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Chapter First Online: 12 January This is a preview of subscription content, log in to check access. Ashtekar, R. Hansen, A unified treatment of null and spatial infinity in general relativity.

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Geroch, E. Kronheimer and R. Penrose, Ideal points in spacetime, Proc. Observe that then our Fermat metric is the addition of the Fermat one-form and the square root of Perlick's Fermat metric. The above computations show that in conformally standard stationary space- times, Fermat's principle relates future-pointing lightlike geodesies of S x R, g as in 4 with geodesies of the Finsler manifold S, F with F given in 6 up to reparametrizations. Let us state the relation including parametrizations. Theorem 3. Let S x R, g be a standard stationary space- time as in 4.

Let us point out that V. Perlick [47] considers a more general case than conformally standard stationary spacetimes. Observe that as the fiber is R, there always exists a section of the bundle see for example [30, page 58]. But the existence of a spacelike section is not guaranteed. In fact, assuming that K is complete, this happens if and only if i M.

Given a section S of the fiber bundle, we can express the metric of M, g as in 4 , but with go not necessarily spacelike. In this case, the global time given by the second coordinate is not necessarily a time function, that is, it does not have to be strictly increasing in causal curves.

As a consequence, the Fermat metric obtained in 6 can be non-positive along some directions of the tangent space. In fact, it is not difficult to see that the Fermat metric 6 is a Finslcr metric with the definition given in Section 2 if and only if the section S is spacelike. It can be helpful to restate the Fermat's principle as follows see [16, Proposition 4. Proposition 3.

Manifolds, classification of surfaces and Euler characteristic - Differential Geometry 25

Let us observe that Fermat metric depends on the spacelike section you choose to obtain the standard splitting which in some references as [25, 46] is called the gauge choice. Last proposition can be used to obtain the relation between two Fermat metrics associated to different splittings of the same stationary spacetime with a fixed timelike Killing vector field K. With the above notation, the Fermat metric associated to the splitting S x R, gj is Ff — F — df , where F is the Fermat metric associated to S x R, g and df is the differential of the smooth function f.

Moreover, as a consequence of Proposition 3.

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See also [16, Proposition 5. Let us call Stat 5 x R the space of standard stationary spacetimes with normalized Killing vector field dt and Rand S' the space of Randers metrics on S. Propo- sition 3. This relation constitutes a very important issue for Randers metrics, because the global invariants in the spacetime must be translated in invariants for the entire class of Randers metrics that differ in the differential of a function.

Causality and Fermat metrics As, by Proposition 3. As a consequence, we can charaterize the causal conditions of a standard stationary spacetime in terms of the Format metric. This relation was established in [16] with some previous partial results in [15].

Recall that we say that two events p and q in a spacetime are chronologically related, and write p -C q resp. Then the chronological future resp. Proposition 4. See [16, Proposition 4.

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  5. For definitions and properties of the different levels of causality we refer to [43]. Theorem 4. Let S x R, g be a standard stationary spacetime as in 4. Then S x R, g is causally continuous. See [16, Theorems 4. For part b see also [16, Proposition 2. Furthermore, an extension of last theorem characterizing the stationary regions that are causally simple in terms of convex regions for the Fermat metric has been achieved in [11].

    In particular we can establish a generalization of the classical Hopf-Rinow theorem. Moreover, these conditions imply the convexity of M,R. The equivalence between the two first conditions is standard and it holds for any Finler metric. For i Hi , first observe that any Randers metric can be obtained as the Fermat metric of a standard stationary spacetime see [8, Propo- sition 3. Now let S x R, g be the standard stationary spacetime having as a Fermat metric R. By Proposition 4. Consider the splitting associated to the Cauchy hypersurface.

    By Proposition 3. Moreover, by Proposition 4. Therefore it is globally hyperbolic and i follows from part b of Theorem 4. Recall that h is defined in 7. Given any Riemannian metric g in S, we will denote by d g the distance in S associated to g. This condition does not depend on xq.

    Let S x R, g be a conformally standard stationary spacetime with g as in 3. For i and iv see [17, Theorem 2]. For ii , see part 1 of Proposition 2 in [17] and for Hi and v , [17, Theorem 4]. We point out that in [51] especially in Corollary 3. As a further relation between Causality of a standard stationary spacetime and Randers metrics, Cauchy developments will be constructed in terms of the Fermat metric.

    The future resp. Causal boundaries and Fermat metrics In General Relativity it is important to complete the spacetime with some kind of boundary.