A variational principle for Newton-Cartan theory

In the framework of a space-time theory of gravitation a variational principle is set up for the gravitational field equations and the equations of motion of matter. The general framework leads to Newton's equations of motion with an unspecified force term and, for irrotational motion, to a restriction on the propagation of the shear tensor along the streamlines of matter. The field equations obtained from the variation are weaker than the standard field equations of Newton-Cartan theory. An application to fluids with shear and bulk viscosity is given.     

Zur Konstanz der Teilchenzahlen in der allgemeinen Relativitätstheorie

This paper suggests that the “spontaneous creation of matter in strong gravitation fields” – which is asserted in some papers – may be a result of an inconsistent approximation for the simultaneous systems of Einstein field equations and of matter field equations. Firstly we shall remark that Einstein's dynamical equation is not only the consequence of the Einstein field equations (via Bianchi identity) but also the consequence of the relativistic matter field equations and of the weak principle of equivalence. Therefore, for all discussions we must start from the dynamical equation and from the conservation laws of charges. Secondly, we find a formal inhomogeneous dynamical equation for the matter tensor by working in a background-field approximation. But, this inhomogenity vanishes according to the matter field equations. Thirdly, we discuss the properties in an evolution-cosmos. In such cosmos a law of conformal conservation of energy is fulfilled. From this conservation law results that the energy is a continuous function of the gauge of time. Such continuous dependence involves the constance of discrete numbers according to the well known argument of Planck.     

New Concepts from the Evans Unified Field Theory. Part One: The Evolution of Curvature,Oscillatory Universe Without Singularity,Causal Quantum Mechanics,and General Force and Field Equations

No Heading The Evans field equation is solved to give the equations governing the evolution of scalar curvature R and contracted energy-momentum T. These equations show that R and T are always analytical, oscillatory, functions without singularity and apply to all radiated and matter fields from the sub-atomic to the cosmological level. One of the implications is that all radiated and matter fields are both causal and quantized, contrary to the Heisenberg uncertainty principle. The wave equations governing this quantization are deduced from the Evans field equation. Another is that the universe is oscillatory without singularity, contrary to contemporary opinion based on singularity theorems. The Evans field equation is more fundamental than, and leads to, the Einstein field equation as a particular example, and so modifies and generalizes the contemporary Big Bang model. The general force and conservation equations of radiated and matter fields are deduced systematically from the Evans field equation. These include the field equations of electrodynamics, dark matter, and the unified or hybrid field.     

Einstein's Equations and Associated Linear Systems

The relationship between Einstein's field equations and classical higher spin field equations is investigated using two-component spinor valued differential forms. Linear systems of equations associated to both the vacuum and coupled gravitational matter field equations are constructed. The latter equations are shown to be the integrability conditions of the linear systems.     

Higher-dimensional vacuum solutions of Einstein's field equations

Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.     

A note on conformal field equations

Conformal geometry is more fundamental than a Riemannian one. Whereas Riemannian geometry determines lengths and angles, a conformal one determines only angles and ratios of length. Equivalently, conformal geometry of space-time determines light cones, or causal structure. No length scale isa priori distinguished. It can be distinguished onlya posteriori, given a particular solution of matter field equations. Einstein's field equations of gravitation can be thought of as describing interaction of causal structure with a matter described by a real scalar massless field of weight 1/4. Electromagnetic field equations need precisely a conformal structure. One can also write down field equations for a spin-1/2 Dirac massless field, given information about light cones only. The energy-momentum tensor density is obtained by vierbeim variations.Supported by the Humboldt Foundation.     

The geometrical unification of gravity with its source

I give a personal retrospect covering the years during which Mashhoon, myself and others developed a unification of the gravitational field and its source (matter). In this geometrical approach, Einstein’s 4D field equations of general relativity with a source are derived from the 5D Ricci equations for apparent vacuum. The main equations are given, along with comments on how they were arrived at. They describe gravity, electromagnetism and a scalar field. This induced-matter or space–time–matter theory is in agreement with observation and invites further development. Mass, Matter and Mashhoon: a Tribute to Bahram Mashhoon on his 60th Birthday.     

The euclidean vacuum: Justification from quantum cosmology

We evaluate the wave function of the universe for a de Sitter minisuperspace with inhomogeneous matter perturbations from a massive scalar field. From the Wheeler-DeWitt equation, we derive Schrödinger equations for the matter modes. We show that the matter part of the Hartle-Hawking wave function is the euclidean vacuum state of quantum field theory in curved spacetime.     

Field equations and contracted Bianchi identities in the generalized Einstein-Cartan theory

The generalized Palatini variational principle for the gravitational field interacting with a matter field is considered. The contracted Bianchi identities are proved and the reduction of the generalized Einstein-Cartan equations to the set of 6 dynamical equations and 4 constraints is presented.     

