment localized to scales much smaller than the Planck length as the 4 2 I = d x√ g Λ(µ)+ κ0(µ)R + κ1(µ)R + Z − − necessary energy would be much greater than ab the Planck energy. The resulting black hole κ2(µ)RabR + Lm ... would be large compared to the Planck length (1)
and so governed by ordinary general relativity. containing a finite set of higher derivative terms This UV/IR connection has been pointed out by Banks among others [7, 11]. Formation of with associated coefficients, κ1,2,3... in addition to the usual cosmological constant and Einstein- such a hole would certainly preclude transmis- Hilbert terms. µ is an energy scale associated sion of any information generated by a local- with an experiment and the couplings are al- ized experiment out to infinity, however it does lowed to run with µ. Since gravity couples to the not a priori imply that an experiment cannot matter stress-energy tensor, the actual strength locally measure arbitrarily small scales. Forbid- of the gravitational coupling in a given exper- ding the measurement of arbitrarily small scales iment is (heuristically) totally controlled by µ. requires the horizon to be inside the region of It is therefore useful to consider the dimension- the experiment. Consider for the moment pure general relativity and spherical symmetry, such less couplingκ ¯I between gravity and matter d that the resulting black hole is Schwarzschild. generated by each term,κ ¯I = µ /κI , where In this case any 2-sphere inside the black hole d = [κI ], i.e the mass dimension of the coeffi- cient (as opposed to the operator itself). So, for necessarily contains an apparent horizon from example, the Einstein-Hilbert term generates a which information cannot escape and there is a 2 minimum length. However, for Planck sized re- dimensionless couplingκ ¯0 = µ /κ0, which is the familiar Gµ2 interaction term. Higher cur- gions we are not in pure general relativity any- vature terms generate dimensionless couplings more - the gravitational constants may run sig- with lower powers of µ. Some of these di- nificantly and higher curvature terms become mensionless couplings could blow up under the relevant. Hence the small scale geometry may change significantly. If, for example, a pair of renormalization group flow of the κI ’s (with re- horizons formed outside a localized experiment spect to the scale µ). In particular, if κ0 did then there would be no trapped surface inside not run,κ ¯0 diverges as µ . In asymptotic safety, the dimensionless couplings→ ∞ remain finite the region of the experiment and therefore no and flow to a UV fixed point instead as µ . minimum length. A highly localized experiment If this is true then metric gravity remains→ ∞ a would still create a black hole as seen by an ob- valid, and in principle predictive description, server at infinity, but that is irrelevant to ar- down to arbitrarily short distances and high en- guing that the experiment can not resolve dis- ergies. While we do not know whether quantum tances below the Planck length. general relativity is asymptotically safe, quite For our simple thought experiment, consisting a few truncations offer evidence of a UV fixed of a ball of uncharged matter inside a sphere point [17–21]. Ω of proper radius L with total energy E Since asymptotic safety requires that quan- 1/L, the necessity of a horizon forming inside∝ tum gravity is a local field theory valid to arbi- Ω as L 0 vanishes for asymptotic safety. trarily high energies, while asymptotic darkness There is→ no theorem that it doesn’t form, and indicates that quantum gravity is not, there is in fact whether it does or not depends on the an apparent tension between these two ideas. size of higher curvature terms which are cur- In this work we investigate this problem by re- rently unmeasured and not uniquely calculated examining the hoop conjecture in the context in asymptotic safety, but the strong argument of the minimum length argument using one of from general relativity and quantum mechan- the results of asymptotic safety - the running of ics no longer holds. Hence the minimum length coefficients such thatκ ¯I remains finite. In par- argument for localized experiments falls apart. ticular we concern ourselves with the necessity We further conjecture that for L << LP lanck of one or more horizons forming for an arbitrar- an interior horizon forms outside of Ω but with ily localized experiment and, more importantly, radius less than LP lanck. While this would where they form. We then leverage a subtlety not necessarily have an effect on local experi- to the minimum length argument that allows ments, it does mean that one cannot transmit us to perhaps reconcile asymptotic safety with information from the local experiment to infin- 3 ity, as we noted above. The L << LP lanck of the conjecture in terms of trapped surfaces states also have a large outer horizon that ap- [25, 26]/apparent horizons [27–29]. Spherical proaches the ordinary Schwarzschild horizon of symmetry and the momentarily stationary as- general relativity. This is similar to the mod- sumption are both used for simplicity. We note, ified Schwarzschild solution of Reuter and Bo- however, that the stationary assumption also nanno [22] that possesses an interior and exte- intuitively matches the idea of scaling a static rior horizon.[35] In this way, the extreme UV system down to try to measure ever smaller dis- behavior remains possibly compatible with both tances in some frame. asymptotic safety and the results of general rela- The proof of Bizon et. al. [25], uses the Hamil- tivity for large black holes where quantum grav- tonian constraint from classical general relativ- ity is not important. A local experiment may ity and constitutes a proof of the minimum not measure a minimum length, but we can’t length argument under certain conditions. We get that result back to our low energy, IR world simply adjust the proof to include higher curva- that lives outside the outer horizon. ture terms and the running of coefficients. One might argue that the classical Hamiltonian con- straint shouldn’t be applied for questions about II. THE HOOP “THEOREM” FOR quantum gravity. Generically, there could be SPHERICAL SYMMETRY quantum corrections or other modifications to the Hamiltonian constraint at the Planck scale Black hole dominance in the UV is a rather that are not captured by this simple semiclassi- vague statement - there are myriad ways one cal approach. However, we take the view that could try and form black holes. However, if since we are asking how asymptotic safety can the hoop conjecture is correct, pretty much any modify the standard minimum length argument, way we concentrate enough matter into a small which is based on just such a semiclassical pic- enough region should form one. Therefore we ture, the right thing to attempt first is to con- choose a specific behavior for gravity, our space- sider a specific prediction of asymptotic safety time topology, and the distribution of matter as on a specific, mathematically precise formula- follows: tion of the argument. A more problematic con- cern is specific to asymptotic safety itself, which 1. Gravity is asymptotically safe. is defined perturbatively. We use the results of a fully quantum mechanical but perturba- 2. The spacetime topology is such that we tive framework (where there is a well-defined S- can foliate spacetime into spacelike slices matrix) to gain insight into a non-perturbative Σ , where t is a time function. t question, and there is no guarantee that non- 3. A spherical ball Ω of proper radius L is perturbative effects can or should be neglected. filled with matter with density ρ on a max- However, this is again exactly what is done in imal spacelike slice Σ0. the usual minimum length argument, where one treats an effective field theory (general relativ- 4. The system is at least momentarily sta- ity) that is only consistent in the low-energy per- tionary. turbative regime classically even in the extreme UV. Hence for now we proceed in the same spirit If an experiment of proper radius L is to ex- and turn to the proof itself. hibit a minimum length, then there must be at Consider a foliation of spacetime with Cauchy inside least an apparent horizon Ω as we shrink hypersurfaces denoted by Σ . Let na be the the size of the experiment. Our question is then t normal to the leaves of the foliation. Choose whether an apparent horizon must be present ab t = 0 and let TM denote the matter stress ten- on Σ0 inside Ω in the limit as L 0. ab → sor and ρ = TM nanb the energy density of a ball We choose this particular setup for a num- of matter in Σ . We assume the matter is in- ber of reasons. First, the hoop conjecture 0 stantaneously at rest inside a volume Ω of proper in its usual form rests on explicitly using the radius L bounded by a 2-surface ∂Ω. The total Schwarzschild radius of a black hole. This poses proper energy of the matter, E, is defined by several different problems. First, the event hori- the following integral over proper volume Ω, zon is a global object. Further, the very ex- istence of a sharply defined event horizon is ab questionable in a theory of quantum gravity. E = TM nanb = ρdv. (2) ZΩ ZΩ This observation has prompted much work on reformulating black hole physics in terms lo- ∂Ω has a unit spacelike normal ra. The spa- cal trapping and/or dynamical horizons [23, 24]. tial metric qab on Σ0 and the extrinsic curvature We therefore use a more local reformulation Kab constitute canonical data. Whatever the 4 particular truncation used, any higher curvature As Bizon et al. argue, this condition is crucial in terms in eq. (1) contribute to the Hamiltonian how the integrated constraints lead to eq. (13) constraint equation. We package the contribu- (see below). This is rather obvious physically, as tion of those terms into a single quantity Hκ. a negative energy density would certainly allow Then the Hamiltonian constraint equation for one to avoid creating a black hole. Integrating ab the pair (qab,K ) is given by the Hamiltonian constraint over Ω we have,

