It is argued that Minkowski space-time cannot serve as the deep structure within a ``constructive'' version of the special theory of relativity, contrary to widespread opinion in the philosophical community.

Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary (...) quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems. (shrink)

Should physicists deal with the question of the reality of Minkowski space (or any relativistic spacetime)? It is argued that they should since this is a question about the dimensionality of the world at the macroscopic level and it is physics that should answer it.

In quantum relativistic Hamiltonian dynamics, the time evolution of interacting particles is described by the Hamiltonian with an interaction-dependent term (potential energy). Boost operators are responsible for (Lorentz) transformations of observables between different moving inertial frames of reference. Relativistic invariance requires that interaction-dependent terms (potential boosts) are present also in the boost operators and therefore Lorentz transformations depend on the interaction acting in the system. This fact is ignored in special relativity, which postulates the universality of Lorentz transformations and their (...) independence of interactions. Taking into account potential boosts in Lorentz transformations allows us to resolve the “no-interaction” paradox formulated by Currie, Jordan, and Sudarshan [Rev. Mod. Phys. 35, 350 (1963)] and to predict a number of potentially observable effects contradicting special relativity. In particular, we demonstrate that the longitudinal electric field (Coulomb potential) of a moving charge propagates instantaneously. We show that this effect as well as superluminal spreading of localized particle states is in full agreement with causality in all inertial frames of reference. Formulas relating time and position of events in interacting systems reduce to the usual Lorentz transformations only in the classical limit (ħ→0) and for weak interactions. Therefore, the concept of Minkowski space-time is just an approximation which should be avoided in rigorous theoretical constructions. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture (...) can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means (...) the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

The purpose of this paper is twofold: a) to explore the compatibility of Minkowski’s space-time representation of the Special theory of relativity with a dynamic conception of space-time; b) to locate its roots in invariant features - like entropic relations - of the propagation of signals in space-time. From its very beginning Minkowski’s four-dimensional space-time was associated with a static view of reality, e.g. a block universe. Einstein added his influential voice to this conception (...) when he wrote: ‘From a “happening” in three-dimensional space, physics becomes (…) an “existence” in the four-dimensional “world”.’ (Einstein, Relativity 1920, 122) Yet it is by no means clear that Minkowski himself was a believer in the block universe. In his 1908 Cologne lecture on ‘Space and Time’ he speaks of a four-dimensional physics but concedes that a ‘necessary’ time order can be established at every world point. Although the conception of the block universe has gained much currency, an alternative view has been in circulation since the 1910s according to which the trajectories of particles constitute histories in space-time. (Robb 1914, Cunningham 1915, Carathéodorys 1924, Schlick 1917, Reichenbach 1924). (shrink)

We calculate the mixing of real and imaginary components of space and time under the influence of superluminal boosts in thex direction. A unique mixing is determined for this superluminal Lorentz transformation when we consider the symmetry properties afforded by the inclusion of three temporal directions. Superluminal transformations in complex six-dimensional space exhibit unique tachyonic connections which have both remote and local space-time event connections.

Minkowski geometry is axiomatized in terms of the asymmetric binary relation of optical connectibility, using ten first-order axioms and the second-order continuity axiom. An axiom system in terms of the symmetric binary optical connection relation is also presented. The present development is much simpler than the corresponding work of Robb, upon which it is modeled.

This paper argues that the Einstein-Minkowski space-time of special relativity provides an adequate model for classical tense logic, including rigorous definitions of tensed becoming and of the logical priority of proper time. In addition, the extension of classical tense logic with an operator for predicate-term negation provides us with a framework for interpreting and defending the significance of future contingency in special relativity. The framework for future contingents developed here involves the dual falsehood of non-logical contraries, only one (...) of which becomes true. This has several methodological, metaphysical and physical advantages over the alternative traditional frameworks for handling future contingents. (shrink)

In this paper, we continue the study of the Killing symmetries of an N-dimensional generalized Minkowski space, i.e., a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the finite structure of the space–time rotations in such spaces, by confining ourselves (without loss of generality) to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski (...) space M_4 (i.e., a pseudoeuclidean space with metric coefficients depending on energy), for which we derive the explicit general form of the finite rotations and boosts in different parametric bases. (shrink)

In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e., a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the translations in such spaces, by confining ourselves (without loss of generality) to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space $$\widetilde M_4 $$ (i.e. (...) a pseudoeuclidean space with metric coefficients depending on energy). (shrink)

