Gravitational forces on a codimension-2 brane
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We compute the gravitational response of six dimensional gauged, chiral
supergravity to localized stress energy on one of two space-filling branes,
including the effects of compactifying the extra dimensions and brane
back-reaction. We find a broad class of exact solutions, including various
black-brane solutions. Several approximate solutions are also described, such
as the near-horizon geometry of a small black hole which is argued to be
approximately described by a 6D Schwarzschild (or Kerr) black hole, with event
horizon appropriately modified to encode the brane back-reaction. The general
linearized far-field solutions are found in the 4D regime very far from the
source, and all integration constants are related to physical quantities
describing the branes and the localized energy source. The localized source
determines two of these, corresponding to the source mass and the size of the
strength of a coupling to a 4D scalar mode whose mass is parametrically smaller
than the KK scale. At large distances the solutions agree with those of 4D
general relativity, but for an intermediate range of distances (larger than the
KK scale) the solutions better fit a Brans-Dicke theory. For a realistic choice
of parameters the KK scale could lie at a micron, while the crossover to
Brans-Dicke behaviour could occur at around 10 microns. While allowed by
present data this points to potentially measurable changes to Newton's Law
arising at distances larger than the KK scale.