Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect

Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the laws of pure mathematics. This is sufficient to prove that Einstein’s unimodular field equations are invalid. Since the unimodular form must hold in the general theory of relativity, it follows that the latter is also physically and mathematically unsound, lacking a proper mathematical foundation.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

Orbital dynamics on invariant sets of contact Hamiltonian systems

<p style='text-indent:20px;'>In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets <inline-formula><tex-math id="M1">\begin{document}$ \Omega_\pm, \Omega_0 $\end{document}</tex-math></inline-formula>. On the invariant sets <inline-formula><tex-math id="M2">\begin{document}$ \Omega_\pm $\end{document}</tex-math></inline-formula>, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set <inline-formula><tex-math id="M3">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula> may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.</p>

Classification of curves on de Sitter plane

In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described.

Differential Invariants of the (2+1)-Dimensional Breaking Soliton Equation

We construct the differential invariants of Lie symmetry pseudogroups of the (2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra. Their syzygies and recurrence relations are classified. In addition, a moving frame and the invariantization of the breaking soliton equation are also presented. The algorithms are based on the method of equivariant moving frames.