In this paper, we introduce the concept of N-dimensional generalized Minkowski space, i.e., a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coordinates. This is the first of a series of papers devoted to the investigation of the Killing symmetries of generalized Minkowski spaces. In particular, we discuss here the infinitesimal-algebraic structure of the space-time rotations in such spaces. It is shown that the maximal Killing (...) group of these spaces is the direct product of a generalized Lorentz group and a generalized translation group. We derive the explicit form of the generators of the generalized Lorentz group in the self-representation and their related, generalized Lorentz algebra. The results obtained are specialized to the case of a 4-dimensional, “deformed” Minkowski space $${\widetilde M}$$ 4, i.e., a pseudoeuclidean space with metric coefficients depending on energy. (shrink)

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of (...) extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup. (shrink)

Under the eternalist hypothesis that objects or events exist temporally, but independently of being present two different views of persistence are on the market: Persisting objects endure if they are multiply located in time, and persisting objects perdure if they are singly located by having numerically different temporal parts. In the framework of the special theory of relativity, the metaphysics of persistence is confronted with peculiar difficulties. Things persist by being “wholly present” at more than one time; but what are (...) times within a temporally non-separated spacetime? Things persist by having different temporal parts at different times; but what are the temporal parts of a four-dimensional object in Minkowski spacetime? Recently, several authors have argued that SR favours perdurantism over its endurantist rival. In this paper, I intend to show that the purported arguments are only those against endurantism. The first simply fails, but the second, more convincing one, is such that with a similar strategy we should argue against perdurantism as well: Enduring and perduring entities are hence both in conflict with SR which undermines the eternalist hypothesis. (shrink)

We discuss the possible breakdown of Lorentz invariance—at distances greater than the Planck length—from both the theoretical and the phenomenological point of view. The theoretical tool to deal with such a problem is provided by a “deformation” of the Minkowski metric, with parameters dependent on the energy of the physical system considered. Such a deformed metric realizes, for any interaction, the “solidarity principle” between interactions and spacetime geometry (usually assumed for gravitation), according to which the peculiar features of every (...) interaction determine—locally—its own spacetinie structure. The generalized theory of relativity, based on the locally deformed Minkowski spacetime, is called “deformed special relativity” (DSR). In the first part of the paper, we give the foundations and the basic laws of DSR. In the second part, we analyze some experimental data, which admit an interpretation in terms of the DSR formalism and are, therefore, candidates for displaying a breakdown of the Lorentz symmetry. They are (i) the superluminal propagation of evanescent electromagnetic waves in waveguides, (ii) the meanlife of the KS 0, (iii) the Bose-Einstein correlation in pion production and (iv) the comparison of clock rates in the gravitational field of Earth. Such analysis provides us with the explicit forms of the related deformed metrics as functions of the energy, thus putting in evidence, in all four cases (and therefore for all four fundamental interactions), departures from the usual Minkowski metric. This preliminary evidence for a broken Lorentz invariance may be regarded as the signature of possible nonlocal effects involved in the processes examined. Moreover, the corresponding deformed metrics obtained by our analysis provide an effective dynamical description of the interactions (at least in the energy range considered). (shrink)

I examine the issue of persistence over time in thecontext of the special theory of relativity (SR). Thefour-dimensional ontology of perduring objects isclearly favored by SR. But it is a different questionif and to what extent this ontology is required, andthe rival endurantist ontology ruled out, by thistheory. In addressing this question, I take theessential idea of endurantism, that objects are whollypresent at single moments of time, and argue that itcommits one to unacceptable conclusions regardingcoexistence, in the context of SR. (...) I then propose anddiscuss a plausible account of coexistence forperduring objects, which is free of these defects. This leaves the endurantist room for some maneuvers. I consider them and show that they do not really helpthe endurantist out. She can accommodate the notionof coexistence in the relativistic framework only atthe cost of renouncing central endurantist intuitions. (shrink)

This volume is dedicated to the centennial anniversary of Minkowski's discovery of spacetime.

