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Axiomatic relativity theory

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1982 Russ. Math. Surv. 37 41 (http://iopscience.iop.org/0036-0279/37/2/R02) View the table of contents for this issue, or go to the journal homepage for more

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Uspekhi Mat. Nauk 37:2 (1982), 39-79

Russian Math. Surveys 37:2 (1982), 41-89

Axiomatic relativity theory A.K. Guts

CONTENTS

Introduction Chapter I. Space-time geometry § 1. The concept of space-time § 2. Notation and definitions § 3. The continuity theorem § 4. Contingency theorems § 5. The mapping of cones § 6. The geometry of space-time § 7. Micro-causality and the geometry of space-time § 8. Time-like Minkowsky space § 9. Lorentz and Galilean kinematics §10. An axiomatic definition of the Galili and Lorentz groups Chapter II. Mappings of families of cones in an affine space §11. Mappings of elliptic cones §12. Conformal space §13. The simplest axiomatizations of space-time §14. Theorems on finitely many light sources §15. Mappings of strictly convex cones §16. How many interial systems of reference? §17. The axiomatics of relativity theory §18. Mappings of arbitrary cones §19. Mappings of discrete cones §20. Mappings of pseudo-Euclidean spaces Chapter III. Connected pre-orders in an affine space §21. Maps of ordered spaces §22. Connected pre-orders §23. Cones with a transitive group Chapter IV. The chronogeometry of spaces §24. Spaces with a non-commutative group §25. Mappings of cones in Lobachevskii space §26. The chronogeometry of Lorentz manifolds §27. An analogue of Aleksandrov's theorem in the class of partially ordered fields §28. Problems References

42 44 44 46 48 49 52 54 56 58 59 60 61 61 63 64 65 66 68 70 70 71 73 74 74 75 76 77 78 80 80 82 83 85

42

Α.Κ. Guts

Introduction The history of science shows that a more complete and deeper understanding of any scientific theory is very closely connected with its axiomatic exposition. The aziomatic method is universally accepted as a tool for mathematicians, who use it whenever it is required either to clarify the essentials of the material under investigation or to express it in a more accessible and rational form. Long ago the successes of this method in mathematics led Hilbert to propose its application to physical disciplines. Not only did Hilbert formulate the problem of the axiomatization of physics in his "Mathematical Problems" [43], but he also axiomatized the phenomenological theory of radiation and his own unified field theory of gravitation and electromagnetism, [63]. Nearer the present time we find numerous well-known examples of the axiomatization of very diverse branches of physics [39], [43], [49], [58], among which axiomatic quantum field theory [32] is especially distinguished. However, there have been very many more attempts to axiomatize special relativity theory [12], [19], [31], [40], [41], [53], [56], [57], [60], [72], [73], [75] than any other physical thoery. This is understandable in view of the fact that this theory is profoundly geometrical and can, therefore, be axiomatized with relative ease. In this survey we study very diverse systems of axioms relating to special relativity theory. We deal principally with research mainly carried out in our own country and due to several authors. The main aim of this paper is to show that the metric structure of Minkowsky space-time (or the geometry of the inhomogeneous Lorentz group), is determined by the fact that some events act upon others. In this way an answer is given to Riemann's problem "on the intrinsic reason for the appearance of metric relations in a space", which was posed by the great geometer in his lecture "Uber die Hypothesen, welche der Geometrie zu Grunde liegen" ([11], 15; [44]). The concept of space-time is the starting point in the creation of an axiomatic theory of relativity. The idea of space-time as a four-dimensional world in space and time was first introduced by Hermann Minkowsky. Minkowsky wrote that "for a true understanding of the Lorentz group Gc the term "relativity postulate" seems to me to be too poor for the requirement of invariance under Gc. According to him, the true fundamental theory of relativity is a "postulate of an absolute world", which asserts that phenomenologically we are given only a four-dimensional world of events (cosmic points): that is, space-time whose projection onto space and onto time can be taken with a certain arbitrariness. The next step was taken in 1914 by Robb [75], who put the concept of a sequence of events at the basis of an axiomatic construction of the theory of relativity.

Axiomatic relativity theory

43

The ideas of Minkowsky and Robb were thoroughly developed in the works of A.D. Aleksandrov [2] - [ 5 ] , [11], [ 12], [14], [ 15], which so far represent the most significant advance in the creation of an axiomatic spacetime theory. Space-time, in Aleksandrov's view, is the set of all cosmic events abstracted from all its properties except those that are determined by relations of the influence of some events on others, that is, expressed less precisely, the space-time structure of the world is none other than its abstract cause-and-effect structure. This survey consists of four chapters. In the first chapter, called "Spacetime geometry", we present results on the axiomatization of Minkowsky geometry based on the representation of cause and effect interactions. Here we understand by Minkowsky geometry the four-dimensional pseudoEuclidean geometry of signature (H ) with the metric defined by the differential form d s2 — dx\ — dx\ — dx\ — dx\.

The group of motions of this geometry is called the inhomogeneous Lorentz group or Poincare group. Therefore, the given geometry is sometimes said to be Lorentz. This term is used in this paper only in Ch. I §§8-9. In Ch. I, § 1, we present the initial propositions on the structure of spacetime, which we take from a paper of Aleksandrov [3]. In §3 we prove an important theorem, which shows in what cases the topology of space-time is generated by causal order. Various axiom systems are presented in § § 6 - 1 0 . In Ch. II we collect results on mappings preserving families of cones in «-dimensional affine space. Historically, many first attempts to axiomatize the theory of relativity were concerned with families of cones, hence, in this chapter we list the corresponding axiom systems (see §§ 13, 14, 17). The second chapter has the most extensive list of references: [ 1] -[3] , [9] -[11 ] ,i [14]-[18], [ 2 0 ] - [ 2 3 ] , [ 3 3 ] - [ 3 5 ] , [47], [50], [ 5 2 ] - [ 5 5 ] , [61], [65]-[68], [70], [71], [74], [76], [77], [79], [80]. In Ch. Ill we study connected pre-orders. We give conditions under which the pre-order is given by a cone (§22). In §23 we give a description of cones admitting a transitive group of transformations. In Ch. IV we present separate results concerning various generalizations of the concept of space-time. In § 26 we formulate an interesting theorem of Malament on maps preserving causal order in a pseudo-Riemannian space. I wish to thank A.D. Aleksandrov and V.Yu. Rovenskii for their invaluable advice, which contributed to an improvement of the manuscript.

44

A.K. Guts CHAPTER I

SPACE-TIME GEOMETRY § 1 . The concept of space-time [3]

(1.1) What is space-time? We need an answer not on a philosophical level, but one that would give grounds for the construction of a theory. The answer must be contained in the theory of relativity, since it is a theory of space-time, but it must be extracted from it. The nature of an object is, ultimately, no more than the set of relation of its parts. Therefore, we are concerned with those material connections of the world that in their entirety determine space-time. The simplest element of the world is what is called an event. This "point" phenomenon is not unlike an instantaneous flash of a point source of light or, in terms of the usual concepts of space and time, one whose extension in space and time is negligible. In a word, an event is an analogue of a point in geometry and, imitating Euclid's definition of a point, we may say that an event is a phenomenon that has no parts; it is 'atomic'. Every phenomenon, every process is represented as a certain connected set of events. From this point of view the whole world is regarded as a set of events. Removing from an event all its properties except that it exists, we represent it as a point, a "cosmic point". And space-time is the set of all cosmic points. However, in such a definition, space-time does not yet possess any structure: it is simply a set of events, in which only the one fact is retained that they exist as distinct events, in abstraction from all other properties and for the time being without any relations between them. One can introduce the concept of continuity of a series of events by borrowing it from the intuitive idea or by giving it some suitable definition. Then space-time turns out to be simply a four-dimensional manifold in the topological sense. We do not dwell on this and define the structure and the very continuity of space-time starting from the most general and fundamental relation of events, such as there is in the world. We have in mind the motion of matter. Every event one way or another acts upon certain other events and is itself subject to the actions of other events. The physical nature of the action can be very diverse; we may think of the propagation of light, the departure of a particle, and so on. Naturally, it need not be direct, but can involve a number of agents. The motion of a small body represents a series of events in which preceding events influence subsequent ones. In the language of physics, action can be defined as the transfer of momentum and energy. These concepts then are primordinal, and this corresponds to the essence of the matter, since momentum and energy are the basic physical characteristics of motion and action. But by abstracting the events

