``Branification:'' an alternative to compactification

Steve Giddings, University of California at Santa Barbara giddings@physics.ucsb.edu
Recent developments have breathed new life into the old idea that the observable Universe is embedded in a spacetime with extra large or even infinite dimensions. This raises the exciting prospect that Planckian physics could be observed in high-energy accelerators, provides interesting new techniques to address hierarchy problems in physics, and could possibly lead to novel phenomena in cosmology and black hole physics.

Obstacles to the viability of such a scenario have included explaining why the matter that we see moves only along the 3+1 dimensional hypersurface, and explaining the observed gravitational 1/r2 force law characteristic of four dimensions. Old ideas on confinement of gauge fields and fermions to a domain wall have been supplemented with new ones from string theory involving D-branes - these address the first issue. Recall that D-branes are surfaces which open string ends stick to; if observable matter consisted of open strings and the Universe was a D3-brane, that could solve the problem. But gravity is harder to ``confine'' to a brane-like structure.

One idea that has been actively pursued by Arkani-Hamed, Dimopoulos, and Dvali [3] is that the brane is immersed in space with d extra large but compact dimensions. If the d+4 dimensional fundamental Planck mass is M, then the effective four-dimensional Planck mass follows in terms of the compact volume Vd by an elementary argument from the Einstein-Hilbert action:

\begin{displaymath}{1\over M^{d+2}} \int dV_{d+4} {\cal R} \sim {V_d \over M^{d+2}} \int
dV_{4} {\cal R} \ ,
\end{displaymath}

giving

\begin{displaymath}M_4^2\sim M^{d+2} V_d\ .\eqno(1)\end{displaymath}

An alternative explanation of the weakness of gravity is thus not that the fundamental Planck mass is so big, but rather that the compact volume is big. This raises the exciting prospect that the fundamental Planck scale may be more readily accessible in accelerator experiments, or that the compact dimensions may be detected through experiments with microgravity (see the next article in this issue of MOG).

A new variant of this scheme of even more theoretical interest was proposed by Randall and Sundrum (RS) [4]. In their picture, the brane is instead the Poincare-invariant boundary of a slice of 4+1 dimensional anti-de Sitter space. RS observed that the negative curvature of anti-de Sitter space plays a very similar role to that of a compact dimension, and effectively binds a graviton mode to the brane. As a result, at low energies matter living on the brane effectively interacts through four-dimensional gravity. The scale at which this ceases to be true, and the underlying infinite fifth dimension is revealed, is set by the anti-de Sitter radius, R. The non-compactness of the extra dimension distinguishes these ``branification'' scenarios from compactification, and has novel consequences such as the existence of a continuum of ``Kaluza-Klein'' modes. In analogy to equation (1), we have

\begin{displaymath}M_4^2\sim RM^3\ ,
\end{displaymath}

again raising the possibility that if the anti-de Sitter radius is large enough, the fundamental Planck scale is commensurately lower and Planckian or extra-dimensional physics may be much more experimentally accessible. Variants of the RS proposal have also been considered, involving either parallel branes in 5 dimensions [5], which may help with the hierarchy problem, or intersecting branes in more dimensions.

Initially there were questions of consistency of this proposal; for example Chamblin, Hawking, and Reall [6] and others observed the existence of black holes arising from matter on the brane with infinitely extended horizons and strong-coupling singularities at the horizon of anti-de Sitter space. However, they also suggested as a possible resolution that these would exhibit a Gregory-Laflamme instability resulting in a solution with horizon confined near the brane. This expectation was confirmed in the case of a 2+1 dimensional brane by Emparan, Horowitz, and Myers [7], and in a linearized analysis by Katz, Randall, and the author [8], who independently found that the horizon of such a black hole is shaped like a pancake. Specifically, its radius along the brane is the familiar r=2m, but the extent transverse to the brane grows only as $R\log m$ with the mass.