3 2 ab −1 1 −5 2 R + K KabK = κ0 Hκ + ρ . dv( 8ϕ ϕ) (9) −  2  ZΩ − ∇ r0 (3) − = 4πϕ6r2dr(8ϕ 5 2ϕ) − Z ∇ Note that the coefficients of the higher curva- 0 −1 ture terms as they appear in the action eq. (1) = (2κ0) dv (ρ +2Hκ) . Z are implicitly absorbed in Hκ in eq. (3). We do Ω not consider the contribution of the cosmologi- The right hand integral is 1/2κ0 times the total cal constant term to the Hamiltonian constraint. proper energy E of the matter contained within The implication of setting Λ = 0 is discussed be- Ω plus the contribution from the higher curva- low when we review the renormalization group ture terms. In spherical coordinates we then running of Newton’s constant. have We first simplify using our assumptions. For r0 a maximal slice that has everything momen- 2 1 −1 tarily stationary, the extrinsic curvature K 32π drϕ∂r(r ∂rϕ)= E + κ0 dvHκ. ab − Z0 2κ0 ZΩ (and any other single time derivative) vanishes. (10) For spherical symmetry, a gauge can be cho- sen such that the metric qab is conformally flat, We can rewrite the left hand side of this equa- 4 qab = ϕ (r)δab. This in turn implies that the 3- tion as 3 −5 2 d Ricci scalar is given by R = 8ϕ ϕ. For r0 − ∇ 2 2 outgoing null rays emanating from any 2-sphere 16π dr ϕ ∂r(r∂r(rϕ )) Z − inside Ω at radius r, the criterion that the 2- 0 −1 2 sphere is not a trapped surface is a condition on +rϕ∂rϕ + ϕ r(∂rϕ)∂r(rϕ ) (11) the null expansion
The first term of the integral is simply 16πL, a θ = r ; a > 0. (4) where L is the proper radius of Ω. We hence have Conversely, this must be less than or equal to zero if the 2-sphere is a trapped surface. With 2 2 4 16πL 16πr∂r(rϕ ) dΩ (12) ra = (r2ϕ6)−1 d(r ϕ ) , the condition eq. (4) − | ;a dr −1 2 +16π dr rϕ∂rϕ + ϕ r(∂rϕ)∂r(rϕ ) that the 2-sphere is not trapped is simply Z
2 θr = ∂r rϕ > 0 , (5) −1 E = κ0 + dvHκ .
 2 ZΩ  where the prime denotes an r-derivative.Finally, the proper radius of ∂Ω is L = ϕ2dr. If there are no trapped surfaces in Ω and ∂Ω Now, fix the 2-surface ∂Ω toR have coordinate is not a trapped surface then by eqs. (5) and r0 2 radius r0 (and hence proper radius 0 ϕ dr). (8) every term except the first on the left hand Note that if the energy density dueR to matter side of (12) is negative. Therefore if there are and that of the higher curvature terms [30–32] no trapped surfaces in Ω we have the condition is non-negative, then from the Hamiltonian con- that straint −1 E 16πL>κ + dvHκ . (13) 3 −5 2 −1 1 0  2  R = 8ϕ ϕ = κ Hκ + ρ (6) ZΩ − ∇ 0  2  The converse is also true - if this inequality is we have that violated there is a trapped surface in Ω. This then is the revised hoop condition that must be 2ϕ 0 (7) ∇ ≤ met by the mass and the integrated density of which in turn implies that higher curvature terms. In the case of pure general relativity, i.e. with dϕ 0. (8) Hκ = 0 and no running of couplings, eq. (13) dr ≤ in conjunction with quantum mechanics implies 5 that a trapped surface will inevitably form as A. The cosmological constant the size of an experiment is shrunk. As we re- duce the proper distance we are trying to re- Before we examine the effect of this behav- solve, E must rise according to the uncertainty ior on the formation of local trapped surfaces, principle. If we trap an object in a region of we would like to justify setting Λ = 0. As dis- −1 proper distance L, E scales as L . In the limit cussed in existing literature [19], when we con- as L 0 the LHS of eq. (13) goes to zero while → sider pure gravity, with say just the Einstein- E diverges. Hence at some point the inequality Hilbert term, Newton’s constant is an inessen- cannot be satisfied and a trapped surface must tial coupling and it can be changed by an overall form. This occurs at energy E = 32πκ0L field redefinition. Such redefinitions will leave −1 ≈ 32πκ E , i.e. at energy E √κ = EP lanck. 0 ≈ 0 the action invariant. Thus in the pure gravity At energies greater than EP lanck the situation sector, the running of Newton’s constant is usu- only gets worse, as a horizon is still present in- ally understood with reference to the running of side Ω. some other fiducial operator in the action. The most commonly adopted choice for this refer- ence operator is the cosmological constant term. III. IMPLICATIONS OF ASYMPTOTIC In detail, the RG flow equation for the dimen- SAFETY sionless Newton’s constant

In the asymptotic safety scenario of quantum ∂g −1 E = (2+ η)g (15) gravity, Newton’s constant, G = (16πκ0) is ∂E itself a running coupling (eq. 1). We first iden- contains an anomalous dimension term η(Λ). tify the energy scale µ of the running with E, For the Einstein-Hilbert truncation this is given the total proper energy in our ball of matter. by This is of course not exactly true. Since we have a localized experiment there is no one particu- gB1(Λ) lar energy we can assign to it (in contrast to a ηEH = (16) 1 gB (Λ) scattering scenario, for example). However, the − 2 total energy E still is acceptable as a rough es- Here B1(Λ) (respectively B2(Λ)) are cosmolog- timate of the energy scale we are probing space- ical constant dependent functions which addi- time at, especially since we are relating E to the tionally also depend in a rather involved way on proper length L via E 1/L, i.e. just interested ∼ the details of how the infrared cut-off for the Ex- in the scaling behavior. act Renomalization Group Equations (ERGE) A number of recent works in the literature [21] is chosen. The anomalous dimension can be have confirmed the existence of a non-gaussian evaluated at zero arguments for these functions, fixed point with various higher curvature trun- B1(0) and B2(0), effectively restricting to the cations and matter contribution included in the case of zero cosmological constant. In the case total effective action. Integrating the flow equa- of the Einstein-Hilbert truncation the so-called tion is an involved process, but the upshot that optimized cut-off due to Litim [20] for instance we will exploit in the present article is that in leads to the extreme UV, the running of Newton’s con- 12g stant is dominated by the fixed point behavior. ηEH = , (17) This is given by 2g 2(Λ 1 )2 − − 2 1 g∗ hence setting Λ = 0 in eq. (16) amounts to = G(E) 2 (14) 16πκ0(E) ≃ E having where g∗ is the value of the dimensionless New- (1 16g)2g ton’s constant at the non-Gaussian fixed point, β = − (18) g 1 4g and we work in the regime where the matter − proper energy exceeds the fiducial Planck energy The behavior of Newton’s constant gleaned from set by the infrared value of Newton’s constant. eq. (18) near the non-Gaussian fixed point 2 1 g∗ Thus in the extreme UV as E increases, New- ∗ g = 16 , for large E, is G E2 . Let us also note ton’s constant as a running coupling gets pro- in passing that when higher≃ curvature trunca- gressively smaller. We also note in our discus- tions are included in the effective action, the ef- sion here and what follows, the energy scale E fect of these terms show up again in the anoma- is to be compared with fiducial Planck scale E pl lous dimension, now written as η(Λ,Hκ). The (used below) which is defined with respect to the fixed point behavior that this leads to has been 1 infrared value of Newton’s constant G0 = 2 . 8πEpl discussed by Benedetti et al. [21]. 6