Axiomatic relativity theory

45

themselves from their concrete properties we are also abstracting the concept of action from its concrete properties, except for the fact that it is a relation between events having the properties of precedence (antisymmetry and transitivity). If we think of an axiomatic construction of the theory of space-time (and this is the basic problem of this paper), then the concepts of an event, that is a cosmic point, and action, that is, precedence are taken to be primary and not subject to definition. Those events that are subject to the actions of a given event a form "the domain Pa of action of the event a". Such domains determine a certain structure in the set of all events. It is, of course, equivalent to the structure determined by the relations of the action themselves. And this is the space-time structure of the world. In other words, space-time itself can be defined as follows. Space-time is the set of all events in the world, abstracted from all its properties except those that are determined by the relations of actions of events on others. The action of an event on another is the elementary form of the causal tie, precisely because an event is an 'atomic' phenomenon. For this reason, what we have just said can be expressed, less accurately but more expressively, as follows: the space-time structure of the world is none other than its cause-and-effect structure, taken to abstraction. This abstraction consists in the removal of all properties from phenomena and their causal ties, except that phenomena are compounded from events and the mutual influences of the actions of some events upon others. The fact that this definition of space-time is possible within the framework of special relativity theory is proved in this paper. The definition of space-time given is nothing but a precise expression, in accordance with modern physics, of the fact that space-time is a form of existence of matter. Matter itself in its motion and so in the interaction of its elements thus defines its own space-time form. (1.2) We now pass onto a purely mathematical definition of space-time. Axiom A t . Space-time is a connected simply-connected locally compact four-dimensional Hausdorff space V on which are defined a family of subsets {Pa : a £ V} ("domains of action") and a transitive commutative group Τ of homeomorphisms of V onto itself, satisfying the following conditions: 1) a subset Pa φ {a} is associated with each point a G V; 2) a G Pa for any a G V; 3) ify^Px, thenPy CPX; 4) t (Pa) = Pt(a)for any homeomorphism t G Τ and any point a G V. This axiom appears not to correspond to the wish to define the topology of space-time starting only from the structure determined by the domains of action Pa. But from the theorems below (see § §3, 6) it will become clear that it is in fact so and that we could have expressed the initial axiom A! in terms using only the domains Pa; but this would have led to a very

46

АЛ. Guts

complicated formulation of the axiom and to the loss of its simplicity and visual clarity. (1.3) From Axiom Ax it follows immediately [42], [59], that one can introduce co-ordinates x1, x2, x3, x* in V in such a way that an element i G T can be represented as an ordinary translation t : (x1, x2, xs, x*) ->-*- (x1 + t1, x2 + f, x3 + t3, x* -f i 4 ), that is V is a four-dimensional affine space, and Τ its group of translations. The postulate that Τ exists is connected with the assumption that spacetime is homogeneous, and when we drop commutativity we are led to a geometry of space-time distinct from that of Minkowsky (see §24). (1.4) From Axiom Аг it follows that the family of domains of action {Pa} is invariant under homeomorphisms of T. Therefore, it is sufficient to determine one domain Pe corresponding to a certain fixed point e, and to obtain the remaining ones by translations in V using T. So we fix the point e and write Ρ instead of Pe. (1.5) We say that the domain of action Ρ defines a pre-order in V if Axiom A 1 ; l)-3) are satisfied for the system of subsets {Pa '• a- 6 V). If, in addition, the pre-order Ρ is such that: χ Φ у implies that Px Φ Py, then we say that Ρ defines an order in V. We introduce the following notation: we write χ < j if and only if у G Px. It is clear that when Ρ defines an order in V, the relation < defines a partial order in V. Physically, the relation χ < у can be interpreted as causality, in which χ is the cause and у the effect, that is, we claim that action (of energymomentum) is transmitted from χ to y. (1.6) Next set Pi = {y ζ V : г/< х}. (1.7) We say that the pre-order Ρ is closed if Ρ is a closed set; pre-order Ρ is open if Ρ \ {e} is open. When Ρ is a cone and Ρ \ {e} is open, we talk of an open cone. (1.8) Axiom A2. Px О fy is bounded for any x, у € V such that у G Px. In the language of physics, axiom A 2 states that the velocity of transmission of action is finite. §2. Notation and definitions (2.1) As we said in (1.3), space-time V is a four-dimensional affine space. We assume this from now on throughout the paper. We denote by A" an «-dimensional affine space. (2.2) We denote points of A" byismall Latin letters. If В is a set, then int В, В, ЪВ, conv В denote, respectively, its interior, closure, boundary, and convex hull.

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47

Let Μ be a set in A". By Mx we denote the set obtained from Μ by using the translation t £ Τ for which t(e) = x. By definition, Μ = Me. (2.3) If x, у G A", then [x, y] is the segment of the straight line in A" with the ends χ and у and (x, у) = [ж, г/] Ч {x, у}- We introduce the Euclidean metric \x~y I in A" with origin e. Next, l(x, y) and / + (x, y) (χ Φ у) denote, respectively, the straight line passing through the points χ and у and the ray from χ passing through y. If χ G A" and r > 0 is a certain number, then 5(x, r) denotes the open ball of radius r with centre at x, that is, B(x, r) = {y £ An : \ χ — у | A" such that f(Px) == PHx) for any χ G A", where Ρ is a pre-order in A", is said to be isotonic. From the definition it follows that the inverse map /- 1 is also isotonic, so that f^(Px) = Pf-nX) for any χ G A". (2.6) A generator of a cone in A" is called extreme if it is not contained in the convex hull of the remaining generators. In the case of a convex cone

48

A.K. Guts

an extreme generator is one that does not lie in the angle between any two other generators. If Μ is a set in A", then a point χ G Μ is called extreme if it is not contained in the convex hull of the remaining points. If Μ is a convex set, then an extreme point is one that is not contained in the interior of a segment with end-points in M. §3. The continuity theorem Suppose that Axiom A! is satisfied. (3.1) Theorem 1. Suppose that the domains of action {Pa} satisfy Axiom A2 and that either Ρ \ {e} is open, or Ρ is closed with interior points. Then any isotonic map f is a homeomorphism. Proof [8], [12]. (a) Let Ρ 4 {e} be open. We consider the non-empty set Μ = (Ρ Π Pa) \ Iе- α}> where a & Ρ is arbitrary. Clearly, Μ is open. Let χ G V be an arbitrary point and U an arbitrary open neighbourhood of f(x). We set Mx = U ЩМ) : e e ЦМ)}. This set is centrally symmetric with respect to e. So is its closed convex hull H. Let b and b be two mutually symmetric extreme points of H. Clearly, b, b G M1. Let t and Τ be transformations such that t(b) = f(x) and T(b ) = f(x). Since b and b are extreme, i(Mx) f| ϊ(Μ{) = {/(χ)}. Hence, there are transformations τ and τ, such that

(1)

/WftWniWcf/.

From f(Pu) = pf(u) and the bijectivity of / it follows that tx(Pu) = Р}-Ци) for any и G V. Consequently, f'1 maps the family of open sets{M a : α ζ V} onto a family of open sets {/-1(Μα) : α ς V}. Each set / " ' ( ¥ , ) can be > c r represented in the form (Pb f) / c)\{^. } f° points b and с with с G P ft . an 1 ΐί Consequently, Β = /^(τ^-^ι)) d ^ = /" ( (Λ^ι)) can be represented as unions of open sets of the form (Pb f\ РГ) \ {b, c}, that is, В and В are open sets containing x. But then (1) can be re-written as (2)

f(x)ef(B

(]B)cz U.

Because χ G Β Π Β and Β Π Ζ? is an open set, (2) demonstrates the continuity of/ at x. But χ is arbitrary, h e n c e / is continuous everywhere. Thus, / is a homeomorphism. This proves (a). (b) Let Ρ be a closed set with interior points. Let us assume that Μ = Ρ Π Ρ~ contains interior points. We choose any у Ε V and an open ball U containing it, small enough so that any set t(M) covers it. Let r be the radius of U. Let us form the intersection W of all

Axiomatic relativity theory

49

) U. It contains no balls equal to U (except U itself), since if U' is a ball and τ is the translation from its centre to the centre of U, then x(W)z> U and so x(W) = W by the definition of W. But then W is unbounded, contradicting the fact that Μ is bounded. Thus, W does not contain balls equal to U. From this it follows that outside the ball IV (concentric with U and of double its radius) W does not contain open balls U' of radius r' ;> r — ε with ε > 0. We choose the ball Uy of radius ri = ε/2 concentric with U. For it we define the set Wx just as W was defined for U. Then, by the choice of ^ it turns out that tVj С 2U. For suppose that и е И7! \ 2U. Then и G W, since Wx С W. There is a point z G ( F \ W ) Π Π S(u, r — ε), where S(u, p) denotes the ball with centre at и and radius p. Let t be the translation of ζ to u. Then t(W)^D f/x and м t\Un), since H^ is the intersection of all t(M)=> Un. But since /''(Pj) =• P r i ( l ) , the set /-4ί(Λί)1 has the form P b fl ^c, where с G P b . Therefore, the /^(W^) are compact. But their intersection is χ = /~1(г/). Hence, for any of the neighbourhood Ox of χ there is a number η such that / ^ ( W J c Ол.. This proves the continuity of f'1 and so of/. This proves (b) and completes the proof of the theorem. (3.2) When Ρ \ {e} is open, the sets of the form (Pa f| ^b)N {a, b) form a base of a topology equivalent to that of V. When Ρ is closed with interior points, the "intervals" Pa f| P- (b 6 Pa) form a base of a topology that is, in general, stronger than that of V. §4. Contingency theorems We consider an affine space A" (n > 2). (4.1) Given a set Μ С A", the contingency cont(M, a) of Μ at a is the cone formed by all limits of rays / + (a, x), where χ G M, as χ tends to a. If a is not a limit point of M, then there is no such cone. But then we can take cont(M, a) == {a} to be the "null cone". All the propositions below are due to Aleksandrov [10]. (4.2) cont(M, a) = cont(M, a). Proof [38]. Since Μ CM, clearly cont(M, a)c= cont(M, a). We claim that (1)

cont(M, α)ςζ сопЦМ, а).

