Black Hole Constraints on Modifications of Gravity
Black Hole Constraints on Modifications of Gravity Gia Dvali CERN Theory Division / CCPP, New York University / Max Planck Institute for Physics
Outline 1) Lesson from the black hole physics: Species inevitably affect gravity. 2) Species determine the fundamental (holographic) gravitational length scale 4) Implications for large (and short) distance modifications of gravity 5) Generalized holography ?
The well-understood large-distance black hole physics + effective field theory consistency gives a powerful tool for understanding short distance gravitational physics.
Existence of species in the low energy theory implies modification of short distance gravity by lowering the gravitational cutoff below MP and by emergence of extra gravitational dimensions. This follows from the consistency of the gravitational physics, independently of our input assumptions about the nature of short distance geometry. Consider a low energy theory with N particle species. Under species we mean weakly interacting particles, with the decay width less than their mass, Γ < m In the opposite case we deal with broad resonances, and our analysis has to be modified accordingly.
It follows from the consistency of Black Hole physics that in any theory with N species the scale of quantum gravity is inevitably lowered, relative to the Planck mass M2 * = M2 P /N ! The fundamental length scale at which classical gravity is getting strong is L* = M-1 * = √N / MP . This can be proven by black hole thought experiments. [G.D.; G.D. and Redi,’07; G.D., Lüst 08]
The fact that in a theory with N particle species, the quantum gravity scale is M2 * = M2 P /N can be seen by number of arguments, perhaps the most elegant being, that black holes of size R < M-1 * , have the lifetime tBH < R and thus, cannot be regarded as semi-classical states of the Hawking temperature TH = 1/R ! Thus, black holes of this size are quantum objects.
Hawking flux with T = 1/R The black holes of size R < L* = √N / MP cannot afford to be quasi-classical, because they half-evaporate faster than their size! Or equivalently, the rate of temperature-change is faster than the temperature-squared dT/dt > T2 L*
Hawking flux of N species with T = 1/R Einsteinian black holes would half-evaporate during the time t = R3 M2 P N And for R2 < N / M2 P , we have t < R, or equivalently dT/dt > T2
Thus, the fundamental length scale below which no quasi-classical black holes can exist in any consistent theory with N species is L* = √N / MP and the corresponding mass scale marks the cutoff.
The well known explicit example, where the relation M2 * = M2 P /N is fixed by the fundamental theory , is extra dimensions. So consider a theory with n extra dimensions compactified on n-torus of radii R .
As a result of the dilution, there is a simple relation between the true quantum gravity scale and the Planck mass measured at large distances: M2 P = M2 * (M*R)n Volume of extra space Notice, that the above relation can be rewritten as, M2 P = M2 * N, Where N is the number of Kaluza-Klein species . This very important, because the latter expression turns out to be more general than the former: What matters is the number of species!
Alternative proof of the bound comes from quantum information. [G.D., Gomez, `08] Species exist as long as we can distinguish them by physical measurements.
Equivalently, we can encode information in the species numbers. With N species, a simple message can be an N-sequence of 1-s and 0-s, each referring to an occupation number of a given species in the message. A typical message = (0,1,0,0,0 . 1, 1,1) The simplest message includes a single 1, and all other 0-s: A simplest Message = ( 0,0, 0,1,0,...0) What is the minimal space-time scale L on which we can decode such a message?
The decoder contains samples of all N species in each pixel . So the spacetime resolution is set by the size of the pixel. How small can this size be? Without gravity, there is no limit to the smallness of L.
Message encoded in a green flavor Pixel of size L, with all the sample flavors
L A processor of information encoded in species contains N species localized within the pixel of scale L . In a theory without gravity, L can be arbitrarily small. However, because of gravity, we must have L > √N / MP , or else the processor itself collapses into a black hole!
M-1 P √N/MP R mrg m-1 Strength of Gravity as Function of Distance
Implication for large distance modification of gravity Consider a 3-brane (a Z2 – orbifold plane) embedded in 5D flat space . Assume that the bulk theory is a pure 5D Einstein gravity with 5D Planck mass M5 . Whereas the world-volume brane theory is a four-dimensional theory with N four-dimensional particle species. Extra dimension Bulk gravity : V(r - 1/(M3 5 r2 ) N species localized on the brane
Consider now the two observers: One (Bob) is the bulk observer , and the other (Alice) is localized on the brane.
Both try to determine their own holographic scales. Because Bob measures pure bulk 5D gravity, for him the holographic scale obviously is the five-dimensional Planck Length, LB = M-1 5 . Extra dimension Bob sees pure bulk gravity : V(r - 1/(M3 5 r2 ) Alice sees N species on the brane But, what is the fundamental holographic scale for Alice?