B. Running Newton’s constant and the where I¯(¯κI ) is a dimensionless number whose formation of trapped surfaces details depend on both the unknown κI and the precise solution for the metric in the presence We now include the running of κ0, eq. (14) of the higher curvature terms with our given into eq.(13). Momentarily restricting to the stress tensor. It suffices for us that it is a lin- Einstein-Hilbert + matter truncation only we ear function ofκ ¯I . In asymptotic safety theκ ¯I find the condition for no trapped surfaces inside coefficients remain finite as E , and hence → ∞ Ω to be I is also finite. Therefore as E there is → ∞ g∗ no proof that a horizon is formed, as each side L> E (19) 2E2 scales the same way with E, and so the argu- ment for a minimum length no longer necessar- Now using the uncertainty relation, we replace ily holds (at least in this formulation). L 1/E. The factors involving the proper en- ∼ Of course, in the above approach, there is no ergy E drop out from both sides of the inequal- proof that a horizon is not formed either. Since ity eq. (19) and the (necessary) condition for no E drops out in asymptotic safety at high ener- trapped surfaces to lie in Ω takes the form gies, whether or not a horizon forms inside Ω depends on both the particulars of the value of g∗ < 2 (20) g∗ at the non-Gaussian fixed point (and there- This condition shows that up to O(1) num- fore on the truncation) and the actual size of bers arising from the order of magnitude esti- theκ ¯I coefficients. Since both eq. (19) and eq. mates/scaling arguments we use in our deriva- (24) are obtained in the extreme UV limit in tion, there is some value of coefficients for which which we invoke the largely scheme independent the minimum length argument falls apart, in [33] fixed point dominated running eq. (14), contrast to ordinary general relativity. Interest- our conclusions (the bound on the fixed point ingly, g∗ in different truncations has been found g∗ < 2, modulo O(1) numbers discussed above) to be quite close to 2. For example, the most are at least essentially insensitive to myriad IR- recent results of Benedetti et al. [21], who con- cutoff schemes in vogue and the precise size of sider both a minimally coupled scalar field with the higher curvature terms at some fiducial in- the Einstein-Hilbert action and a higher curva- frared scale. ture truncation involving the Ricci and Weyl scalars, are IV. GENERAL RELATIVITY AND THE g∗ =0.860 (For Einstein-Hilbert truncation) INFRARED LIMIT 2 2 g∗ =2.279 (For R + C truncation) (21) In general the inclusion of higher curvature in- While the question of whether or not a hori- variants in the effective action, constructed out zon forms inside Ω for Planck sized experiments of Ricci, Riemann and Weyl tensors, pushes the becomes muddled with asymptotic safety, in the high energy fixed point towards values higher limit of energy much greater than EPl there than their Einstein-Hilbert counterparts. still must be a trapped surface at large radius, Consider now the impact of higher curvature as this corresponds to a large black hole with terms with running couplings on our argument. low horizon curvature (hence governed by gen- Returning to eq. (13) we find that no trapped eral relativity). To reach such a spacetime pic- surfaces form inside Ω if ture in the semiclassical limit, one considers quantum states with low enough energy density g∗ E that gravitational backreaction can be neglected L > + dvHκ (22) E2  2 Z  propagating in a spacetime generated by some

−1 separate matteer distribution. For our super- With L replaced by E eq. (22) becomes Planckian experiments, we have no such lux- ury as we specifically are exploring a change in −1 g∗ E E > + Hκdv (23) the gravitational action, the running of coeffi- E2  2 Z  cients, via the energy scale of the experiment. We now observe that the integral involving the However, we can analyze the “horizon” struc- higher curvature terms is dimensionally an en- ture generated by a matter distribution within ergy and so the integral must be a combination our framework by considering a background dis- of E multiplied by the dimensionless coefficients tribution of matter with total energy EB in Ω κ¯I . We therefore have, and ask when an experiment at a length scale L (or equivalent energy scale E ) encounters −1 −1 −1 ¯ E E E >g∗ (2E) + E I(¯κI ) (24) a trapped surface.
7

If we wish to run our experiment all the way back to the infrared, we can no longer consider 2 2 2 2 just the UV fixed point behavior for G. We g∗EBJB g∗EBJB 4g∗MPl(1 g∗JE) LE = ± −2 − . therefore will take a specific form for G(E) [22, p 2g∗MPl 34], (29) We now consider the “large black hole limit” in the infrared by assuming EB = NMPl with g∗ N >> 1, i.e. there’s a lot of mass in our back- G(E)= 2 2 (25) E + g∗MPl ground spacetime. In this limit we have

and return to equation (13), but modify the scenario. We define EB JB LE+ = 2 (30) MPl 1 g∗JE − EE + EB LE = − . (31) F (EE)= LE G(EE ) + dvHκ g∗EBJB −  2 Z  (26) LE+ is R GEB , i.e. it corresponds to the ≈ F (E) > 0 is the condition for no trapped sur- Schwarzschild radius for a large black hole. LE− face in the region. As EE it dominates is the location of the inner horizon. We can this expression, so we return→ to ∞ our original for- get a more intuitive picture by setting all the mulation of the hoop conjecture for a single ex- higher curvature terms equal to zero. In this periment. We assume that g∗ and the κI are case JE = JB = 1/2 and the inner horizon lo- such that there is no trapped surface within Ω cation becomes LE− = (2 g∗)/(g∗EB ). In the − as EE . Given this, as we EE towards the previous section, for just one experiment, our → ∞ IR eventually F (EE) may cross zero. This signi- limit was roughly g∗ < 2 to avoid the minimum fies that at some radius LE we have crossed into length argument. Here we have the additional a region that contains trapped surfaces. The in- background energy EB in addition to EE , so ner boundary of this region can be thought of we must further decrease g∗ to avoid a horizon as an “inner horizon”, similar to the Reissner- forming in the domain of an experiment. In Nordstrom inner horizon, although we caution other words, if we chose EE = EB for exam- that it is defined only as a function of the exper- ple, then there is twice the energy density as we iment energy EE and so is not a true spacetime had in the previous section in a region of size quantity. Similarly, at very small EE , F (EE) LE = LB so g∗ would need to be less than one can cross zero again, signaling the outer bound- to prevent formation of a horizon. (Note that ary of our trapped surface region. This outer this experiment dependence of the limit on g∗ boundary surface corresponds to the black hole is symptomatic of this approach - whether and event horizon in the infrared. where horizons are formed depends strongly on We now solve for the location of these two how you plan to probe the spacetime as that boundary surfaces. The presence of a horizon is influences the running of couplings.) If we con- when sider g∗ < 1, such that this bound is satisfied, then we see that the inner horizon is reached when LE 2/(g∗EB ). We therefore have the ≈ EE + EB following picture for g∗ < 1. We have a back- LE G(EE) + dvHκ = 0 (27) −  2 Z  ground energy density EB >> MPl in a region of radius LB =1/EB. We probe the system at EE and LE are related via EE = 1/LE, but length scales smaller than LB and we see no EB is to be considered fixed. The contribu- trapped surface. As we approach the length tion of the higher curvature terms can be re- scale (2 g∗)LB/g∗ we see a trapped surface − expressed as IE(κI )EE and IB (κI )EB , where form inside our experiment. This behavior per- each I is again a dimensionless number depen- sists until we reach the large radius GEB , at dent on the κI ’s. If we define JE = 1/2+ IE which point we return to semiclassical physics and JB =1/2+ IB we can rewrite (27) in terms outside an ordinary Schwarzschild horizon. of the J’s and LE as

