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Philosophy of Mathematics

Mathematical Models of Spacetime in Contemporary Physics and Essential Issues of the Ontology of Spacetime

Maciej Gos

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ABSTRACT: The general theory of relativity and field theory of matter generate an interesting ontology of space-time and, generally, of nature. It is a monistic, anti-atomistic and geometrized ontology — in which the substance is the metric field — to which all physical events are reducible. Such ontology refers to the Cartesian definition of corporeality and to Plato's ontology of nature presented in the Timaeus. This ontology provides a solution to the dispute between Clark and Leibniz on the issue of the ontological independence of space-time from distribution of events. However, mathematical models of space-time in physics do not solve the problem of the difference between time and space dimensions (invariance of equations with regard to the inversion of time arrow). Recent research on space-time singularities and asymmetrical in time quantum theory of gravitation will perhaps allow for the solution of this problem based on the structure of space-time and not merely on thermodynamics.

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Since the General Theory of Relativity is a fundamental model of spacetime, most often used as a starting point for a physicists' research, I will begin my analysis with an attempt to formulate the consequences of the GTR for spacetime ontology and, generally, ontology of nature. The preliminary remark, however, has epistemological character. The formulation of the GTR resulted in definite fall of a dogmatic thesis of distinguished value of the 3-dimensional Euclidean geometry, as the only geometrical structure, adequate for a description of nature. This thesis was formulated explicitly by Kant, who considered this structure to be apriori form of inspection, and as such it was to validate the science. Kant's epistemology was created mainly to validate science, whose best developed part was embodied in the classical mechanics. However, the attitude of the authors of the field theory of matter towards Kant's epistemology was not negative at all, which can be proved by the words of Weyl, who, although confessed that in himself a philosopher had been dominated by a mathematician, began his work Space, Time, Matter with philosophical considerations devoted to the question of time and space, considerations comparable to those of Kant.

The formulation of the General Theory of Relativity was of major importance for the spacetime ontology, mainly because it ended the famous debate between Clarke and Leibniz concerning the ontological status of spacetime. In this debate Clarke defended the Newton's position in which he emphasized ontological independence of time and space with regard to physical events. Time and space were to serve only as an arena for the physical events, unlike in the Leibnizian conception, according to which time and space were considered to be relations between events, and so time and space were devoid of the substantial, independent from the events and autonomous being. The GTR by connecting the spacetime geometry with the distribution of energy-momentum (so with physical events) initially seemed to solve definitely the problem of the relationship between space and time and physical events along with the Leibnizian conception. But then, de Sitter published the solution of the GTR equations that described the world with well-defined spacetime geometry and, at the same time, zero density of energy-matter. Although the ontology of spacetime generated by the GTR is coherent with the Leibnizian conception due to the fundamental Einstein-Hilbert's equation, this "empty" solution creates compromise with the Newton- Clarke's theory. While discussing the relationship between spacetime and events, very important conclusions for ontology can be drawn from the notion of vacuum in the quantum theory of field. These conclusions are coherent with those drawn directly from the GTR. In the quantum theory of field spacetime totally independent from the events cannot exist. Vacuum — the "empty" spacetime is merely a state of the quantum field with the lowest energy, which, however, manifests itself in the creation of virtual particles. It results directly from the Heisenberg's Principle of Uncertainty. Therefore, both, a theory of macrocosm (the GTR) and a theory of microcosm lead to the same conclusion about the ontological relationship between spacetime and events.