Already in 1835 Lobachevski entertained the possibility of multiple geometries of the same type playing a role. This idea of rival geometries has reappeared from time to time but had yet to become a key idea in space-time philosophy prior to Brown's _Physical Relativity_. Such ideas are emphasized towards the end of Brown's book, which I suggest as the interpretive key. A crucial difference between Brown's constructivist approach to space-time theory and orthodox "space-time realism" pertains to modal (...) scope. Constructivism takes a broad modal scope in applying to all local classical field theories---modal cosmopolitanism, one might say, including theories with multiple geometries. By contrast the orthodox view is modally provincial in assuming that there exists a _unique_ geometry, as the familiar theories have. These theories serve as the "canon" for the orthodox view. Their historical roles also suggest a Whiggish story of inevitable progress. Physics literature after c. 1920 is relevant to orthodoxy primarily as commentary on the canon, which closed in the 1910s. The orthodox view explains the spatio-temporal behavior of matter in terms of the manifestation of the real geometry of space-time, an explanation works fairly well within the canon. The orthodox view, Whiggish history, and the canon have a symbiotic relationship. If one happens to philosophize about a theory outside the canon, space-time realism sheds little light on the spatio-temporal behavior of matter. Worse, it gives the _wrong_ answer when applied to an example arguably _within_ the canon, a sector of Special Relativity, namely, _massive_ scalar gravity with universal coupling. Which is the true geometry---the flat metric from the Poincare' symmetry group, the conformally flat metric exhibited by material rods and clocks, or both---or is the question faulty? How does space-time realism explain the fact that all matter fields see the same curved geometry, when so many ways to mix and match exist? Constructivist attention to dynamical details is vindicated; geometrical shortcuts can disappoint. The more exhaustive exploration of relativistic field theories in particle physics, especially massive theories, is a largely untapped resource for space-time philosophy. (shrink)

We here present explicit relational theories of a class of geometrical systems (namely, inner product spaces) which includes Euclidean space and Minkowski spacetime. Using an embedding approach suggested by the theory of measurement, we prove formally that our theories express the entire empirical content of the corresponding geometric theory in terms of empirical relations among a finite set of elements (idealized point-particles or events) thought of as embedded in the space. This result is of interest within the (...) general phenomenalist tradition as well as the theory of space and time, since it seems to be the first example of an explicit phenomenalist reconstruction of a realist theory which is provably equivalent to it in observational consequences. The interesting paper "On the Space-Time Ontology of Physical Theories" by Ken Manders, Philosophy of Science, vol. 49, number 4, December 1982, p. 575-590, has significant affinities to this one. We both, in a sense, try to formally vindicate Leibniz's notion of a relational theory of space, by constructing theories of spatial relations among physical objects which are provably equivalent to the standard absolutist theories. The essential difference between our approaches is that Manders retains Leibniz's explicitly modal framework, whereas I do not. Manders constructs a spacetime theory which explicitly characterizes the totality of possible configurations of physical objects, using a modal language in which the notion of a possible configuration occurs as a primitive. There is no doubt that this is a more accurate realization of Leibniz's own conception of space than the embedding-based approach developed here. However, it also remains open to objections (such as those cited here from Sklar) on account of the special appeal to modal notions. Our approach here, by contrast, aims to avoid the special appeal to modal notions by giving directly a set of laws which are satisfied by a configuration individually, if and only if it is one of the allowable ones. One thus avoids the need for reference to possible but not actual configurations or objects, in the statement of the spacetime laws. We may then take this alternative set of laws as the actual geometric theory, and do away with the hypothetical entity called 'space'. Yet at the same time there is no invocation of modality, except in the ordinary sense in which every physical theory constrains what is possible. So that a relationalist is not forced to utilize a modal language (though Leibniz certainly does.). (shrink)

In Minkowski spacetime, because of the relativity of simultaneity to the inertial frame chosen, there is no unique world-at-an-instant. Thus the classical view that there is a unique set of events existing now in a three dimensional space cannot be sustained. The two solutions most often advanced are that the four-dimensional structure of events and processes is alone real, and that becoming present is not an objective part of reality; and that present existence is not an absolute notion, (...) but is relative to inertial frame; the world-at-an-instant is a three dimensional, but relative, reality. According to a third view, advanced by Robb, Capek and Stein, what is present at a given spacetime point is, strictly speaking, constituted by that point alone. I argue here against the first of these views that the four-dimensional universe cannot be said to exist now, already, or indeed at any time at all; so that talk of its existence or reality as if that precludes the existence or reality of the present is a non sequitur. The second view assumes that in relativistic physics time lapse is measured by the time co-ordinate function; against this I maintain that it is in fact measured by the proper time, as I argue by reference to the Twin Paradox. The third view, although formally correct, is tarnished by its unrealistic assumption of point-events. This makes it susceptible to paradox, and also sets it at variance with our normal intuitions of the present. I argue that a defensible concept of the present is nonetheless obtainable when account is taken of the non-instantaneity of events, including that of conscious awareness, as that region of spacetime comprised between the forward lightcone of the beginning of a small interval of proper time t and the backward lightcone of the end of that interval. This gives a serviceable notion of what is present to a given event of short duration, as well as saving our intuition of the “reality” or robustness of present events. (shrink)