V. CONCLUSIONS 2 g∗LE JE LE 2 2 + JB EB =0. (28) − 1+ g∗LEMPl LE  There’s a rather obvious tension between black hole dominance and the asymptotic safety Solving for LE yields scenario of quantum gravity. The black hole 8 dominance arguments indicate that the very the fixed points are thus believed to be quite high spectrum is dominated by states whose en- robust. However, since our result depends on tropy density, via the Bekenstein-Hawking for- the fixed points in such a sensitive way, it is mula, does not scale as an extensive quantity perhaps of general interest to extend the fixed with volume, which is a characteristic feature of point calculations to more realistic choices of any local conformal field theory. Since the high matter Lagrangian and compare to a more de- energy spectrum of renormalizable field theories tailed gedanken experiment. are that of a conformal field theory, the impli- Finally compatibility with general relativity cation is that gravity must be an exception to can still be maintained even in our scenario. this rule. On the other hand, the asymptotic First, even if one can localize an experiment safety approach claims that precisely at those without formation of a trapped surface inside high energies where black holes dominate in the the region of the experiment, that is no guar- usual picture, a sensible local field theory based antee that such microscopic information can be on the metric exists, and provides a reliable fun- transmitted out to an observer at spatial infin- damental framework for quantum gravity. ity with physics dictated by general relativity. Motivated by this, we re-examined the hoop In fact, we found that no information about the conjecture in light of asymptotic safety. Adopt- local physics can be communicated to an ob- ing an approach relatively independent of any server at infinity, even if a local experiment in- specific truncation of the effective action, we side that horizon can in principle measure ar- explored how in particular one key feature of bitrarily small length scales. If we interpret asymptotic safety, the running of couplings, af- asymtotic darkness as a statement about lim- fects standard arguments for a minimum length itations on information transfer from a local ex- based on the hoop conjecture when the size of periment to an asymptotic observer owing to an experiment is shrunk below a critical length. horizon formation, then our observations show For a gedanken experiment consisting of a ball that it may be possible to satisfy asymptotic of matter inside a volume Ω of proper size L darkness within the asymptotic safety scenario. and energy E 1 , the fixed point behavior of Furthermore, for spacetimes with a background ∼ L the running couplings conspire such that the ne- total energy E Epl, where the physics of gen- cessity of forming a trapped surface inside Ω as eral relativity dictate≫ that there must exist a we shrink the size of an experiment vanishes. large horizon at the classical Schwarzschild ra- Hence the standard minimum length argument dius, we argue that as one probes at lower and fails to necessarily be true in this case. However lower energy scales (from the UV to the IR) one there is no proof in this approach that a trapped eventually encounters a trapped surface. The surface does not form inside Ω - whether it does resulting horizon structure consists of an exper- or not depends on specifics of quantum gravity imentally dependent “inner horizon” and the that we do not know. classical Schwarzschild horizon in the extreme The qualitative features of the fixed point infrared. dominated running and indeed the existence of a non-Gaussian fixed point has been argued by Percacci [33] to be a scheme independent fea- ture of the the Wilsonian ERGE. He further Acknowledgments notes that the position of the fixed point has a weak dependence on parameters in the choice We thank Jishnu Battacharyya and Keisuke of the IR cut-off function and so the values for Juge for comments and stimulating discussions.

[1] K. S. Thorne, in Magic Without Magic, ed. 77, 085025 (2008) J. Klauder, (W. H. Freeman, San Francisco, [6] T. Banks and W. Fischler, eprint 1972). hep-th/9906038. [2] M. Choptuik and F. Pretorius, 0908.1780 [gr- [7] O. Aharony and T. Banks, JHEP, 9903:016 qc] (1999). [3] P. D.D Eath´ and P. N. Payne, Phys. Rev. D46 , [8] N. Kaloper and J. Terning, Gen. Rel. Grav, 658 (1992); Phys. Rev. D46, 675 (1992); Phys. 39, 1525 (2007)[Int. J. Mod.Phys, D 17, 665 Rev. D 46, 694 (1992). (2008)). [4] P. C. Aichelburg and R. U. Sexl, Gen. Rel. [9] D. M. Eardley and S. B. Giddings, Phys.Rev. Grav, 2, 303 (1971) D66, 044011 (2002). [5] S. B. Giddings and M. Srednicki, Phys. Rev. D [10] L. Crane and L. Smolin, Nucl. Phys. B267, 714 9

