Abstract
The recent concept of relativistic positioning system (RPS) has opened the possibility of making Relativity the general standard frame in which to state any physical problem, theoretical or experimental. Because the velocity of propagation of the information is finite, epistemic relativity proposes to integrate the physicist as a truly component of every physical problem, taking into account explicitly what information, when and where, the physicist is able to know. This leads naturally to the concept of relativistic stereometric system (RSS), allowing to measure the intrinsic properties of physical systems. Together, RPSs and RSSs complete the notion of laboratory in general relativity, allowing to perform experiments in finite regions of any space-time. Epistemic relativity incites the development of relativity in new open directions: advanced studies in RPSs and RSSs, intrinsic characterization of gravitational fields, composition laws for them, construction of a finite-differential geometry adapted to RPSs and RSSs, covariant approximation methods, etc. Some of these directions are sketched here, and some open problems are posed.
I want the organizers of this seminar, Dirk Pützfeld and Claus Lämmerzahl, to know how much I appreciated their inviting me to talk about my ideas on this subject. It is also a pleasure to thanks the Wilhelm und Else Heraeus-Stiftung for their kind hospitality.
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Notes
- 1.
The differential operator associated to the \(n(n+1)/2\) Einstein equations is not hyperbolic, but degenerate. Nevertheless, because n of these equations (constraint equations) are involutive with respect to the other \(n(n-1)/2\) ones (evolution equations), one can supplement this last system with n suitable additional equations (coordinate conditions). It is the differential operator of this new system of \(n(n+1)/2\) equations for the metric coefficients which may be made hyperbolic with a suitable choice of coordinate conditions (for example, with harmonic ones).
- 2.
The influence domain for an initial instant (local spatial hypersurface) is the domain where the solution exists and is unique for every initial (Cauchy) data on this initial instant.
- 3.
Retrodictive, as opposed to predictive, means here that one cannot but verify afterwards that the physical quantities measured in the influence domain agree with the initial data received at or after the cusp of this domain.
- 4.
In Newtonian physics, the contents of the three-dimensional space at every instant is supposed known or knowable at that instant, so that the evolutive equations describe the dependence in time of this three-dimensional contents.
- 5.
Epistemic relativity was first presented at the GraviMAS FEST workshop, in honor of Lluís Mas, Mallorca, Spain, 2008, http://www.uib.es/depart/dfs/GRG/GraviMAS_FEST/. See also [1].
- 6.
For the genesis of the concept of relativistic positioning systems, see for example [1].
- 7.
They were relativistic stereometric systems which, joined with relativistic positioning systems, suggested the idea of epistemic relativity. This is why relativistic stereometry and epistemic relativity were conceived conjointly.
- 8.
Some subjects related to epistemic relativity have been the object of interesting developments (e.g. positioning systems, relativistic geometric optics, intrinsic characterization of non-vacuum metrics, Regge calculus) not directly related to the basis of this theory. For this reason the corresponding bibliography is absent here.
- 9.
In all this text, the worlds ‘physicist’, ‘observer’ or ‘user’ denote any person or device able to receive the pertinent information, to record and to analyze it and to perform the actions and computations needed for the problem in question. For short, we shall refer to any of them as ‘it’.
- 10.
Remember that relativity is retrodictive.
- 11.
From the Greek ‘ontos’, being, with the meaning of ‘what is’ as opposed to ‘how one knows it’.
- 12.
From the Greek ‘episteme’, knowledge, with the meaning of ‘how we obtain it’.
- 13.
It is to be noted, because frequently forgotten, that the local charts on a differentiable manifold may be structural, i.e. belonging to the atlas defining its differentiable structure, or not. Only in the first case, the region \(\mathcal {R}\) has to be an open set. Nonstructural local charts may be of lesser differentiable class than structural ones. In particular, they may be simply continuous, although in this case, the natural frame being absent, one may be lead to complete the local chart with an independent field of vector tetrads in order to construct a basis for the tensor algebra.
- 14.
It is understood that such regions have the appropriate accessibility characteristics to make gravimetry.
- 15.
Generically. Some distributions of clocks may associate same coordinates to different events, whatever the region considered (see [2]).
- 16.
Or fourfold. Generically there does not exist a privileged synchronization of the form time = constant in relativistic positioning systems, but emission coordinates being constituted by four times \(\tau ^A\), one could say that there exist four synchronizations \(\tau ^A\) \(= constant.\)
- 17.
Euphemism for “Newtonian defaults”.
- 18.
The pseudo-ranges are only considered as parameters able to calculate conventional coordinates on the Earth surface (WGS84, ITRS or others). The appellation ‘GPS coordinates’ refers to these conventional coordinates provided by the GPS.
- 19.
The grid \(\mathcal G\) of emission coordinates is the Cartesian product of the segments \([\tau ^A]\) of the times broadcast by the relativistic positioning system: \(\mathcal G\) \(\equiv \) \([\tau ^1] \times [\tau ^2] \times [\tau ^3] \times [\tau ^4]\).
- 20.
The field of light-cones defines all the possible coordinate hypersurfaces of the coordinate systems associated to all the possible relativistic positioning systems. Accordingly, the particular trajectories of the clocks select the specific ones for the auto-locating system and, consequently, its region of validity or domain of definition in the grid.
- 21.
The time broadcasted by every clock may be any convenient time, not necessarily its proper time, although proper time simplifies in general the theoretical analysis.
- 22.
More precisely, a \(4\pi \)-wide hypergon eye or \(4\pi \) steradian eye.
- 23.
In principle, a suitable choice of stars, in e.g. Hipparcos star catalogue, could help a spacecraft in the Solar system to estimate its velocity with respect to the Barycentric Celestial Reference System.
- 24.
Of course, by broadcasting their proper times, a relativistic stereometric system may also act as a relativistic positioning system. But because of their dual ways of working, I believe it is clearer, for the moment, to study them separately.
- 25.
There is no matter here what instant-identifier is used: a clock associate to the point, measuring any time, non-necessarily proper, a flash, or any other pertinent one.
- 26.
Formulation of Maxwell equations, Cauchy problem for the permanence of electromagnetic waves, shock, detonation and deflagration waves in hydrodynamics, and some others.
- 27.
This notion of finite-differential geometry was first presented as part of a lesson at the International School on Relativistic Coordinates, Reference and Positioning Systems, Salamanca, 2005. The mathematical results also appeared in [1].
- 28.
The concept is due to Fréchet. Haussdorff named them ‘metric spaces’ (‘metrischer Raum’) but in our context it is better to call them ‘distance spaces’.
- 29.
I am grateful to Abraham Harte for a pertinent observation on this fact.
- 30.
At points out of the data of \(\Lambda \).
- 31.
See Sect. 5 for this notion.
- 32.
On a metric manifold a distance function is generically local (normal geodesic domain), so that a (global) metric cannot be generically interchanged by a sole distance function, but by a suitable atlas of normal geodesic domains with distance functions submitted, in the intersection of charts, to compatibility conditions. In our case, nevertheless, the constellation \(\mathcal C\) is contained clearly in a normal geodesic domain. Anyway, what we want here is to show the interest of the concept.
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Acknowledgements
This work has been supported by the Spanish “Ministerio de Economía y Competitividad”, MICINN-FEDER project FIS2015-64552-P.
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Coll, B. (2019). Epistemic Relativity: An Experimental Approach to Physics. In: Puetzfeld, D., Lämmerzahl, C. (eds) Relativistic Geodesy. Fundamental Theories of Physics, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-030-11500-5_8
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