The dynamical structure of the Einstein-Cartan-Sciama-Kibble theory of gravity

It is shown that in the SO(3)-covariant Hamiltonian formulation the system of the ECSK equations can be reduced to 7 gravitational constraints, 18 gravitational dynamical equations, and a system of matter field equations. The geometric meaning of the canonical (symplectic) and gauge variables is also explained. Moreover, a general method of how to analyse degenerate matter field lagrangians in the framework of the ECSK theory is discussed. The exposition is given in the language of SO(3)-covariant differential operators on 3-dimensional slices of spacetime.     

The interaction of spin and torsion. II. The principle of general covariance

The principle of general covariance, as formulated by V. Guillemin and S. Sternberg, is used to derive the passive equations of motion for a spinning matter field in the presence of an external gravitational field with torsion. In the case where the matter field becomes concentrated along a curve, the equations of motion for a spinning particle studied by D. Rappaport and Sternberg are recovered. The results of A. Einstein, L. Infeld, and B. Hoffmann and of J. M. Souriau are thus generalized to include spin and torsion.     

Equilibrium between anisotropic normal and pion-condensed nuclear matter

The properties of the pion-condensed phase of nuclear matter are investigated at finite temperatures in the framework of a relativistic field theory. The solution of the field equations and the expectation value of the energy-momentum tensor are calculated in the mean-field approximation. It is observed that the self-consistent set of equations for the amplitudes of the mesonic fields obtained directly from the field equations are identical with the conditions of thermodynamical equilibrium. The pressure of the pion-condensed phase is found to be isotropic in thermodynamical equilibrium.

The possibility of phase equilibrium between pion-condensed and anisotropic normal nuclear matter is studied. The nuclear matter produced in heavy-ion collisions is anisotropic and it is far from thermodynamical equilibrium. During the collision process the anisotropy is decreasing and the system approaches thermodynamical equilibrium. It is shown that non-equilibrated pion- condensed nuclear matter may have the same anisotropy as the normal one and they may be in phase equilibrium during the whole collision process. This circumstance allows us to draw the following conclusion: if there is a chance at all for the phase transition from normal to pion- condensed phase then the anisotropy inevitably produced in heavy-ion collisions does not prevent this transition.     

Dynamical mechanism for varying light velocity as a solution to cosmological problems

A dynamical model for varying light velocity in cosmology is developed, based on the idea that there are two metrics in spacetime. One metric g_μν describes the standard gravitational vacuum, and the other describes the geometry through which matter fields propagate. Matter propagating causally with respect to can provide acausal contributions to the matter stress-energy tensor in the field equations for g_μν, which, as we explicitly demonstrate with perfect fluid and scalar field matter models, provides a mechanism for the solution of the horizon, flatness and magnetic monopole problems in an FRW universe. The field equations also provide a ‘graceful exit' to the inflationary epoch since below an energy scale (related to the mass of ψ_μ) we recover exactly the standard FRW field equations.     

f(R,L m ) gravity

We generalize the f(R) type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the matter Lagrangian L _m . We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy density of the matter only. Generally, the motion is non-geodesic, and it takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert–Einstein Lagrange density are also derived.     

Dynamic field theory and equations of motion in cosmology

We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays the role of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmological manifold defined as a solution of Einstein’s equations in the form of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations with an effective matter density contrast δρ/ρ≤1$\delta \rho /\rho \le 1$. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ?1$\delta \rho /\rho ?1$. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equations while the small scale perturbations are described by a particular solution of these equations with the bare stress–energy tensor of the baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars, galaxies and their clusters.     

The many-body problem within the Lee model

The structure of a field theoretical many-body problem is studied within the (non-static) Lee model. The explicit solvability of the renormalization problem allows the investigation of renormalization corrections in many-particle systems. Herefore, the renormalized equations are worked out for the N-V scattering and for the binding-energy problem of “N-V matter” — these cases taken in analogy to nucleon-nucleon scattering and nuclear matter. The N-V matter equations are obtained from a cluster expansion suitably defined for the field theoretical case. The ansatz for the correlated wave functions is chosen in such a way as to generate a two-hole-line expansion of the binding energy. The renormalized form of this field theoretical extension of Brueckner theory is discussed in detail revealing the medium effects on renormalization.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.     

Can the Local Energy-Momentum Conservation Laws be Derived Solely from Field Equations?

The vanishing of the divergence of the matter stress-energy tensor for General Relativity is a particular case of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. This identity, holding for any covariant theory of gravitating matter, relates the divergence of the stress tensor with a combination of the field equations and their derivatives. One could thus wonder if, according to a recent suggestion [1], the energy-momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that this can be done only in particular cases, while in general it leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique.     

The Universe Without a Singularity

A rigorous solution of the field equations with a finite density of matter in the Universe is obtained in the context of classical gravitation theory.