50

AX. Guts

If cont(M, α) = {α}, then also cont(Af, a) = {a}. Let b ζ cont(Af, a), b Φ a. To prove (1) it is sufficient to establish that the ray / = l + (a, b) is contained in cont(M, a). With any ε > 0 we consider the set l+ (a, x).

I)

W =

|χ-6| 0 we consider the set W=

U

l+(a, и)·

|и-зс| 2, be a continuous P-isotonic map and Ρ a closed pre-order satisfying Axiom A 2 ; then f(Cx) = CHx) for any χ Ε A", where С = cont(P, e).

52

A.K. Guts

Proof [10]. By Theorem (4.5), Cx = Sx, where Sx is the union of all directed curves starting from x. Since/ is a homeomorphism, it takes directed curves to directed curves. Hence, f(Sx) = Sf(X) or f(C~) = Cf^x). This proves the theorem. Note: In [38] it is shown that A 2 can be omitted in Theorem 2. §5. The mapping of cones (5.1) Let Ce be a cone with vertex e in the affine space A", that is, Ce is a set of points consisting of rays starting from e. We write С = Ce. We assume that С is a closed convex set with a cusped vertex (that is, С does not contain a straight line). Theorem 1. // С Φ L χ Κ, where L is a ray with origin e and К a cone with L Π Κ = {e}, then every homeomorphism f: A" -*• A" (n > 2) such that f(Cx) = СцХ) is an affine transformation. Proof [ 5 ] , [ 1 0 ] . (A) Let у G ЪСС, у Φ χ.

We define a set

тху = и cz{ze cz, ye dcz\. Txy consists of rays emanating from у and passing through all the points of Cx. If its closure Τxy is a half-space, then dTxy is the tangent plane to Cx at y. In general, Txy is a convex cone with vertex у containing the straight line through χ and y. Let Hxy be a maximal plane passing through у and contained in Txy. Then Tx-y- = Τxy for a pair of points χ , у' if and only if x', У' 6 Rxy Hence, Rxy is the set of all x' for which there is a y' such that ±

χ V

M

xy

From this representation of the Txy we see that / maps these sets into themselves. Consequently, / maps Rxy into planes of the same type, with preservation of dimension. In particular, / maps the tangent planes Rxy = дТху into tangent planes. We take η tangent planes 7} to С bounding an «-faced solid angle We. Since / takes tangent planes to tangent planes and parallels to parallels, the edges of the angles Wx map into parallel edges of WKx). We take any edge λ of W. The cone С has tangent planes other than the 7}, for if not, then С = W and we would have a contradiction to the condition С Φ L χ Κ. Not all such tangent planes can pass through λ, since otherwise we should again have С = L χ Κ. Hence, there is a tangent plane Τ that does not pass through λ and is distinct from the planes Tt opposite to it. Hence, in addition to λ there is at least one other edge X1 not contained in T. The plane Q spanned by λ and \ l intersects Γ in a straight line / = Q Π Τ. Thus, on Q we have three families of striaght lines parallel to λ, Хг and /, respectively. Under the map / straight lines parallel to λ and X b go into parallel lines. Thus, the planes Qx go into the parallel planes and the tangent planes Tx into parallel planes, so that the {lx} go into parallel (straight) lines.

53

Axiomatic relativity theory

Thus, Q is mapped onto a certain plane Q' such that to the families ίΚ), {λιχ}, {Ι χ) there correspond parallel lines. Since/is continuous, it maps Q onto Q' affinely (see, for example, Ch II, §19, Theorem 1). Consequently, / i s affine on λ and on the whole straight line extending it. Now λ is arbitrary, hence / is affine on straight lines extending edges of W. This is also true for edges of the Wx. When we now take a coordinate system with axes along edges of W, we see that / is affine on A". This proves Theorem 1. (5.2) Theorem 2. Let С = Lx χ ... χ Lp χ Κ where Κ Φ L χ Ku and Κ, Κλ are cones, Lu ..., Lp, L rays. Then every homeomorphism /: A" -*• A" (n > 2) such that f(Cx) = CHx), can be represented in the form (1)

/ = /o ° dx о . . . о dp,

where / 0 is an affine map and the dt are displacements dE.L. and homeomorphisms of A" onto itself. Here, f(C) is always an affine image of С and any dt = dE L.are admissible in (1), their order being immaterial. Proof f 10]. Each ray Z,,- in the decomposition (2)

С = Lv X . . . X Lv Χ Κ

is an edge of С and is the intersection of all tangent planes, except the Et opposite to Lj. Under / tangent planes go into tangent planes (see the proof of Theorem 1). Therefore,/(L,·) is also an edge of Cf(e). The plane Ε spanned by К is the intersection of the Et. Therefore, it maps into a plane (and the planes Ex parallel to it map into parallel planes). In Ε we have a system of cones Kx that map into cones superimposed by translations. Since Κ Φ L χ Klr by Theorem 1, / is affine on E. From this it follows that there exists an affine map from f0A" onto A" taking С to С / ( е ) and such that /0(1,·) = /(Ζ,,·) for every edge Z,,·, fo(K) =f(K), and / coincides with / 0 on E. Therefore, putting /ό1 ° / = h, we have (3)

h(C) = C, h(Lt) = Li (i = 1, . . ., p),

h(E) ~ E,

that is, h is the identity on E. Of course, the same conclusion holds for any cone Cx with the difference that h(Cx) = Chx, and so on. Let di be a map of dElLl that agrees with h on the straight line TVj containing the ray Z-j. Then d"1 о h is the identity on Nl and, in addition, satisifes relations similar to (3) (because d^C) — t(C), while / is the identity, since the vertex e of С is invariant under h and so under the chosen c/j). Now d~\ is equivalent on each cone Cx to its translation so that relations 1 similar to (3) hold for d" о h and any Cx. But d~\ о h is the identity on 7Уг=) Ζ/χ, hence, becomes a translation on all straight lines Nlx parallel to it.

54

A.K. Guts

We set d~x ° h — h1 and define for hx a displacement d2 just as άχ was defined for h: d'1 agrees with h2 on the straight line N ZD L2. Continuing this process we arrive at a map hv — dp1 о . . . 0 d"1 о h that is the identity on all the lines Nx, ..., Np and, what is more, reduces to a translation on straight lines Nix parallel to them. But h is the identity on K, preserves the same straight lines N( and translates the Nlx. From this it follows that hp is the identity on the whole of A". Therefore, recalling that h = /"J о /, we find that d~\, ° . . . ° d~\° j~\o f — hv is the identity. Hence, / = /0 о dx о . . . о dP, as required. This proves Theorem 2. 2

§6. The geometry of space-time Here we give an account of the system of axioms contained in Aleksandrov's papers "On the foundations of the geometry of space-time" [12] - [ 1 3 ] . (6.1) Let Ga be the group of all bijections g of V onto itself having the properties that g(a) = a and g(Px) = Pg(x) for any χ e V. Axiom A3. For any x, у ζ d P a 4 {α} there exists a bijection g G Ga such that g(x) = y. Axiom A4. For any x, у ζ int Pa = Pa\ g G Ga such that g(x) = y.

dPa there exists a bijection

Axiom A s . int Ρ Φ 0 . ( 1 ) Axioms A 3 and A 4 are the conditions of maximal homogeneity and isotropy of space-time. (6.2) Theorem 1 [12]. Suppose that the axioms Aj-As are satisfied. Then Ρ is a closed or an open elliptic cone, and Ge is the homogeneous Lorentz group with dilations; that is, there is a Cartesian coordinate system x i> X2, хз, X4 in V s u c n t n a t either α=1

or 2

,

3 i

)

,

\Σ α=1

i

} ] { }

where e has the coordinates (0, 0, 0, 0) and Ge is the direct product of the group of homogeneous linear transformations preserving the form x\ — x\ — x\ — x\ excluding the transformation x[ = — xit and the group of similarities x\ = λ^; (г = 1, 2, 3, 4), where λ is an arbitrary positive number. Thus, Minkowsky geometry can be axiomatized by the Axioms Ai-A 5 . In [12] A5 denotes another condition.