The General Theory of Relativity was just a starting point for perhaps the most ambitious attempt in the history of science — the attempt to formulate pure field theory of the whole matter (nature). And, however such a theory in its complete shape has not been created yet, its idea and a temporary shape generate not only a certain spacetime ontology, but, moreover, this ontology sets immediately the ontology of nature (the world of physical events). The core of this conception is constituted by Weil's definition of the world as a 3+1 dimensional manifold, with a defined metric field ( metric field is defined by the components of a geometrical object-metric tensor). All physical events which constitute the world are reduced in this conception to the manifestations of a metric field on manifold (which means simply the manifestations of geometrical structure of spacetime). A similar field theory of matter, developed by Mie, was characterised by K. Maurin as the one refering to the first field conception of matter — the stoic view of Poseidonios. If, however, all phenomena that create nature can be eventually reduced to manifestations of disturbances of geometrical structure of spacetime (manifold with a metric field defined on it), then, it indicates the return to the fundamental ontological idea — the idea of substance. In the field theory of nature, spacetime considered a manifold with a defined metric field is precisely the substance, the primary substratum of events. In that way defined geometrical structure of spacetime is substantial, because it is invariant as a pseudo-Riemannian space with affine relationship — all physical events are accidental and fully reducible to the manifestations of the substratum i. e. metric field. Weyl expressed this idea when he compared metric tensor to the notion of ether in XIX century physics.

A crucial ontological consequence here is the return to the Cartesian definition of corporeality i.e. belonging to the world of nature exclusively by extension (defined precisely by the GTR as pseudo-Riemannian manifold). That is how the borderline between physics and geometry becomes unclear and no longer exists. It is not the only ontological implication of the theory of field. Another, equally important issue is strong, anti-atomistic meaning of the ontology of nature, in which the substance is identified with spacetime (metric field). It is obvious that ontology that assumes one substance, so a monistic one, excludes the possibility to consider physical events to be ontological atoms of nature. As the manifestations of the substance events are merely accidental — they are fully reducible to the substance (as the components of a metric tensor). The General Theory of Relativity itself generates the ontology of nature, in which there is no room for the ontological atomism; every event is dependent — according to Mach's rule — on all the other physical events that constitute the world, an event exists only in a system of relationships with other events. If one wanted to define this ontology of nature generated by the field theory of matter as concisely as possible, one should call it physical, geometrized and monistic (so incosistent with ontological atomism) ontology. It is not anthropocentric — its starting point is a concideration concerning nature and not the content of human mind. Such a geometrical conception of nature reaches farther back than the Cartesian thought. One cannot help associatons with Plato's dialogue "Timaios", where the Plato's ontology of nature is displayed. And despite a lapse of more than twenty five centuries there is a striking similarity between his ontology and the one generated by the field theory of matter. The core of both is geometrization of the substratum of nature. In Plato's conception, the substance is composed of tiny triangles which build symmetrical bodies called today Plato's bodies — cube, octahedron, dodecahedron and icosahedron which serve as a frame for four elements and the whole physical world. In the field theory of matter this substantial role is played by a different geometrical object-metric tensor of spacetime or, to say it by means of quality notions, variable geometical structure of spacetime, manifestations of which are all events constituting nature — the events are fully reducible to the structure.

"Timaios" appears to be timely in the contemporary ontology of spacetime not only with regard to the GTR and the field theory of matter, but also to the latest works concerning mathematics and mathematical physics, especially theories of spacetime based on the complex analysis, worked out by Manin and Penrose. Penrose' conception as well as Manin's stem from the notions of spaces of spinors and twistors. Manin's construction is complex space of spinors, which is a base by means of which 3-dimensional Euclidean space of the classical mechanics and 4-dimensional Minkowski's space can be defined. The similarity to Plato's world can be seen through the essential issue of the philosophy of nature, the issue of a relationship between a mathematical model described by a physical theory and the world. In contemporary physics, this relationship poses some subtle problems due to the fact that mathematical models themselves are constructions with abstract, intricate and many level structures. That is the case when we take into consideration the relationship between unitary space of spinors and the structure of physical spacetime. Here the relationship is indirect to a large extent. The Euclidean space and Minkowski's space are the intermediate structures between fundamental mathematical structure — complex space of spinors — and physical spacetime. That the unitary complex space is fundamental means that it allows to define both, the Euclidean and Minkowski's spaces. On the other hand, the indispensable intermediary role of those classical structures is played through their relation with experiments and measurement, that can be made only in their categories. Here again we find an analogy with Timaios' ontology. Plato's triangles and bodies, geometrical substratum of the world, correspond to deep structure of spacetime — abstract, complex mathematical structure that allows to define models of spacetime of the classical mechanics. The significance of the latter models is not weakened since they describe adequately — to use the expression taken from the domain of linguistics — surface structure of the physical spacetime, as they enable us make concrete measurements, that serve as a base of verification of a physical theory. Therefore, they connect the ideal Plato's world with the world of phenomena, similarly like in "Timaios" a description of this surface structure of nature, i. e. concrete events, was made by means of the four elements, the frame of which was the actual geometrical substratum of nature. It is remarkable that Penrose, whose contribution to the examination of complex spaces of spinors and twistors was the most valuable in our times, shares the view of the strong mathematical Platonism concerning ontology as well as epistemology.

