Gergely Székely

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we “derive” an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

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 the 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.

The Existence of Superluminal Particles is Consistent with the Kinematics of Einstein's Special Theory of Relativity

Within an axiomatic framework of kinematics, we prove that the existence of faster than light (FTL) particles is logically independent of Einsteins special theory of relativity. Consequently, it is consistent with the kinematics of special relativity that there might be faster than light particles.

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.

The existence of superluminal particles is consistent with relativistic dynamics

Abstract Within an axiomatic framework, we prove that the existence of faster than light (FTL) particles is consistent with (does not contradict) the dynamics of Einsteins special relativity. The proof goes by constructing a model of relativistic dynamics where FTL particles can move with arbitrary speeds. To have a complete picture, we not only construct an appropriate model but explicitly list all the basic assumptions (axioms) we use.

Faster than light motion does not imply time travel

Seeing the many examples in the literature of causality violations based on faster-than- light (FTL) signals one naturally thinks that FTL motion leads inevitably to the possibility of time travel. We show that this logical inference is invalid by demonstrating a model, based on (3+1)-dimensional Minkowski spacetime, in which FTL motion is permitted (in every direction without any limitation on speed) yet which does not admit time travel. Moreover, the Principle of Relativity is true in this model in the sense that all observers are equivalent. In short, FTL motion does not imply time travel after all.

Existence of faster than light signals implies hypercomputation already in special relativity

Within an axiomatic framework, we investigate the possibility of hypercomputation in special relativity via faster than light signals. We formally show that hypercomputation is theoretically possible in special relativity if and only if there are faster than light signals.

Comparing Relativistic and Newtonian Dynamics in First-Order Logic

In this paper we introduce and compare Newtonian and relativistic dynamics as two theories of first-order logic (FOL). To illustrate the similarities between Newtonian and relativistic dynamics, we axiomatize them such that they differ in one axiom only. This one axiom difference, however, leads to radical differences in the predictions of the two theories. One of their major differences manifests itself in the relation between relativistic and rest masses, see Thms. 5 and 6.

A note on ‘Einstein's special relativity beyond the speed of light by James M. Hill and Barry J. Cox’

We show that the transformations Hill and Cox introduce, between inertial observers moving faster than light with respect to each other, are consistent with Einsteins principle of relativity only if the space–time is two dimensional.

CLOSED TIMELIKE CURVES IN RELATIVISTIC COMPUTATION

In this paper, we investigate the possibility of using closed timelike curves (CTCs) in relativistic hypercomputation. We introduce a wormhole based hypercomputation scenario which is free of the common worries, such as the blueshift problem. We also discuss the physical reasonability of our scenario, and why we cannot simply ignore the possibility of the existence of spacetimes containing CTCs.