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Proof of Theorem 1. (a) From A 2 and A 4 it follows that e 6 Ρ \ {e}. For int Ρ Φ φ by A 5 . Let a G int P. If 6 is such that eb = 2ea, then a < b and Z> € int P. By A4 there is a g G Ge such that α = g{b); hence, g(6) < & and n < g"(i). From A2 it follows that there exist points gn(b) and gm(Z>) g +\b) arbitrarily close to one another. Therefore, the points χ ζ Ρ \ {e} are arbitrarily close to e. (b) The maps in Ga are continuous. For it follows from A 3 that either Pa = Pa or int Ρa = Pa\ {a}. But then (b) follows from A s and the continuity theorem (§3). (c) Let С = cont(/>, e). By Theorem (4.6), g(Ca) = Ca for any a G F and any g & Ga. Either С lies in a ^-dimensional hyperplane where к < 4 or int С Φ φ. The first possibility contradicts Axioms A 3 and A 4 . Therefore, int С Φ φ. (d) All g G Ga are affine. For g preserves the family of closed convex cones {Ca : α ζ V} having cusped vertices. Therefore, either g is affine, by (5.1) Theorem 1, or С is a quasi-cylinder and я = Яо ° ^i ° · · · ° dr, where g0 is an affine transformation and dt (i = 1, ..., p) are displacements, in accordance with (5.2). In the second case, by repeating the arguments in 6.3-6.8 of [10] we see that either g is affine or Ρ is a quasi-cylinder. But Ρ cannot be a quasi-cylinder, by Axiom A 3 . Therefore, g is affine. (e) We claim that Pa = Ca is an elliptic cone. This then will prove the theorem. Let Ka be the cone projecting Pa from a. From (d) it follows that g(Ka) = Ka for g G Ga. If Ka Φ Ca, then by adding to Ga similarities with centre at a we obtain an affine group acting transitively on Ka \ Ca. But, as is not hard to see, there can be no such group. Therefore, Ka = Ca, hence, Pa = Ca. But then Ga acts transitively on ЪСа, and by [ 6 ] , Theorem 2,. Ca is an elliptic cone (see also Ch.III, §23, Theorem 2). Theorem 1 is now proved. (6.3) The axiomatic system is not the only one possible. To state others we introduce the following conditions. Dj. Pa does not contain a ray with origin at a that is invariant under Ga. D 2 . There exists a point b £ Pa and a map g G Ga such that g{b) G Pb. D 3 - αζΡα\ {a}. D 4 . Either Pa \ {a} is open or Pa is closed but int Pa Φ φ. D 5 . For any b G ЪРа the set dPa f] dPl lies on a straight line or is linearly ordered in the pre-order defined by P. Theorem 2 [ 12], [ 13] . ( 1 ) Minkowsky geometry is axiomatized by any of the following sets of axioms: (A,, A2, A3, Dj, D 2 >, (Aj, A 2 , A 3 , Dj, D 3 >, rp, with the point e = (0, 0, 0, 0) added. For ρ = 1 Theorem 1 holds, and for ρ < 1 the group Ge is generated by transformations of the form x\ = %Pxk, x'a = λχα (a = 1, 2, 3) and by orthogonal transformations leaving x4 fixed. §7. Micro-causality and the geometry of space-time So far we have presented an axiom system in which the domains of action Pa contain events arbitrarily close to a. This means that the cause and effect structure is propogated equally for microsopic and macroscopic phenomena. Moreover, in this approach it is assumed that the transmission of action is continuous; a jump transmission of energy/momentum is forbidden. However, physicists quite frequently question the validity of the ideas of cause-and-effect connections for microscopic phenomena. For this reason it would be interesting to study a case in which the domain of action Pa is "isolated" from the event a, that is, a is an isolated point of Pa. In 1973 Aleksandrov indicated the desirability of such an investigation. Below we present results contained in [13], [27], and [29]. (7.1) Axiom Gl. The domain of action Ρ satisfies the following conditions: (1) Ρ == {e} U Q, where Q is an open or closed set with interior points, whose closure does not contain e; (2) Ρ lies within a convex cone with a cusped vertex e {which means that the cone does not contain straight lines). Definition 1. We put C=

U

l+(e, x).

This С with vertex e is called an exterior cone of P. Definition 2. The domain of action Ρ is said to be k-ruled, where к = 1, 2, 3, 4, if there exist rays Ц_(е, хг), . . ., /J(e, χ*) that do not lie in a single (k- l)-dimensional space and are such that (a) l\{e, x,)cz С (i = 1, . . ., A); (b) for any straight line λ parallel to any one of the rays l\{e, xj) (i = 1, . . ., k) the set λΠ β is either empty or is a ray. A set Ρ that is not Лг-ruled for any к = 1, 2, 3, 4, is called non-ruled. Axiom G2.

The domain of action Ρ is 4-ruled.

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Theorem 1 [27]. Suppose that the axioms A b G b and G 2 are satisfied. Then either any P-isotonic map f: V -> V is affine or Ρ is a quasi-cyUnder and f has the form (1)

/ = /о о d, о . . . о dp,

where f0 is an affine transformation and the d{ are displacements dBiit or dE. (1? Moreover, the order in which the dt occur in (1) is immaterial. Theorem 2 [ 13], [29]. Suppose that the axioms A b Gu and G 2 hold. Then Ρ defines an order with the condition A 3 if and only if Q is the section of the cone С with vertex at e and with an affine group acting transitively inside it, by planes cutting out a constant volume in С Here Ge is an unimodular subgroup of this affine group. Theorem 2 can be deduced from Theorem 1; it is not hard to see that under the conditions of Theorem 2 Ge preserves the exterior cones Ca. (7.2) Axiom G 3 . There is no m-dimensional plane (1 < m < 3) passing through e that is mapped onto itself by an isotonic mapping g G Ge. Theorem Г [29]. Suppose that the axioms A b G b and G 3 hold. Then the conclusion of Theorem 1 is valid. Theorem 2' [29]. Theorem 2 remains valid if Axiom G 2 is replaced by Axiom G 3 . (7.3) We now formulate the main result of this section. Theorem 3 [29]. Suppose that the axioms A l 5 G t , A 3 , and G 2 hold and that the exterior cone С is not a quasi-cylinder or a 4-faced angle. Then (1) С is a closed elliptic cone; (2) Ge is the homogeneous Lorentz group; (3) there is a Cartesian coordinate system xx, x2, x3, xi, in which dQ is defined by the relations x\ — x\ — x^ — x\ = μ2, xk > 0: where μ = const Φ 0, and С — {χ ζ V : x\ — x\ — x\ — x\ =--- 0, xi > 0}, e having the coordinates (0, 0, 0, 0). Axiom G 2 may be replaced by Axiom G 3 . (7.4) Thus, to establish that space-time geometry is pseudo-Euclidean it is by no means obligatory to assume the presence of cause-and-effect connections for microphenomena. The Lorentz group is a consequence of causality relations in the macrocosmos, while the structure of the microcosmos is to a certain extent only the result of this fundamental symmetry of space-time. 'onto' mappings.

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§8. Time-like Minkowsky space (8.1) In [56] Busemann has constructed a theory of time-like G-spaces. A particular case of this theory are the time-like Minkowsky spaces, which are directly related to axiomatic relativity theory. A time-like Minkowsky space is a non-empty set V for which in addition to Axiom A l 5 in which it is assumed that the {Pa} define an order in V, the following axioms hold. Axiom B t . The set of all pairs (x, y) G V χ V such that χ 3) satisfies the condition f(Mx) -- Mf(Xs, for any point χ Ε A", where Mx is a set of one of the six types (l)-(3+), then f is a Lorentz transformation to within a similarity [ 14]. (11.2) Physical content of Theorem 1. The relations (I+)-(III+) and (I)-(III) can be interpreted as the existence of action and causal linkage, respectively, between events χ and у proceeding by: (I) direct propagation of light; (II) propagation of light and mechanical means (that is, of sending a particle with non-zero rest mass from χ to у or from у to χ); (Ill) mechanical means or reflected light. In the cases (I+)-(III+) we talk of the action of χ on y, thus underlining the causal character of this relation between the events χ and у. The study of the relations (I)-(III+) is connected with the existence of inertial frames of reference. The invariance of the relations under bijections of space-time V expresses the law of the constancy of the velocity of light. Consequently, the physical content of Theorem 1 lies in the fact that Lorentz transformations (the core of special relativity theory) can be derived from the law of the constancy of the velocity of light in any of its usual forms (I)-(III+) and from the Euclidean nature of space, without invoking the principle of relativity [2]. This latter, as is known, speaks of the equivalidity of all interial frames of reference, that is, those in which the law of propagation of light is given by (I), or of the fact that the expressions of physical laws are invariant under Lorentz transformations. Since the relations (11+) and (III+) define a partial order in V, in these cases Theorem 1 amounts to the fact that "causality implies the Lorentz group" [80], [14]. (11.3) Theorem 1 was first established for the case I in 1949 by Aleksandrov [1]. The cases (1+) and (11+) were investigated in 1953 in the paper [2] by Aleksandrov and Ovchinnikova [ 2 ] . In 1964 Zeeman [80] published a paper dealing with the cases (1+) and (III+). Finally, in 1972 Borchers and Hegerfeldt [55] repeated Theorem 1 for the case (I) and studied the case (II), which had been established in [6].

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(11.4) We remark that the relations (I)-(III) are preserved under the following mappings other than Lorentz transformations: (1) the homothetic transformations Η: χ = x + a (λ Φ 0 and possibly λ < 0; by allowing λ = 1 we subsume translations under homothetic transformations); (2) the inversions Ia: χ - > - ρ — ^ + a; (x

a)

(3) the singular double inversions /

· τ

^ х

а + с(х

а)

г

where the vector с Φ 0 is such that c2 = 0, but is otherwise arbitrary. Here the inversion la is not defined on the cone Ca: (χ —a)2 — 0, and J fails to be defined on the plane Ρca: 1 + 2c(x — a) = 0. Therefore, these ca maps must be considered not on the whole of A" but on the subset D С А" where they are defined. Theorem 2 [ 15]. Let f: D -> A" be a bijective mapping of a domain D С A" that preserves one of the relations (I+)-(III+) together with its inverse. Then f is either a homogeneous Lorentz transformation L with homothetic part H, or it can be represented as such a transformation with the addition of an inversion I or a singular double inversion J. In other words, f takes one of the three forms HL, HLI, or HLJ, and in the last two cases it can also be expressed in the forms IHL and JHL, respectively. Theorem 2a [15]. Let f: D ->· A" be a bijective mapping of a domain D С A" such that for every cone Mx of one of the six types ( l ) - ( 3 + ) and any χ Ε D f(Mx П D) = Mm

П f(D).

