The Rich Structure of Minkowski Space Domenico Giulini Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute) Am M¨uhlenberg 1 D-14476 Golm/Potsdam, Germany [email protected] Abstract Minkowski Space is the simplest four-dimensional Lorentzian Mani- fold, being topologically trivial and globally flat, and hence the sim- plest model of spacetime—from a General-Relativistic point of view. But this does not mean that it is altogether structurally trivial. In fact, it has a very rich structure, parts of which will be spelled out in detail in this contribution, which is written for Minkowski Spacetime: A Hundred Years Later, edited by Vesselin Petkov, to appear in 2008 in the Springer Series on Fundamental Theories of Physics, Springer Verlag, Berlin. Contents 1 General Introduction 2 2 Minkowski space and its partial automorphisms 4 2.1 Outlineofgeneralstrategy. 4 2.2 Definition of Minkowski space and Poincar´egroup . 5 2.3 Frommetrictoaffinestructures. 8 2.4 Fromcausaltoaffinestructures . 10 2.5 Theimpactofthelawofinertia. 12 2.6 Theimpactofrelativity ......................... 15 2.7 Localversions............................... 20 arXiv:0802.4345v1 [math-ph] 29 Feb 2008 3 SelectedstructuresinMinkowskispace 24 3.1 Simultaneity................................ 24 3.2 The lattices of causally and chronologically complete sets . 29 3.3 Rigidmotion ............................... 36 A Appendices 45 A.1 Setsandgroupactions.......................... 45 A.2 Affinespaces ............................... 47 A.3 Affinemaps................................ 48 A.4 Affine frames, active and passive transformations . 50 1 1 General Introduction There are many routes to Minkowski space. But the most physical one still seems to me via the law of inertia. And even along these lines alternative approaches exist. Many papers were published in physics and mathemat- ics journals over the last 100 years in which incremental progress was re- ported as regards the minimal set of hypotheses from which the structure of Minkowski space could be deduced. One could imagine a Hesse-diagram- like picture in which all these contributions (being the nodes) together with their logical dependencies (being the directed links) were depicted. It would look surprisingly complex. From a General-Relativistic point of view, Minkowski space just models an empty spacetime, that is, a spacetime devoid of any material content. It is worth keeping in mind, that this was not Minkowski’s view. Close to the beginning of Raum und Zeit he stated:1 In order to not leave a yawning void, we wish to imagine that at every place and at every time something perceivable exists. This already touches upon a critical point. Our modern theoretical view of spacetime is much inspired by the typical hierarchical thinking of mathemat- ics of the late 19th and first half of the 20th century, in which the set comes first, and then we add various structures on it. We first think of spacetime as a set and then structure it according to various physical inputs. But what are the elements of this set? Recall how Georg Cantor, in his first article on transfinite set-theory, defined a set:2 By a ‘set’ we understand any gathering-together M of deter- mined well-distinguished objects m of our intuition or of our thinking (which are called the ‘elements’ of M) into a whole. Do we think of spacetime points as “determined well-distinguished objects of our intuition or of our thinking”? I think Minkowski felt a need to do so, as his statement quoted above indicates, and also saw the problematic side of it: If we mentally individuate the points (elements) of spacetime, we—as physicists—have no other means to do so than to fill up spacetime with actual matter, hoping that this could be done in such a diluted fash- ion that this matter will not dynamically affect the processes that we are going to describe. In other words: The whole concept of a rigid background spacetime is, from its very beginning, based on an assumption of—at best— approximate validity. It is important to realise that this does not necessarily 1 German original: “Um nirgends eine g¨ahnende Leere zu lassen, wollen wir uns vorstellen, daß allerorten und zu jeder Zeit etwas Wahrnehmbares vorhanden ist”. ([39], p. 2) 2 German original: “Unter einer ‘Menge’ verstehen wir jede Zusammenfassung M von bes- timmten wohlunterschiedenen Objecten m unserer Anschauung oder unseres Denkens (welche die ‘Elemente’ von M genannt werden) zu einem Ganzen.” ([12], p. 481) 2 refer to General Relativity: Even if the need to incorporate gravity by a variable and matter-dependent spacetime geometry did not exist would the concept of a rigid background spacetime be of approximate nature, provided we think of spacetime points as individuated by actual physical events. It is true that modern set theory regards Cantor’s original definition as too na¨ıve, and that for good reasons. It allows too many “gatherings- together” with self-contradictory properties, as exemplified by the infamous