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How many points do we need to approximate a given metric space S (e.g., a ball in the Euclidean space) with a given accuracy ε > 0? To be more precise, how many points do we need to reproduce the metric ρ(X, Y ) on S with an accuracy ε? This problem is known to be equivalent to the following geombinatoric problem: find the smallest number of balls of given radius ε that cover a given set S. A similar approximation problem is also important for spacetimes. In this case, instead of a regular metric ρ(X, Y ) that describes distance between points X and Y , we have a kinematic metric τ(X,Y ) that describes the proper time between events X and Y . It turns out, rather surprisingly, that this space-time analogue of the above geombinatoric problem require much more points to approximate: e.g., to approximate a compact set in a 4-D Euclidean space, we need ≈ ε−4 points, while to approximate a similar compact in a 4-D space-time, we need ≈ ε−8 points, approximately the square of the previous number. 1 This work was partially supported by NSF grant No. EEC9322370, and by NASA Research Grant No. 9-757. The author is thankful to all the participants of the Novosibirsk seminar on chonogeometry (especially to A. D. Alexandrov) and Leningrad seminar on mathematical logic and constructive mathematics for valuable discussions. 1. INFORMAL DESCRIPTION OF THE PROBLEM: HOW MANY POINTS DO WE NEED TO DESCRIBE SPACE-TIME WITH A GIVEN ACCURACY? According to the models used in modern physics, space-time consists of infinitely many points (events). If we fix a coordinate system, then each event can be characterized by a pair (x0, x), where x0 is a real number (that describes the time of the event), and x is an element of the 3-D space (that describes the spatial location of the event). This subdivision into time and space essentially depends on our choice of a coordinate systems, and thus, does not have any direct physical meaning. To get a physically meaningful quantity, we must consider two different events X = (x0, x) and Y = (y0, y). Some pairs of events are related by the causality relation X ≤ Y (meaning that X can influence Y ). If X can influence Y , then we can define proper time τ(X, Y ) as a time that a (geodesic) particle will “feel” when moving from X to Y (according to relativity theory, all processes like decay of a particle, aging of a human astronaut, etc., depend on proper time). This function τ is only defined on causally related pairs. In geometry of space-time, this function is often extended to arbitrary pairs so that τ(X,Y ) = 0 if X 6≤ Y (see, e.g., (Busemann 1967), (Pimenov 1970)). This extended function is called the kinematic metric. In particular, when gravity is negligible (and it is negligible for the majority of Earth measurements of space-time), we can use the formulas of special relativity, in which X ≤ Y ↔ y0 − x0 ≥ ρ(x, y), where ρ(x, y) is the standard Euclidean metric on R, and τR(X,Y ) = √ (y0 − x0) − ρ2(x, y). In this paper, we will also consider two generalizations of this formula: • The above formulas are consistent with all known measurement results. However, this does not mean that formulas of special relativity are necessarily absolutely precise: Indeed, measurements are never 100% precise; therefore, formulas that lead to close expression for τ(X, Y ) are also consistent with all measurement results. In particular, Busemann (Busemann 1967) proposed the following formula τα(X,Y ) = ((y0 − x0) − ρ(x, y)) with a parameter α that can take any value from 1 to ∞. For α ≈ 2, the “Finsler-type” space described by this formula is consistent with the results of all known experiments of special relativity. In this paper, we will consider generalizations that correspond to different value of α. • Traditionally, physicists considered 4-dimensional space-time (one temporal dimension + 3 spatial ones). However, in modern physics, space-time is often assumed to be of different dimension (see, e.g., (Brink et al. 1988) and (Siegel 1988)). In view of this possibility, in the following text, we will consider space-time models of arbitrary dimension d. Since there are infinitely many events, an ideal (exhaustive) description of space-time would consist of describing these infinitely many events. At any given moment of time, we can only have records about the finite number of events. The more events we record, the better is our knowledge of the space-time; in particular, the better is our approximation of the kinematic metric τ (that describes the space-time). It is, therefore, natural to ask the following question: How many events do we need to record in order to know the space-time (i.e., to be precise, the kinematic metric) with a given accuracy ε > 0? 2. FORMAL DESCRIPTION OF THE PROBLEM Definition 1. Let S be a set, and let τ : S × S → R 0 be a function that assigns a non-negative number to every pair (X, Y ) of elements of S. Let ε > 0 be a real number. We say that a finite set {X1, . . . , Xn} is an ε−approximation to S iff there exists a function π : S → {X1, . . . , Xn} such that for every X,Y ∈ S, the following three inequalities hold: • |τ(X,π(Y ))− τ(X, Y )| ≤ ε. • |τ(π(X), Y )− τ(X, Y )| ≤ ε. • |τ(π(X), π(Y ))− τ(X,Y )| ≤ 2ε. In these terms, the problem is: For a given area of space-time, to find the smallest possible number of point that form an ε−approximation to this area. In this paper, we will solve this problem for the space-time of special relativity and for its generalization proposed by Busemann. These results partially appeared in (Kreinovich 1979). 3. ANALOGY WITH METRIC SPACES REVEALS THE GEOMBINATORIC NATURE OF THIS PROBLEM Many method of space-time geometry first appeared as a natural generalization of the traditional geometry (that is intended to describe only space). It is, therefore, natural, before we start solving a space-time problem, to try to solve a similar problem for a normal metric space, i.e., for the case in which τ(X, Y ) is a normal metric ρ(X, Y ) (with triangle inequality). In this case, Definition 1 is reduced to the well-known notion of a ε−net (see, e.g., (Lorentz 1966)): Definition 2. A set {X1, . . . , Xn} of points from a metric space S with a metric ρ is an ε−net for S iff for every X ∈ S, there exists an i for which ρ(X, Xi) ≤ ε. PROPOSITION 1. For a metric space, a set is an ε−approximation iff it is a ε−net. (For reader’s convenience, all the proofs are delayed until the special Proofs section.) For metric spaces, the definition of an ε−net can be reformulated in purely geometric terms: a set {X1, . . . , Xn} is an ε−net for a metric space S iff S can be covered by n balls of radius ε > 0 with centers in Xi. Thus, for metric spaces, the problem of finding the ε−approximating set with the smallest possible number of elements can be reformulated as a geombinatoric problem: For a given metric space S and a given ε > 0, how many balls of radius ε do we need to cover S? Comments. 1. Since for metric spaces, approximability can be reformulated as a geombinatoric problem, our main problem (approximability of space-time) can be thus viewed as a natural space-time analogue of this geombinatoric problem. 2. For sets in the Euclidean space R, at least the asymptotics of the smallest number Nε of such balls is known: e.g., for a compact set S with a non-zero interior, we have Nε(S) ∼ C/ε (Lorentz 1966). In fact, this formula is so well known, that in fractal theory it is often used as a definition of the dimension d (see, e.g., (Mandelbrot 1982)): if for some set S, we have the above asymptotics for some real number d, then we say that this set is of dimension d. To determine d from the experimentally determined values of Nε(S), we take logarithms of both sides and determine d as Hε(S)/ log2 ε, where Hε(S) = log2 Nε(S) is called an ε−entropy of the set S. (For sets in Euclidean space, Hε(S) ∼ d · log2 ε.) To use the analogy with metric spaces, we will use a similar logarithmic measure for space-times: Definition 3. (Kreinovich 1974) For a given set S with a metric τ , by a kinematic ε−entropy H̃ε(S), we mean a binary logarithm of the smallest possible number of elements in a ε−approximation to S.