A system of axioms for Minkowski spacetime
aa r X i v : . [ phy s i c s . h i s t - ph ] J u l A system of axioms for Minkowski spacetime ∗Lorenzo Cocco Joshua Babic
Abstract
We present an elementary system of axioms for the geometry ofMinkowski spacetime. It strikes a balance between a simple andstreamlined set of axioms and the attempt to give a direct formalizationin first-order logic of the standard account of Minkowski spacetime in[Maudlin 2012] and [Malament, unpublished]. It is intended for futureuse in the formalization of physical theories in Minkowski spacetime.The choice of primitives is in the spirit of [Tarski 1959]: a predicateof betwenness and a four place predicate to compare the square ofthe relativistic intervals. Minkowski spacetime is described as a fourdimensional ‘vector space’ that can be decomposed everywhere into aspacelike hyperplane - which obeys the Euclidean axioms in [Tarskiand Givant, 1999] - and an orthogonal timelike line. The length ofother ‘vectors’ are calculated according to Pythagoras’ theorem. Weconclude with a
Representation Theorem relating models M of oursystem M that satisfy second order continuity to the mathematicalstructure x R , η ab y , called ‘Minkowski spacetime’ in physics textbooks. The aim of this paper is to provide an elementary system of axioms thatcharacterizes the geometry of Minkowski spacetime. It will be pursued inthe style of Tarski; that is, with a primitive predicate of betweenness and aquaternary predicate to compare the relativistic intervals between points.A system of this sort is needed, first of all, for certain investigations on thefoundations of relativity. One question that we believe deserves attention isthat of the theoretical equivalence of two types of formulations of relativity.There are ‘dynamical’ formulations of relativity, framed in terms of observers,coordinates systems and the like [Andr´eka, N´emeti et al. 2011]. We can ∗ We are grateful to John Burgess, Dino Calosi, Harold Hodes and Chris W¨uthrich fordiscussion and comments on parts of this paper. Unfortunately all of these systems are rather unwieldy to work with, when oneattempts to extract physics from them. Just to account for the descriptionof spacetime along the lines of [Maudlin 2012] and [Malament, unpublished]requires several pages of definitions and derivations. On the other hand,our preferred standard of theory equivalence is a modification of one due to[Barrett & Halvorson 2016]. Any reasonable definition requires a ‘dictionary’between talk of coordinates and spacetime notions. We also need to derivethe translation of the axioms of [Andr´eka, N´emeti et al. 2011] from thegeometric theories and vice versa . In future work, we plan to describe such atranslation and consider some of its philosophical implications. But we havefound it more convenient to give first an equivalent but more manageabletheory, with simpler extralogical primitives, to act as an intermediate. A second, intrinsic justification for our system is that it allows a straightforwardproof of the
Representation Theorems of [Tarski 1959] and [Tarski and Sczerba, The systems of Mundy [1986a, 1986b] are notable examples. Mundy [1986a] is closeto that of Robb[1936] and is based on lightlike connectibility. Mundy [1986b] is themost similar to ours, but requires five primitives: three primitive notions of betweenness,timelike, spacelike and lightlike betweenness, and two primitive notions of congruence,temporal and spatial congruence. Other systems worth mentioning are that of [Ax 1978]and [Schutz 1997], although they both heavily rely on set-theoretic machinery. [Ax 1978] isa ‘dynamical’ system (in our terminology). It employs variables of two sorts: one rangingover particles and one ranging over signals. It construes segments as sets of ‘particles’. They themselves modify an earlier proposal of [Quine 1975]. The book [Halvorson2019] surveys several such notions of equivalence for scientific theories and argues thata plausible candidate should be intermediate in strength between mutual interpretabilityand bi-interpretability. In a future paper, we will propose an ulterior refinement of [Barrett& Halvorson 2016][Quine 1975] [Spector 1958] and defend that it is the best criterion ofequivalence. We need to allow for the translation of theories with different domains ofdiscourse, as in all the generalized notions of interpretation described in [Alscher, 2016;chap. 1] and several natural examples of reconstrual in mathematics [Halvorson 2019, pp.143-145] The variables of the system of [Andr´eka, N´emeti et al. 2011] range over bodies,observers and real numbers. Their primitive predicates are those of the theory ofreal-closed fields and a primitive predicate Cooordinatization obxyzt that applies to anobserver, a body and four coordinates in the obvious circumstances. The calculus ofsegments seems to be needed to translate this talk of localization relative to coordinates. We use the results of [Tarski, 1959] forEuclidean space to show that: (1) every model of a [second order version]of our theory admits of a coordinatization into R and (2) any two suchcoordinatizations f and f are equivalent up to rescaling U and a Poincar´etransformation L (sec. VI). In addition, it is plausible that the systembelow can be be more easily supplemented to axiomatize a field theory, forexample electrodynamics. It acts as a useful ‘buffer’ between ‘dynamical’and geometric formulations of the theory. A proof of the equivalence of oursystem to our target system - in the sense of W.V.O Quine [1975] - will ipsofacto carry over to other geometric systems of axioms that are interderivable.It will be evident enough how to derive from our system all the axioms inthe appendix to Goldblatt [1987]. Derivability in the reverse direction canbe established by more theoretical considerations. Goldblatt sketches in theappendix to his book a proof that his own system is complete and decidable,and he demonstrates that his primitive of orthogonality is interdefinable withthat of causal connectibility. He derives his result from quantifier eliminationfor the theory of real closed fields. [Pambuccian 2006] constructs an explicitdefinition of betweenness and congruence in terms of causal connectibility. Since the system of Goldblatt [1987] and ours are almost self-evidently sound,we get that a derivation must exist without having to go through the hurdleof providing one. This closes the circle. Our system, that of Goldblatt [1987],and a proper formalization of [Robb 1936] must all be equivalent.
Remark.
The system that is most similar to what we are about to propose is The price to pay is that our axioms cannot be stated simply in primitive notation.Tarski and Givant [1999, p.192/f], and most logicians working on geometry, attach muchimportance to avoiding defined symbols. This does not appear to us to be a decisivedefect. In axiomatic set theory, nobody would take the pains to write down the axiom‘
V=L ’, or
Martin’s axiom , or the
Proper Forcing Axiom only in terms of quantifiers, truthfunctions and the epsilons. This does not disqualify them as possible additions to ZFC[Jensen 1972]. Consider the problem of formalizing Maxwell’s theory on the systems of Goldblatt[1987] and Mundy [1986a, 1986b]. To formulate a nominalistic analog to a system of partialdifferential equations - in the style of [Field 1980] - and set up an initial value problem, weare forced to introduce by definition the apparatus to describe a foliation and employ itin the axioms. This means that the main advantage that the systems of Goldblatt [1987]and Mundy [1986a] have over ours, the fact that they can be stated elegantly withoutabbreviative definitions, disappears when we come to relativistic electrodynamics. Beth’s definability theorem and a first order strengthening of the Alexandrov-Zeeman’stheorem - according to which every automorphism of Minkowski spacetime preservescongruence relations - already imply that such a definition must exist. [Sklar 1985] saysthat Malament proved a similar theorem in his PhD thesis; [Malament 2019] attributes aversion of the theorem to Robb. Pambuccian [2006] has explicitly found such an adequatedefinition in terms of lightlike connectibility. he axiomatization of Galilean spacetime sketched by Hartry Field in chapt.4of [Field 1980]. We use the same methods to form a theory for relativisticspacetimes. The main idea is to employ the already existing systems foraffine spaces of dimension four [Tarski and Sczerba, 1979] and for Euclideangeometry [Tarski and Givant, 1999] as basic building blocks of our account.The system that we propose is nominalistic. We will return to the connectionwith [Field 1980] and the nominalization of physics at the end. As in Tarski’s system for Euclidean geometry [Tarski and Givant 1999],we assume only one type of entity in the range of the variables: points.The logical vocabulary consists of the identity symbol ‘=’, negation ‘ ’,conjunction ‘ ^ ’ the existential quantifier ‘ D ’, and auxiliary symbols. Thevariables are x , y , z ... x , y ... In defiance of the usual conventions, we use v , v , v as metavariables ranging over variables to state some schemata.The two extralogical primitives are a ternary predicate of betweenness:(1) Bet p x, y, z q and a quaternary predicate to compare lengths:(2) ă ” p x, y, z, w q that holds of four points x , y , z and w when the square of the relativisticintervals between x and y is less than that between z and w. By therelativistic interval between two points we mean the geometric quantity thatis measured, under appropriated coordinates, by the algebraic expression: a p t ´ t q ´ p x ´ x q ´ p y ´ y q ´ p z ´ z q The square of the interval is, therefore, the real valued quantity p t ´ t q ´ p x ´ x q ´ p y ´ y q ´ p z ´ z q We stress that, for reasons of simplicity, we work with the square of theinterval rather than the interval. This partitions pairs of points into threecategories: those such that the term above is negative, those such that theterm above is positive and those such that the term above is zero. This ofcourse embodies a convention about signs. It means that spacelike separatedpoints, for example, will count as having negative ‘length’, since the square4f the above quantity is a negative number. As we have mentioned, we donot even attempt to formulate the axioms in primitive notation and, for thisreason, the next section is devoted to a battery of definitions.
