We begin by reviewing the special relativistic properties of a rotating system of coordinates. These were considered by Einstein in the spinning disk thought experiment during his initial considerations of time and length changes in gravity. Using a novel extension of this approach, we identify a variation in an object’s internal energy within a gravitational field, through employing the equivalence principle. We then find an additional gravitational lensing and redshift prediction over and above current theory from the Schwarzschild metric. Implications for rapid clumping in the early universe and ultramassive blackhole formation are also considered. Correlations to recent James Webb findings are also discussed. Experiments to test this principle in the terrestrial domain are also proposed.

We develop an analytical Stackelberg game framework for optimal resource allocation in a sequential attacker–defender setting with a finite set of assets and probabilistic attacks. The defender commits to a mixed protection strategy, after which the attacker best-responds via backward induction. Closed-form expressions for equilibrium protection and attack strategies are derived for general numbers of assets and defensive resources. Necessary constraints on rewards and costs are established to ensure feasibility of the probability distributions. Three distinct payoff regimes for the defender are identified and analysed. An eight-asset numerical example illustrates the equilibrium structure and reveals a unique Pareto-dominant attack configuration.

We present a new framework for creating a quantum version of a classical game, based on Fine's theorem. This theorem shows that for a given set of marginals, a system of Bell's inequalities constitutes both necessary and sufficient conditions for the existence of the corresponding joint probability distribution. Using Fine's theorem, we re-express both the player payoffs and their strategies in terms of a set of marginals, thus paving the way for the consideration of sets of marginals -- corresponding to entangled quantum states -- for which no corresponding joint probability distribution may exist. By harnessing quantum states and employing Positive Operator-Valued Measures (POVMs), we then consider particular quantum states that can potentially resolve dilemmas inherent in classical games.

The four dimensional spacetime continuum, as first conceived by Minkowski, has become the dominant framework within which to describe physical laws. In this paper, we show how this four-dimensional structure is a natural property of physical three-dimensional space, if modeled with Clifford geometric algebra C ℓ ( R 3 ) . We find that Minkowski spacetime can be embedded within a larger eight dimensional structure. This then allows a generalisation of the invariant interval and the Lorentz transformations. Also, with this geometric oriented approach the fixed speed of light, the laws of special relativity and a generalised form of Maxwell’s equations, arise naturally from the intrinsic properties of the algebra without recourse to physical arguments. We also find new insights into the nature of time, which can be described as two-dimensional. Some philosophical implications of this approach as it relates to the foundations of physical theories are also discussed.

Referring to the behavior of accelerating objects in special relativity, and applying the principle of equivalence, one expects that the coordinate acceleration of point masses under gravity will be velocity dependent. Then, using the Schwarzschild solution, we analyze the similar case of masses moving on timelike geodesics, which reproduces a little known result by Hilbert from 1917, describing this dependence. We find that the relativistic correction term for the acceleration based on general relativity differs by a factor of two from the simpler acceleration arguments in flat space. As we might expect from the general theory, the velocity dependence can be removed by a suitable coordinate transformation, such as the Painlev{e}-Gullstrand coordinate system. The validity of this approach is supported by previous authors who have demonstrated vacuum solutions to general relativity producing true flat space metrics for uniform gravitational fields. We suggest explicit experiments could be undertaken to test the property of velocity dependence.

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein-Podolsky-Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.

The principle of equivalence is used to argue that the known law of decreasing acceleration for high speed motion, in a low acceleration regime, produces the same result as found for a weak gravitational field, with subsequent implications for stronger fields. This result coincides with Hilbert's little explored equation of 1917, regarding the velocity dependence of acceleration under gravity. We derive this result, from first principles exploiting the principle of equivalence, without need for the full general theory of relativity.

This paper presents an approach to classification of substances based on their terahertz spectra. We use geometric algebra to provide a concise mathematical means for attacking the classification problem in a coordinate-free form. For the first time, this allows us to perform classification independently of dispersion and, hence, independently of the transmission path length through the sample. Finally, we validate the approach with experimental data. In principle, the coordinate-free transformation can be extended to all types of pulsed signals, such as pulsed microwaves or even acoustic signals in the field of seismology. Our source code for classification based on geometric algebra is publicly available at: https://github.com/swuzhousl/Shengling-zhou/blob/geometricalgebra-classifier/GAclassifier/.

The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which {\it `flows equably without relation to anything external'}. In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that they could provide a unified representation of space and time. With the publishing of Einstein's theory of special relativity these ideas then lead to the generally accepted Minkowski spacetime formulation in 1908. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted rather an approach using four-vectors. The Minkowski framework is indeed found to provide a versatile formalism for describing the relationship between space and time in accordance with Einstein's relativistic principles, but nevertheless fails to provide more fundamental insights into the nature of time itself. In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.