Then / is as in Theorem 2. §12. Conformal space [15] The space V can be completed by a "cone at infinity" to which any inversion maps its singular null cone. Then we obtain a space С in which the transformations L, H, I, and / act as bijective maps (and are continuous in the natural topology on C). The space С is said to be conformal, because the conformal maps of domains in A" are the transformations L, H, I, J and their combinations HL, HLI, HLJ, so that the completion of A" to С regularizes the conformal maps. The fact that HL, HLI, HLJ, are conformal maps is well known when dim A" < °° (see, for example, [45]). Under the completion of A" to С the cones Cx and Kx are also completed, so that in С we have in mind these completed cones. Every isotropic straight line is completed by a "point at infinity" and thus becomes closed.

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The space С with an arbitrary cone Cx deleted becomes A". A map that is planar in A" turns out to be an inversion in C. The distinction of the singular double inversions loses its meaning, since inversions have no singularities*1 > on С Thus, it is natural to distinguish on С only three forms of maps: (1) the homotheties H0, (2) the Lorentz transformations Lc without reflections, and (3) the inversions Iе, that is, such (homeomorphic) maps of С onto itself that after deletion of a suitable cone become maps of the forms named. Theorem 1 [ 15]. A map f: D -»· С ofa domain D С С satisfying the same conditions as in § 11, Theorem 2a, is either HCLC or HCLC with the addition of one of the two inversions Iе. Note: In Theorem 1 we remember the relations (I)-(III) (see, however, [15], 9). The space С is homeomorphic to the product of a sphere and a circle. Therefore, there is a covering space С homeomorphic to the product of a sphere and a straight line. In it there is a naturally induced geometry that coincides locally with the geometry in C. Isotropic lines in С are no longer closed so that we can distinguish the ordinary cones C£, К% and Q j . The mappings Hc, Lc, Iе extend to C, and Theorem 1 has the following consequence. Theorem 2 [15]. The content of Theorem 1 is applicable to C. Here the condition for ordinary cones means that it can be considered not only locally, but globally. §13. The simplest axiomatization of space-time (13.1) Theorems 1 and 2 of § § 11 and 12 indicate that when any one of the six conditions (I)-(III+) is preserved together with its inverse, this alone, without any additional conditions, ensures a definite character of the possible transformations of space-time. Under the natural requirement that these transformations form a group, it is not even necessary to postulate that with a given relation its inverse is also preserved, since this is ensured by its preservation under the inverse transformation. If we confine ourselves only to unrestrictedly extendable transformations, then the inversions Ia and Jcn in the plane Minkowsky space-time V drop out and only Lorentz transformations with homotheties remain. But in the case of С and С any locally admissible transformations are unrestrictedly extendable and accordingly the groups of all conformal transformations act in these spaces. The requirement that the transformations form a group leads to the same results as that of their extendability. The latter does not mean that we distinction of the inversions Jca in A" had the meaning that they are defined on n half-spaces, while the inversions Ia are defined on domains, into which A is split by their singular cones.

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65

consider the world as a whole, since the extendability of the natural numbers does not mean that it is to be regarded as actually infinite. If we assume, in accordance with the familiar general view of geometry, that the geometry of space-time is determined by its group of transformations, then we can conclude from Theorems 1 and 2 in § § 11 -12 that each of the six relations (I)-III+) determines in this sense the geometry of space-time. Apart from Minkowsky space V we have two models С and С for spacetime in which the geometry is determined by one of the relations (I)-(III+). These models have been studied by Segal in [78] in connection with cosmology. (13.2) Thus, we introduce the following axioms. Axiom I v (1) On V there is given the relation (I), which is preserved by a set G of bijections of V forming a group; (2) G contains all the bijections preserving (I). The axioms I I V — Illy, I c — H I C , Ig — HI'c a r e formulated similarly. Consequently, the Minkowsky geometry is axiomatized by any one of the axioms I v , . . ., Illy, the conformal space С by any one of the axioms I c , II C , I I I C , and finally, С by the axioms I~, . . ., I I I ~ . ( 1 ) (13.3) The axiomatics of Berger [53]. The relations (I)—(III) are symmetrical. One of them, namely (II), was taken by Berger as the basis for an axiomatic construction of Minkowsky geometry. His axiom system contains quite a number of propositions that force the author to talk of the fact that the metric structure of the world is defined both by its causal and its topological structure. This latter does not play an essential part in I I V . Therefore, the Berger axioms must be regarded as a complicated version of I I V . §14. Theorems on finitely many light sources (14.1) In 1974, in a discussion of §11, Theorem Г at Aleksandrov's seminar on chronogeometry at the University of Novosibirsk, Kopylov made the conjecture that in formulating the theorem one can avoid imposing conditions on all the cones parallel to C+ (see § 11). Subsequently he proved the following. Theorem 1. Let F be a subset of the hyperplane # 0 = {xn = 0} such that for any sphere Sn~2 (n > 3) lying in Ho with centre at (0, ..., 0) G A" the set |[F\conv Sn~2] (where ξ is the central projection of # 0 \ { ( 0 , . . .,0)} from (0, ..., 0) onto Sn~2) is everywhere dense in Sn~2. We denote by D the set of points of all straight lines parallel to the axis xn such that D Π Ho = F. (1)

One of the axioms listed is taken. We note that the question of the axiomatization of the geometries of С and С remains open (see § §28, Problem 2).

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Then every bijective map f: A" -+ A" (n> 3) such that /(CJ) = C){x>for every point χ G D is a Lorentz transformation to within a similarity.

(The proof is unpublished.) This theorem was generalized by Kuzminykh, whose results are presented in this section. (14.2) Theorem 2 [54]. There is a set Dx С A" (n > 3), the union of ( ( 9 n - 1)12) {n odd) and ( ( 9 и - 4)/2) {n even) straight lines parallel to the xn-axis, such that every bijective map f: A" -> A" with f(C%) = Cj(x) for any χ G Dx is a Lorentz transformation to within a similarity. Theorem 3 [34]. Let To С Ho, where Ho is the hyperplane xn = 0, be a set consisting of η points in general position such that for three of these points x, y, ζ the ratio of the distances \x~y\/ \x-z\(in the Euclidean metric in A") is irrational. We denote by D2 a set that is the union of straight lines parallel to the xn-axis such that D2 Π Ho = To. Then every bijective map f: A" -» A" (n> 3) with condition f (Cx) = CKx) for any χ G D2 is a Lorentz transformation to within a similarity. Note i. In [34] a method is given for constructing such a set Д . Note 2. The number of straight lines in Theorem 3 cannot be decreased: for every pair of straight lines /t and l2 in A3 parallel to the xn-axis there is a discontinuous map φ: A3 -> A3 such that fp(Cx) = С ф ( ж ) for every χ G / t U l2.

(14.3) Kuzminykh has given the following physical interpretation of Theorem 2. It "means that a bijective transformation of space-time onto itself preserving the constancy of the velocity of light emitted from some sixteen sources and at rest in a certain inertial system of reference is a Lorentz transformation". [34]. (14.4) If we denote by (,Iy I.Di) a n d Iv \D2) the restrictions of the first parts of relations ( I + ) and (I) to Д and D2, respectively, then we can regard the axiomatic schemes I v I-Di and I v \D2 as defining Minkowsky geometry. §15. Mappings of strictly convex cones [11] (15.1) We consider an affine space A of dimension greater than 2, possibly infinite, in other words, a topological linear space with translations and convex neighbourhoods. By a cone we mean a set formed by rays (generators) emanating from a point (the vertex). A generator is said to be extreme if in any section of the cone containing it by a 2-dimensional plane there are no other generators in the convex angle between which it could lie.

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A cone is called strictly convex if it has the following properties: (a) The section of the cone by every 2-plane containing at least one generator is a convex angle, and its sides belong to it, that is, are extreme generators of the cone. All such generators are extreme. (b) The cone has both extreme and non-extreme generators. Theorem 1 [ 11 ] . Let К be a strictly convex cone in an affine space A. Let f be a bijective map from A onto an affine space A' such that f(Kx) for any χ EL A is a cone K'x>; suppose also that the cones K'x, are obtained from one another by translations. Then f maps straight lines of A into straight lines of A'. Therefore, in the case of a finite-dimensional space the map is affine. (15.2) Let Kx be the cone symmetric to Kx with respect to its vertex x. We say that a topology in A is defined by К if the intersections Ky Π Κ~χ, form a neighbourhood base, where χ lies on a non-extreme generator of Ky. The cone can then be said to be strictly convex if it is a closed convex set with interior point and every support plane contains at most one generator (it is assumed that the cone does not extend over the whole space). It is easy to see that a cone that is strictly convex in this sense has the properties (a) and (b). The converse (that a cone with the properties (a) and (b) is strictly convex in this sense) is self-evident in a finite-dimensional space. When dim A — °°, this may or may not be the case. Theorem la [ 1 1 ]. // under the conditions of Theorem 1 the cone К is strictly convex in the sense just given, then the map takes straight lines of A into straight lines of A' and is continuous in the topology of A' defined by K'. If the topology in A is defined by K, then f is a homeomorphism. Since straight lines map into straight lines under /, the cone K' has the properties (a) and (b). Theorem 2 [11]. Theorem 1 remains true if the cone is such that its convex hull satisfies (a) and (b). (15.3) By a double cone with vertex χ we mean the set Kx U Kx. cones Kx and K~ are the "halves" of the double cone.