Relation: local-global is of great importance not only in ontology, but also in entire science and philosophy. On the one hand, ontology tends to be defined as knowledge concerning the notion of the whole — the notion of global nature, indeed. On the other hand, the contrast: local-global is often used to define and contrast scientific knowledge and philosophy. Such a view is expressed by Thom, who thinks that the basic feature of a scientific theory is its locality expressed as the possibility to geometrize it. This view is also shared by Maurin, who states that a category of the whole is specifically philosophical, strictly religious. In the domain of the ontology of spacetime I will try to prove that the latest mathematical models that use the methods of global analysis on complex manifolds let us obtain important results concerning the connection between local homogeneity of spacetime and its global homogeneity. The former one, well proved by the whole classical physics and through the Noether's theorem connected with the principles of conservation in the classical mechanics, has purely scientific nature, the latter, on the other hand, left without any justification would be only an arbitrary metaphysical postulate commonly assumed, since it provides "comfortable" universality of physical laws in the whole Universe. And here, contemporary mathematics can give a kind of solution. The crucial significance for a demonstration of the global homogeneity of spacetime resulting from the local homogeneity of spacetime has Penrose' postulate which defines spacetime as a 2-dimensional complex, i. e. 4-dimensional real, holomorphic manifold. For such manifolds the principle of identity binds. According to this principle, for any two holomorphic or meromorphic functions, if they are identical in optionally small neighbourhood, they are also identical on the whole manifold. Such a geometrical model of spacetime lessens remarkably the arbitrariness of the metaphysical postulate of global homogeneity of spacetime. What follows, is the connection between what is local, so scientific, and what is global, so ontological, — the means that enables this connection is mathematics, strictly speaking, global analysis.

Contemporary culture for at least two centuries seems to become more and more shizophrenic, if we consider the origin of the notion (Greek schizein — to split). This split of culture is marked by a division between the whole humanistic culture, on the one hand, and science, mathematics and nature science in particular, on the other hand. The latter are often perceived as the ones that play an ancillary role for technics. Only few intellectuals of 20 century — among them Weyl and Penrose — have realised how absurd and harmful such a split is for the development of culture. Varying approaches to time and space as notions considered in science as well as the Arts are good example of that split of culture. The basic issue, in which it can well be seen, is a question of so called time arrow. This issue is an important subject of examination in mathematical physics as well as ontology of spacetime and philosophical anthropology. It reveals crucial contradiction between the knowledge about time, provided by mathematical models of spacetime in physics and psychology of time and its ontology. The essence of the contradiction lies in the invariance of the majority of fundamental equations in physics with regard to the reversal of the direction of the time arrow (i. e. the change of a variable t to -t in equations). Neither metric continuum, constituted by the spaces of concurrency in the spacetime of the classical mechanics before the formulation of the Particular Theory of Relativity, the spacetime not having metric but only affine structure, nor Minkowski's spacetime nor the GTR spacetime (pseudo-Riemannian), both of which have metric structure, distinguish the categories of past, present and future as the ones that are meaningful in physics. Every event may be located with the use of four coordinates with regard to any curvilinear coordinate system. That is what clashes remarkably with the human perception of time and space. Penrose, who is an outstanding mathematician and physicist as well as a philosopher, realizes and understands the necessity to formulate such theory of spacetime that would remove this discrepancy. He remarked that although we feel the passage of time, we do not perceive the "passage" of any of the space dimensions. Theories of spacetime in mathematical physics, while considering continua and metric manifolds, cannot explain the difference between time dimension and space dimensions, they are also unable to explain by means of geometry the unidirection of the passage of time, which can be comprehended only by means of thermodynamics. The theory of spaces of twistors is aimed at better and crucial for the ontology of nature understanding of the problem of the uniqueness of time dimension and the question of time arrow. There are some hypotheses that the question of time arrow would be easier to solve thanks to the examination of so called spacetime singularities and the formulation of the asymmetric in time quantum theory of gravitation — or the theory of spacetime in microscale.