Relativistic quantum theories are equipped with a background Minkowski spacetime and non-relativistic quantum theories with a Galilean space-time. Traditional investigations have distinguished their distinct space-time structures and have examined ways in which relativistic theories become sufficiently like Galilean theories in a low velocity approximation or limit. A different way to look at their relationship is to see that both kinds of theories are special cases of a certain five-dimensional generalization involving no limiting procedures or approximations. When one (...) compares them, striking features emerge that bear on philosophical questions, including the ontological status of the wave function and time reversal invariance. (shrink)

Under the eternalist hypothesis that objects or events exist independently of being present two different views of persistence are on the market: Persisting objects endure if they are multiply located in time, and persisting objects perdure if they are singly located by having numerically different temporal parts. Recently, several authors have argued that special relativity favours perdurantism over its endurantist rival. In my talk, I want to show that in fact the purported arguments are only those against endurantism, and that (...) with similar ones we should argue against perdurantism, as well: Enduring and perduring entities are both in conflict with SR which undermines the eternalist hypothesis. For arguing in favour of perdurantism Yuri Balashov, on the one hand, considers spatially unextended objects in Minkowski space-time and claims that for the endurantists there are unwelcome consequences from an adequate concept of their coexistence. On the other hand, spacelike extended objects are under investigation. Concerning point-like objects, Cody Gilmore has convincingly shown that Balashov’s arguments fail and, therefore, I will confine me to extended object. My paper has two parts following the two different strategies – namely, concerning the problem of the endurantist “explanatory deficiency” according to Balashov, and the problem of criss-crossing hyperplanes according to Gilmore. (shrink)

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a “canonical” quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i $h =$. The two cases will be considered simultaneously. In that case the event (...) class='Hi'>space is the Hilbert space L2(ℝ3). Unitary symmetries consist of Poincaré-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is “confined” in an $h =$ size region of Minkowski space $\mathbb{M}^4$ at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugovečki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugovečki's extended elementary particles. When $h =$ →0, these particles shrink to point particles and $\mathbb{M}^4$ is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i $h =$, and applications in hadron physics give the fit $h =$ ⋍2/5 fermi/GeV. (shrink)

An often repeated account of the genesis of special relativity tells us that relativity theory was to a considerable extent the fruit of an operationalist philosophy of science. Indeed, Einstein’s 1905 paper stresses the importance of rods and clocks for giving concrete physical content to spatial and temporal notions. I argue, however, that it would be a mistake to read too much into this. Einstein’s operationalist remarks should be seen as serving rhetoric purposes rather than as attempts to promulgate a (...) particular philosophical position --- in fact, Einstein never came close to operationalism in any of his philosophical writings. By focussing on what could actually be measured with rods and clocks Einstein shed doubt on the empirical status of a number of pre-relativistic concepts, with the intention to persuade his readers that the applicability of these concepts was not obvious. This rhetoric manoeuvre has not always been rightly appreciated in the philosophy of physics. Thus, the influence of operationalist misinterpretations, according to which associated operations strictly define what a concept means, can still be felt in present-day discussions about the conventionality of simultaneity. The standard story continues by pointing out that Minkowski in 1908 supplanted Einstein’s approach with a realist spacetime account that has no room for a foundational role of rods and clocks: relativity theory became a description of a four-dimensional ‘absolute world’. As it turns out, however, it is not at all clear that Minkowski was proposing a substantivalist position with respect to spacetime. On the contrary, it seems that from a philosophical point of view Minkowski’s general position was not very unlike the one in the back of Einstein’s mind. However, in Minkowski’s formulation of special relativity it becomes more explicit that the content of spatiotemporal concepts relates to considerations about the form of physical laws. If accepted, this position has important consequences for the discussion about the conventionality of simultaneity. (shrink)

