Perfect hypermomentum fluid: variational theory and equations of motion

The variational theory of the perfect hypermomentum fluid is developed. The new type of the generalized Frenkel condition is considered. The Lagrangian density of such fluid is stated, and the equations of motion of the fluid and the Weyssenhoff-type evolution equation of the hypermomentum tensor are derived. The expressions of the matter currents of the fluid (the canonical energy-momentum 3-form, the metric stress-energy 4-form and the hypermomentum 3-form) are obtained. The Euler-type hydrodynamic equation of motion of the perfect hypermomentum fluid is derived. It is proved that the motion of the perfect fluid without hypermomentum in a metric-affine space coincides with the motion of this fluid in a Riemann space.Comment: REVTEX, 23 pages, no figure

The Spinning Particles as a Nonlinear Realizations of the Superworldline Reparametrization Invariance

The superdiffeomorphisms invariant description of $N$ - extended spinning particle is constructed in the framework of nonlinear realizations approach. The action is universal for all values of $N$ and describes the time evolution of $D+2$ different group elements of the superdiffeomorphisms group of the $(1,N)$ superspace. The form of this action coincides with the one-dimensional version of the gravity action, analogous to Trautman's one.Comment: 4 pages, RevTe

Comments on the tethered galaxy problem

In a recent paper Davis et al. make the counter intuitive assertion that a galaxy held `tethered' at a fixed distance from our own could emit blueshifted light. Moreover, this effect may be derived from the simplest Friedmann-Robertson-Walker spacetimes and the (0.3,0.7) case which is believed to be a good late time model of our own universe. In this paper we recover the previous authors' results in a more transparent form. We show how their results rely on a choice of cosmological distance scale and revise the calculations in terms of observable quantities which are coordinate independent. By this method we see that, although such a tethering would reduce the redshift of a receding object, it would not do so sufficiently to cause the proposed blueshift. The effect is also demonstrated to be much smaller than conjectured below the largest intergalactic scales. We also discuss some important issues, raised by this scenario, relating to the interpretation of redshift and distance in relativistic cosmology.Comment: 6 pages, 3 figures, submitted to Am.J.Phy

Maxwell Fields in Spacetimes Admitting Non-Null Killing Vectors

We consider source-free electromagnetic fields in spacetimes possessing a non-null Killing vector field, $\xi^a$. We assume further that the electromagnetic field tensor, $F_{ab}$, is invariant under the action of the isometry group induced by $\xi^a$. It is proved that whenever the two potentials associated with the electromagnetic field are functionally independent the entire content of Maxwell's equations is equivalent to the relation \n^aT_{ab}=0. Since this relation is implied by Einstein's equation we argue that it is enough to solve merely Einstein's equation for these electrovac spacetimes because the relevant equations of motion will be satisfied automatically. It is also shown that for the exceptional case of functionally related potentials \n^aT_{ab}=0 implies along with one of the relevant equations of motion that the complementary equation concerning the electromagnetic field is satisfied.Comment: 7 pages,PACS numbers: 04.20.Cv, 04.20.Me, 04.40.+

The Motion of a Body in Newtonian Theories

A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1), (1975)] provides the sense in which the geodesic principle has the status of a theorem in General Relativity (GR). Here we show that a similar theorem holds in the context of geometrized Newtonian gravitation (often called Newton-Cartan theory). It follows that in Newtonian gravitation, as in GR, inertial motion can be derived from other central principles of the theory.Comment: 12 pages, 1 figure. This is the version that appeared in JMP; it is only slightly changed from the previous version, to reflect small issue caught in proo

Lattice calculations on the spectrum of Dirac and Dirac-K\"ahler operators

We present a matrix technique to obtain the spectrum and the analytical index of some elliptic operators defined on compact Riemannian manifolds. The method uses matrix representations of the derivative which yield exact values for the derivative of a trigonometric polynomial. These matrices can be used to find the exact spectrum of an elliptic operator in particular cases and in general, to give insight into the properties of the solution of the spectral problem. As examples, the analytical index and the eigenvalues of the Dirac operator on the torus and on the sphere are obtained and as an application of this technique, the spectrum of the Dirac-Kahler operator on the sphere is explored.Comment: 11 page

A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation.Comment: 20 page

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan (TNC) geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.Comment: 79 pages, 7 figures; v2: added material on affine structure of connection space, former Section 4 postponed to 3rd paper of the serie

Generalized Misner-Sharp quasi-local mass in Einstein-Gauss-Bonnet gravity

We investigate properties of a quasi-local mass in a higher-dimensional spacetime having symmetries corresponding to the isomertries of an $(n-2)$-dimensional maximally symmetric space in Einstein-Gauss-Bonnet gravity in the presence of a cosmological constant. We assume that the Gauss-Bonnet coupling constant is non-negative. The quasi-local mass was recently defined by one of the authors as a counterpart of the Misner-Sharp quasi-local mass in general relativity. The quasi-local mass is found to be a quasi-local conserved charge associated with a locally conserved current constructed from the generalized Kodama vector and exhibits the unified first law corresponding to the energy-balance law. In the asymptotically flat case, it converges to the Arnowitt-Deser-Misner mass at spacelike infinity, while it does to the Deser-Tekin and Padilla mass at infinity in the case of asymptotically AdS. Under the dominant energy condition, we show the monotonicity of the quasi-local mass for any $k$, while the positivity on an untrapped hypersurface with a regular center is shown for $k=1$ and for $k=0$ with an additional condition, where $k=\pm1,0$ is the constant sectional curvature of each spatial section of equipotential surfaces. Under a special relation between coupling constants, positivity of the quasi-local mass is shown for any $k$ without assumptions above. We also classify all the vacuum solutions by utilizing the generalized Kodama vector. Lastly, several conjectures on further generalization of the quasi-local mass in Lovelock gravity are proposed.Comment: 13 pages, no figures, 1 table; v4, new results added in the asymptotically AdS case, accepted for publication in Physical Review

Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation

We present a simple method to derive the semiclassical equations of motion for a spinning particle in a gravitational field. We investigate the cases of classical, rotating particles (pole-dipole particles), as well as particles with intrinsic spin. We show that, starting with a simple Lagrangian, one can derive equations for the spin evolution and momentum propagation in the framework of metric theories of gravity and in theories based on a Riemann-Cartan geometry (Poincare gauge theory), without explicitly referring to matter current densities (spin and energy-momentum). Our results agree with those derived from the multipole expansion of the current densities by the conventional Papapetrou method and from the WKB analysis for elementary particles.Comment: 28 page