(1986) Rev.Rel.7:10,2004. eprint gr-qc/0407042. [11] Lecture available online at [25] P. Bizon, E.Malec and N O’Murchadha, Phys. hep.phys.uoc.gr/mideast2/lectures/BanksLecture.pdf Rev. Lett 61, 1147 (1988) [12] A. Shomer, eprint Arxiv: 0709.3555. [26] E. Flanagan, Phys. Rev. D15, 1429 (1992) [13] X. Calmet, M. Graesser and S. D. H. Hsu, Int. [27] C. Barrab´es, V. P. Frolov and E. Lesigne, Phys. J. Mod. Phys. D14,2195,(2005). Rev. D69, 19501(R) (2004) [14] L. J. Garay, Int. J. Mod. Phys, A10, 145 (1995). [28] H. Yoshino and Y. Nambu, Phys. Rev. D67, [15] S. Weinberg, in General Relativity: An Ein- 024009 (2003) stein Centenary Survey ed. S. W. Hawking and [29] T. Chiba, T. Nakamura, K. Nakao and M. W. Israel, Cambridge University Press, Cam- Sasaki, Class. Quant. Grav. 11, 431 (1994). bridge, 1979, pp. 790-831. [30] A. Strominger, Phys. Rev. D30, 2257 (1984) [16] S. Weinberg, arXiv:0908.1964 [hep-th]. [31] S. Deser and B. Tekin, Phys. Rev. D67, 084009 [17] O. Lauscher and M. Reuter, JHEP 0510 050 (2003) (2005), eprint hep-th/0508202. [32] S. Deser and B. Tekin, Phys. Rev. D75, [18] M. Niedermaier, Class. Quant. Grav. 24, R171 084032(2007) (2007), eprint gr-qc/0610018. [33] G. Narain and R. Percacci, eprint:0910. 5390 [19] M. Niedermaier and M. Reuter, Living. Rev. [hep-th]. Relativity, 9, 5 (2006) [34] JoAnne Hewett and Thomas Rizzo, JHEP [20] D. F. Litim, Phys. Rev. Lett. 92, 201301 (2004), 0712: 009 (2007) eprint hep-th/0312114. [35] We caution the reader that in our approach the [21] D. Benedetti, P. Machado and F. Sauressig, “inner horizon” is not a true spacetime inner Nucl.Phys.B824, 168 (2010), eprint ArXIv: horizon in the sense of a Reissner-Nordstrom in- 0902.4630. ner horizon. It is instead rather experiment de- [22] A. Bonanno and M. Reuter, Phys.Rev.D62, pendent, in that the location of the inner hori- 043008 (2000). zon depends on the total energy of the space- [23] A. Ashtekar, C. Beetle and S. Fairhurst, time and the specific total energy of an experi- Class.Quant.Grav.16:L1-L7,1999. eprint mental probe. gr-qc/9812065. [24] A. Ashtekar and B. Krishnan, Living arXiv:1006.0718v2 [hep-th] 24 Jan 2011 hc mle h cln fsz ihenergy with size wavelength, of Broglie scaling De the the implies be which to taken and experiment fgaiy h cln fbakhl nrp iharea with entropy hole flat black be- asymptotically of in tension that scaling non-renormalizability implies obvious The in and an theory gravity. dominance is field of hole there ordinary black an particular, tween as as In well gravity UV. as of the 11], status theory[10, the field for quantum of [5–9]. structure the scattering particle on This work other and hole. Aichelburg-Sexl 4] black the [3, from a result derived is insights at the indeed scattering vindicates parameters particle two impact of small outcome rel- generic general classical the in en- ativity that at numerically experiment verified any recently for forms surface ergies trapped a then if fcaatrsi size characteristic of ruetrle nTon’ opcnetr 1:I an If [1]: The conjecture matter or hoop energy mechanics. of Thorne’s amount quantum on solely relies and argument argument quan- relativity an of of make general Proof theory can with one particular so however one. a gravity, holes, requires old tum black conjecture an by a is dominated such darkness”, is “asymptotic scale called Planck the near † ∗ conje hoop the and Darkness, Asymptotic Safety, Asymptotic lcrncades [email protected] address: Electronic lcrncades sbasu@pacific.edu address: Electronic R < R smttcdrns a neetn osqecsfor consequences interesting has darkness Asymptotic h ojcueta h pcrmo atcestates particle of spectrum the that conjecture The E > E S 2 = n oa nomto rmrahn nnt n oteei s is there so and experiment. infinity the infinity. reaching Pl at outside the from observers form than information the still smaller local of much must any version be surface to local trapped purely localized one any is longer experiment the an no of values when is numerical perf there can specific sense, observer the this an on t not depends or proof regions whether o contrary small approach any f domain this there the proof in is within modified Instead, neither surface the However, that trapped context. c show a this We of higher formation couplings. include mandatory running to as relativity e well general or as pr in a observer modifying systems of by gravity symmetric independent safe fairly asymptotically minimu in be fundamental conclusion a to for considered arguments usually in invoked frequently is surface trapped a of formation h tepst efr oaie xeieti narbitra an in experiment localized a perform to attempts who GE lanck P suigteho ojcuei lsia eea relativi general classical in conjecture hoop the Assuming hsc eatet nvriyo h aic30 aicAv Pacific 3601 Pacific the of University Department, Physics .INTRODUCTION I. If . hpukadPeois[]have [2] Pretorius and Choptuik . L L rpe ufc fsize of surface trapped a , hsc eatet nvriyo e aphr,Dra,N Durham, Hampshire, New of University Department, Physics sietfidwt h sz”o an of “size” the with identified is E scmrse noaregion a into compressed is d dmnin,tehigh the -dimensions, within R h pta oano h xeiet hstogtexperime thought This experiment. the of domain spatial the L ∝ ai Mattingly David aadbBasu Sayandeb forms 1 /E , hoyi h V ic ntelte aeteuniversal as the goes case states latter of the density field in of conformal since scaling local UV, a the to in related theory be cannot gravity tum opig tagvnsaermi finite. remain whose scale operators, given relevant a by at theory perturbation couplings a field be conformal to a considered of be be can cannot renormal- theory gravity a field that since izable is theory, [12], field renormalizable ref. ordinary in an out pointed as tion, soesrnsa xeietsc that such experiment an shrinks the size is one of logic min- surface as The a trapped a for physics. 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GL tr nteeteeUV extreme the in cture − 03824 H 1 hrfr hr sn observer no is there Therefore . † e E L d − ∗ d eesrl forms necessarily 1 h implica- The . nt L < L L lanck P lanck P and is L , 2 in the UV as well as the IR? Such a scenario is envisioned generated by a localized experiment out to infinity, how- by the asymptotic safety program, originally championed ever it does not a priori imply that an experiment cannot by Weinberg [15, 16]. Asymptotic safety can be described locally measure arbitrarily small scales. As mentioned in the following heuristic way. Consider a truncated (for above, forbidding the measurement of arbitrarily small technical reasons, see [19]) effective action for 4-d metric scales for all observers requires a trapped surface form at gravity with cosmological constant coupled to matter the scale of the very localized experiment and this simply may fail in asymptotic safety. For Planck sized regions we 4 2 I = d x√ g Λ(µ)+ κ0(µ)R + κ1(µ)R + are not in pure general relativity - the gravitational con- Z − − stants may run significantly and higher curvature terms ab κ2(µ)RabR + Lm ... (1) become relevant. Hence the small scale geometry may
change significantly and one may not have trapped sur- containing a finite set of higher derivative terms with as- faces all the way from a large outer horizon all the way sociated coefficients, κ in addition to the usual cos- 1,2,3... down to the experiment size L when L<