The standard coordinate-based formulation of the space-time theory of special relativity (Minkowski geometry) is philosophically unsatisfactory for various reasons. We here present an explicit axiomatic formulation of that theory in terms of primitives with a definitive physical interpretation, prove its equivalence to the standard coordinate formulation, and draw various philosophical conclusions concerning the physical content and assumptions of the space-time theory. The prevalent causal interpretation of physical Minkowski geometry deriving from Reichenbach is criticised on the basis (...) of the present formulation. (shrink)

Cosgrove here applies the historical findings of the previous two chapters to the concept of Minkowski spacetime. While the Minkowski spacetime interval is often called a “generalization” of the Pythagorean Theorem, Einstein himself always more correctly referred to a formal analogy between the four-dimensional spacetime continuum and the three-dimensional continuum of Euclidean space. Thus the physical reality of Minkowski spacetime depends on whether the squared terms in the expression c2dt2 − dx2 designate actual geometrical quantities. Cosgrove (...) concludes that the aforementioned algebraic terms do not designate geometrical quantities but rather represent symbolically abbreviated compound ratios. The Minkowski spacetime interval and all four-vectors constructed upon it are thereby revealed as symbolic artifacts. (shrink)

In this paper we argue that structural explanations are an effective way of explaining well known relativistic phenomena like length contraction and time dilation, and then try to understand how this can be possible by looking at the literature on scientific models. In particular, we ask whether and how a model like that provided by Minkowski spacetime can be said to represent the physical world, in such a way that it can successfully explain physical phenomena structurally. We conclude by (...) claiming that a partial isomorphic approach to scientific representation can supply an answer only if supplemented by a robust injection of pragmatic factors. (shrink)

The quantum transition probability bears the symmetry of a process “evolving” around a symplectic area-bounding loop in the projective Hilbert space that essentially underlies the notion of a global geometric phase. The basic idea is that this symmetry can be associated with a synchronization procedure in the quantum phase space, which is only implicit due to the insistence of interpreting the temporal variable as a classical one, overlooking the subtle interrelation of the complex with the symplectic structure. This (...) leads to the qualification of quantum amplitudes in terms of symplectic spinorial objects which doubly cover the corresponding complex vectors obtained through squaring. Their interrelation can be simply formulated in terms of null vectors of a 3-d Minkowski space, which proves to be instrumental for the Hermitian and unitary representations of the symplectic group. In this light, we show that the quantum transition probability assignment constitutes an equiareal transformation from the annulus of spinors to the disk of complex vectors, which makes it equivalent to the equiareal measure-preserving transformation of Archimedes. This realization ignited a process of re-evaluating the original works of Archimedes in light of their conceptual significance. It turns out that the symplectic method of equiareal projection on a disk pertains to a precise synchronization procedure that can be applied to the quantum phase space. From this viewpoint, we examine the pertinent notions of time, entangled symplectic area, objective indistinguishability, and qualify physically the non-squeezing theorem of symplectic geometry. (shrink)

This book is a ricochet against mainstream physics. It sprang out of the idea that outer symmetries of space-time are the same as inner symmetries of matter. In other words, the standard model of physics is a space-time group. This book is about structures and phenomena that are lying hidden underneath the surface of space-time. It begins with a few biographic events, Majoranas legacy, the philosophy of Gerhard Frey and some related anthropological topics which have to do (...) with high energy physics. It continues with a reconstruction of the theorem by Banach and Tarski in Minkowski space. We are making acquaintance with the standard model as a property of space-time. So we are challenging quite unusual actions such as penetration of quarks by a probe. We propose to apply a penetrating function D. Then, measure and basis are connected with the axiom of choice. (shrink)

A new theory is considered according to which extended objects in n-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of center of mass coordinates. While the usual center of mass is a point, by generalizing the latter concept, we associate with every extended object a set of r-loops, r=0,1,...,n−1, enclosing oriented (r+1)-dimensional surfaces represented by Clifford numbers called (r+1)-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they (...) are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called C-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but C-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in C-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of C-space. (shrink)

If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segré classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized (...) eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity. (shrink)