The

Theorem 3 [11]. Let К be a double cone in a space A with convex halves satisfying (a) and (b). Let fbe an infective map of A into A' taking Kx into the double cone K'x,, obtained by translation from some cone K'. Then f maps straight lines of A into straight lines of A', hence, is affine if A is finite-dimensional. The map / takes the halves of Kx into the halves of K'i(X). In particular, the continuity (homeomorphism) of/ is ensured as in Theorem la. We emphasize that the map / in Theorem 3 is not assumed to be onto A'. Theorem 2 can be similarly strengthened.

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Theorem 2a [11]. Theorem 2 remains true if f is assumed to be into, but not necessarily onto, A'. (15.4) Theorem 4 [11]. The cone К in Theorem 3 can be taken to be the boundary of a double cone with strictly convex halves. Theorems 1-4 generalize Theorems 1.1 of §11. §16. How many inertial systems of reference? (16.1) We have assumed that the propagation of light is described in a certain frame of reference by a system of equal and parallel cones. In passing to another frame of reference we again have a system of equal and parallel cones. Consequently, there must be more than one inertial frame of reference. In this context the following physical law is formulated in [2] : every Lorentz transformation is possible. However, in our view, this is a strong formulation of an actual physical law. A weaker form would be to postulate only the possibility of motion with respect to an inertial system of reference, that is, to admit that there is a transformation taking a system of equal and parallel elliptic cones into another system of cones, not necessarily equal, not necessarily parallel. At the same time, this would mean abandoning the law of the constancy of the velocity of light as an axiom for the derivation of Lorentz transformations. Thus, could we abandon not only the principle of relativity (see § 11 (11.2)), but also the law of the constancy of the velocity of light? An affirmative answer to this question became possible after Shaidenko in 1974 showed that one can give up the requirement of parallellism of the elliptic cones in the image and still obtain Lorentz transformations as before. In this section we present results due to Aleksandrov, Kopylov, Kuzminykh, and Shaidenko, which generalize substantially the theorem of Shaidenko. (16.2) We consider an affine и-dimensional space A" (n > 3). Let К be a strictly convex cone. Its extreme generators form the "surface" cone dK. The generators that are not extreme form the "open" strictly convex cone о

К U {e}, where e is the vertex of K. In the usual topology of A", K is о

closed, dK is the boundary of K, and К is open. Therefore, in what follows we call these cones closed, open, and surface without hesitation. If a cone consists of straight lines, then we call them its generators and talk of a double cone. If К is closed and strictly convex and K~ is the cone symmetric to К with respect to the vertex, then К U K~ is a closed double cone; we also call it strictly convex. The double surface cone ЪК U ЪК~ and the double open о

cone К U K~ U {е) are defined correspondingly. In contrast to a double cone, we call a cone ordinary if its generators are rays.

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69

Thus, we consider six types of strictly convex cones, three of them ordinary: (I o ) closed K, (II0) surface ЬК, о

(III0) open К U {e} (where e is the vertex of K) and three double: (IV) closed К U K~, (V) surface дК [}дК~, (VI) open К U К- U {e}. Theorem 1 [17]. Let Q be a strictly convex cone of one of the six types. Let f: A" -> A" {n> 3) be a bijection such that for every point χ the set f(Qx) is a strictly convex cone homeomorphic to Q. Then f is affine. It turns out that in this theorem we can dispense with the requirement that/((λ,.) and Q are homeomorphic. In fact, the following theorem holds. Theorem 2 [47]. Let Q С A" (n > 3) be a strictly convex cone of one of the types (I0)-(VI) and f: A" -> A" a bijection such that f(Qx) for every χ G A" is a strictly convex cone; when Q is a double surface cone we assume, in addition, that f(x) is the vertex of f(Qx). Then f is affine. (16.3) The following question arises naturally. What can one say about the map / is we do not assume that f(Qx) is a cone? If we assume that the sets f(Qx) are obtained from one another by translations, then the following theorem holds: Theorem 3 [47]. Let Q С A" {n > 3) be a double solid closed strictly convex cone, that is, of type IV. Let MCA" be a set for which: (a) M = int M; (b) Μ is connected and there is a point e Ε Μ such that M\ {e} splits into two connected components; moreover an e such that M\{e) is disconnected is unique. (c) there exists a double solid closed strictly convex cone Q' such that MCQl Then a bijection f: A" -*• A" such that f(Qx) for each point χ is obtained from Μ by a translation is affine. (16.4) Finally, as a supplement to the question of (16.3) one can ask whether there is a postulate that the images of all the cones obtained by translations of К can also be obtained by translations of f{K) (see § 14). These two questions are answered by the following theorem. Theorem 4 [35]. There is a set D С A" (n > 3) such that: (1) for every pair Hx and H2 of hyperplanes parallel to xn = 0 the set D Π convttfj U Я2) is finite; (2) for every closed strictly convex double cone Q С A" every bijective

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map f: A" -*• A" such that f(Qx) for any χ G D is the union of two closed convex intersecting sets is affined Note 1. For ordinary cones the analogue of Theorem 4 is not true (see [35]). Note 2. The set D can be constructed as follows. We denote by Bk (fc = 1, 2, ...) the setKzj, . . ., xn): хл ---= IJm^ . . ., xn — Zn/wn, where lx, ..., ln, mx, ..., mn are integers; \mx I < k, ..., \mn \ k). We denote by B' the set {(zi, . . ., xn): x\ + . . . + xl_, < z2}. Then

/)=5'П( G Bh). §17. The axiomatics of relativity theory (17.1) Let D С V be the set of § 16 Theorem 4. Axiom Ry. On the space D χ V there is a relation II[Z>] such that (x, y) G II[£>] if and only if{x-yf > 0. Axiom H2. (1) There exists a group G of bijections of V onto itself such that iff G G and Kx = {y ζ V: (χ, у) 6 II W]}, where x G Д //геи / ( ^ ) и ί/ге union of two closed convex intersecting sets; (2) G contains all such bijections. Theorem. This system of axioms defines the Minkowsky geometry, and G is the Lorentz group {excluding similarities). This theorem amounts only to a reformulation of Theorem 4 in § 16 for the case of the cones Kx defined by (II) in §11. (17.2) In our view the axiom system ( H b H 2 ) is the most significant achievement in the solution of the problem of constructing an axiomatic relativity theory. In the idea of abandoning the law of the constancy of the speed of light (see § 16) is combined with that of postulating the principle that in each finite interval of time only finitely many short burst effective (point) sources of light are available for observation. (17.3) The axiom system 3 and that there is an χ Φ 0 for which (x, x) = 0. Let f: X -+ X be a bijection such that f(Cx) =; Cf(x) for any χ Ε X, where Cx = {y 6 X: (y - x, у - x) = 0}. Then f(x) = Lx+f(0), where the pair (L, τ) is a semilinear bijection such that (\)L(x + y) = Lx + Ly; T (2) L(ax) — et Lx where τ: F -*• F is an automorphism o / F ; (3) (Lx, Ly) = λ(χ, y)T for some fixed non-zero λ e F and for all x, у G X. Note. If F = R, then τ = id R (since R has no non-trivial automorphisms) and for some μ G R either λ = μ 2 , or λ = - μ 2 . In the first case г = μ-1/, satisfies (ix, iy) = (x, y), that is, i is an isometry on X. In the second case / = μ - 1 £ satisfies (jx, jy) = -(x, y). The latter means that the bilinear forms (,) and - ( , ) have the same signature. Consequently, the first case comprises Theorems 1 of § 11 for the relation (I) [ 2 ] , [54].

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A.K. Guts CHAPTER III

CONNECTED PRE-ORDERS IN AN AFFINE SPACE

From the classical point of view it is a natural assumption that action is continuously transferred from an event χ to an event у through a continuum of intervening events. The energy-momentum from χ must be transmitted at the initial stage of the action process to an event arbitrarily close to χ and then gradually from point to point must reach у, thus causing the occurrence of y. In mathematical language this means that a point a must be a limit point of its domain of action Pa. An even stronger requirement can be made: if b G Pa, then a and b can be joined by a continuous path lying in Pa, and the points of this path are related by cause-and-effect. In this chapter we study pre-orders in an affine space, formalizing in a certain sense the idea of continuous transmission of action. It is convenient to call such orders connected. In Ch. II and also in §5 we have studied axiomatics of relativity theory based on connected orders. Here we study some results concerning maps of a connectedly ordered space A" and the structure of connected pre-orders. § 2 1 . Maps of ordered spaces [10] (21.1) We consider an affine space A" in which there is defined a pre-order Ρ that is invariant under parallel translations. We use the notations and concepts of (1.4)-(1.7) and of §2. By a map / w e mean in this chapter a bijective map from A" into Am such that each set f(Px) can be obtained from f(P) by a translation. Setting P1 = f{P) we can express this condition as f(I\) ~ P'x·. Thus, a pre-order P' is induced in /04"), which extends uniquely to all Am as a pre-order invariant under translations (see [10], §3, Lemma 1). (21.2) We impose the following conditions on P. A) There exists a neighbourhood of e in which Ρ and P~ have no common points other than e; Β) Ρ contains a cone with vertex e having interior points. Theorem [10]. Let Ρ be a pre-order in A" satisfying A) and B). Let /: A" ->· Am (n > 2) be a continuous map. If Ρ is not a quasi-cyUnder, then f is affine. But if Ρ is a quasi-cy Under and we represent it by means of Q(EU /Д ..., ..., Q{Ek, lk), then

where / 0 is an affine transformation from A" into Am, and the dt are commuting displacements dEil. {where /,· may be a ray of Z,,·).

