Interpreting spacetimes of any dimension using geodesic deviation

We present a general method that 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 the local effect of the gravitational field on particles, as described by the equation of geodesic deviation with respect to a natural orthonormal frame, can always be decomposed into a canonical set of transverse, longitudinal and Newton–Coulomb-type 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 generic higher-dimensional gravitational waves observed in such physical subspace need not be trace-free.

Figure 1

In a curved $D$-dimensional spacetime, nearby test particles moving along geodesics accelerate toward or away from each other, as given by the equation of geodesic deviation (1). Here, $\mathit{u}$ is the velocity vector of a reference particle, and $\mathit{Z}$ is the separation vector that represents actual relative position of the second test particle at a given proper time $\tau$.

Figure 2

Natural choice of the interpretation null frame and the related orthonormal frame (12, 13). Up to spatial rotations of ${\mathit{m}}_i={\mathit{e}}_{(i)}$, they are uniquely given by the velocity vector $\mathit{u}$ of the observer and the WAND $\mathit{k}$ at any event $\mathcal{P}$ of the spacetime.