If an experiment of proper radius L is to exhibit a ab minimum length, then there must be at least a trapped E = TM nanb = ρdv. (2) ZΩ ZΩ surface inside Ω as we shrink the size of the experiment. Our question is then whether a trapped surface must be ∂Ω has a unit spacelike normal ra. The spatial met- ab present on Σ0 inside Ω in the limit as L 0. ric qab on Σ0 and the extrinsic curvature K constitute We choose this particular setup for a→ number of rea- canonical data. Whatever the particular truncation used, sons. First, the hoop conjecture in its usual form rests on any higher curvature terms in eq. (1) contribute to the explicitly using the Schwarzschild radius of a black hole. Hamiltonian constraint equation. We package the contri- This poses several different problems. First, the event bution of those terms into a single quantity Hκ. Then the ab horizon is a global object. Further, the very existence of Hamiltonian constraint equation for the pair (qab,K ) a sharply defined event horizon is questionable in a the- is given by ory of quantum gravity. This observation has prompted much work on reformulating black hole physics in terms 3 2 ab −1 1 R + K KabK = κ0 Hκ + ρ . local trapping and/or dynamical horizons [23, 24]. We −  2  therefore use a more local reformulation of the conjec- (3) ture in terms of trapped surfaces [25, 26]/apparent hori- zons [27–29]. Spherical symmetry and the momentarily Note that the coefficients of the higher curvature terms as stationary assumption are both used for simplicity. We they appear in the action eq. (1) are implicitly absorbed note, however, that the stationary assumption also intu- in Hκ in eq. (3). We do not consider the contribution itively matches the idea of scaling a static system down of the cosmological constant term to the Hamiltonian to try to measure ever smaller distances in some frame. constraint. The implication of setting Λ = 0 is discussed The proof of Bizon et. al. [25], uses the Hamiltonian below when we review the renormalization group running constraint from classical general relativity and consti- of Newton’s constant. tutes a proof of the minimum length argument under cer- We first simplify using our assumptions. For a max- tain conditions. We simply adjust the proof to include imal slice that has everything momentarily stationary, higher curvature terms and the running of coefficients. the extrinsic curvature Kab (and any other single time One might argue that the classical Hamiltonian con- derivative) vanishes. For spherical symmetry, a gauge straint shouldn’t be applied for questions about quantum can be chosen such that the metric qab is conformally 4 gravity. Generically, there could be quantum corrections flat, qab = ϕ (r)δab. This in turn implies that the 3-d or other modifications to the Hamiltonian constraint at Ricci scalar is given by 3R = 8ϕ−5 2ϕ. For outgoing the Planck scale that are not captured by this simple null rays emanating from any 2-sphere− ∇ inside Ω at radius semiclassical approach. However, we take the view that L, the criterion that the 2-sphere is not a trapped surface 4 is a condition on the null expansion If there are no trapped surfaces in Ω and ∂Ω is not a trapped surface then by eqs. (5) and (8) every term ex- θ = ra > 0. (4) ; a cept the first on the left hand side of (12) is negative. Conversely, this must be less than or equal to zero if the 2- Therefore if there are no trapped surfaces in Ω we have 2 4 a 2 6 −1 d(r ϕ ) the condition that sphere is a trapped surface. With r ;a = (r ϕ ) dr , the condition eq. (4) that the 2-sphere is not trapped is −1 E simply 16πL>κ + dvHκ . (13) 0  2 Z  2 Ω θr = ∂r rϕ > 0 , (5) The converse is also true - if this inequality is violated where the prime denotes an
r-derivative.Finally, the there is a trapped surface in Ω. This then is the revised proper radius of ∂Ω is L = ϕ2dr. hoop condition that must be met by the mass and the Now, fix the 2-surface ∂ΩR to have coordinate radius r0 r0 integrated density of higher curvature terms. (and hence proper radius ϕ2dr). Note that if the en- 0 In the case of pure general relativity, i.e. with H =0 ergy density due to matter and that of the higher curva- κ R and no running of couplings, eq. (13) in conjunction with ture terms [30–32] is non-negative, then from the Hamil- quantum mechanics implies that a trapped surface will tonian constraint inevitably form as the size of an experiment is shrunk. As 3 −5 2 −1 1 we reduce the proper distance we are trying to resolve, R = 8ϕ ϕ = κ0 Hκ + ρ (6) − ∇  2  E must rise according to the uncertainty principle. If we have that we trap an object in a region of proper distance L, E scales as L−1. In the limit as L 0 the LHS of eq. (13) 2 ϕ 0 (7) goes to zero while E diverges. Hence→ at some point the ∇ ≤ which in turn implies that inequality cannot be satisfied and a trapped surface must −1 form. This occurs at energy E = 32πκ0L 32πκ0E , dϕ ≈ i.e. at energy E √κ0 = EP lanck. At energies greater 0. (8) ≈ dr ≤ than EP lanck the situation only gets worse, as a trapped As Bizon et al. argue, this condition is crucial in how the surface is still present inside Ω. integrated constraints lead to eq. (13) (see below). This is rather obvious physically, as a negative energy density would certainly allow one to avoid creating a black hole. III. IMPLICATIONS OF ASYMPTOTIC Integrating the Hamiltonian constraint over Ω we have, SAFETY

−5 2 dv( 8ϕ ϕ) (9) In the asymptotic safety scenario of quantum gravity, ZΩ − ∇ −1 r0 Newton’s constant, G = (16πκ0) is itself a running − = 4πϕ6r2dr(8ϕ 5 2ϕ) coupling (eq. 1). We first identify the energy scale µ of − Z0 ∇ the running with E, the total proper energy in our ball −1 of matter. This is of course not exactly true. Since we = (2κ0) dv (ρ +2Hκ) . ZΩ have a localized experiment there is no one particular energy we can assign to it (in contrast to a scattering The right hand integral is 1/2κ times the total proper 0 scenario, for example). However, the total energy E still energy E of the matter contained within Ω plus the con- is acceptable as a rough estimate of the energy scale we tribution from the higher curvature terms. In spherical are probing spacetime at, especially since we are relating coordinates we then have E to the proper length L via E 1/L, i.e. just interested r0 ∼ 2 1 −1 in the scaling behavior. 32π drϕ∂r(r ∂rϕ)= E + κ0 dvHκ. (10) − Z0 2κ0 ZΩ A number of recent works in the literature [21] have We can rewrite the left hand side of this equation as confirmed the existence of a non-gaussian fixed point r0 with various higher curvature truncations and matter 2 2 16π dr ϕ ∂r(r∂r(rϕ )) contribution included in the total effective action. In- Z − 0 tegrating the flow equation is an involved process, but −1 2 +rϕ∂rϕ + ϕ r(∂rϕ)∂r(rϕ ) (11) the upshot that we will exploit in the present article is
that in the extreme UV, the running of Newton’s con- The first term of the integral is simply 16πL, where L is stant is dominated by the fixed point behavior. This is the proper radius of Ω. We hence have given by 2 16πL 16πr∂r(rϕ ) dΩ (12) − | 1 g∗ −1 2 = G(E) 2 (14) +16π dr rϕ∂rϕ + ϕ r(∂rϕ)∂r(rϕ ) 16πκ (E) ≃ E Z 0
−1 E where g∗ is the value of the dimensionless Newton’s con- = κ0 + dvHκ .  2 ZΩ  stant at the non-Gaussian fixed point, and we work in 5 the regime where the matter proper energy exceeds the curvature truncations are included in the effective action, fiducial Planck energy set by the infrared value of New- the effect of these terms show up again in the anomalous 2 ton’s constant. Thus in the extreme UV as E increases, dimension, now written as η(Λ,Hκ). The fixed point be- Newton’s constant as a running coupling gets progres- havior that this leads to has been discussed by Benedetti sively smaller. We also note in our discussion here and et al. [21]. what follows, the energy scale E is to be compared with fiducial Planck scale Epl (used below) which is defined with respect to the infrared value of Newton’s constant B. Running Newton’s constant and the formation 1 of trapped surfaces G0 = 2 . 8πEpl