In conclusion, it must be emphasized that a unifying role of mathematics in nature science has never been so important as it is today, when — to quote Maurin — it became "the main tool of Logos" and the unity of mathematics and physics is seen particularly well. This role of mathematics has a great significance for the philosophy of nature — it allows for precise formulation of its problems and consequently makes their examination easier. Although the GTR provided the conclusion of Carke's and Leibniz' dispute, the reconstruction of geometrical model of spacetime in the classical mechanics with and without gravitation, by means of the methods provided by contemporary mathematics, leads to denial of the existence of absolute, substantial and independent from a distribution of events spacetime regardless the GTR. The transition from the geometric model of spacetime in the classical mechanics to the General Theory of Relativity is just completing it with metric. The reconstruction started by Lange and Cartan and reformulated after the introduction of the notion of affine relationship by Weyl, was made together with the philosophical comment by Heller. The essence of this reconstruction is the analysis of the basic geometrical models of spacetime with associating kinematics and dynamics and there appears a tendency to move from the models with which many arbitrary assumptions are connected (like these in which there are distinguished frames of reference such as the system of absolute state of rest in Aristotle dynamics or the inertial system in the Newtonian dynamics), to the models that are logically easier. The continuity and logic of evolution of physics of spacetime reconstructed by Heller is apparent, owing to the use of the language of contemporary mathematics. Logical development of the physics of spacetime in Heller's view is exposed through the use of the theory of fibre bundle and the theory of differential manifolds. The first model deconstructed in that way is the spacetime of the Aristotle dynamics — which is Cartesian product of 3-dimensional space and 1-dimensional time, both space and time have defined metric (Pitagorean) and topology generated by the metric. The spacetime of the classical mechanics (before the formulation of the Particular Theory of Relativity) must be considered separately for the case with gravitation and the case without it. While moving from the spacetime of the Aristotle dynamics to the spacetime of the Newtonian dynamics without gravitation, spacetime loses the structure of Cartesian product, but the vector bundle of directed bases on the spacetime has a product structure (it is a fibre bundle, since the set of directed bases on any point of spacetime is a fibre on this point), it is a trivial bundle. In the presence of gravitation a bundle of directed bases also loses the structure of Cartesian product on the spacetime. Additionally, the use of notions of affine relationship and metric tensor (by means of which the coordinates of affine relationship are expressed) in the reconstruction of spacetime in the classical mechanics before the Particular Theory of Relativity with gravitation leads immediately to discovery of curvature of such spacetime, that is not assigned uniquely to the GTR. In the case of transition to the spacetime of the GTR there is a characteristic connection of the metric structure of spacetime with its curvature, whereas the spacetime of the mechanics before the Partcular Theory of relativity does not have metric as the whole — the metric spaces are only spaces of concurrency and time, as the whole, the spacetime has only affine structure, which results from the Newtonian dynamics associated with this model of spacetime i. e. with a distinguished status of inertial reference frames. A cognitive advantege of the above reconstruction is the fact that it explains the way to the GTR from the spacetime of the Aristotle dynamics emphasizing the continuity and logic of this evolution.

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