Remark.
Our primitive vocabulary contains the predicate ă ” p x, y, z, w q inlieu of the usual congruence predicate ” p x, y, z, w q [Tarski, 1959; Tarskiand Givant, 1999]. It is natural to ask whether we could have based oursystem on congruence instead. The predicate ‘ ă ” p x, y, z, w q ’ is simply notdefinable in terms of ‘ Bet p x, y, z q ’ and ‘ ” p x, y, z, w q ’ in plane geometry.The Minkwoski two dimensional plane admits of an automorphism of thesystem of congruence - a bijection that sends congruent segments to congruentsegments - but inverts relationships of shorter and longer. Anticipating a biton our account of representation, we can specifiy it in coordinates as thetransformation (x,t) ÞÑ (t,x) (swapping of space and time coordinates). InMinkowski spacetime a definition is possible. We can distinguish spacelikesegments by the fact that they have congruent orthogonal segments and define‘shorter than’ as usual. The chain of definitions is cumbersome and we havepreferred to adopt ‘ ă ” p x, y, z, w q ’ as an undefined predicate. Our plan is to describe the geometry of a flat spacetime by specifying axiomsthat (a) characterize it as a four dimensional vector space and (b) fix the‘length’ of arbitrary segments. We fix their length by decomposing theminto a basis. The length of our initial segment is expressed as a function ofthose of its projections or components. This requires the machinery of linearalgebra. We also need the notion of orthogonality and a development of thetheory of proportions; essentially of a device to mimic algebraic computationswithin the theory. The crucial definition is that of the orthogonality of twosegments . The development of linear algebra depends on orthogonality ratherthan orthogonality being defined as in linear algebra. One cannot just startfrom a given ‘chosen’ basis and define the dot product - or a particular linearform - as a linear function of the components relative to the ‘preferred’ basis.The definitions that follow build up the conceptual tools that we need:
The first definition introduces the usual congruence predicate ‘ ” ’.( D0): ” p x, y, z, w q Ø df ă ” p x, y, z, w q ^ ă ” p z, w, x, y q D1): L p x, y q Ø df ” p x, y, x, x q x y Figure 1: The points x and y are lightlike separated (yellow).A spacelike segment is a segment of negative ‘length’ (in blue).( D2): S p x, y q Ø df ă ” p x, y, x, x q x y Figure 2: The points x and y are spacelike separated.A timelike segment is a segment of positive ‘length’ (in red).( D3): T p x, y q Ø df ă ” p x, x, x, y q x y Figure 3: The points x and y are timelike separated. The following definitions are imported wholesale from the literature onthe axiomatization of geometry and need no further explanation: they6efine collinearity in terms of betweenness, coplanarity of four pointsand a preliminary definition of parallelism between the lines on which xy and zw stand. Later we will settle on another definition.( D4):
Coll p x, y, z q Ø df Bet p x, y, z q _ Bet p x, z, y q _ Bet p y, x, z q ( D5):
Copl p x, y, z, w q Ø df D v pp Coll p x, y, v q ^ Coll p z, v, w qq _ p Coll p x, z, v q ^ Coll p y, v, w qq _ p Coll p y, z, v q ^ Coll p x, v, w qqq ( D6):
P ar W p x, y, z, w q Ø df Copl p x, y, z, w q ^ pp Coll p x, y, z q ^ Coll p x, y, w qq _ D v p Coll pp x, y, v q ^ Coll p z, w, y qq ( D7):
Intersect p x, y, z, w q Ø df D v p Bet p x, v, y q ^ Coll p z, w, v qq The points x , y , z and w form a parallelogram when the segments thatunite them are pairwise parallel.( D8):
P arallelogram p x, y, z, w q Ø df P ar W p x, y, z, w q ^ P ar W p x, w, y, z q The main business of this section is to provide a definition of the ternarypredicate orthogonality in terms of congruence and betweenness: thesegment from x to y is orthogonal to that from x to z. The definitionthat we give is a definition by cases. The three cases we need totreat separately are (1) the segment from x to y is lightlike, (2) thesegment from x to y is spacelike or (3) the segment from x to y istimelike. The strategy is easily grasped by considering how one mightdefine orthogonality in Euclidean geometry. In Euclidean geometry, theorthogonal projection of a point z on a line passing through x and y issimply the closest point on the line . This definition can be reproducedwholesale in the case when the segment from x to y is timelike . Whenthe segment from x to y is not timelike, the state of affairs is reversed ormore complicated. The presence of null and negative lines complicatesthe business. In all scenarios, a vector from z to some v that falls onthe line determined by x and y will give us a right triangle if and onlyif the segments are orthogonal. The segment from z to v is going to bethe hypotenuse of it. Pythagoras has taught us that the square of thehypotenuse is a sum of squares: if the basis of the triangle is spacelike,then the cathetus from x to z is going to contribute negatively to thelength of the hypotenuse. This means that the path from z to x is goingto be the longest straigth path to the line xy.7
D9):
Case p x, y, z q Ø df S p x, y q ^ @ v p Col p v, y, x q Ñ pă ” p v, z, x, z q _ v “ x qq x yz v Figure 4: Case 1 of orthogonalityIf the basis is timelike, we get the reverse situation. This puts us back,as we noted, in the old Euclidean case. The orthogonal projection of zonto the line xy is the closest point on the line:(
D10):
Case p x, y, z q Ø df T p x, y q ^ @ v p Col p v, y, x q Ñ pă ” p x, z, v, z q _ v “ x qq x yz v Figure 5: Case 2 of orthogonalityThe last case that needs to be treated is when the base of the triangleis lightlike. There are two ways to deal with it. With the two notions oforthogonality at hand, we have enough material to define an orthogonalbasis and the arithmetic of segments (cf. next section). This apparatusis enough to develop linear algebra. We can then define a nominalisticproxy of the Lorentzian form between two segments. Two orthogonalsegments are going to be two segments such that the form gives zerowhen applied to them. The approach we adopt is more elegant andconsists in reducing the third case to the former two. Let us assumeagain that xy is lightlike and that xz is a candidate to orthogonality.Either (a) z is collinear to x and y or (b) xz is spacelike. We candecompose xy in a spacelike component xy and a timelike componentxw so that xw is orthogonal to xz. At this point, by the distributivityof the Lorentzian product, we see that xz is orthogonal to xy if and onlyif it is orthogonal to xy . This means that a spacelike xz is orthogonal8o a lightlike xy just in case there is a decomposition of xy such thatxz is orthogonal to both components in the senses already treated:( D11):
Case p x, y, z q Ø df L p x, y q ^ r Coll p x, y, z q _ pD w D y p T p w, x q ^ S p w, y q ^ Case p x, w, z q ^ P arallelogram p x, w, y, y q ^ Case p x, z, y qqs x y zw y Figure 6: Case 3 of orthogonalityWe can now define orthogonality by a disjunction:(
D12):
Orth p x, y, z q Ø df Case p x, y, z q _ Case p x, y, z q _ Case p x, y, z q ( D13):
Orth p x, y, z q Ø df Orth p x, y, z q _ x “ y _ x “ z and give another definition of parallelism in terms of orthogonality: ( D14):
P ar p x, y, z, w q Ø df [ P ar W p x, y, z, w q^D z p Coll p z, w, z q^ Orth p x, y, z q^ Orth p z , x, z qqs _ x “ y _ z “ w We are ready to define the apparatus of linear algebra. When is asegment, or vector, generated from other vectors? To generate a vector ~ox (that stems from a given origin o ) from other vectors ~oy , ~oz and ~ow , means that we can reach the ‘top’ x from the ‘tail’ o by travellingalong directions that are parallel to the vectors ~oy , ~oz , ~ow .( D15):
Gen D p o, x, y, z, v q Ø df D x D y p Coll p o, x, x q^ P ar p x , y , o, y q^ P ar p o, z, y , v qq It is useful to stipulate that a point x - that is, a degenerate segment - is respectivelyorthogonal to lines through x and parallel to lines that do do not pass through x . xy x y z v Figure 7: The point v is a linear combination of ~ox , ~oy , ~oz in three dimensions.Notation for the generation of more than one vector is easily introduced.( D16):