This essay explores Kaila's interpretation of the special theory of relativity. Although the relevance of his work to logical empiricism is well-known, not much has been written on what Kaila calls the ‘Einstein-Minkowski invariance theory’. Kaila's interpretation focuses on two salient features. First, he emphasizes the importance of the invariance of the spacetime interval. The general point about spacetime invariance has been known at least since Minkowski, yet Kaila applies his overall tripartite theory of invariances to space, (...) time and spacetime in an original way. Second, Kaila provides a non-conventionalist argument for the isotropic speed of electromagnetic signals. The standard Einstein synchrony is not a mere convention but a part of a larger empirical theory. According to Kaila's holistic principle of testability, which stands in contrast to the theses of translatability and verification, different items in the theory cannot be sharply divided into conventional and empirical. Kaila's invariantism/non-conventionalism about relativity reflects an interesting case in the gradual transition from positivism to realism within the philosophy of science. (shrink)

Conformal group of Minkowski space-time M is considered as a group of bundle automorphisms of a vector bundle U over M. 4-component spin-vectors (4-spinors) are sections of a subbundle of the tangent bundle over U. Isotropic 4-vectors are images of 4-spinors under projection. This leads to a particularly clear interpretation of the spin properties of Nature.

Special relativity is most naturally formulated as a theory of spacetime geometry, but within the spacetime framework probability appears to be a purely epistemic notion. It is possible that progress can be made with rather different approaches - covariant stochastic equations, in particular - but the results to date are not encouraging. However, it seems a non-epistemic notion of probability can be made out in Minkowski space on Everett's terms. I shall work throughout with the consistent histories formalism. (...) I shall start with a conservative interpretation, and then go on to Everett's. (shrink)

The chapter shows how local indeterminism underlying BST combines with relativistic space-times. First it defines particular BST structures in which histories are isomorphic to Minkowski space-times. It further argues that many general relativistic space-times are one-history structures of BST. It introduces the notion of non-Hausdorff differential manifolds and investigates if they can be interpreted modally, as structures of BST with multiple histories. It investigates bifurcating curves in non-Hausdorff manifolds, which are natural representations of alternative evolutions of (...) point-like objects. As the required bifurcating curves are unlikely in General Relativity, whereas there are cases of indeterministic general relativistic space-times, the chapter concludes that General Relativity is globally indeterministic, but locally deterministic. (shrink)

In the usual description of space and time, particles are represented by continuous world lines. We replace these world lines by discrete rows of points, obtaining a locally finite, partially ordered set. The “distances” between points along these discrete world lines, and also the “distances” between different world lines, are measured not simply as the distances within the space-time manifold in which the partially ordered set happens to be embedded, but rather in terms of the partially ordered set (...) itself. It is found that variations in the density of points of the partially ordered set produce corresponding variations in the scale of “distances” within the set. These variations produce a gravitational effect which is similar to that of the general theory of relativity. (shrink)

Part I. Space and Time in the History of Philosophy. The Concept of Space in Antiquity / Max Jammer. -- Aristotle and the Sea Battle / G.E.M. Anscombe. -- Questions About Time / St. Augustine. -- Space and Matter / Renè Descartes. -- Absolute Space and Time / Isaac Newton. -- The Relational Theory of Space and Time / Gottfried Leibniz. -- Place, Extension and Duration / John Locke. -- Transcendental Ideality of Space and (...) Time / Immanuel Kant. -- Mirror Images / Immanuel Kant. -- Newton's Views of Time, Space and Motion / Ernst Mach. -- The Unreality of Space and time / F.H. Bradley. -- Duration and Intuition / Henri Bergson. -- The Problem of Infinity Considered Historically / Bertrand Russell. -- Part II. Geomoetry and Physics. What is Geometry? / A.S. Eddington. -- Space and Geometry / Ernest Nagel. -- Non-Eucledian Spaces / Hans Reichenbach. -- Geometry and Definition of Congruence / Hans Reichenbach. -- Geometry as a Branch of Physics / H.P. Robertson. -- Part III. Space-Time and Relativity. Relativity / R.E. Peierls. -- Autobiographical Notes / Albert Einstein. -- The Four-Dimensional World / Moritz Schlick. -- Space and Time / H. Minkowski. The Philosophical Retention of Absolute Space in Einstein's General Theory of relativity / Adolf Grünbaum. -- Part IV. Recent Philosophical Analyses. Ostensible Temporality / C.D. Croad. -- Time : A Treatment of Some Puzzles / J.N. Findlay. -- Time and Language, and the Passage fo Time / Nelson Goodman. -- Time / W.V. Quine. -- Geometrical Objects / W.V. Quine. -- Spatial and Temporal Ananlogies and the Concept of Identity / Richard Taylor. -- Time, Irreversible Processes, and the Physical Status of Becoming / Adolf Grünbaum. (shrink)