We now include the running of κ0, eq. (14) into eq.(13). A. The cosmological constant Momentarily restricting to the Einstein-Hilbert + mat- ter truncation only we find the condition for no trapped surfaces inside Ω to be Before we examine the effect of this behavior on the g∗ formation of local trapped surfaces, we would like to jus- L> E (19) tify setting Λ = 0. As discussed in existing literature 2E2 [19], when we consider pure gravity, with say just the Now using the uncertainty relation, we replace L 1/E. Einstein-Hilbert term, Newton’s constant is an inessen- The factors involving the proper energy E drop out∼ from tial coupling and it can be changed by an overall field both sides of the inequality eq. (19) and the (necessary) redefinition. Such redefinitions will leave the action in- condition for no trapped surfaces to lie in Ω takes the variant. Thus in the pure gravity sector, the running form of Newton’s constant is usually understood with refer- ence to the running of some other fiducial operator in g∗ < 2 (20) the action. The most commonly adopted choice for this reference operator is the cosmological constant term. In This condition shows that up to O(1) numbers arising detail, the RG flow equation for the dimensionless New- from the order of magnitude estimates/scaling arguments ton’s constant we use in our derivation, there is some value of coef- ficients for which the minimum length argument falls ∂g E = (2+ η)g (15) apart, in contrast to ordinary general relativity. Inter- ∂E estingly, g∗ in different truncations has been found to be quite close to 2. For example, the most recent results of contains an anomalous dimension term η(Λ). For the Benedetti et al. [21], who consider both a minimally cou- Einstein-Hilbert truncation this is given by pled scalar field with the Einstein-Hilbert action and a higher curvature truncation involving the Ricci and Weyl gB1(Λ) ηEH = (16) scalars, are 1 gB (Λ) − 2 g∗ =0.860 (For Einstein-Hilbert truncation) Here B1(Λ) (respectively B2(Λ)) are cosmological con- 2 2 stant dependent functions which additionally also depend g∗ =2.279 (For R + C truncation) (21) in a rather involved way on the details of how the infrared cut-off for the Exact Renomalization Group Equations In general the inclusion of higher curvature invariants in (ERGE) is chosen. The anomalous dimension can be the effective action, constructed out of Ricci, Riemann evaluated at zero arguments for these functions, B (0) and Weyl tensors, pushes the high energy fixed point to- 1 wards values higher than their Einstein-Hilbert counter- and B2(0), effectively restricting to the case of zero cos- mological constant. In the case of the Einstein-Hilbert parts. truncation the so-called optimized cut-off due to Litim Consider now the impact of higher curvature terms [20] for instance leads to with running couplings on our argument. Returning to eq. (13) we find that no trapped surfaces form inside Ω 12g if ηEH = , (17) 1 2 2g 2(Λ 2 ) g∗ E − − L > + dvHκ (22) E2  2 Z  hence setting Λ = 0 in eq. (16) amounts to having With L replaced by E−1 eq. (22) becomes (1 16g)2g βg = − (18) 1 4g −1 g∗ E − E > + Hκdv (23) E2  2 Z  The behavior of Newton’s constant gleaned from eq. (18) 1 ∗ near the non-Gaussian fixed point g = 16 , for large E, We now observe that the integral involving the higher g∗ is G 2 . Let us also note in passing that when higher curvature terms is dimensionally an energy and so the ≃ E 6 integral must be a combination of E multiplied by the dimensionless coefficientsκ ¯ . We therefore have, g∗ I G(E)= (25) E2 + g∗M 2 −1 −1 −1 Pl E >g∗ (2E) + E I¯(¯κI ) (24) and return to equation (13), but modify the scenario.
We define where I¯(¯κI ) is a dimensionless number whose details de- pend on both the unknown κI and the precise solution for the metric in the presence of the higher curvature terms EE + EB F (EE)= LE G(EE ) + dvHκ (26) with our given stress tensor. It suffices for us that it is a −  2 Z  linear function ofκ ¯ . In asymptotic safety theκ ¯ coeffi- I I F (E) > 0 is the condition for no trapped surface in the cients remain finite as E , and hence I is also finite. region. As E it dominates this expression, so Therefore as E there→ ∞ is no proof that a trapped E we return to our→ original ∞ formulation of the hoop con- surface is formed,→ as ∞ each side scales the same way with jecture for a single experiment. We assume that g∗ and E, and so the argument for a minimum length no longer the κ are such that there is no trapped surface within necessarily holds (at least in this formulation). I Ω as EE . Given this, as we EE towards the IR Of course, in the above approach, there is no proof that eventually→F ( ∞E ) may cross zero. This signifies that at a trapped surface is not formed either. Since E drops out E some radius LE we have crossed into a region that con- in asymptotic safety at high energies, whether or not a tains trapped surfaces. The inner boundary of this region trapped surface forms inside Ω depends on both the par- can be thought of as an “inner horizon”, similar to the ticulars of the value of g∗ at the non-Gaussian fixed point Reissner-Nordstrom inner horizon, although we caution (and therefore on the truncation) and the actual size of that in this framework it is defined only as a function of theκ ¯I coefficients. Since both eq. (19) and eq. (24) the experiment energy E and so is not a true spacetime are obtained in the extreme UV limit in which we invoke E quantity. Similarly, at very small EE , F (EE) can cross the largely scheme independent [33] fixed point domi- zero again, signaling the outer boundary of our trapped nated running eq. (14), our conclusions (the bound on surface region. This outer boundary surface corresponds the fixed point g∗ < 2, modulo O(1) numbers discussed to the usual black hole event horizon in the infrared. above) are at least essentially insensitive to myriad IR- We now solve for the location of these two boundary cutoff schemes in vogue and the precise size of the higher surfaces. The presence of a horizon is when curvature terms at some fiducial infrared scale.

EE + EB LE G(EE ) + dvHκ = 0 (27) IV. GENERAL RELATIVITY AND THE −  2 Z  INFRARED LIMIT EE and LE are related via EE = 1/LE, but EB is to be considered fixed. The contribution of the higher While the question of whether or not a trapped sur- curvature terms can be re-expressed as IE(κI )EE and face forms inside Ω for Planck sized experiments becomes IB(κI )EB, where each I is again a dimensionless number muddled with asymptotic safety, in the limit of energy dependent on the κI ’s. If we define JE = 1/2+ IE and much greater than EPl there still must be a trapped sur- JB = 1/2+ IB we can rewrite (27) in terms of the J’s face at large radius, as this corresponds to a large black and LE as hole with low horizon curvature (hence governed by gen- eral relativity). To reach such a spacetime picture in the 2 g∗LE JE semiclassical limit, one considers quantum states with LE + JBEB =0. (28) − 1+ g∗L2 M 2 L  low enough energy density that gravitational backreac- E Pl E tion can be neglected propagating in a spacetime gen- Solving for LE yields erated by some separate matter distribution. For our super-Planckian experiments, we have no such luxury as 2 2 2 2 g∗EB JB g∗EB JB 4g∗MPl(1 g∗JE) we specifically are exploring a change in the gravitational LE = ± − − . p 2g∗M 2 action, the running of coefficients, via the energy scale of Pl (29) the experiment. However, we can analyze the “horizon” We now consider the “large black hole limit” in the structure generated by a matter distribution within our infrared by assuming E = NM with N >> 1, i.e. framework by considering a background distribution of B Pl there’s a lot of mass in our background spacetime. In matter with total energy E in Ω and ask when an ex- B this limit we have periment at a length scale LE (or equivalent energy scale EE) encounters a trapped surface. If we wish to run our experiment all the way back to EB JB LE+ = 2 (30) the infrared, we can no longer consider just the UV fixed MPl point behavior for G. We therefore will take a specific 1 g∗JE LE− = − . (31) form for G(E) [22, 34], g∗EB JB 7