Gen p o, x, y, z, a, b, c, d q Ø df Gen D p o, x, y, z, a q ^ Gen D p o, x, y, z, b q ^ Gen D p o, x, y, z, c q ^ Gen D p o, x, y, z, d q It is useful to take into account the intermediate steps that are madewhen moving from o to a given w along one of the specified directions.A specific trajectory may be called a development of w from ~ox , ~oy , ~ox and ~ot . There are of course multiple ways to reach w from o , dependingwith which direction one starts from, and also on the different waysof proceeding. We will later impose an axiom that makes paralleltrajectories in two separate developments congruent to each other. Toexpress it, we need to refer to these intermediate steps. We read thenext predicates as ‘w can be reached from x , y , z and t via x , y , z ’.This is the standard fashion of reaching a point:( D17):
Reached p v , v , v , v , v , x , y , z , w q Ø df Coll p v , v , x q ^ P ar p v , v , x , y q^ P ar p v , v , y , z q ^ P ar p v , v , w, z q ^ Orth p x , v , y q ^ Orth p y , x , z q^ Orth p z , y , w q ( D18):
Reached p o, x, t, v, w q Ø df Coll p o, x, v q ^ Orth p v, o, w q ^ P ar p o, t, v, w q An arbitrary development is defined by permuting the order of directions:(
D19):
Develop p v , v , v , v , v , x , y , z , v q Ø df Ž Reached p v , v σ p q , v σ p q , v σ p q , v σ p q , x , y , z , v q [where σ ranges over permutations of the set of indices t , , , us .10eneration in four dimension is defined in terms of development:( D20):
Gen D p o, x, y, z, t, v q Ø df D x D y D z Develop p o, x, y, z, t, x , y , z , v q A basis is for us a quintuple of points. It consists of an origin o and fourpoints that determine four mutually orthogonal directions in spacetime.(
D21):
Basis p o, x, y, z, t q Ø df T p o, t q^ S p o, x q^ S p o, y q^ S p o, z q^ Orth p o, t, x q^ Orth p o, t, y q ^ Orth p o, t, z q ^ Orth p o, x, y q ^ Orth p o, x, z q ^ Orth p o, y, z q The next definition plays a crucial role in the economy of our system.It is a key ingredient of all our main axioms. It defines the relationthat obtains between a timelike and spacelike vector when the squareof the interval, or the ‘length’ of these segments, differ only in term of‘sign’: when they are of equal absolute value. To define the notion wehave to transport one of the segments to a congruent one orthogonalto the other. At this point we can call them of opposite length if theirsum is a null vector: their contributions to the hypotenuse cancel out.(
D22):
Opp p x, y, z, w q Ø df D w D v p Orth p x, y, w q^ ” p x, w , z, w q^ P ar p w , v, x, y q^ ”p w , v, x, y q ^ L p x, v qq x yw vzw Figure 8: The points x, y, z and w are opposite. Following the ideas of Hilbert [1899], one can define algebraic operations onthe points of a line. Given a line in Euclidean space, fix two arbitrary pointsto play the role of the null element 0 and the neutral element 1. We can usethe method to define addition and multiplication of line segments that are11ollinear to 0 and 1 in such a way that the line satisfies the axioms of a realclosed field. Of course lines in different models will lead to different fields orrings. Each of them, however, simulates well enough the familiar field of realnumbers (an informal presentation of the construction is in [Hartshorne 2000,ch.4]). Segment arithmetic is of crucial importance for Field’s Program. Itallows us to translate numerical statements about real numbers into purelygeometrical ones. It also plays an important role in our attempt to constructcoordinate systems within the geometrical theory. Tarski and Sczerba [1979]use it construct and classify the coordinatizations of various spaces modulothe ‘passive transformations’ between them; they fix an origin in an affinespace A with certain properties, find a line ℓ living inside it that satisfies thefield axioms, they define vector operations among the points and make thestructure ă A, F ą into a vector field V over a field F .The arithmetic of segments is now needed to compute within the theorythe ‘length’ of the hypotenuse of a right triangle relative to the ‘length’ ofthe sides. It will allow us to postulate the existence of a third segmentwhose ‘length’ is the sum or the product of the length of any two givensegments. It will also allow us to postulate a segment on any given linesuch that its ‘length’ is the square or the square root of the length of anygiven segment. The apparatus is imported as a block from [Schwabh¨auser,Szmielew and Tarski 1983]. But their definitions are meant in the context ofEuclidean geometry. We will therefore need to restrict the variables so thatall the segments involved are spacelike. Some of their initial definitions canbe restated more simply for our purposes in terms of congruence:( D20):
Add p x, y, z, w, v, l q Ø df S p x, y q ^ S p z, w q ^ D v D v D v p Bet p v , v , v q ^” p x, y, v , v q^ ” p z, w, v , v q^ ” p v, l, v , v qq ( D21):
Dupl p x, y, z, w q Ø df Add p x, y, x, y, v, l q ( D22):
Square p x, y, z, w q Ø df D v D v D v p Bet p v , v , v q ^ Orth p v , v , v q ^” p x, y, v , v q^ ” p x, y, v , v q ^ Dupl p z, w, v , v qq ( D23):
Sqrt p x, y, z, w q Ø df Square p z, w, x, y q The next definition formalizes of that of Hartshorne [2000, p. 170] :( D24):
P roduct p o, e, x, y, z, w, l, v q Ø df D y D w D v ( Orth p e, o, y q ^ Orth p w , o, v q^ ” p o, w , z, w q ^ ” p e, y , x, y q ^ ” p l, v, w , v q An alternative approch, purely in terms of betweenness, can be found in the treatiseof [Schwabh¨auser, Szmielew and Tarski 1983, p. 160] (SUM1) . It isincluded only to avoid cluttering the axiom and to improve readability.(
D25):
Remterm p o, e, x, y, w, v q Ø df D v D v D v D v pp Square p w, y, x, v q ^ P rod p o, e, x, w, w, y, x, v q ^ Dup p x, v , x, v q ^ Add p x, v , x, v , x, v q ^ Sqrt p x, v , x, v qq To extend the calculus of segments to timelike vectors the simplestapproach is to move back and forth using opposites. For instance, thesum and product of two points on a timelike line is the opposite of thesum and product of two opposite segments on a spacelike line.(
D26):
Add p x, y, z, w, v, l q Ø df D x D y D z D w D v D l p Opp p x , y , x, y q ^ Opp p z , w , z, w q^ Opp p v , l , v, w q ^ Add p x , y , z , w , v , l qq ( D27):
P rod p o, e, x, y, z, w, v, l q Ø df D x D y D z D w D v D l p Opp p x , y , x, y q^ Opp p z , w , z, w q^ Opp p v , l , v, l q ^ P rod p o, e, x, y, z, w, v, l qq ( D28):
Remterm p o, e, x, y, w, v q Ø df D x D y D w D v p Opp p x , y , x, y q ^ Opp p x , v , x, v q^ Remterm p o, e, x, y, w, v qq To define a product operation on a timelike and a spacelike segment weproceed in a similar fashion using the notion of opposites.(
D29):
P rod p o, e, x, y, z, w, v, l q Ø df T p x, y q ^ S p z, w q ^ D x D y D v D l p Opp p x , y , x, y q^ P rod p o, e, x , y , z, w, v , l q ^ Opp p v, l, v , l qq The system can be divided into six groups of axioms. The first axioms governthe notion of betweenness on a line. This part consists of the axioms for afour dimensional affine space, as formalised in [Tarski and Sczerba, 1979] orin [Schwabh¨auser, Szmielew and Tarski, 1983, p. 415-416]. We omit figuresfor them. We call the second part dimensionality axioms: they assert theexistence of a basis for every choice of an origin; that every point can bereached or ‘generated’ through alternative paths; and finally, that o, x, y,and z form a basis for a Euclidean subspace. Three segments in this basisare spacelike; a fourth is timelike. The set of points that is spanned by theorthogonal spacelike ones must always form a three dimensional Euclideanspace. This requirement is ensured by postulating that these points obey the13xioms of Euclidean geometry in [Tarski and Givant, 1999]. A third groupof axioms constrains the length of arbitrary segments in terms of the lengthsof the components. The fourth group consists of construction axioms: theypostulate the existence of segments on a given line that match any other line- either in the sense that they are congruent or opposites. We then have afifth group of axioms concerning formal properties of the relations employed.We conclude with the axiom schema of continuity and an axiom for density.
Definition 1. x T arski y abbreviates the conjunction of Tarski’s axioms forthree dimensional Euclidean geometry in [Tarski and Givant, 1999] with theexception of (a) the axiom schema of continuity, (b) the axioms of affinegeometry, (c) the Five-Segment Axiom [Ax. 5] and (d) [Ax.23] and [Ax.24] . Definition 2.