Galilean space-time plays the same role in nonrelativistic physics that Minkowski space-time does in relativistic physics. In this paper, the fundamental concepts (velocity, momentum, kinetic energy, etc.) and principles (laws of motion and conservation laws) of classical physics are formulated in the language of Galilean space-time. Much of the development closely parallels the development of similar concepts and principles in the theory of special relativity.

Physical Relativity explores the nature of the distinction at the heart of Einstein's 1905 formulation of his special theory of relativity: that between kinematics and dynamics. Einstein himself became increasingly uncomfortable with this distinction, and with the limitations of what he called the 'principle theory' approach inspired by the logic of thermodynamics. A handful of physicists and philosophers have over the last century likewise expressed doubts about Einstein's treatment of the relativistic behaviour of rigid bodies and clocks in motion in (...) the kinematical part of his great paper, and suggested that the dynamical understanding of length contraction and time dilation intimated by the immediate precursors of Einstein is more fundamental. Harvey Brown both examines and extends these arguments (which support a more 'constructive' approach to relativistic effects in Einstein's terminology), after giving a careful analysis of key features of the pre-history of relativity theory. He argues furthermore that the geometrization of the theory by Minkowski in 1908 brought illumination, but not a causal explanation of relativistic effects. Finally, Brown tries to show that the dynamical interpretation of special relativity defended in the book is consistent with the role this theory must play as a limiting case of Einstein's 1915 theory of gravity: the general theory of relativity.Appearing in the centennial year of Einstein's celebrated paper on special relativity, Physical Relativity is an unusual, critical examination of the way Einstein formulated his theory. It also examines in detail certain specific historical and conceptual issues that have long given rise to debate in both special and general relativity theory, such as the conventionality of simultaneity, the principle of general covariance, and the consistency or otherwise of the special theory with quantum mechanics. Harvey Brown' s new interpretation of relativity theory will interest anyone working on these central topics in modern physics. (shrink)

A-theoretic presentness is commonly regarded as non-solipsist and non-relative. The non-solipsism of a non-relative, A-theoretic presentness requires at least two space-like separated things to be present simpliciter together – this co-presentness further implies the global, non-relative, non-conventional simultaneity of them. Yet, this implication clashes with the general view that there is no global, non-relative, non-conventional simultaneity in Minkowski space-time. In order to resolve this conflict, this paper explores the possibility that the non-solipsism of a non-relative, A-theoretic presentness (...) does not require at least two space-like separated things to be present simpliciter together. This can be done by holding exclusive disjunctivism –that mutually space-like separated things are present simpliciter exclusively disjunctively, and each one of them gets to be present simpliciter in a non-successive way (just like mutually time-like related things are present simpliciter exclusively disjunctively, and each one of them gets to be present simpliciter, but in a successive way). (shrink)

_ Source: _Volume 5, Issue 2, pp 290 - 322 Einstein's special theory, as interpreted by Herman Minkowski, suggests that an understanding of space and time requires the replacement of three-dimensional space and one dimensional time with a four-dimensional spacetime continuum, as a natural kind of thing with a characteristic, geometrical, structure. Issues of space and time in general, and of special relativity in particular, are not addressed in Bhaskar's _A Realist Theory of Science_, and their (...) treatment in subsequent realist writings has been patchy and indecisive. Some of Bhaskar's observations in later writings, including his defence of fundamental ontological asymmetries between space and time, and between present, past and future, appear incompatible with a relativistic perspective. Equally problematic, from a critical realist perspective, is the apparent ontological prioritisation of events, and causation as action-by-contact, within the Minkowski interpretation of special relativity. This paper argues that the four-dimensional spacetime continuum of special relativity occupies its own level at the base of the ontological hierarchy of natural kinds, providing lower-order components and laws for higher-order physical, chemical, biological and social structures. While providing no definitive reconciliation of relativistic and realist perspectives, this paper does suggest that A. N. Whitehead's idea of atomic events, or ‘actual occasions’, could pave the way for some possible future reconciliation. (shrink)