Causal Structures in Linear Spaces

2019

To a significant extent, the metrical and topological properties of spacetime can be described purely order-theoretically. The K relation has proven to be useful for this purpose, and one could wonder whether it could serve as the primary causal order from which everything else would follow. In that direction, we prove, by defining a suitable order-theoretic boundary of K(p), that in a K-causal spacetime, the manifold-topology can be recovered from K. We also state a conjecture on how the chronological relation I could be defined directly in terms of K.

We present a deductive theory of space-time which is realistic, objective, and relational. It is realistic because it assumes the existence of physical things endowed with concrete properties. It is objective because it can be formulated without any reference to knowing subjects or sensorial fields. Finally, it is relational because it assumes that space-time is not a thing, but a complex of relations among things. In this way, the original program of Leibniz is consummated, in the sense that space is ultimately an order of coexistents, and time is an order of successives. In this context, we show that the metric and topological properties of Minkowskian space-time are reduced to relational properties of concrete things. We also sketch how our theory can be extended to encompass a Riemannian space-time.

Bulletin of the Australian Mathematical Society, 1991

An axiomatic approach to relativity theory is proposed which is based entirely on a single reflexive relation called signalling. This generalises and simplifies earlier schemes of Kronheimer and Penrose and of Carter which are based on causality relations. Many of the features of space-times, such as photon paths and topology, have their counterparts in general signal spaces.

We present a deductive theory of space-time which is realistic, objective, and relational. It is realistic because it assumes the existence of physical things endowed with concrete properties. It is objective because it can be formulated without any reference to cognoscent subjects or sensorial fields. Finally, it is relational because it assumes that space-time is not a thing but a complex of relations among things. In this way, the original program of Leibniz is consummated, in the sense that space is ultimately an order of coexistents, and time is an order of succesives. In this context, we show that the metric and topological properties of Minkowskian space-time are reduced to relational properties of concrete things. We also sketch how our theory can be extended to encompass a Riemmanian space-time.

The article discusses the concept of causality. The issues of the passage of time determining the sequence of events in the chain of cause-consequence relationship, quantization of continuous physical processes into events, the process of transformation of cause into effect and the topology of space within the elementary link of cause-consequence relationship are considered. The problems of connectivity of the event space are analyzed. The problems of modern physics and the expediency of returning to the idea of ether as the basis of matter in all its manifestations are discussed.

The Frontiers Collection, 2015

The distinction between a theory's kinematics and its dynamics, that is, between the space of physical states it posits and its law of evolution, is central to the conceptual framework of many physicists. A change to the kinematics of a theory, however, can be compensated by a change to its dynamics without empirical consequence, which strongly suggests that these features of the theory, considered separately, cannot have physical significance. It must therefore be concluded (with apologies to Minkowski) that henceforth kinematics by itself, and dynamics by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. The notion of causal structure seems to provide a good characterization of this union.

2012

We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, when restricted to compact spaces satisfying a simple separability condition, it is a distance. In particular, this provides a quantitative, covariant way of describing how close two manifolds with Lorentzian metrics are, or how manifoldlike a causal set is.