We define the restriction x φ E p o,x,y,z q y of φ to the space generatedby o, x, y and z by induction on the complexity/construction of formulae:(1) if φ is atomic, then φ E p o,x,y,z q is φ .(2) x φ y E p o,x,y,z q is x φ E p o,x,y,z q y (3) x φ ^ ψ y E p o,x,y,z q is x φ E p o,x,y,z q y ^ x ψ E p o,x,y,z q y (4) x D v φ y E p o,x,y,z q is D v p Gen D p o, x, y, z, v q ^ x φ E p o,x,y,z q y q We write $ φ to mean that the universal closure of φ is an axiom. Axioms for affine space ( AFF0): $ Bet p x, y, x q Ñ x “ y ( AFF1): $ Bet p x, y, z q ^ Bet p y, z, u q ^ y ‰ z Ñ Bet p x, y, u q ( AFF2): $ Bet p x, y, z q ^ Bet p x, y, u q ^ x ‰ y Ñ Bet p y, z, u q _ Bet p y, u, z q ( AFF3): $ D x p Bet p x, y, z q ^ x ‰ y q An alternative approach would be an axiom that says that the set of points which havea fixed positive distance to an origin satisfy the axioms of hyperbolic geometry, i.e. theaxioms of Euclidean geometry where Euclid axiom is replaced by its negation (for detailsof the construction see the last chapter of [Malament, unpublished]). AFF4): $ Bet p x, t, u q ^ Bet p y, u, z q Ñ p Bet p x, v, y q ^ Bet p z, t, v qq ( AFF5): $ Bet p x, u, t q^ Bet p y, u, z q^ x ‰ u Ñ D v D w p Bet p x, y, v q^ Bet p x, z, w q^ Bet p v, t, w qq Dimension axioms
The following axioms ensure that the entire space is a four dimensional vectorspace. The first axiom says that, for any choice of a point o as the origin,there are other four points such that they form a basis of the space.( A0): $ @ o D x D y D z D t Basis p o, x, y, z, t q Any orthogonal segments ~ox and ~ot can be supplemented to a basis:(
A1): $ T p o, t q ^ Orth p o, x, t q Ñ D y D z Basis p o, x, y, z, t q The next axiom asserts that every basis generates every point.(
A2): $ Basis p o, x, y, z, t q Ñ Gen D p o, x, y, z, t, v q The space spanned by the spatial subbasis obeys the axioms of Tarski.It is a three dimensional Euclidean space. The schema of continuityand the axioms for betweenness are assumed later for all lines.(
A3): $ Basis p o, x, y, z, t q Ñ T arski E p o,x,y,z q Typical axioms of Eucliden geometry postulate the congruence of certaintriangles under hypotheses about the congruence of certain angles andcertain sides. In the system described in [Tarski and Givant, 1999] thesecriteria of congruence are derived from a single Five-Segment Axiom[Ax. 5]. It is convenient to adapt it to our system by assuming thatthe two triangles to be compared can come from different spacelikehyperplanes. The abbreviation below is self-explanatory.(
A4): $ p
Basis p o, x, y, z, t q ^ Basis p o , x , y , z , t q ^ Gen p o, x, y, z, t, a, b, c, d q ^ Gen p o , x , y , z , t , a , b , c , d qq Ñ Five-Segment Axiom p a, b, c, d, a , b , c , d q This axiom asserts that alternative paths to the same point consist ofcongruent segments. This implies that the lengths of the componentsof a segment depend only on the basis and not on the development.15
A5): $ Reached p o, x, y, v, w q Ñ D v p Reached p o, y, x, v , w q ^ ” p o, v , v, w q ^” p o, v, v , w qq o xy vv w Figure 9: Axiom (A5)Linear algebra requires that the sum of two vectors be unique. Thenext axiom imposes that a vector have a unique decomposition. ( A6): $ p
Reached p o, x, t, v, w q ^ Reached p o, x, t, v , w q ^p Coll p o, x, w q _ Intersect p v, v , o, x qq Ñ v “ v o xt v “ v ww Figure 10: Axiom (A6)To extend (A5) and (A6) to the uniquess of sums of more than twovectors, that is of developments in three or four steps, we need (A7):(
A7): $ p
Orth p o, x, z q ^ Orth p o, y, z q ^ D vReached p o, x, y, v, w qq Ñ Orth p o, z, w q The analogy between
Reached p o, x, t, v, w q and the operation of vector sum isimperfect because it does not distinguish bewteen ~a ` ~b and ~a ´ ~b . y xz w v Figure 11: Axiom (A7)Minkowski spacetime cannot be accurately described unless we relatethe ordering on a generic line with our foliations into a timelike line anda spacelike hyperplane. Betwenness on a lines corresponds to another basicnotion of linear algebra: scalar multiplication. Two points are on the sameline if and only if the components of one are scaled with respect to thecomponents of the other by the same factor λ . Axiom (A8) reads:( A8): $ Reached p o, x, t, x, r q ^ Reached p o, t, x, t, r q ^ Reached p o, x, t, x , r q^ Reached p o, t, x, t , r q ^ T p o, t q ^ o ‰ e Ñ r
Bet p o, r, r q Ø p Bet p o, x, x q^ Bet p o, t, t q ^ D z D z p S p z, z q ^ P roduct p o, e, x, z, z , o, x q ^ P roduct p o, e, t, z, z , o, t qqq ] o x x t t r r Figure 12: Axiom (A8)The last axiom of the present section postulates that to two orthogonalvectors can indeed always be associated a sum.(
A9): $ Orth p o, x, t q Ñ D w Reached p o, x, t, x, w q o xt w Figure 13: Axiom (A9)17 emark.
The relative ugliness of the axioms in this section can be remediedsomewhat by introducing the notation of linear algebra. This may improvetheir readability as well. For example, axioms (A7) and (A8) assert theexistence and uniqueness of the sum ~ox ` ~oy of the orthogonal vectors ~ox and ~oy . Axiom (A4) asserts the familiar axiom of a vector space: ~ox ` ~oy “ ~oy ` ~ox (commutativity of addition). The axiom (A7) is a basic consequence of thedistributivity of the Lorentzian product: the statement : x p ~oz ‚ ~ox “ q ^p ~oz ‚ ~oy “ q Ñ p ~oz ‚ p ~ox ` ~oy q “ q y . (A8) concerns scalar multiplication. Summation axioms
We have postulated axioms that assert the existence of bases and permit adecomposition of arbitrary segments into orthogonal components. We nowneed axioms for the metrical structure. We want a segment extending aspacelike segment to be spacelike and shorter and a segment extending atimelike segment to be timelike and longer. We want, moreover, to be ableto compute the length of a segment from that of its components.Two segments that are both opposite to a third are congruent.(
SUM0): $ Opp p x, t, z, w q ^ Opp p x, t, z , w q Ñ ” p z, w, z , w q xt wzz w Figure 14: Axiom (SUM0)The following two axioms employ the arithmetic of segments that wehave defined in section to calculate the length of a segment fromits decomposition onto a given basis. Every vector ~xv can be construedas the sum of a spacelike component and a timelike component. Wetreat separately the case in which (1) the spacelike segment is longer inabsolute value (SUM1) and that in which (2) the timelike segment islonger in absolute value (SUM2) . The basis ~xy is longer in absolutevalue than the timelike component ~xz if and only if there is a point18etween x and y that is of opposite length to ~xz . If the spacelike sideis longer in absolute value, then the hypothenuse ~yz of the right trianglexyz is spacelike. If the timelike side is longer in absolute value, then thehypothenuse ~yz is timelike. To quantify more precisely the length of thehypothenuse ~yz in all cases we need a calculation. Suppose the spacelikesegment xy is orthogonal to a timelike segment xz (see figure 11). Callthe length of xz A and the length of xy B . Suppose a segment xw isopposite to xz , whose length we call D . Call E the length of wy . Then,the length C of the resultant vector xv is conguent to the hypotenuse xz . By Pythagoras’s theorem, the length of the hypothenuse is: (*) C “ A ` B “ A `p D ` E q “ ✚✚ A ` ✚✚ D ` E ` DE “ E ` DE .( SUM1): $ p S p o, v q ^ S p o, x q ^ T p o, t q ^ Reached p o, x, t, x, v q ^ o ‰ e ÑD w D v p Bet p o, w, x q^ Opp p o, w, o, t q^ Remterm p o, e, o, x, w, v q^ ” p o, v, o, v qq o xt vw Figure 15: Axiom (SUM1)A similar calculation can be made when ~xv is timelike.(
SUM2): $ p T p o, v q ^ T p o, t q ^ S p o, x q ^ Reached p o, x, t, x, v q ^ o ‰ e ) Ñ D w D v p Bet p o, w, t q^ Opp p o, w, o, x q^ Remterm p o, e, o, t, w, v q^ ” p o, v, o, v qq The proportion between xv and wy that results from the calculation is exactly what isexpressed by the predicate Remterm p x, y, w, v q introduced without explanation in (D28) t xvw Figure 16: Axiom (SUM2)The sum of segments of opposite length gives lightlike vectors:(
