``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/*r*^{2} 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 *V*_{d} by an elementary argument from the Einstein-Hilbert action:

giving

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

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*=2*m*, but the
extent transverse to the brane grows only as
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 , 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 . 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`.