Time is a monotonic strictly increasing single valued real parameter that exists in spacetime. Here we consider an observer in his rest frame belonging to the Minkowskian spacetime.The order of the sequence of events on his World line is strictly preserved in the sense that the order of the sequence of events remains invariant under Lorentz transformations in Minkowskian spacetime: because the world line of the observer is always time-like. 1.INTRODUCTION Time is awake when all things sleep. Time stands straight when all things fall. Time shuts in all and will not be shut. Is,was,and shall be are Time`s children. O Reasoning, be witness, be stable. [1] VYASA,the Mahabharata [ca.A.D 400] This universe has basic temporal structure. The fundamental nature of TIME in relation to human consciousness is evident as soon as we think that our judgements related to time and events in time appear themselves to be IN TIME. Our analysis concerning SPACE do not appear in any obvious sense to be IN SPACE. But SPACE seems to be appeared to us all of a piece, whereas TIME comes to us only BIT by BIT. The Past exists only in our memory and the Future is hidden from us. Only the Present is the physical reality experienced by us. Thus TIME is always an ONE-WAY membrane. We cannot go from Present to the Past; while one can perform backward and forward motion in SPACE. The free mobility in SPACE leads to the idea of transportable rigid rods. The absence of free mobility in TIME leads to the concept ONE-WAY membrane TIME is a monotonic strictly increasing single valued real parameter corresponding to a non - spatial dimension represented by a straight line in Minkowskian spcetime and the SPACE is three dimensional. Minkowski unified space and time to a single entity called spacetime which is absolute. Einstein used the concept of spacetime for constructing spacetime geometry so that physics becomes part and parcel of geometry in Minkowskian spacetime. Einstein introduced the concept square of the distance between two events ds2 = -dx2-dy2-dz2+dT2 [2] here ds is distance between two events P(x,y,z,T) and Q(x+dx,y+dy,z+dz,T+dT).If ds2 is greater than zero the separation between events is called time-like;if ds2= 0, the separation between events is called null-like leading to the concept of Light Cone Structure in Special Theory of Relativity([3] &[4]) and if ds2 is less than zero, the separation between events is called space-like. Time-like events are causally connected and also null-like events are causally connected; there is no causal connection between events separated by space-like interval. All real particles trace curves in space time. These curves are called time-like curves. Light rays travel along null curve in spacetime.Here we are concerned only with time-like curve so that the order of sequence of occurrence of events shall be the same for every observer under admissible co-ordinates transformations. The world view proposed by Minkowski is often termed as Minkowskian spacetime [5] or M-space. It is said to have a (3+1) description of spacetime. Here “3” represents the Three Dimensional Euclidean space and “1” the One Dimensional time. We introduce spacetime co-ordinates to order events. In Mspace, the co-ordinates of an event can be represented by an ordered set of four real numbers, <x1,x2,x3,x4>. Here the numbers x1, x2, x3 and x4 are taken to be PURE real numbers. 1,2,3,4 are superscripts used to specify the co-ordinates. It is always convenient to consider a Lorentz frame with orthonormal basis vectors e1,e2,, e3, and e4 [1]. Relative to the origin of this frame the time-like worldline of a particle with real non-zero restmass has a co-ordinates description

Foundations of Science, 2020

axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and topology in a fully operational manner. Curiously enough, primitive concepts such as points, spaces, straight lines, planes are all defined within our formalism. Our construction breaks down with the usual literature, as our axioms have deep connection with nature. Besides that, we hope this operational approach could also be of pedagogical interest.

Communications in Mathematical Physics, 1990

This is the first of a planned series of investigations on the theory of ordered spaces based upon four axioms. Two of these, the order (1.1.1) and the local structure (II.5.1) axioms provide the structure of the theory, and the other two [the identification (1.1.11) and cone (1.2.7) axioms] eliminate pathologies or excessive generality. In the present paper the axioms are supplemented by the nontriviality conditions (1.1.9) and a regularity property (II.4.2). The starting point is a nonempty set M and a family of distinguished subsets, called light rays, which are totally ordered. The order axiom provides the properties of this order. Positive and negative cones at a point are defined in terms of increasing and decreasing subsets and are used to extend the total order on the light rays to a partial order over all of M. The first significant result is the polygon lemma (1.2.3) which provides an essential constructive tool. A non-topological definition is found for the interiors of the cones; it leads to a "more homogeneous" partial order relation on M. In Sect. II, subsets called D-sets (Def. Π.2.2), possessing certain desirable properties, are studied. The key concept of perpendicularity of light rays is isolated (Def. II.3.1) and used to derive the basic "separation properties," provided that the interiors of cones are nonempty. It is shown that, in a D-set, "good" properties of one cone can be transported along light rays, so that the structure of a D-set is homogeneous. In particular, if one cone has nonempty interior, so have all others. However, the existence of even one cone with nonempty interior does not follow from the axioms, but has to be imposed as an additional regularity condition. The local structure axiom now states that every point lies in a regular D-set. It is proved that the family of regular D-sets is closed under finite intersections. The order topology is defined as the topology which has this family as a base. This topology is Hausdorlf, and coincides with the usual topology for Minkowski spaces. 0. Introduction Ordered spaces play a central role in theoretical physics. The ordering of time, for instance, induces an ordering of the space of events by the future light cone.