Reference frames and rigid motions in relativity.

*(English)*Zbl 1076.83007In relativity, any massive object moves through spacetime on a timelike world line. Given a 4-dimensional spacetime \((V,g)\), we call “space” any congruence of such curves, i.e. smooth foliations by timelike curves, given by the integral curves of a timelike unit vector field \(u\) on \(V\). The space of integral curves then forms a 3-dimensional manifold \(E\), and the canonical projection \(\pi : V \to E\) sending each point \(v \in V\) to the congruence curve passing through \(v\) is a submersion. A proper time scale of the congruence is a 1-form \(\theta\) with \(\langle\theta,u\rangle = -1\). The distribution \(\ker\theta\) is transversal to \(\ker d\pi\) and thus forms a horizontal bundle for the submersion \(\pi\). Any Riemannian metric \(\bar g\) on \(E\) can be canonically lifted to a semi-definite symmetric 2-form on \(V\) which we also denote by \(\bar g\). The triple \((E,\bar g,\theta)\) is called a reference frame.

In the present paper these notions and their relevance in physics are discussed. The main result (p. 3087) reads as follows: For any spacetime \(V\) we can find locally a reference frame \((E,\bar g, \theta)\) such that \(\bar g\) is a metric of constant sectional curvature on \(E\) which is related to the spacelike part \(g+\theta^2\) of the given spacetime metric \(g\) as follows: \[ \bar g = \phi\cdot(g+\theta^2 \pm \mu^2) \] where \(\phi\) is a smooth function and \(\mu\) a 1-form on \(E\). In fact, there is much freedom in the choice of \((E,\bar g,\theta)\): One can prescribe arbitrary initial data (satisfying the constraints) on any 3-dimensional sub-spacetime.

In the present paper these notions and their relevance in physics are discussed. The main result (p. 3087) reads as follows: For any spacetime \(V\) we can find locally a reference frame \((E,\bar g, \theta)\) such that \(\bar g\) is a metric of constant sectional curvature on \(E\) which is related to the spacelike part \(g+\theta^2\) of the given spacetime metric \(g\) as follows: \[ \bar g = \phi\cdot(g+\theta^2 \pm \mu^2) \] where \(\phi\) is a smooth function and \(\mu\) a 1-form on \(E\). In fact, there is much freedom in the choice of \((E,\bar g,\theta)\): One can prescribe arbitrary initial data (satisfying the constraints) on any 3-dimensional sub-spacetime.

Reviewer: Jost-Hinrich Eschenburg (Augsburg)