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Remark. The requirement of continuity in this theorem is superfluous for a wide class of pre-orders, for example, if Ρ satisfies Axiom A 2 and is closed or open (see (1.7), (1.8)). This follows from the continuity theorem of (§3). §22. Connected pre-orders (22.1) If by a connected pre-order we mean a pre-order Ρ for which the set Ρ is only connected, then we can hardly expect a very satisfactory description. It may be quite varied and it is easy to construct relevant examples. However, in axiomatic relativity theory, orders defined by convex cones play an important part. Therefore, it is quite natural to aim for conditions that characterize a pre-order given by a convex cone in terms of the idea of connectedness of the pre-order. This problem was posed by Aleksandrov and was solved by Levichev in [36] - [ 3 8 ] . (22.2) A pre-order Ρ is said to be strongly connected (or respectively, quasiconnected) if for any a and b such that a < b (that is, when b G Pa) the points a and b belong to a certain connected set Μ contained in Pa f] PI (or Pa Π PI)· Clearly, a strongly connected pre-order is quasi-connected. It is easy to check that a pre-order is strongly connected if and only if the sets Pa Π Рь are connected for all a and b such that a < b. The simplest example of a quasi-connected order that is not strongly connected is the set Ρ of all non-negative rational numbers on the real axis. (22.3) A convex cone Ρ defines a strongly connected pre-order in A". It is also clear that if Ρ is a convex cone and Ρ determines a pre-order, then this is quasi-connected. Theorem 1. If Ρ defines a quasi-connected pre-order, then Ρ is a convex cone [ 3 6 ] ,

[37].

Corollary 1. A closed quasi-connected (in particular, strongly pre-order is determined by a convex cone.

connected)

Theorem 2. If a pre-order is quasi-connected and e ζ int P, then Ρ contains the interior of its convex hull. Corollary 2. An open quasi-connected (in particular, strongly pre-order is given by a cone.

connected)

(22.4) Before demonstrating interesting examples of pre-orders illustrating Theorems 1 and 2, we introduce one more concept. A pre-order is said to be linearly connected, if for any a and b such that a < b these points belong to a certain linearly ordered connected set Μ contained in Pa (] Рь (that is, either л: < у or у < χ when χ, у €Ξ Μ). Clearly, a linearly connected pre-order is always strongly connected. Linearly connected orders and pre-orders can be defined not by a cone. For ([36]), there is a function φ: R -• R such that φ(χ + у) =• ψ(χ) -f

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Α. Κ. Guts

for any x, у G R, the graph Γ φ of φ in R 2 is connected, but φ is not continuous. We set P= Γ φ and Q = {(χ, ψ(χ)): χ > 0}. Then Ρ defines a linearly connected pre-order in R 2 , and Ρ f] Ρ* = Ρ for all x G i , while Q defines a linearly connected order in R 2 . But neither Ρ nor Q is a cone. In this connection we have the following results. Theorem 3 [38]. If a pre-order is linearly connected and there is a neighbourhood U of e such that the family of sets {P fl Pi- x 6 Ρ Π U) is bounded, then Ρ is a cone. Theorem 4 [38]. If a closed pre-order Ρ satisfies the axiom A 2 , then Ρ is a cone. (22.5) A pre-order given by a set Ρ is said to be locally compactly connected if for any χ G Ρ and any ε > 0 the set Ρ Π Ρχ contains a connected compact set K=

m

(J Kt such that all the Kt are connected compact sets

i=l

whose diameters do not exceed ε, where e, χ G K. Theorem 5 [38]. If Ρ defines a locally compactly connected pre-order, then Ρ is a convex cone. A pre-order Ρ is said to be compactly connected if for any χ Ε Ρ the set Ρ Π Ρχ contains a connected compact set joining e and x. Clearly, a locally compactly connected pre-order is compactly connected. Theorem 6 [38]. Suppose that Ρ defines a compactly connected pre-order in A". If one of the following three conditions is satisfied: (1) conv Ρ = A"; (2) conv Ρ does not contain a straight line; (3) dim conv Ρ «ξ 2, then Ρ is a convex cone. §23. Cones with a transitive group (23.1) The definitions in §22 of connected, strongly connected, quasiconnected, and linearly connected orders may be used as axioms characterizing the orders given by a cone. In relativity theory the light cone is elliptic. Consequently, the problem arises of finding conditions that enable us to single out elliptic cones among all convex cones, and what would be even more interesting, among all possible cones in an affine space A". Such additional conditions or hypotheses about the structure of space-time can be discovered without difficulty and have long been postulated in classical physics. We are talking about the hypotheses of homogeneity of the world in space-and-time and of spatial isotropy. Mathematically this leads to the idea of studying the structure of all homogeneous cones, that is, cones on which some group of transformations act transitively [6], [56], [95].

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77

(23.2) Let С be a cone in an affine space A" (n > 2) in the sense of § 18. We denote by G the group of all bijective maps of A" onto itself preserving the family of cones {Cx: χ ζ Л"}, so that if / Ε G, then f(Cx) = Cfix) for any χ G A", f(e) = e. Axiom T. G acts transitively on C, that is, for any x, у G С ч {е} there is an /EG such that f(x) = y. Theorem 1 [6]. If С is a {n- \)-dimensional closed set contained in a halfspace and is not a plane and if Axiom Τ is satisfied, then С is an elliptic cone and G is the Lorentz group (with similarities). For a convex surface cone С this theorem was proved by Busemann ( [ 5 6 ] , 34) in 1967. Theorem 2 [6]. Suppose that Axiom Τ be satisfied and let С be a cone with a cusped vertex such that An\C is a disconnected set and one of the sets C\{e}or An\C has at most countably many connected components. Then С is an elliptic cone and G is the Lorentz group (with similarities). Theorem 3 [6]. Suppose that Axiom Τ is satisfied and that the cone С has a cusped vertex and contains interior points. Then (1) С is a convex cone and C\ {e} is an open set; (2) С is an elliptic cone and G is the Lorentz group (with similarities) if one of the following conditions holds: (a) dC is a smooth surface (with e deleted); (b) a section of ЬС by a plane intersecting all generators contains at least one point with an osculating paraboloid (not degenerating to a cylinder). Busemann ([56], 38) proved Theorem 3 under the assumption that С is convex.

CHAPTER

IV

THE CHRONOGEOMETRY OF SPACES

In the preceding chapters we have understood by space-time a connected, simply-connected, locally compact Hausdorff space V with a transitive commutative group Τ of homeomorphisms of V onto itself preserving the domains of action {Pa}. Such is, for example, the Minkowsky space, the space-time of special relativity theory. However, in modern cosmology other geometric models have been used very successfully. Friedmann's spatially homogeneous and isotropic models have much to recommend them; other models of de Sitter, Godel, and others have also been used. Therefore, it is quite natural to try to construct a general axiomatic theory of space-time based on the idea of causal order. Such theories were proposed by Busemann [56], Pimenov [40], [41], Kronheimer and Penrose [64]. An account of these theories falls outside the framework of this survey, and in

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this chapter we give only results that concern maps preserving families of "light cones" in spaces closely connected with the models of Godel and de Sitter, in Lobachevskii spaces, and in Lorentz manifolds. §24. Spaces with a non-commutative group (24.1) In Ch. I and II, to arrive at Minkowsky geometry we spread a double light cone С around an affine space V by means of the group of parallel translations T. Instead of Τ we consider now the following non-commutative Lie group:

where α, β, у, δ are real numbers. The family of elliptic surface double cones x

x

xJtV·

{z

ci)2

α)2

(a

δ °* (ζ

α)2

"* ^ i —

is invariant under this group. Theorem 1 [28]. Any homeomorphism f: of the form _

V -*• V such that f(Ca) = C/ (a)

{ zl = (±)xl + af ->-2 — ""8

/л\

1 ' P»

:

4=±

//ze symbol (±) indicates that the same sign must precede xx and x2 but the one preceding x 4 can be chosen arbitrarily. It is easy to verify that the transformations (1) are motions of Godel space-time with the metric 2

ds* = a ( dx\ - dx\— ~ e*** da*—dx* + 2e** dXl dx3 ) . Theorem 1 can be regarded as a generalization of the theorem of Aleksandrov and Ovchinnikova (see § 11, Theorem 1) [28]. (24.2) Now we consider the non-commutative group X\