antinomies of classical set theory. Also, modern set theory deliberately stands back from any characterisation of elements in order to not confuse the axioms themselves with their possible interpretations.3 However, ap- plications to physics require interpreted axioms, where it remains true that elements of sets are thought of as definite as in Cantors original definition. Modern textbooks on Special Relativity have little to say about this, though an increasing unease seems to raise its voice from certain directions in the philosophy-of-science community; see, e.g., [11][10]. Physicists some- times tend to address points of spacetime as potential events, but that always seemed to me like poetry4, begging the question how a mere potentiality is actually used for individuation. To me the right attitude seems to admit that the operational justification of the notion of spacetime events is only approximately possible, but nevertheless allow it as primitive element of the- orising. The only thing to keep in mind is to not take mathematical rigour for ultimate physical validity. The purpose of mathematical rigour is rather to establish the tightest possible bonds between basic assumptions (axioms) and decidable consequences. Only then can we—in principle—learn any- thing through falsification. The last remark opens another general issue, which is implicit in much of theoretical research, namely how to balance between attempted rigour in drawing consequences and attempted closeness to reality when formulat- ing once starting platform (at the expense of rigour when drawing conse- quences). As the mathematical physicists Glance & Wightman once for- mulated it in a different context (that of superselection rules in Quantum Mechanics): The theoretical results currently available fall into two cate- gories: rigorous results on approximate models and approximate results in realistic models. ([48], p. 204) 3 This urge for a clean distinction between the axioms and their possible interpretations is contained in the famous and amusing dictum, attributed to David Hilbert by his student Otto Blumenthal: “One must always be able to say ’tables’, ‘chairs’, and ‘beer mugs’ instead of ’points, ‘lines’, and ‘planes”. (German original: “Man muß jederzeit an Stelle von ’Punkten’, ‘Geraden’ und ‘Ebenen’ ’Tische’, ‘St¨uhle’ und ‘Bierseidel’ sagen k¨onnen.”) 4 “And as imagination bodies forth The forms of things unknown, the poet’s pen Turns them to shapes, and gives to airy nothing A local habitation and a name.” (A Midsum- mer Night’s Dream, Theseus at V,i) 3 To me this seems to be the generic situation in theoretical physics. In that respect, Minkowski space is certainly an approximate model, but to a very good approximation indeed: as global model of spacetime if gravity plays no dynamical rˆole, and as local model of spacetime in far more general situ- ations. This justifies looking at some of its rich mathematical structures in detail. Some mathematical background material is provided in the Appen- dices. 2 Minkowski space and its partial automorphisms 2.1 Outline of general strategy Consider first the general situation where one is given a set S. Without any further structure being specified, the automorphisms group of S would be the group of bijections of S, i.e. maps f : S S which are injective (into) and surjective (onto). It is called Perm(S), where ‘Perm’ stands for ‘permutations’. Now endow S with some structure→∆; for example, it could be an equivalence relation on S, that is, a partition of S into an exhaustive set of mutually disjoint subsets (cf. Sect. A.1). The automorphism group of (S,∆) is then the subgroup of Perm(S | ∆) Perm(S) that preserves ∆. Note ⊆ that Perm(S | ∆) contains only those maps f preserving ∆ whose inverse, f−1, also preserve ∆. Now consider another structure, ∆′, and form Perm(S | ∆′). One way in which the two structures ∆ and ∆′ may be compared is to compare their automorphism groups Perm(S | ∆) and Perm(S | ∆′). Comparing the latter means, in particular, to see whether one is contained in the other. Containedness clearly defines a partial order relation on the set of subgroups of Perm(S), which we can use to define a partial order on the set of structures. One structure, ∆, is said to be strictly stronger than (or equally strong as) another structure, ∆′, in symbols ∆ ∆′, iff5 the ≥ automorphism group of the former is properly contained in (or is equal to) the automorphism group of the latter.6 In symbols: ∆ ∆′ Perm(S | ≥ ∆) Perm(S | ∆′). Note that in this way of speaking a substructure (i.e. ⊆ one being defined by a subset of conditions, relations, objects,⇔ etc.) of a given structure is said to be weaker than the latter. This way of thinking of structures in terms of their automorphism group is adopted from Felix Klein’s Erlanger Programm [34] in which this strategy is used in an attempt to classify and compare geometries. This general procedure can be applied to Minkowski space, endowed with its usual structure (see below).