SUM3): $ Opp p o, x, o, t q ^ Orth p o, x, t q ^ P ar p o, t, x, v qq Ñ L p o, v q o xt v Figure 17: Axiom (SUM3)The following two axioms assure us that continuing on a spacelike linewe traverse progressively shorter segments, as we move towards infinity.(
SUM4): $ Bet p x, y, z q ^ S p x, z q Ñ ă ” p x, z, x, y q x y z Figure 18: xz is shorter than xy (identical figure for (SUM5)(
SUM5): $ Bet p x, y, z q ^ S p x, y q Ñ ă ” p x, z, x, y q We can obtain a similar result for timelike segments. Continuing on atimelike line, we traverse longer and longer segments. We can derivethis result from principles relating opposites. Let us remind ourselvesthat orthogonal opposites cancel i.e., they give a lightlike segment whensummed. We postulate (SUM6) that the opposite of a longer timelikesegment must be shorter - more in the negative - and vice versa .20
SUM6): $ă ” p x, x , z, w q ^ Opp p x, x , t, t q ^ Opp p z, w, z , w q Ñ ă ” p z , w , t, t q x x wztt z w Figure 19: If xy is shorter than zw , then the opposite of zw is shorter thanthe opposite of xy .(SUM 6) tells us little about the arrangement of opposite segments ona spacelike and a timelike line. Using our primitive of betweenness,we need to postulate (SUM 7) and (SUM 8) that the ordering of theopposites on a segment mirrors that of the original segment:( SUM7): $ p
Opp p x, y, x , y q^ Bet p x, y, z qq Ñ D z p Bet p x , y , z q^ Opp p x , z , x, z qq x y zx y z Figure 20: SUM7 (same figure for SUM8)(
SUM8): $ p
Opp p x, z, x , z q^ Bet p x, y, z qq Ñ D y p Bet p x , y , z q^ Opp p x , y , x, y qq The following axiom tells us that summing a null or lightlike line doesnot change the length: it gives back a congruent segment.21
SUM9): $ L p x, y q ^ Orth p x, y, z q ^ P ar p x, z, y, v q ^ P ar p x, y, z, v q Ñ” p x, z, x, v q x y vz Figure 21: Axiom (SUM9)
Segment construction axioms
We now want axioms that guarantee the existence of segments of a givenlength. They are adapted from the Euclidean context. Given two spacelikesegments, we can find a third on the second line congruent to the first.(
CONST0): $ S p x, y q ^ S p z, w q Ñ D v p Coll p v, z, w q^ ” p z, v, x, y qq x yz wz v Figure 22: Space-like segments construction.The following two axioms guarantee that, given a spacelike and atimelike segment, we can find a third segment on the line determinedby the second that is of opposite length to the first, and vice versa .( CONST1): $ T p x, t q ^ S p z, w q Ñ D v p Coll p v, z, w q ^ Opp p z, v, x, t qq t z w v Figure 23: Construction of opposite segments 1(
CONST2): $ T p x, t q ^ S p z, w q Ñ D v p Coll p v, x, t q ^ Opp p x, v, z, w qq xt z wv Figure 24: Construction of opposite segments 2The last axiom of this section postulates that a timelike line is infinitein both directions. Time has no beginning and no end.(
CONST3): $ T p x, t q Ñ D w p Bet p w, x, t q ^ ” p w, x, x, t qq xtw Figure 25: Axiom (CONST3)23 ormal properties
The relation of orthogonality is symmetric in the second and third term.(
F0): $ Orth p x, y, z q Ñ Orth p x, z, y q The following axioms guarantee that the relation of congruence is anequivalence relation and that the relation of being shorter than inducesa linear order on the equivalence classes of congruent segments.(
F1): $ă ” p x, y, z, w q ^ ” p x, y, x , y q Ñ ă ” p x , y , z, w q ( F2): $ă ” p x, y, z, w q ^ ” p z, w, z , w q Ñ ă ” p x, y, z , w q ( F3): $ ă ” p x, y, x, y q Degenerate segments are congruent:(
F4): $ ă ” p x, x, y, y q These standard axioms describe the relative length between two segmentsthat are the sum of respectively (a) congruent segments, (b) smallersegments or (c) some combination of the two. We can derive, forexample, that (a) sums of congruent segments are congruent.(
F5): $ p
Bet p x, y, z q ^ Bet p x , y , z q ^ ă ” p x, y, x , y q ^ă ” p y , z , y, z qq Ñ ă ” p x , z , x, z q ( F6): $ p
Bet p x, y, z q ^ Bet p x , y , z q ^ ă ” p x , z , x, z q ^ ă ” p x , y , x, y qqÑ ă ” p y , z , y, z q Continuity and density
The first axiom of continuity states that a line ℓ divides every plane in whichit lies in two half-planes: the points whose connecting segments intersect ℓ and the points such that their connecting segment does not.(INT): $ p Copl p x, y, z, w q^ Copl p t, x, z, w q^ Intersect p x, y, z, w q^ Intersect p y, t, z, w qq Ñ Intersect p x, t, z, w q wx yt Figure 26: Axiom (INT)These axioms are imported from [Tarski and Givant, 1999]. The continuityschema constrains the ordering of the points on a line to be as Dedekindcomplete as possible, without quantifying over sets of points. Density is theusual fact that between every two distinct points there is a third.(ASC): $ D x @ y @ z p φ ^ ψ Ñ Bet p x, y, z qq Ñ D x @ y @ z p φ ^ ψ Ñ Bet p x , y, z qq where φ and ψ are formulae of L , the first of which does not containany free occurrences of x , x , z , the second of which does not containany free occurrences of x , x , y .(DENS): $ x ‰ z Ñ D y p y ‰ x ^ y ‰ z ^ Bet p x, y, z qq x y z Figure 27: Axiom (DENS)This completes the presentation of our system for Minkowski spacetime,which will be denoted by M . Its adequacy can now be briefly investigated. A Second-order Continuity Axiom
Minkowski spacetime is the ‘intended’ model of the system M . It is thephysical spacetime that is postulated by the theory of Special Relativity(SR). It can be singled out, up to isomorphism, as an uncountable model M of M such that the lines ℓ in M are true continua. In a line ℓ in M everybounded set of points has a least upper bound. An equivalent method is tolook at models of the following second order continuity axiom (ASC):(CONT): @ X @ Y D x @ y @ z p X p x q ^ Y p y q Ñ Bet p x, y, z qq Ñ D w @ y @ z p X p x q ^ Y p y q Ñ Bet p w, y, z qq M by replacing all instances of (ASC) with(CONT) will be denoted by M and we will now consider its models. In a physics textbook, ‘Minkowski Spacetime’ refers to a certain mathematicalstructure x R , η ab y . It is assumed that calculations performed on x R , η ab y reflect certain physical state of affairs in the physical manifold of pointsin which physical objects are located, and on which physical fields assumevalues. Let us call Minkowski spacetime a spacetime obeying the axioms of M and Minkowski spacetime the following mathematical structure: Definition 3.
The tensor η ab is the covariant tensor on R such that, for all p, q ǫ R , η ab p p, q q “ a p t p ´ t q q ´ p x p ´ x q q ´ p y p ´ y q q ´ p z p ´ z q q When p, q ǫ R , we have denoted by x p the first number in the quadruple p , by y q the second number in the quadruple q and so on. Definition 4.
The function distance is the tensor on R such that, for all p, q ǫ R , distance ( p, q ) “ a p x p ´ x q q ` p y p ´ y q q ` p z p ´ z q q ` p t p ´ t q q Definition 5.
Minkowski spacetime is x R , η ab , distance y The sense in which x R , η ab , distance y can be used used to ‘represent’Minkowski spacetime , and the role of frames of reference, is clarified byproving a Representation Theorem . A frame or coordinatization is a bijection f : Minkowski spacetime Ñ Minkowski spacetime such that spacetimeevents satisfy intrinsic geometry relations if and only if their images satisfycorresponding algebraic relationships. The passive symmetries of the theory M emerge as the transformations f : x R , η ab y Ñ x R , η ab y that can becomposed with an arbitrary coordinatization to leave a coordinatization. Tarski and his students have constructed a simple system of axioms E forEuclidean geometry in three dimensions. We have already mentionned it andwe have exploited it in the formulation of our axioms. A model F of E isrepresented by the mathematical structure x R , distance s y : Definition 6.
