Kaluza–Klein equations, Einstein’s equations, and an effective energy-momentum tensor

Abstract

Following earlier work, it is inquired how far the 5-D Kaluza–Klein equations without sources may be reduced to the Einstein equations with sources. It is shown by algebraic means that this can be done, provided the extra part of the 5-D geometry is used appropriately to define an effective 4-D energy-momentum tensor. The latter has reasonable properties, but will require further detailed study.

Hydrostatic Equilibrium of a Perfect Fluid Sphere with Exterior Higher-Dimensional Schwarzschild Spacetime

“…Interior solutions for higher dimensional perfect-fluid have been discussed by Krori et al [4] and Shen et al [5]. Other interior solutions, as well as their physical properties, have been discussed in the context of "matter-from-geometry" theory, in which four-dimensional matter is interpreted as a manifestation of five-dimensional geometry [6,7].…”

Geodesics, the equivalence principle and singularities in higher-dimensional general relativity and braneworlds

“…Concerning the field equations, the usual and simplest practice 2 is to take n = 4 and employ the 4 + 1 Einstein field equations. While this would provide a simple choice for the 4 + 1 field equations, it would cause the 3 + 1 apparent-world field equations to differ from the 3 + 1 GR Einstein field equations, as a result of projections of higher-d matter and 'matter' induced terms from the higher-d geometry [1,5].…”

“…We recall that the conventional formulation of GR has three main postulates. (i) 3 + 1 spacetime is a pseudo-Riemannian manifold, which is assumed to be the arena in which real physical events and interactions take place, (ii) Einstein field equations hold and (iii) light rays and freely-falling non-rotating, test-mass particles respectively follow the null and timelike geodesics of the spacetime 1 . This allows contact to be made with observations, and is intimately connected with the equivalence principle, according to which freely-falling non-rotating test particles follow timelike geodesics irrespective of their composition (see [13,14] for a thorough discussion).…”

“…Examples of such higher-d GR theories include (noncompactified) (n + 1)-d GR [1] (where the choice of the 3 + 1 apparent world is arbitrary), Kaluza-Klein theory [2] (where one dimension is compactified leading to a 3 + 1 apparent world described by GR, electromagnetism and a scalar), low energy string cosmology [3] (based on generalized compactifications), and some of the simpler braneworlds [4,5,6,7,8] (in which the apparent world consists of matter and tension restricted to a special brane hypersurface and the actual world is this hypersurface together with a surrounding bulk spacetime).…”

Classical confinement of test particles in higher-dimensional models: Stability criteria and a new energy condition

“…Such "braneworld" scenarios have been extensively analyzed in the literature, and have been used to address issues such as the hierarchy problem of particle physics [1,2,3,4], as well as the idea that the post-inflationary epoch of our universe was preceded by the collision of D3-branes [5,6]. In all fairness, it should be mentioned that the current flurry of interest in braneworld scenarios has been preceded by numerous other models making use of large or infinite extra dimensions [7,8,9,10,11,12].…”