In this paper we develop a framework allowing a natural extension of the Lorentz transformations. To begin, we show that by expanding conventional four-dimensional spacetime to eight-dimensions that a natural generalization is indeed obtained. We then find with these generalized coordinate transformations acting on Maxwell's equations that the electromagnetic field transformations are nevertheless unchanged. We find further, that if we assume the absence of magnetic monopoles, in accordance with Maxwell's theory, our generalized transformations are then restricted to be the conventional ones. While the conventional Lorentz transformations are indeed recovered from our framework, we nevertheless provide a new perspective into why the Lorentz transformations are constrained to be the conventional ones. Also, this generalized framework may assist in explaining several unresolved questions in electromagnetism as well as to be able to describe quasi magnetic monopoles found in spin-ice systems.

There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, and quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of 19th century science was to suitably describe vectors in 3-D space. This situation also has the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford's multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science.

We demonstrate and validate Geometric Algebra (GA) based terahertz (THz) signal classification of various powders in tablet form of various thicknesses, and compare the results with a conventional Support Vector Machine (SVM) approach. By using geometric algebra we can perform classification independently of dispersion and hence independently of the transmission path length through the sample. In principle, it may be possible to extend the GA coordinate-free transformation to other types of pulsed signals, such as pulsed microwaves or even acoustic signals in such fields as seismology. The classifier is available for download at Github, https://github.com/swuzhousl/Shengling-zhou/blob/geometric-algebra-classifier/GAclassifier/.

A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the outcome of such games. We demonstrate how the standard version of a quantum game can be analysed as a non-factorizable game and determine the limitations of our approach.

The four dimensional spacetime continuum, as originally conceived by Minkowski, has become the default framework within which to describe physical laws. Due to its fundamental nature, there have been various attempts to derive this structure from more fundamental physical principles. In this paper, we show how the Minkowski spacetime structure arises directly from the geometrical properties of three dimensional space when modeled by Clifford geometric algebra of three dimensions $ C\ell(\Re^3) $. We find that a time-like dimension, as well as three spatial dimensions, arise naturally, as well as four additional degrees of freedom that we identify with spin. Within this expanded eight-dimensional arena of spacetime, we find a generalisation of the invariant interval and the Lorentz transformations, with standard results returned as special cases. The power of this geometric approach is shown by the derivation of the fixed speed of light, the laws of special relativity and the form of Maxwell's equations, without any recourse to physical arguments. We also produce a unified treatment of energy-momentum and spin, as well as predicting a new class of physical effects and interactions.

Following Minkowski's formulation of special relativity, it is generally accepted that we live in a four-dimensional world consisting of three space and one time dimension. Due to its fundamental importance, a variety of arguments have been proposed over the years attempting to derive this spacetime structure from underlying physical principles. In our approach, we show how Minkowski spacetime arises from the geometrical properties of three dimensional space. We demonstrate this through modeling physical space with Clifford's geometric algebra of three dimensions. We indeed find using this representation that a time-like dimension arises naturally within this space but also extends spacetime to eight dimensions through incorporating four spin degrees of freedom. This expanded arena of spacetime produces a generalized group of Lorentz transformations and provides a natural description of fundamental particles. Nearly all standard results are returned in this expanded structure and so its main achievement is that it provides an efficient alternative formalism to the conventional four-vector formalism but we were also able to identify several areas where new physics may be indicated.

Following Minkowski's formulation of special relativity, it is generally accepted that we live in a four-dimensional world consisting of three space and one time dimension. Due to its fundamental importance, a variety of arguments have been proposed over the years attempting to derive this spacetime structure from underlying physical principles. In our approach, we show how Minkowski spacetime arises from the geometrical properties of three dimensional space. We demonstrate this through modeling physical space with Clifford's geometric algebra of three dimensions. We indeed find using this representation that a time-like dimension arises naturally within this space but also extends spacetime to eight dimensions through incorporating four spin degrees of freedom. This expanded arena of spacetime produces a generalized group of Lorentz transformations and provides a natural description of fundamental particles. Nearly all standard results are returned in this expanded structure and so its main achievement is that it provides an efficient alternative formalism to the conventional four-vector formalism but we were also able to identify several areas where new physics may be indicated.

Understanding the nature of time remains a key unsolved problem in science. Newton in the Principia asserted an absolute universal time that {\it `flows equably'}. Hamilton then proposed a mathematical unification of space and time within the framework of the quaternions that ultimately lead to the famous Minkowski formulation in 1908 using four-vectors. The Minkowski framework is found to provide a versatile formalism for describing the relationship between space and time in accordance with relativistic principles, but nevertheless fails to provide deeper insights into the physical origin of time and its properties. In this paper we begin with a recognition of the fundamental role played by three-dimensional space in physics that we model using the Clifford algebra multivector. From this geometrical foundation we are then able to identify a plausible origin for our concept of time. This geometrical perspective also allows us to make a key topological distinction between time and space, with time being a point-like quantity. The multivector then allows a generalized unification of time and space within a Minkowski-like description.

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that Cl (R-3) provides a particularly versatile algebraic framework.