The function distance s is the function on R such that, for all p, q ǫ R , distance s ( p, q ) “ a p x p ´ x q q ` p y p ´ y q q ` p z p ´ z q q
26n analogue system of axioms E for one dimensional temporal geometrycan be constructed. A model F of E is mirrored by the real line, that is bythe mathematical structure x R , distance t y , where: Definition 7.
The function distance t is the tensor on R such that, for all p, q ǫ R , distance t ( p, q ) “ a p t p ´ t q q “ | t p ´ t q | The proof of our
Representation Theorem , relating the system M tothe structure x R , η ab , distance y , will follow from two theorems of [Tarski,1959] and a theorem of [Suppes, 1959]. The first theorem of [Tarski, 1959]is simply the appropriate Representation Theorem for his own axiom system E for Eucliden geometry and for the Cartesian space x R , distance s y . Theorem 1. (Tarski 1959). M is a model of E if and only if there is abijection f : U( M q Ñ R such that, for all choices of a, b, c, d ǫ U( M ):(1) M |ù Bet p x, y, z q r a, b, c ] if and only if distance s ( f p a q , f p c q ) =distance s ( f p a q , f p b q ) ` distance s ( f p b q , f p c q )(2) M |ù x ” p x, y, z, w q y r a, b, c, d ] if and only if distance s ( f p a q , f p b q ) =distance s ( f p c q , f p d q )and, for every two functions f and f that satisfy (1)-(2), there exists anisometry I: R Ñ R and a function U: R Ñ R that multiplies each entryby a fixed constant such that f = U ˝ I ˝ f .A similar theorem can be proven for an appropriate system of one dimensionalgeometry [cf. Tarski and Givant 1999, pp. 204-209]. Theorem 2. (Tarski 1959). M is a model of E if and only if there is abijection f : U( M q Ñ R such that, for all choices of a, b, c, d ǫ U( M ):(1) M |ù x ” p x, y, z, w q y r a, b, c, d ] if and only if distance t ( f p a q , f p b q ) =distance t ( f p c q , f p d q )and, for every two functions f and f that satisfy (1)-(2), there exists atranslation b : R Ñ R and a function k : R Ñ R that consists of multiplyingall components by a constant, such that f = k ˝ b ˝ f .The theorem of [Suppes, 1959] is a basic result characterizing the relationbetween the relativistic intervals and the Poincar´e transformations on x R , η ab y : Suppes [1959] proves in fact a stronger results. He only assumes that f and f agreeon lightlike and timelike connected points. Note that Suppes [1959] refers to the Poincar´etransformations - the composition b ˝ L of a translation b and a linear transformation L corresponding to a Lorentz matrix - as the ‘Lorentz transformations’. heorem 3. (Suppes 1959). For any two bijective functions f : M Ñ R and f : M Ñ R from the same uncountable set M into R such that, for all p, q ǫ M , η ab p f p p q , f p q qq = η ab p f p p q , f p q qq .i.e, they agree on the relativisticinterval, there exists a Poincar´e transformation L such that f = L ˝ f . The following theorem is the main result of this paper:
Theorem 4. ( Representation theorem ). M is a model of M if and onlyif there is a bijection f : U( M q Ñ R such that, for all a, b, c, d ǫ U( M ):(1) M |ù Bet p x, y, z q r a, b, c ] if and only if distance ( f p a q , f p c q ) =distance ( f p a q , f p b q ) ` distance ( f p b q , f p c q )(2) M |ù x ă ” p x, y, z, w q y r a, b, c, d ] if and only if η ab p f p a q , f p b q ) ď η ab p f p c q , f p d q )and, for every two functions f and f that satisfy (1)-(2), there existsa Poincar´e transformation L: R Ñ R , and a function U: R Ñ R thatmultiplies each coordinate by a positive constant, such that f = L ˝ U ˝ f . The main idea behind the proof of the existence part is to start from abasis with a given time axis L and and a spacelike hyperplane E , and thenextend coordinatizations f and g of E and of L , given by Theorem 1. and
Theorem 2. , to a coordinatization f of the entirety of M . The specificationof how to extend f and g can be done in a uniform way. In all the definitionsthat follow, let M be a model of M and let o , x , y , z , and t determine abasis in the model M . Let E be the associated spacelike hyperplane in M and L be the timelike line through o and t . We will use subscripts to denotecomponents. For example, if f( p )= x , , y , then f ( p )= 15 and f ( p )= 2. Definition 8.
Let us assume that f : E Ñ R satisfies condition (1)-(2) in Theorem 1 when restricted to E and that g : L Ñ R satisfies (1)-(2) in Theorem 2 when restricted to L. Assume η ab p g p o q , g p t q ) = - η ab p f p o q , f p x q ).A function f : U( M q Ñ R is determined by f and g if and only if:1. f ( p )= x f ( p ), f ( p ), f ( p ), 0 y if p ǫ E Field [1980, p. 50/f] has noticed the addition of U to the group of symmetries is dueto the conventionality of the choice of measuring units. It marks the difference betweene.g., measuring the relativistic interval in second, minutes or hours. E is the set of elements of the domain U p M q that are generated by o, x, y, z in M and L is the set of elements of the domain U p M q that collinear to o, t in M . f ( p )= x
0, 0, 0, g ( p ) y if p ǫ L f ( p )= x f ( q ), f ( q ), f ( q ), g ( t ) y if ~op is the sum of ~ot ǫ L and ~oq ǫ E . Remark on notation : We follow the conventions of Shoenfield [1971]for models of set theory and use superscripts to form predicates for thesatisfaction of object language predicates in a model. For example, we have
Orth M p b, a, c q if and only if a , b , c ǫ U( M ) and a , b , c are orthogonal in M .The first preliminary lemma tells us that in M a quadrilateral with tworight angles at the base and the other two sides parallel and congruent is arectangle: opposite sides are congruent and all angles are right. a bc d “ ed Figure 28: Lemma 1
Lemma 1.
For any a , b , c , d ǫ U( M ), if Orth M p b, a, d q , Orth M p a, b, c q , ” M p a, c, b, d q and P ar M p a, c, b, d q , then ” M p a, b, c, d q and Orth M p d, b, c q . Proof.
Orth M p a, b, c q and P ar M p a, c, b, d q by hypothesis. It follows by definitionthat Reached M p b, a, d, a, c q . By axiom (A5), there exists an e in U( M ) suchthat Reached M p b, d, a, e, c q and we have the two congruences ” M p e, c, b, a q and ” M p b, e, a, c q . The hypothesis ” M p a, b, c, d q and the transitivity ofcongruence (F1)-(F3) imply that ” M p b, e, b, d q . By definition again, thefact that Reached M p b, d, a, e, c q implies that Coll M p b, d, e q and Orth M p e, b, c q .Axioms (SUM4) to (SUM5) and the fact that ” M p b, e, b, d q , reduce now thechoice to either e “ d or e “ d , where d is the reflection of d over the line ℓ through a and b . But Axiom (INT) excludes that e “ d . So e “ d , andtherefore we have ” M p a, b, c, d q and Orth M p d, b, c q . If Reached M p o, q, t, q, p q and the segment ~tq does not intersect the hyperplane E .2.1 Lemmata on Opposites The coordinatizations f and g are worth combining together only if η ab p g p o q , g p t q )= - η ab p f p o q , f p x q ). This obviously implies that η ab p f p o q , f p t q ) = - η ab p f p o q , f p x q ).In general, two segments ~ot ( t ǫ L ) and ~ow ( w ǫ E ) are of opposite lengthin M if and only if η ab p f p o q , f p t q ) = - η ab p f p o q , f p w q ). Lemma 2.
For any t , t ǫ L and x , x ǫ E , if Bet M p o, t, t q and Bet M p o, x, x q , η ab p f p o q , f p t q ) = - η ab p f p o q , f p x qq and η ab p f p t q , f p t q ) = - η ab p f p x q , f p x qq ,then we have that η ab p f p o q , f p t q ) = - η ab p f p o q , f p x qq . Proof. η ab p f p o q , f p t qq “ | g p t q ´ g p o q| (By definition 8 and t ǫ L ) “ |p g p t q ´ g p t qq ` p g p t q ´ g p o qq|“ | g p t q ´ g p t q| ` | g p t q ´ g p o q| (By Bet M p o, t, t q and condition 1 of Theorem 2) “ η ab p f p t q , f p t qq ` η ab p f p o q , f p t qq (By definition 8) “ iη ab p f p o q , f p x qq ` iη ab p f p x q , f p x qq (By hypothesis)An analogous argument shows that: η ab p f p o q , f p x qq “ η ab p f p o q , f p x qq ` η ab p f p x q , f p x qq . Let us now prove the existence of a rectangle with two given sides.
