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Reference frames and rigid motions in relativity. (English) Zbl 1076.83007
In 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.

##### MSC:
 83C10 Equations of motion in general relativity and gravitational theory 53B50 Applications of local differential geometry to the sciences 53B30 Local differential geometry of Lorentz metrics, indefinite metrics
##### Keywords:
congruence of curves, time scale, free movability
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