What is the fundamental (holographic) length scale detected by Alice? If Alice would also measure 5D gravity, then her holographic scale, LA , would be given by: LA = N1/3 / M5 The mass of a pixel of size LA, with N localized species is at least M = N/ LA, Thus, in 5D gravity, the critical size of a pixel saturating its gravitational Radius is given by the condition, N/ (M5 LA Any pixel, with N localized species, smaller than the above length collapses Into a black hole. Thus, if Alice were measuring 5D gravity, her fundamental length scale would be larger than the one measured by Bob!
How can the two observers in the same theory have different fundamental Scales? What if we require that LA = LB a consistency condition? Then we are forced to admit that Alice and Bob cannot measure the same gravitational law at all scales.
Alice must measure 4D gravitational law V(r - 1/ [(M3 5 rc ) r] , for distances r < rc = N M-1 * The effective for dimensional Planck mass That Alice measures at short distances is M2 P = M3 5 rc Thus, from Alice’s perspective gravity is 4D at short distances, and becomes 5D in far infrared!
What is the physics behind this modification? Perturbatively, the effect may be explained as coming from the quantum effects of particle Species on the brane. Quantum loops of these particle generate 4D Einstein-Hilbert term for the induced metric (G.D., Gabadadze; Veneziano): Extra dimension N species loops on the brane generate 4D Einstein term (N M2 5 ) ∫ d4 X√-g4 R4
So the effective action is (N M2 5 ) ∫ d4 X√-g4 R4 + M3 5 ∫ d5 X√-g5 R5 The physical consequences of this effective action are well known (DGP gravity ): it gives a crossover behavior from 4D to 5D gravitational law exactly at the scale rc = N M-1 5 What we are learning here is: Localization of species + BH holographic bound implies IR modification of effective world-volume gravity to higher dimensional regime! This is pretty remarkable, since without knowledge of the holographic bound, we could have naively assumed that contribution to 4D Einstein-Hilbert action from species loops can be simply fine-tuned to an arbitrary value.
But , black hole holographic bound (which is fully non-perturbative and exact) tells us that this is impossible. Theory has to guarantee the change of the gravitational regime, out of consistency.
The above suggests that the higher dimensional gravities with crossover behavior are equivalent to 4D theories with species. Some sort of generalization of Maldacena’s ADS/CFT. Cosmological applications for large distance modifications of gravity are clear. One can try to understand such cosmologies in terms of cosmologies of 4D theories with many species. Explicit duality in case of DGP gravity was demonstrated recently by Barvinsky, Deffayet and Kamenshchik .
Classical modification of graviton propagator in the interval M2 P /N > p2 > 1/R2 , implies that there is a tower of (Spin-2) states.
This can be seen from the spectral representation, G(p2 ∫ ds ρ(s) /(p2 + s) . Since G(p2 ) ≠ p-2 , there must be new poles within the interval M2 P /N > p2 > 1/R2 . These poles can be regarded as Kaluza-Klein tower of the new dimensions.
The statement that species are separated in a true dimension, is also supported by the locality properties in the species flavor . Consider a microscopic black hole of mass ~ M * , produced in a particleantiparticle annihilation of i-th flavor of species at energies ~ M *. . By unitarity decay rate of such a black hole back to i-th species is Γ ~ M * And the decay rate into all other flavors j ≠ i must be suppressed by 1/N. i i j j BHi
Although at no point did we assumed any input geometric substructure, the species flavor leads us to the existence of the gravitational extra dimensions.
The model-independent non-Einsteinianity interval for the black hole mass is al least MP /√N < MBH < MP √N at the lower end of this interval black holes are maximally non-democratic in the species flavor, and conserve it. From what we’ve learned, the flavor violating processes mediated by the virtual microscopic black holes, must be suppressed by some form factors , sd sd exp(- E/ M*) where E is a characteristic energy of the interaction.
Micro Black Hole Non-Democracy Puzzle. GD, Pujolas ‘08. We have seen that the microscopic black holes, by unitarity cannot evaporate democratically in all the particle species. Thus, different species must ``see’’ different horizons and be produced at different thermal rates. Thus, the micro black holes can be labeled by species. Micro black holes carry the species hair. But, how is this possible for quasi-classical micro black holes? The resolution of the puzzle is in democratic transition, during which the microscopic black holes become democratic and loose the species hair !
Black hole Extra dimension After democratic transition all branes share the same black hole
Black hole stabilized by the stretched string
String shortens and brane gets connected to black hole
The resolution of black hole non-democracy puzzle, shows that microscopic black holes have two characteristic time scales: 1) Quantum Hawking evaporation time; 2) Classical democratization time, set by N. For the smallest black holes, the evaporation is the dominant effect, and such black holes therefore conserve the species flavor for all the practical purposes. For larger ones, the democracy sets in, and correspondingly they can mediate flavor-violating interactions.
The latter processes are however suppressed because creation of the democratic (heavy) black holes is very costly.