These and other checks in the linearized analysis (properties of propagators have been worked out in [8]; other linearized analysis appears in [1] support the consistency of RS branification. Moreover, they raise some interesting possibilities. For example, we, as four-dimensional observers, would see processes through their projection onto the brane. Therefore motion of an object flying around the pancake-shaped black hole through the fifth dimension could be interpreted by four-dimensional observers as motion into one side of the horizon and out the other!

More novelties in cosmology arise because of the extra degrees of freedom associated to motion of the brane or other five-dimensional perturbations of the metric. Initially concerns were raised that the Hubble law came out to be $H\propto \rho$, but more recent work [9,10] has shown that in the presence of extra dynamics that stabilizes the brane's motion we recover the familiar $H\propto \sqrt\rho$. More subtle consequences for early Universe physics are being explored, and there have been suggestions that these and related scenarios address the cosmological constant problem [12,13,14]

Finally, the proper setting for branification proposals is presumably string theory, and direct connection has been made to the celebrated AdS/CFT correspondence by Maldacena, Witten, Gubser [2] and [8]. In particular, H. Verlinde [11] has given a closely related proposal within string theory compactified (or perhaps noncompactified?) on a noncompact manifold with an AdS region. Verlinde's scenario deserves more close scrutiny.

Beyond the need to extend understanding of examples of branification in string theory, a number of interesting problems remain both in phenomenology (with a realistic model in hand, what would be the first observable consequence of this picture?); in cosmology, black hole physics and other aspects of the gravitational dynamics in its subtle interplay between four and five dimensions; and finally, with luck, in experiment.

References:

[1] J. Garriga and T. Tanaka, ``Gravity in the brane world,'' hep-th/9911055.

[2]S.S. Gubser, ``AdS/CFT and gravity,'' hep-th/9912001.

[3] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, ``The hierarchy problem and new dimensions at a millimeter,'' hep-ph/9803315 Phys. Lett. B429 263 (1998); ``Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity,'' hep-ph/9807344, Phys. Rev. D59:086004 (1999).

[4]L. Randall and R. Sundrum, ``An alternative to compactification,'' hep-th/9906064, Phys. Rev. Lett. 83 (99) 4690.

[5] J. Lykken and L. Randall, ``The shape of gravity,'' hep-th/9908076.

[6] A. Chamblin, S.W. Hawking, and H.S. Reall, ``Brane-world black holes,'' hep-th/9909205.

[7] R. Emparan, G.T. Horowitz, and R.C. Myers, ``Exact description of black holes on branes,'' hep-th/9911043.

[8] S.B. Giddings, E. Katz, and L. Randall, ``Linearized gravity in brane backgrounds,'' (to appear); for preliminary accounts see S.B. Giddings, talk at ITP Santa Barbara Conference ``New dimensions in field theory and string theory,'' and L. Randall, talk at Caltech/USC conference ``String theory at the millennium,''
http://www.itp.ucsb.edu/online/susy_c99/giddings/
http://quark.theory.caltech.edu/people/rahmfeld/Randall/fs1.html.

[9] C. Csaki, M. Graesser, L. Randall, and J. Terning, ``Cosmology of brane models with radion stabilization,'' hep-ph/9911406.

[10] P. Kanti, I.I. Kogan, K.A. Olive, M. Pospelov, ``Single brane cosmological solutions with a stable compact extra dimension,'' hep-ph/9912266.

[11] H. Verlinde, ``Holography and compactification,'' hep-th/9906182.

[12] J. de Boer, E. Verlinde, H. Verlinde, ``On the holographic renormalization group'',
hep-th/9912012; E. Verlinde and H. Verlinde, ``RG flow, gravity and the cosmological constant,'' hep-th/9912018; E. Verlinde, ``On RG flow and the cosmological constant,'' hep-th/9912058.

[13] N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, and R. Sundrum, ``A small cosmological constant from a large extra dimension,'' hep-th/0001197.

[14]S. Kachru, M. Schulz, and E. Silverstein, ``Self-tuning flat domain walls in 5-d gravity and string theory,'' hep-th/0001206.


Jorge Pullin
2000-02-06