(2)

=

= λχς, + α,

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where λ > Ο, α, β, γ are real numbers, acting transitively on the half-space {хл > 0}. The system of elliptic double cones 3

Ka=--{zi{xl>0):

( Z i - α , ) 2 - Σ ( * α - Ο 2 = 0}, a=l

α = (alf a2, α3, a4), a t > 0 ,

is invariant under the group (2). Theorem 2 [23], [28]. // /: ( ^ > 0} ^- {i, > 0} is a such that f(Ka) = Λ^(α)( then f has the form

(3)

f(x)=Uo

U\x+\t\*

λ >

homeomorphism

°'

where U is an orthogonal matrix. The transformations (3) are motions in de Sitter space-time with the metric [28]

Similar results can be obtained for the Einstein spaces of maximal mobility f2t5 and Тзл. (24.3) The transformations in Theorems 1 and 2 are affine. It would be interesting to clarify whether a map preserving a fairly arbitrary family of cones in A" (n > 2) is affine. So far no exhaustive answer to this problem has been obtained, although there is a quite general theorem due to Shaidenko, which we now present. We consider the family of ordinary cones {Kx: χ 6 An) in A". By txy we denote the translation for which txy(x) = y. We say that the cone Kx is "less than", "equal t o " , or "greater than" the cone Ky according as the following relations hold: (1) txy(Kx)\{y} с int conv Ky, (2) tlv(Kx) = Ky, (3) Ky\{y} cz int conv txy(Kx). Two cones Kx and Ky are said to be comparable if one of these relations is satisfied. We assume that all the cones Kx of our family are either ordinary strictly convex surface cones or that they all satisfy the following condition: for every point χ there exists a solid closed strictly convex ordinary cone Cx with vertex χ such that int Cx С Кх С Сх (then they are said to be general). We also assume that every pair of cones of our family is comparable. Clearly, there exists a ray l+(e,x0) such that l*(e, я о ) \ { е } c i n t conv txe(Kx). for any x. We say that the cones {Kx} depend continuously on χ if for any point χ and any ε > 0 there is a neighbourhood U of χ such that | %P η tyx(Ky) — — XP Π Κχ Ι < ε for every у Ε U and any two-dimensional plane

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A.K. Guts

Ρ гэ [l+(e,x0)]x(htre χΡ fl Кг denotes the angle Ρ f] Kz). It is easy to see that this definition is equivalent to the following more standard one. The cones Kx depend continuously on χ if К = lim Кx

for every χ and every

m-*-oo

sequence of points {xm} with xm

> χ (here lim denotes the topological

limit). Theorem [47]. Let{Kx: χ ς Αη) be a family of cones in A" (n > 3) such that either all Kx are strictly convex surface ordinary cones or all are general strictly convex solid ordinary cones. We also suppose that (1) conv Kx define an order in A"; (2) for arbitrary x, у the cones Kx and Ky are comparable; (3) {Kx} depends continuously on x. Let f: A" -»• A" be a bijective map such that f(Kx) for every χ is a cone of the same type as Kx, and suppose that the cones f{Kx) satisfy (2). Thenf is affine. §25. Mappings of cones in Lobachevskii space Let L" (n > 2) be «-dimensional Lobachevskii space. We consider the family L —{lx: χ G L") of all straight lines parallel to / in a certain direction. Let {Cx} be a family of double circular cones (x is the vertex of the cone) in L" with axes belonging to the family L and with a fixed opening angle a (0 < α < π). This family of cones is invariant under a certain subgroup of isometries acting transitively on L" and is isomorphic to the group (2) of §24, generalized to the «-dimensional case, that is, (xx, . . . xn,) ->• (λ^, λχ2 + au .. ., λχη + ап_г) (хи\> 0) [22], [23]. Theorem 1 [23]. / / / : Ln ->· L" {n > 2) is a bijective map such that f(Cx) = С/(Л), then f is an isometry. §26. The chronogeometry of Lorentz manifolds (26.1) The theorems on maps of an ordered affine space have a fairly natural generalization to Lorentz manifolds. Before presenting results in this direction we need some definitions. By space-time we mean a connected four-dimensional smooth manifold Μ without boundary, together with a smooth pseudo-Riemannian metric g of signature (H ). We also assume that Μ is time-oriented (that is, Μ has a nowhere vanishing time vector field) and that a certain time orientation defining the future and the past has been fixed on M. Let A, U a M, and U r> A be open. By I+(A, U) we denote the set of points χ G U such that there is a smooth time curve, directed to the future 7: I -*• U (where / С R is connected) and points tlt t2 £ I such that tx Μ' a bijection. We say that / is a conformal mapping if/ and f'1 are smooth and there exists a map Ω: M' -*• R vanishing nowhere and such that f*g = £l2g'.

A bijection/: M-*M' is said to be causal if f(x) M ( / C R being a connected set) be a continuous curve. We say that it is temporal and directed to the future if for any t0 G / and any open convex sets U containing j(t0) there is an open subinterval Τ С I containing t0 such that

J t G Г and t < t0 implies that y(t) < y(to)(U), \ t G Г and t0 < t implies that y(t0) < y(t)(U). Finally, 7 is a continuous light geodesic directed to the future if the conditions (*) in the definition above are replaced by tu i 2 € 1 and tx < f2 implies that γ(ί2) ς /+[γ(ίι), ί/Κ/+(γ(*ι), U), where 7 + [γ(ί χ ), U] is the causal future of the point y(ti) with respect to U. By definition, it consists of those у G U for which there is a smooth curve in U joining y{ti) to у with nowhere vanishing non-spatial vectors directed to the future. Theorem 2 (Hocking-Malament). Let (M, g) and (M', g') be two space-times and f: Μ -*• Μ' a homeomorphism such that f and f'1 preserve continuous light geodesies directed to the future. Then f is a conformal transformation [62], [69].

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In the case of a Minkowsky space this theorem was first proved by Borchers and Hegetsfeldt [55]. Theorem 3 [69]. Let {M, g) and Ш', g') be two spafie-times and f. M-+M' a bijection such that f and f'1 preserve continuous time curves directed to the future. Then f is a conformal transformation. §27. An analogue of Aleksandrov's theorem in the class of partially ordered fields (27.1) Let К be a commutative field, X = K" (n > 3), and Q an indefinite quadratic form on X of Witt index 1. This means that X splits into an orthogonal direct sum U J_ H, where U is a subspace that does not contain non-zero light vectors (that is, non-zero χ with Q(x, x) — 0), and Η is the subspace generated by two non-orthogonal light vectors [68]. We set Ca = {χ ζ X: Q(x — α, χ — a) = 0}. Theorem 1 [81 ]. / / / : X •+ X is a bijection such that f(Ca) = CHah then f is the composition of a translation and a similarity g, that is, Q(g(x), ц(х)) = = с- μ(?(χ, χ), for which с Φ 0, where μ is an automorphism of K. If the characteristic of К is not 2, then the requirement that vertices of cones go into vertices can be omitted. Note. The condition char(K) Φ 2 is necessary, as is shown by a counterexample in [81]. (27.2) Now let К be a commutative field equipped with a non-trivial partial order 0}. We define on X = Κ" (η > 3) the form n-l

= (xu . . . , xn). We introduce the notation =< and -< on X: (1) у = yn; (2) у < χ if and only if Q{x~y, χ—y) > 0 and xn > yn. Note that χ ц; у does not mean that χ -< у or χ — у. Let

= {xe X:

Theorem 2 [81 ] . We let (K, < ) be a pair such that: (a) P + PczP; (b) P-P cz P;

( c ) i ( l (-P) = {0}; 2 (d) K cr P.

Axiomatic relativity theory

83

Then the relation =4 is a partial order on X, and any bijection f: X -* X preserving the cones of one of the following types; Pa, PI, Р„, Pa U (—Pa), PI у (—PI), PI U (—PI) {including the vertex Φ 0) is the composition of a translation and of a semilinear map. §28. Problems In conclusion we list a few problems whose solution would lead to further progress in axiomatic relativity theory. 1 (Aleksandrov, [100]). To construct an axiomatics for special relativity theory starting from the following fundamental principles: (a) Space-time is the manifold of all events, regarded only from the point of view of its structure determined by the system of relations of precedence (of the domains of action {Pa}, see § 1), in abstraction from all other properties; (b) Space-time is a four-dimensional manifold; (c) Space-time is maximally homogeneous, that is, the group of its transformations preserving the relation of precedence is as large as possible. It is easy to see that the axiom systems in Ch. I and II do not completely reflect c), since the homogeneity of space-time and not the maximality of the group of isotonic transformations is essentially assumed. The latter can evidently yield much. For example, among pseudo-Riemannian 4-dimensional manifolds the requirement of maximality of the group of motions immediately distinguishes spaces of constant curvature, among which the Minkowsky space has a maximal commutative subgroup. 2. (Aleksandrov). Can one obtain the conformal space С (see § 12) starting out from Problem 1 a)-c)? We recall that С is a planar pseudo-Riemannian manifold with a topology equivalent to that of the space S3 χ S1. The group in question is then a 15-parameter group of conformal transformations. 3. We denote by A5 the following statement: Ρ is not a ray. Is it possible to deduce axiom A S ) that is, int Ρ Φ φ from the system of axioms