LE is R GEB , i.e. it corresponds to the any specific truncation of the effective action, we explored + ≈ Schwarzschild radius for a large black hole. LE− is the how in particular one key feature of asymptotic safety, location of the inner horizon. We can get a more intuitive the running of couplings, affects standard arguments for picture by setting all the higher curvature terms equal to a minimum length based on the hoop conjecture when the zero. In this case JE = JB = 1/2 and the inner hori- size of an experiment is shrunk below a critical length. zon location becomes LE− = (2 g∗)/(g∗EB). In the For a gedanken experiment consisting of a ball of matter previous section, for just one experiment,− our limit was inside a volume Ω of proper size L and energy E 1 , ∼ L roughly g∗ < 2 to avoid the minimum length argument. the fixed point behavior of the running couplings conspire Here we have the additional background energy EB in ad- such that the necessity of forming a trapped surface at dition to EE , so we must further decrease g∗ to avoid a the experimental scale vanishes. There is no proof in this trapped surface forming in the domain of an experiment. approach that a trapped surface does not form inside Ω In other words, if we chose EE = EB for example, then - whether it does or not depends on specifics of quan- there is twice the energy density as we had in the previous tum gravity that we do not know. However the standard section in a region of size LE = LB so g∗ would need to be minimum length argument, that the Planck length is a less than one to prevent formation of a trapped surface. minimum length scale, independent of observer or exper- (Note that this experiment dependence of the limit on iment, fails to necessarily be true in this case. g∗ is symptomatic of this approach - whether and where Compatibility with general relativity can still be main- trapped surfaces form depends strongly on how you plan tained in this scenario. First, even if one can localize an to probe the spacetime as that influences the running of experiment without formation of a trapped surface inside couplings.) If we consider g∗ < 1, such that this bound the region of the experiment, that is no guarantee that is satisfied, then we see that the inner horizon is reached such microscopic information can be transmitted out to when LE 2/(g∗EB ). an observer at spatial infinity where the physics is dic- ≈ We therefore have the following picture for g∗ < 1. tated by general relativity. In fact, we found precisely We have a background energy density EB >> MPl in this result: no information about the local physics can be a region of radius LB = 1/EB. We probe the system communicated to an observer at infinity due to the for- at length scales smaller than LB and we see no trapped mation of an enshrouding trapped surface. The resulting surface. As we approach the length scale (2 g∗)LB/g∗ horizon structure consists of an energy dependent “inner we see a trapped surface form. This behavior− persists horizon” and the classical Schwarzschild horizon in the until we reach the large radius GEB , at which point extreme infrared. This difference between local measure- we return to semiclassical physics outside an ordinary ments and asymptotic information necessitates caution Schwarzschild horizon. Although current truncations when making blanket statements about the presence of a have g∗ of O(1), it is interesting to note that if g∗ << 1, minimum length within the asymptotic safety approach the radius of the inner horizon can be much larger than [35]. One can, for example, instead address this question the radius of the experiment, i.e. there is a relatively in a complementary way for scattering experiments[36] large region outside the experiment where no trapped where the in/out states are at spatial infinity. surfaces exist. In this picture local observers perform- Finally,the qualitative features of the fixed point domi- ing local experiments see no trapped surfaces, and there nated running and indeed the existence of a non-Gaussian is no minimum length for them. However, they cannot fixed point has been argued by Percacci [33] to be a transmit any measurements all the way to spatial infin- largely scheme independent feature of the the Wilsonian ity. Hence for observers at asymptotic infinity (as would ERGE. He further notes that the position of the fixed be the case in a scattering experiment for example) there point has a weak dependence on parameters in the choice is still a fundamental minimum observable length. of the IR cut-off function and so the values for the fixed points are thus believed to be quite robust. However, since our result depends on the fixed points in such a sen- V. CONCLUSIONS sitive way, it is perhaps of general interest to extend the fixed point calculations to more realistic choices of mat- There’s a rather obvious tension between the stan- ter Lagrangian and compare to a more detailed gedanken dard minimum length argument and asymptotic safety. experiment. Asymptotic safety predicts that gravity is renormalizable and hence well behaved into the extreme UV, whereas the minimum length argument predicts a fundamental cut- Acknowledgments off to anny quantum field theory. Motivated by this, we re-examined the hoop conjecture in light of asymptotic We thank Jishnu Battacharyya and Keisuke Juge for safety. Adopting an approach relatively independent of comments and stimulating discussions.

[1] K. S. Thorne, in Magic Without Magic, ed. J. Klauder, [2] M. Choptuik and F. Pretorius, 0908.1780 [gr-qc] (W. H. Freeman, San Francisco, 1972). 8

[3] P. D.D Eath´ and P. N. Payne, Phys. Rev. D46 , 658 Nucl.Phys.B824, 168 (2010), eprint ArXIv: 0902.4630. (1992); Phys. Rev. D46, 675 (1992); Phys. Rev. D 46, [22] A. Bonanno and M. Reuter, Phys.Rev.D62, 043008 694 (1992). (2000). [4] P. C. Aichelburg and R. U. Sexl, Gen. Rel. Grav, 2, 303 [23] A. Ashtekar, C. Beetle and S. Fairhurst, (1971) Class.Quant.Grav.16:L1-L7,1999. eprint gr-qc/9812065. [5] S. B. Giddings and M. Srednicki, Phys. Rev. D 77, 085025 [24] A. Ashtekar and B. Krishnan, Living Rev.Rel.7:10,2004. (2008) eprint gr-qc/0407042. [6] T. Banks and W. Fischler, eprint hep-th/9906038. [25] P. Bizon, E.Malec and N O’Murchadha, Phys. Rev. Lett [7] O. Aharony and T. Banks, JHEP, 9903:016 (1999). 61, 1147 (1988) [8] N. Kaloper and J. Terning, Gen. Rel. Grav, 39, 1525 [26] E. Flanagan, Phys. Rev. D15, 1429 (1992) (2007)[Int. J. Mod.Phys, D 17, 665 (2008)). [27] C. Barrab´es, V. P. Frolov and E. Lesigne, Phys. Rev. [9] D. M. Eardley and S. B. Giddings, Phys.Rev. D66, D69, 19501(R) (2004) 044011 (2002). [28] H. Yoshino and Y. Nambu, Phys. Rev. D67, 024009 [10] L. Crane and L. Smolin, Nucl. Phys. B267, 714 (1986) (2003) [11] Lecture available online at [29] T. Chiba, T. Nakamura, K. Nakao and M. Sasaki, hep.phys.uoc.gr/mideast2/lectures/BanksLecture.pdf Class. Quant. Grav. 11, 431 (1994). [12] A. Shomer, eprint Arxiv: 0709.3555. [30] A. Strominger, Phys. Rev. D30, 2257 (1984) [13] X. Calmet, M. Graesser and S. D. H. Hsu, Int. J. Mod. [31] S. Deser and B. Tekin, Phys. Rev. D67, 084009 (2003) Phys. D14,2195,(2005). [32] S. Deser and B. Tekin, Phys. Rev. D75, 084032(2007) [14] L. J. Garay, Int. J. Mod. Phys, A10, 145 (1995). [33] G. Narain and R. Percacci, eprint:0910. 5390 [hep-th]. [15] S. Weinberg, in General Relativity: An Einstein Cente- [34] JoAnne Hewett and Thomas Rizzo, JHEP 0712: 009 nary Survey ed. S. W. Hawking and W. Israel, Cambridge (2007) University Press, Cambridge, 1979, pp. 790-831. [35] R. Percacci and G.-P. Vacca, arXiv:1008.3621 [hep-th] [16] S. Weinberg, arXiv:0908.1964 [hep-th]. [36] S. Giddings, M. Schmidt-Sommerfeld and J.R. Andersen [17] O. Lauscher and M. Reuter, JHEP 0510 050 (2005), 1005.5408 [hep-th]. eprint hep-th/0508202. [37] We caution the reader that in our approach the “inner [18] M. Niedermaier, Class. Quant. Grav. 24, R171 (2007), horizon” is not a true spacetime inner horizon in the eprint gr-qc/0610018. sense of a Reissner-Nordstrom inner horizon. It is instead [19] M. Niedermaier and M. Reuter, Living. Rev. Relativity, rather experiment dependent, in that the location of the 9, 5 (2006) inner horizon depends on the total energy of the space- [20] D. F. Litim, Phys. Rev. Lett. 92, 201301 (2004), eprint time and the specific total energy of our experimental hep-th/0312114. probe. [21] D. Benedetti, P. Machado and F. Sauressig,