11 research outputs found
On the Possibility and Consequences of Negative Mass
We investigate the possibility and consequences of the existence of particles having negative relativistic masses, and show that their existence implies the existence of faster- than-light particles (tachyons). Our proof requires only two postulates concerning such particles: that it is possible for particles of any (positive, negative or zero) relativistic mass to collide inelastically with 'normal' (i.e. positive relativistic mass) particles, and that four-momentum is conserved in such collisions
Groups of worldview transformations implied by isotropy of space
Given any Euclidean ordered field, Q, and any 'reasonable' group, G, of (1+3)-dimensional spacetime symmetries, we show how to construct a model MG of kinematics for which the set W of worldview transformations between inertial observers satisfies W=G. This holds in particular for all relevant subgroups of Gal, cPoi, and cEucl (the groups of Galilean, PoincarĂŠ and Euclidean transformations, respectively, where câQ is a model-specific parameter orresponding to the speed of light in the case of PoincarĂŠ transformations).
In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set W of worldview transformations satisfies either WâGal, WâcPoi, or WâcEucl for some c>0. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are PoincarĂŠ transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of Q to the field of reals.
As part of this work, we also prove the rather surprising result that, for any G containing translations and rotations fixing the time-axis t, the requirement that G be a subgroup of one of the groups Gal, cPoi or cEucl is logically equivalent to the somewhat simpler requirement that, for all gâG: g[t] is a line, and if g[t]=t then g is a trivial transformation (i.e. g is a linear transformation that preserves Euclidean length and fixes the time-axis setwise)
Investigations of isotropy and homogeneity of spacetime in first-order logic
We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in Image 1, homogeneity of time. In turn, homogeneity of time implies homogeneity of space in general, and the converse also holds true in Image 2
An important innovation in our approach is that formulations of physical properties are simultaneously empirical and axiomatic (in the sense of first-order mathematical logic). In this case, for example, rather than presuppose the existence of spacetime metrics â together with all the continuity and smoothness apparatus that would entail â the basic logical formulas underpinning our work refer instead to the sets of (idealised) experiments that support the properties in question, e.g., isotropy is axiomatized by considering a set of experiments whose outcomes remain unchanged under spatial rotation. Higher-order constructs are not needed
Groups of worldview transformations implied by Einsteinâs special principle of relativity over arbitrary ordered fields
In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle; and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincare transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisovâs original four axioms within an intuitively simple, but strictly formal, first-order logic framework, and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisovâs theorem depends on the structure of quantities.
We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincare models of Borisovâs axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields.
As an application, we note that there is a model of Borisovâs axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning timeâs arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisovâs axioms and neither the assumption of timeâs arrow, nor the rejection of instantaneous motion, can eliminate them
Axiomatizing relativistic dynamics without conservation postulates
A part of relativistic dynamics (or mechanics) is axiomatized by simple and
purely geometrical axioms formulated within first-order logic. A geometrical
proof of the formula connecting relativistic and rest masses of bodies is
presented, leading up to a geometric explanation of Einstein's famous .
The connection of our geometrical axioms and the usual axioms on the
conservation of mass, momentum and four-momentum is also investigated.Comment: 21 pages, 7 figure
Twin Paradox and the logical foundation of relativity theory
We study the foundation of space-time theory in the framework of first-order
logic (FOL). Since the foundation of mathematics has been successfully carried
through (via set theory) in FOL, it is not entirely impossible to do the same
for space-time theory (or relativity). First we recall a simple and streamlined
FOL-axiomatization SpecRel of special relativity from the literature. SpecRel
is complete with respect to questions about inertial motion. Then we ask
ourselves whether we can prove usual relativistic properties of accelerated
motion (e.g., clocks in acceleration) in SpecRel. As it turns out, this is
practically equivalent to asking whether SpecRel is strong enough to "handle"
(or treat) accelerated observers. We show that there is a mathematical
principle called induction (IND) coming from real analysis which needs to be
added to SpecRel in order to handle situations involving relativistic
acceleration. We present an extended version AccRel of SpecRel which is strong
enough to handle accelerated motion, in particular, accelerated observers.
Among others, we show that the Twin Paradox becomes provable in AccRel, but it
is not provable without IND.Comment: 24 pages, 6 figure
No faster-than-light observers (GenRel)
We have previously verified, in the first order theory SpecRel of Special Relativity, that inertial observers cannot travel faster than light. We now prove the corresponding result for GenRel, the first-order theory of General Relativity. Specifically, we prove that whenever an observer m encounters another observer k (so that m and k are both present at some spacetime location x), k will necessarily be observed by m to be traveling at less than light speed
A Geometrical Characterization of the Twin Paradox and its Variants
The aim of this paper is to provide a logic-based conceptual analysis of the
twin paradox (TwP) theorem within a first-order logic framework. A geometrical
characterization of TwP and its variants is given. It is shown that TwP is not
logically equivalent to the assumption of the slowing down of moving clocks,
and the lack of TwP is not logically equivalent to the Newtonian assumption of
absolute time. The logical connection between TwP and a symmetry axiom of
special relativity is also studied.Comment: 22 pages, 3 figure
Does negative mass imply superluminal motion? An investigation in axiomatic relativity theory
Formalization of physical theories using mathematical logic allows us to discuss the assumptions on which they are based, and the extent to which those assumptions can be weakened. It also allows us to investigate hypothetical claims, and hence identify experimental consequences by which they can be tested. We illustrate the potential for these techniques by reviewing the remarkable growth in First Order Relativity Theory (FORT) over the past decade, and describe the current state of the art in this field. We take as a running case study the question âDoes negative mass imply superluminal motion?â, and show how a many-sorted first-order theory based on just a few intuitively obvious, but rigorously expressed, axioms allows us to formulate and answer this question in mathematically precise terms