Lemma 3.
For all o , t , x in U( M ), if Orth M p o, t, x q then there exists a w inU( M ) such that Reached M p o, x, t, x, w q and Reached M p o, t, x, t, w q . Proof.
By axiom (A9) there exists a w in U( M ) such that Reached M p o, x, t, x, w q .Axioms (CONST3), together with the definitions of orthogonality and parallelism,implies that also the reflection w over the line ℓ through ~ox is such that Reached M p o, x, t, x, w q . (SUM6)(SUM7)(SUM8) imply that there are noothers. Similarly t and t are the only points t in U( M ) such that Coll M p o, t, t )and ” M p o, t, o, t q . Either Reached M p o, t, x, t, w q or Reached M p o, t, x, t, w q .In the second case, we get that Reached M p o, t, x, t, w q by (INT) and (A6).The sums of segments opposite length are of opposite length. Lemma 4.
For every t , t , x , x , if Bet M p o, x, x q , Bet M p o, t, t q , Opp M p o, x, o, t q and Opp M p t, t , x, x q , then we have that Opp M p o, x , o, t q .30 roof. Lemma 3 gives us r and r in U( M ) such that Reached M p o, x, t, x, r q and Reached M p o, x , t , x , r q and also the alternative developments: that is Reached M p o, t, x, t, r q and Reached M p o, t , x , t , r q . Let v and v be such thatsimilarly Reached M p o, x, t , x, v q , and Reached M p o, x, t, x , v q , and analogouspermutations. Various applications of Lemma 1 to all the different rectanglesin Fig.28 entail that Reached M p r, v, v , v , r q and the two congruences: ” M p r, v, x, x q and ” M p r, v , t, t q . Axiom (SUM 3) implies that L M p r, r q . Bycontinuity, for all choices of unit e , there must be some segment ~ww such thatProduct( o, e, o, x, w, w , o, x q . The definition of P roduct p o, e, o, t, w, w , o, t q and the hypotheses Opp M p o, x, o, t q and Opp M p t, t , x, x q imply that all theconditions in axiom (A9) are satisfied. This means that Bet M p o, r, r q . Bythe degenerate cases of axioms (F4) and (F5), it follows that L M p o, r q . Bydefinition of Opp , we obtain immediately that
Opp M p o, t , o, x q . o x x t t r r v v Figure 29: Lemma 4
Lemma 5.
For every t , t , x , x ǫ U p M q , if Bet M p o, x, x q , Bet M p o, t, t q , Opp M p o, x, o, t q and Opp M p o, x , o, t q , then Opp M p t, t , x, x q . Proof.
The proof is similar to that of Lemma 4.
Lemma 6.
For any o , t ǫ L and w ǫ E , Opp M p o, t , o, w q if and only if η ab p f p o q , f p t q ) = - η ab p f p o q , f p w q ). Proof.
The hypothesis is that η ab p g p o q , g p t q ) = - η ab p f p o q , f p x q ). Lemma 4and Lemma 2 imply by induction that the statement holds for all integermultiples of the above segments n ¨ ~ox and n ¨ ~ot . Lemma 5 and Lemma 2imply the result for integer submultiples n ¨ ~ox and n ¨ ~ot . By continuity,for all reals k , we have that k ¨ ~ox and k ¨ ~ot are opposites in M and the If Opp M p o, k ¨ t, o, p´ k q¨ x , it will follow from (SUM 1) and (SUM 2) and the continuityof lines that there is some real r such that Opp M p o, k ¨ t, o, ´ r ¨ x q or Opp M p o, r ¨ t, o, ´ k ¨ x q .It suffices, then, to pick a rational pq such that r ă pq ă r and notice that correspondingmultiples of the segments are of opposite length and in-between two opposite irrationalsegments. This contradicts basic consequences of axioms (SUM0) and (SUM6)(SUM8)about the ordering of opposites. η ab p f p o q , f p k ¨ t q ) = - η ab p f p o q , f p k ¨ x q ). Axiom (SUM0) statesthat segments of opposites length to a given segment are congruent. Axiom(SUM6)(SUM7)(SUM8) imply that congruent segments on the lines ℓ o,x and ℓ o,t are of the form k ¨ ~ox and p´ k q ¨ ~ox , or of the form k ¨ ~ot and p´ k q ¨ ~ot .Euclidean geometry (A3) and Theorem 1 imply that every segment in E iscongruent to and of same interval (relative to f ) as a segment in ℓ o,x . Streckenrechnung
The lemmata in this section consist merely in a verification of the adequacyof the ‘calculus of segments’ of Hilbert [1899].
Lemma 7. If φ is a formula of the calculus of segments (D20)-(D29), for any o, e, v , ..., v n , v , ..., v n P U( M ) such that (1) for all i ă n , ” M p v i , v i ` , v i , v i ` q and (2) φ M p o, e, v , ..., v n q and (3) φ M p o, e, v , ..., v n q , ” M p v n ´ , v n , v n ´ , v n q . Lemma 8.
For all o , e, x, y, z, w, v, l P E , if η ab p f p o q , f p e qq “
1, then
P roduct M p o , e, o, x, w, y, v, z q iff η ab p f p v q , f p z qq “ η ab p f p o q , f p x qq η ab p f p w q , f p y qq .Let us fix two points o and e ǫ E such that η ab p f p o q , f p e qq “ Lemma 9.
For all x, y, z, w, v P E , Remterm M p o , e, o, x, w, y, v, z q if andonly if η ab p f p v q , f p z qq “ η ab p f p o q , f p x qq ` η ab p f p o q , f p x qq η ab p f p w q , f p y qq . Lemma 10.
For all x, y, z, w, v P L , Remterm M p o , e, o, x, w, y, v, z q if andonly if η ab p f p v q , f p z qq “ η ab p f p o q , f p x qq ` η ab p f p o q , f p x qq η ab p f p w q , f p y qq . Proof.
These results can be derived from the theory of proportions in anEuclidean space [Hartshorne 2000, Schwabh¨auser, Szmielew and Tarski 1983]and details are omitted. Note that our formulation of (A4) allows us to applythe usual congruence criteria for triangles across different hyperplanes.
A basic property of the model M is that pairs of segments that decomposeinto congruent components on an orthogonal basis are congruent. Lemma 11.
For all o, x, t, v, o , x , t , v in U( M ), if Reached M p o, x, t, x, v q , Reached M p o , x , t , x , v q , ” M p o, x, o , x q , ” M p o, t, o , t q , then ” M p o, v, o , v q . Proof.
This is proven by cases. If L M p o, v q , then by definition Opp M p o, x, o, t q .By axiom (SUM6), it follows that Opp M p o , x , o , t q . By axiom (SUM3), wehave that L M p o , v q . By axiom (F1) to (F4), every two lightlike segments arecongruent. If S M p o, v q or T M p o, v q , the result follows from axioms (SUM1)and (SUM2) and Lemma 6 on the Streckenrechnung.32ur formal verification that the function f in Definition 8 satisfies conditions (1) and (2) of Theorem 4 requires that we be able to restrict ourselves tothe case of segments ~ox and ~oy stemming from the same origin o . The nextlemma shows that to an arbitrary segment ~pq we can associate a congruentvector ~or at the origin such that f assigns to them the same interval. Lemma 12.
For all p , q in U( M ), there is an r ǫ U p M q such that ” M p p, q, o, r q and η ab p f p p q , f p q qq “ η ab p f p o q , f p r qq . Proof.
Let t, x, t , x be the points such that Reached M p o, x, t, x, p q and Reached M p o, x , t , x , q q ,as guaranteed by Lemma 4. Let t and w be the points: t “ : f p f p t q ´ f p t qq x “ : f p f p x q ´ f p x qq Let r be such that Reached M p o, x , t , x , r q and Reached M p o, t , x , t , r q .The identity η ab p f p p q , f p q qq “ η ab p f p o q , f p r qq is obvious.Theorem 1 gives us ” M p x, x , o, x q . Theorem 2 implies that ” M p t, t , o, t q ).Let v be the point such that Reached M p o, x , t, x , v q and and v be thepoint such that reached Reached M p o, x, t , x, v q . Applications of Lemma1 to the different rectangles in Fig. 30 establish that ” M p p, v, x, x q andthat ” M p p, v , t, t q . We also get that Orth M p p, v , v q . The transitivity ofcongruence imply that ” M p p, v, o, x and ” M p p, v , o, t q ). By definition Reached M p p, v, v , v, q q . By the preceding lemma, we obtain that ” M p o, r, p, q q . o xx x tt t p qr v v Figure 30: Lemma 12
This concludes the preliminary results needed to prove the existence of acoordinatization. The proof that two coordinatizations f and f are equivalent33p to a rescaling and a Poincar´e transformation will follow from Theorem 3. ,if we manage to show that there exists a rescaling U such that f and f ˝ U agree on the relativistic interval between any two points. We prove first thatthey agree on a basis. We then show in a sequence of steps that, if f and U ˝ f agree on a basis, then they must agree on the whole of M . Lemma 13.
