Hyperdimensional unified field theories date back to 1921, when Kaluza reported that the Einstein equations of general relativity written in five dimensions (5D) would yield both the standard four-dimensional Einstein equations as well as the Maxwell equations, thus unifying general relativity and electrodynamics. Subsequent research focused on the field equations, with little attention paid to...

We show that Bose-Einstein condensates in a honeycomb optical lattice are described by a nonlinear Dirac equation in the long wavelength, mean field limit. Unlike nonlinear Dirac equations posited by particle theorists, which are designed to preserve the principle of relativity, i.e., Poincaré covariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks this symmetry. We presen...

The pulsation equations for spherically symmetric black hole and soliton solutions are brought into a standard form. The formulae apply to a large class of field theoretical matter models and can easily be worked out for specific examples. The close relation to the energy principle in terms of the second variation of the Schwarzschild mass is also established. The use of the general expressions...

We apply Lindstedt's method and multiple scale perturbation theory to analyze spatio-temporal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasi-one-dimensional Bose-Einstein condensates with mean-field interactions. We determine the dependence of the amplitude of modulated amplitude waves on their wave number. We also explore the band structure of Bose-Einste...

In a space-time M with a Killing vector field ξa which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ξa gives a 3-dimensional space S. Besides the symmetry-reduced action from that of Einstein-Hilbert, an alternative action of the fields on S is also proposed which gives the same fields equations as those reduced from the vacuum Einstein equation o...

We show that solutions of the Bach equation exist which are not conformal Einstein spaces. In connection with fourth order gravitational field equations, cf. e.g. [1, 2] where the breaking of conformal invariance was discussed, the original Bach equation, Bij = 0 , (1) enjoys current interest. Eq. (1) stems from a Lagrangian L = 1 2 √ −gCijklC , and variation gives, cf. Bach [3], 1 √−g δL/δg ij...

Consistency of Einstein’s gravitational field equation Gμν ∝ Tμν imposes a “conservation condition” on the T -tensor that is satisfied by (i) matter stress tensors, as a consequence of the matter equations of motion, and (ii) identically by certain other tensors, such as the metric tensor. However, there is a third way, overlooked until now because it implies a “nongeometrical” action: one not ...

In this paper, we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in Hs, s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric g, we manage to implement the commuting vector field approach to prov...

The paper develops a (2 + 2)-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein-Hilbert action, and it should find a variety of applications. It is ap...