The generally covariant meaning of space distances
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The generally covariant meaning of space distances S. Capozziello1,2,3 , A. Chiappini4 , L. Fatibene4,5,a
, A. Orizzonte4,5
1 2 3 4
Department of Physics “E. Pancini”, University of Napoli “Federico II”, via Cinthia, 80126 Napoli, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli, via Cinthia 9, 80126 Napoli, Italy Scuola Superiore Meridionale, Largo S. Marcellino 10, 80138 Napoli, Italy Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino, Italy 5 Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Torino (Italy), via P. Giuria 1, 10125 Torino, Italy Received: 3 November 2020 / Accepted: 25 November 2020 © The Author(s) 2020
Abstract We propose a covariant and geometric framework to introduce space distances as they are used by astronomers. In particular, we extend the definition of space distances from the one used between events to non-test bodies with horizons and singularities so that the definition extends through the horizons and it matches the protocol used to measure them. The definition we propose can be used in standard general relativity although it extends directly to Weyl geometries to encompass a number of modified theories, extended theories in particular.
1 Introduction In a generally covariant theory as general relativity (GR), there are no Dirac–Bergman observables (other than constants); see, for example, [1]. While distances on spacetime computed with the covariant metric g are generally invariant, they are not endowed with a direct physical meaning. Although the length of a geodesics can be defined for space-like and time-like geodesics, there is no covariant agreement among observers about which geodesic has to be used to define the space distance between two events or their time separation. This is a direct consequence of relativity of contemporaneity, which is where GR takes its name from. Two different observers will generally define different space-like synchronisation hypersurfaces, and consequently, they would disagree on which geodesic to use to define space distances, as it happens already in Minkowski spacetime and special relativity (SR). So, what do astronomers mean when they say the Moon orbits at 380, 000 km from the Earth? What do cosmologists mean when they say the universe is 14 B years old? One answer to these questions is that it does not really matter since whatever (reasonable) choice we made, the difference would be below observation precision, anyway. Another answer is describing a specific observer and its conventions to obtain such a determination, possibly so that the definition reflects the measurement protocols used for it and it is at least accurate since no observation can be absolutely precise. The second strategy is well established; it is usually based on ADM splittings of spacetime; see [2,3]. If one accepts that, only details about how the observer defines the ADM
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:948
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