Holography and 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 modify gravity.

2) Species determine the fundamental (holographic) gravitational length scale

3) Implications for large (and short) distance modifications of gravity

4) Whenever a 4D theory emerges as a theory on the brane embedded in 5D flat space, the 5D holographic scale penetrates in 4D in form of the strong coupling.

5) Generalized holography.

Items 2 - 5 are based on work with Cesar Gomez

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 lessthan 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

* = M2P /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.

[GD; GD & Redi ’07; GD & 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 M2P / N

And for R2

< N / M2P , we have t < R,

or equivalently

dT/dt > T2

Thus, the fundamental length scale below whichno 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:

M2P = M2

* (M*R)n

Volume of extra space

Notice, that the above relation can be rewritten as,

M2P = 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 space-time 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-1P √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 localizedon 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 scaleobviously 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 ) 3 = 1 .

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/ [(M35 rc ) r] ,

for distances r < rc = N M-15 The effective for dimensional Planck mass

That Alice measures at short distances is

M2P = M3

5 rc

Thus, from Alice’s perspective gravity is 4D at short distances, and becomes5D 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 (GD, Gabadadze; Veneziano):

Extra dimension

N species loops on the brane generate 4D Einstein term (N M2

5 ) ∫ d4X√-g4 R4

So the effective action is

(N M25 ) ∫ d

4X√-g4 R4 + M35 ∫ d

5X√-g5 R5

The physical consequences of this effective action are well known(DGP gravity ): it gives a crossover behavior from 4D to 5D gravitational lawexactly 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.

rc = N

LB = N1/3

LA = 1

Hierarchy of 4D and 5D boxes:

Let us summarize what happened.

By stretching the modified gravity scale far beyond the bulk holographic limit,

(rc = N) >> (LB = N1/3 ),

4D observer (Alice) manages to store N bits of information in a 4D box that is much smaller than the 5D holographic counterpart ,

(LB = N1/3 ) >> (LA = 1) So it seems as if at distances r << rc in the 4D theory all memory about the bulk gravity has been erased.

However, some memory about high-dimensions persists because of strong coupling.

On the phenomenological front: this is why modified gravity theories are not ruled out by solar system observations, and that’s why their short distance tests are possible.

The origin of the strong coupling scale in Alice’s 4D world is the following.

Effective 4D graviton that Alice interact with, in reality is a wave-packet (a resonance) of continuum of the massive Kaluza-Klein gravitons. These carry 5 degrees of freedom, and so does their composite 4D-graviton. These helicities are: 5 = 2 ( tensor) + 2 (vector) + 1 (scalar)

It is known for some time that the scalar helicity ( call it π), is strongly-coupled at the scale Λstr

= N-1/2 M5

Deffayet, GD, Gabadadze, Vainshtein ‘01 Luty, Porrati , Rattazzi, ’03

(∂π)2 □π

Λstr3

rc

r*

rg

The r* phenomenon:

For any source of Schwarszchild radius rg a new radius appears:

r* = (rc2 rg )

1/3

The scale r* = (rc2 rg )

1/3

is the distance at which the scalar polarization of 4D graviton becomes strongly -coupled and non-linear . The 5D holographic scale penetrates in 4D exactly through the r* - scale ! In order to understand this let us summarize the story.

The mass of the elementary information bit is mbit = M5 .

Now, in 5D gravity the Scwarszchild radius of this information bit is

rg 5 = L5 ,

and this is why 5D observer can encode only asingle information bit within this radius. Now, for the 4D observer the story is different, because 4D gravity is much weaker, and the Schwarszchild radius of he same information bit is much smaller

rg = L5 / N << rg 5

This is why 4D observer can strore N bits of information within the pixel of size which much is smaller than the 5D holograpic length.

However, the memory about the 5D holography is not lost.

Computing scale or an elementary information bit we get exactly 5D holographic scale!

r* = (rc2 rg )

1/3 = N1/3 L5 Thus, the 4D strong coupling is the way 5D holography manifests itself in 4D theory!

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.

In 4D theory the memory about the higher-dimensional theory persists in form of the strong coupling.

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 .