Abstract
In this article we construct and analyze new classes of wormhole and flux tubelike solutions for the 5D vacuum Einstein equations. These 5D solutions possess generic local anisotropy which gives rise to a gravitational running or scaling of the Kaluza–Klein “electric” and “magnetic” charges of these solutions. It is also shown that it is possible to self-consistently construct these anisotropic solutions with various rotational 3D hypersurface geometries (i.e., ellipsoidal, cylindrical, bipolar and toroidal). The local anisotropy of these solutions is handled using the technique of anholonomic frames with their associated nonlinear connection structures [S. Vacaru, Ann. Phys. (N.Y.) 256, 39 (1997); Nucl. Phys. B 434, 590 (1997); J. Math. Phys. 37, 508 (1996); J. High Energy Phys. 09: 011 (1998); Phys. Lett. B 498, 74 (2001)]. Through the use of the anholonomic frames the metrics are diagonalized, in contrast to holonomic coordinate frames where the metrics would have off-diagonal components. In the local isotropic limit these solutions are shown to be equivalent to spherically symmetric 5D wormhole and flux tube solutions.
Highlights
The first solutions describing black holes and wormholes in 4D and higher dimensional gravity were spherical symmetric solutions with diagonal metrics.1 Later Salam, Strathee and Perracci2 showed that including off-diagonal components in higher dimensional metrics is equivalent to including gauge fields
In this article we extend the investigation of Ref. 5 by applying the anholonomic frames method to construct anisotropic wormhole and flux tube solutions to 5D Kaluza-Klein theory which possess a range of different symmetrieselliptic, cylindrical, bipolar, toroidal
The holonomic parts of an object are indicated with indices of type i, j,k, . . . , while the anholonomic parts have indices of type a,b,c, . . . . Tensors, metrics and linear connections with coefficients defined with respect to anholonomic frames9͒ and10͒ are distinguisheddby N-coefficients into holonomic and anholonomic subsets and are called d-tensors, d-metrics and d-connections
Summary
The first solutions describing black holes and wormholes in 4D and higher dimensional gravity were spherical symmetric solutions with diagonal metrics. Later Salam, Strathee and Perracci showed that including off-diagonal components in higher dimensional metrics is equivalent to including gauge fields. References 3 and 4 examined locally isotropic solutions with off-diagonal metric components for 5D vacuum Einstein equations These solutions were similar to spherically symmetric 4D wormhole or flux tube metrics with ‘‘electric’’ and/or ‘‘magnetic’’ fields running along the throat of the wormhole. Wormholes in 5D gravity 2487 magnetic’’ parameterse.g., the ‘‘electric’’ and ‘‘magnetic’’ charges associated with the solutions This variation of the ‘‘electromagnetic’’ charges, which here occurs in the context of a higher dimensional gravity theory, can be likened to the variation or running of electric charge that occurs when a real electric charge is placed into some dielectric medium or in a quantum vacuum where quantum fluctuations produce a scale dependent electric charge. We will sometimes loosely refer to this gravitational variation of the ‘‘electromagnetic’’ parameters of the solutions as the gravitational running, scaling or renormalization of the charges of the solutions
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