Interpreting spacetimes of any dimension using geodesic deviation

Abstract

We present a general method which can be used for geometrical and physical interpretation of an arbitrary spacetime in four or any higher number of dimensions. It is based on the systematic analysis of relative motion of free test particles. We demonstrate that local effect of the gravitational field on particles, as described by equation of geodesic deviation with respect to a natural orthonormal frame, can always be decomposed into a canonical set of transverse, longitudinal and Newton-Coulombtype components, isotropic influence of a cosmological constant, and contributions arising from specific matter content of the universe. In particular, exact gravitational waves in Einstein's theory always exhibit themselves via purely transverse effects with D(D − 3)/2 independent polarization states. To illustrate the utility of this approach we study the family of pp-wave spacetimes in higher dimensions and discuss specific measurable effects on a detector located in four spacetime dimensions. For example, the corresponding deformations caused by a generic higher-dimensional gravitational waves observed in such physical subspace, need not be tracefree.

Cosmology on a cosmic ring

“…We will now briefly review the aforementioned decomposition of the Weyl tensor in d spacetime dimensions [35,36], and then apply it to the concrete metric that we are interested in. In this section, capital Latin indices A, B, .…”

Geodesic Deviation in Kundt Spacetimes of any Dimension

“…In our recent work [6] we demonstrated that the equation of geodesic deviation, which describes relative motion of nearby free test particles, can in any D-dimensional spacetime be expressed in the invariant form…”

“…known from the D = 4 case [4,6]. In equations (21), (22), only the "electric part" of the Weyl tensor represented by the scalars in the left column of (23) occurs, and there are various constraints and symmetries, for example…”

The Local and Global Geometrical Aspects of the Twin Paradox in Static Spacetimes: I.\nobreakspace {}Three Spherically Symmetric Spacetimes

“…For their metrics one has two classes of physically distinguished and analytically simple timelike geodesics: radial and circular (if exist) ones. Staticity not only considerably simplifies all calculations, moreover it allows for a physically meaningful notion of rest 1 . In an SSS spacetime we introduce three twins: twin A remains at rest on a non-geodesic worldline, twin B revolves on a circular orbit (geodesic or not) around the centre of spherical symmetry and twin C moves upwards and downwards following a radial geodesic.…”

“…In the local problem one considers a bundle of nearby (infinitesimally close) timelike curves and seeks for the longest one in the bundle. Again it is well known that there is a well defined procedure for solving the local problem in terms of the curvature tensor, which physically determines the behaviour of geodesic worldlines of nearby free test particles both in four and in a larger number of spacetime dimensions [1]. A locally maximal timelike curve is always a geodesic and is determined by solving the geodesic deviation equation.…”