The embedding of General Relativity in five dimensions

We argue that General Relativistic solutions can always be locally embedded in Ricci-flat 5-dimensional spaces. This is a direct consequence of a theorem of Campbell (given here for both a timelike and spacelike extra dimension, together with a special case of this theorem) which guarantees that anyn-dimensional Riemannian manifold can be locally embedded in an (n+1)-dimensional Ricci-flat Riemannian manifold. This is of great importance in establishing local generality for a proposal recently put forward and developed by Wesson and others, whereby vacuum (4+1)-dimensional field equations give rise to (3+1)-dimensional equations with sources. An important feature of Campbell's procedure is that it automatically guarantees the compatibility of Gauss-Codazzi equations and therefore allows the construction of embeddings to be in principle always possible. We employ this procedure to construct such embeddings in a number of simple cases.     

Towards a theory of macroscopic gravity

By averaging out Cartan's structure equations for a four-dimensional Riemannian space over space regions, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space. The averaged space is characterized by a metric, Riemannian and non-Rimannian curvature 2-forms, and correlation 2-, 3- and 4-forms, an affine deformation 1-form being due to the non-metricity of one of two connection 1-forms. Using the procedure for the space-time averaging of the Einstein equations produces the averaged ones with the terms of geometric correction by the correlation tensors. The equations of motion for averaged energy momentum, obtained by averaging out the contracted Bianchi identities, also include such terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (the non-Riemannian one is then the field tensor), a theorem is proved which relates the algebraic structure of the averaged microscopic metric to that of the induction tensor. It is shown that the averaged Einstein equations can be put in the form of the Einstein equations with the conserved macroscopic energy-momentum tensor of a definite structure including the correlation functions. By using the high-frequency approximation of Isaacson with second-order correction to the microscopic metric, the self-consistency and compatibility of the equations and relations obtained are shown. Macrovacuum turns out to be Ricci non-flat, the macrovacuum source being defined in terms of the correlation functions. In the high-frequency limit the equations are shown to become Isaacson's ones with the macrovauum source becoming Isaacson's stress tensor for gravitational waves.     

Averaging out the Einstein equations

A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.     

Averaged Lagrangians and MacCallum-Taub's limit in macroscopic gravity

The derivation of MacCallum-Taub's averaged high-frequency Lagrangian [1] is analysed with special attention paid to the assumptions made along the derivation. It is shown that averaged high-frequency Lagrangians of the same form as MacCallum-Taub's Lagrangian can be derived by applying the Brill-Hartle and macroscopic gravity averaging schemes. A procedure for the derivation of a Lagrangian of macroscopic gravity (an averaged Hilbert action) is proposed and its high-frequency limit (namely, its high-frequency perturbation expansion up to the second order terms in perturbations, which is referred to as MacCallum-Taub's limit) is calculated. There is disagreement [2] in the expressions for MacCallum-Taub's averaged high-frequency Lagrangian and the high-frequency limit of the macroscopic gravity Lagrangian. Possible reasons for such disagreement are analysed. The origin of the difference is shown to consist in using the propagation equation for perturbations, i.e. the linearized Ricci tensor vanishes, during the derivation (averaging) carried out in [1]. A new derivation of an averaged high-frequency Lagrangian without assuming the propagation equation to hold and by taking into account the proper correlation functions is given. The newly derived expression is shown to coincide with MacCallum-Taub's limit of the macroscopic gravity Lagrangian, which resolves the disagreement.     

Self‐diffusion in a hyaluronic acid–albumin–water system as studied by NMR

Hyaluronic acid (HA) is an anionic biopolymer that is present in many tissues and can be involved in cancerous neoformations. HA can form complexes with proteins (particularly, serum albumin) in the body. However, HA structures and processes involving HA have not been extensively studied by NMR because the molecule's rigid structure makes these studies problematic. In the current work, self‐diffusion of HA and bovine serum albumin (BSA), and water in solutions was measured by ¹H pulsed field gradient NMR (PFG NMR) with a focus on the HA‐BSA‐D₂O systems at various concentrations of BSA and HA. It was shown that in the presence of even a small amount of HA, the self‐diffusion coefficient (SDC) of BSA decreases. To explain this fact, three hypotheses were proposed and analyzed. The first one was based on the effect of slowing down of water mobility in the presence of HA. The second hypothesis suggested an effect of mechanical collisions of BSA with HA molecules. The third hypothesized that BSA and HA molecules form a complex where BSA molecules reduced in mobility. It was shown that the third mechanism is the most likely. The state of the BSA molecules in the BSA‐HA‐D₂O system corresponds to a ‘fast exchange’ condition from the NMR point of view: BSA molecules reside in the ‘free’ and ‘bound’ (with HA) states for much shorter time than the diffusion time of the PFG NMR experiment, 7 ms. The fractions of ‘bound’ BSA molecules in the BSA‐HA complex were estimated. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Domain of Applicability of General Relativity

We consider the domain of applicability of general relativity (GR), as a classical theory of gravity, by considering its applications to a variety of settings of physical interest as well as its relationship with real observations. We argue that, as it stands, GR is deficient whether it is treated as a microscopic or a macroscopic theory of gravity. We briefly discuss some recent attempts at removing this shortcoming through the construction of a macroscopic theory of gravity. We point out that such macroscopic extensions of GR are likely to be nonunique and involve non-Riemannian geometrical frameworks.     

On general and restricted covariance in general relativity

We reconsider the principle of general covariance and give a rigorous formulation of a principle ofrestricted covariance. We give a number of examples of preferred coordinate systems, considered in the literature, and in each case demonstrate the applicability of the notion of restricted covariance proposed.     

Peculiarities of NMR relaxation in micellar gels of Pluronic F-127

Based on the ¹H relaxation of transverse nuclear magnetization of triblock-copolymer Pluronic F-127 in D₂O, we proposed a model of the associated pluronic structure in which the polyethylene oxide of molecules in neighboring micelles are intertwined in regions of overlapping micellar coronas, while the polypropylene oxide cores of the micelles play a role of nodes in the 3D network.     

Lithium and Cesium Ion-Pair Acidities of Some Terminal Acetylenes and Aggregation in Tetrahydrofuran(1)

The equilibrium lithium acidities in THF have been determined for 4-ethynylbiphenyl, EB (pK 21.5-22.3), 3,3,3-triphenylpropyne, TPP (pK 22.3-22.7), and 1-ethynyladamantane, EA (pK 23.7). Ion pairs of (4-ethynylbiphenylyl)lithium are aggregated in the concentration range from 10(-)(4) to 10(-)(3) M, with an average aggregation number of 2.5. (3,3,3-Triphenylpropynyl)lithium ion pairs are partially aggregated at concentrations from 10(-)(5) to 10(-)(3) M; the average aggregation number is 1.2. Cesium acidities in THF have been determined for 3,3,3-triphenylpropyne (pK 29.1-29.9) and 1-ethynyladamantane (pK 31.56). The average aggregation number of (3,3,3-triphenylpropynyl)cesium ion pairs is 6.2 at concentrations of 10(-)(4)-10(-)(3) M.