For all p , q , r ǫ R , if distance ( p, r ) = distance ( p, q ) ` distance ( q, r ), then η ab p p, r q “ η ab p p, q q ` η ab p q, r q . Proof.
See [Suppes, 1959, p. 294].
Lemma 14. If f : U( M q Ñ R satisfies conditions (1)-(2) in Theorem 4 and
Opp M p o, t , o, x q , then η ab p f p o q , f p t q ) = - η ab p f p o q , f p x q ). Proof. f and f satisfy the conditions of Lemma 6. Lemma 15. If f : U( M q Ñ R and f : U( M q Ñ R are bijections satisfyingconditions (1)-(2) in Theorem 4 and f and f agree on the relativisticinterval between two points p and q i.e., η ab p f p p q , f p q q ) = η ab p f p p q , f p q qq ,then they agree on all the points that are collinear to p and q . Proof.
Lemma 13 and condition (1) of Theorem 4 imply that η ab p f p p q , f p n ¨ q q )= n ¨ η ab p f p p q , f p q q ) and η ab p f p p q , f p n ¨ q q ) = n ¨ η ab p f p p q , f p q qq for integermultiples of the segment ~pq . The same holds for submultiples n . The resultextends by continuity to all multiples of the segment ~pq . Lemma 16.
Let f : U( M q Ñ R and f : U( M q Ñ R like in Lemma15. Suppose that Reached M p o, p, q, p, r q for some o, p, q, r ǫ U( M ). If f and f agree on the components, that is η ab p f p o q , f p p q ) = η ab p f p o q , f p p qq and η ab p f p o q , f p q q ) = η ab p f p o q , f p q qq , then η ab p f p o q , f p r q ) = η ab p f p o q , f p r qq . Proof.
This is proven by cases. If L M (o,r), we have that: η ab p f p o q , f p r qq “ η ab p f p o q , f p o qq p By Axiom (F4) q“ p Obvious q“ η ab p f p o q , f p r q p similarly q In all other cases Axiom (SUM1) (SUM2) and a form of Pythagora’stheorem for spacelike vectors imply that there is a point r that lies on either ℓ oq or ℓ op and ” M p o, r, o, r q . Condition (2) and Lemma 15 imply the result.34 roof of Theorem 4. Existence:
Let M be a model of M in which the ordering of points on a line is acontinuum. By (A1) it has elements o, x, y, z, t such that Basis M p o, x, y, z, t q .By the affine axioms and the axioms (A3)(A4)(DENS)(F5)(F6), it followsthat the structure E with congruence and betweenness restricted to theelements v such that Gen M D p o, x, y, z, v q is a model of E . By Theorem 1 it hasa coordinatization f . An analogous statement is true for the structure on theline L with the same relations restricted to points l such that Coll M p o, t, l q .By Theorem 2 it admits of a coordinatization g . Fix a total coordinatization f as specified in Definition 8. Axiom (A9) and the bijectivity of f and g imply that f is onto. Axiom (A2) implies that it is one-to-one.Let us fix a point e ǫ E such that η ab p f p o q , f p e qq “ p , q , r P U p M q , Bet M p p, q, r q if and only if there is a positive constant λ P R such that f p r q ´ f p p q “ λ ¨ p f p q q ´ f p p qq . An analysis of (A8) and an appeal toLemma 8 are sufficient to verify that this is the case. Let us now turn to (2).Choose four points p , q , l , r P U p M q such that ” M p l, r, p, q q . By Lemma 12,two of the points can be chosen to be the origin o (= l = p ). The proof thatthe relativistic interval, as computed by f , is the same on the two segmentsproceeds by cases. ( Case 1 ) L M p o, q q implies L M p o, r q via (F1)(F2)(F3).Lemma 6 implies that η ab p f p o q , f p r qq = 0 = η ab p f p o q , f p q q . ( Case 2 ) If S M p o, q q we can assume by (SUM 1) that r P E and that there exist points w , x , v P E and t P L such that Reached M p o, x, t, x, q q , Bet M p o, w, x q , Opp M p o, w, o, t q , Remterm M p o, e, o, x, w, v q and ” M p o, p, x, v qq .Lemma 6 implies equation (1). Lemma 13 and the fact that Bet M p o, w, x q justify equation (2) below. Lemma 9 on the calculus of segments impliesequation (3). Equation (4) is from the definition of the interval η ab .( η ab p f p o q , f p w qq “ ´ η ab p f p o q , f p t qq ( η ab p f p o q , f p x qq “ η ab p f p o q , f p w qq` η ab p f p w q , f p x qq` η ab p f p o q , f p w qq η ab p f p w q , f p x qq ( η ab p f p o q , f p v qq “ η ab p f p w q , f p x qq` η ab p f p o q , f p w qq η ab p f p w q , f p x qq ( η ab p f p o q , f p q qq “ η ab p f p o q , f p t qq ` η ab p f p o q , f p x qq
35y substituting in (4) the two terms for their equivalents in (2) and in (1)leaves the expression that figures on the right-hand side of (3). This provesthat η ab p f p o q , f p q qq = η ab p f p o q , f p v qq . The transitivity of congruence implies ” M p o, r, x, v qq . Theorem 1, and the fact that o , r , v P E , imply that also η ab p f p o q , f p r qq = η ab p f p o q , f p v qq . ( Case 3 ) when T M p o, q q is analogous.To prove the conditionals in the other direction, it suffices to note (Case 1) that lightlike segments are congruent by (F4). By (4), lemma 6 and (SUM3)it follows that, if η ab p f p o q , f p q qq “ “ η ab p f p o q , f p r qq , then L M p o, r q and L M p o, q q . (Case 2) and (Case 3) follow from Theorem 1 and (SUM1) (SUM2)and the transitivity of congruence. The biconditionals in (2) of Theorem 4 interms of ‘ ă ” ’ follow readily from the biconditionals in terms of ‘ ” ’. We havealready noted the fact that segments lightlike in M have null interval relativeto f . Timelike and spacelike segments are congruent to segments in E and L respectively by (SUM1)(SUM2) and the square of the interval is respectivelynegative and positive between points in E and L by the construction of f . Uniquess up to a rescaling and a Poincar´e transformation:
Let f : U p M q Ñ R and f : U p M q Ñ R be two bijective functionsthat satisfy conditions (1) and (2) of Theorem 4 . Fix the basis o , x , y , z and t that is associated by f to the canonical basis of R . There is a real k such that η ab p f p o q , f p x qq “ k and (by Lemma 6) η ab p f p o q , f p t qq “ ´ k .Let U : R Ñ R be the rescaling function such that U p ~x q = k ¨ ~x . Itwill suffice to show, by Theorem 3, that U ˝ f “ f and f agree on therelativistic interval between all p and q in U p M q . By condition (2) andLemma 6 it will suffice check the case when p “ o . U ˝ f “ f and f agree on the basis o , x , y , z and t by construction. By axiom (A2) wehave that Gen M D p o, x, y, z, t, q q . By analysing the definition and noticingaxiom (A7) we get a sequence x , y , z such that Reached M p o, x, y, x , y q , and Reached M p o, y , z, y , z q and finally that Reached M p o, z , t, z , q q . The firstequality η ab p f p o q , f p x q ) = η ab p f p o q , f p x qq follows from Lemma 15. The factthat η ab p f p o q , f p y q ) = η ab p f p o q , f p y qq and ultimately that η ab p f p o q , f p x q )= η ab p f p o q , f p q qq follows by successive applications of Lemma 16. We have proposed a formalization of a small fragment of physical theory fora specific purpose, but let us conclude with some other uses that it might Within the three main categories the relation ‘ ă ” ’ is definable in terms of ‘ ” ’. eferences [1] Alscher, D., 2016, Theorien Der Reellen Zahlen Und Interpretierbarkeit ,Berlin, Boston: De Gruyter[2] Andr´eka, H., Madar´asz, J. X., N´emeti, I. and Sz´ekely, G., 2011, Onlogical analysis of relativity theories,
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