A new derivation of the Minkowski metric

2015

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 Cℓ(ℜ 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 eightdimensional 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.

arXiv (Cornell University), 2015

The four dimensional spacetime continuum, as originally conceived by Minkowski, has become the default framework for describing physical laws. Due to its fundamental importance, there have been various attempts to find the origin of this structure from more elementary principles. In this paper, we show how the Minkowski spacetime structure arises naturally from the geometrical properties of three dimensional space when modeled by Clifford geometric algebra of three dimensions Cℓ(ℜ 3). We find that a time-like dimension along with the 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 value of this geometric approach is shown by the emergence of a fixed speed for light, the laws of special relativity and the form of Maxwell's equations, without recourse to any physical arguments.

Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension ict, with the unit imaginary producing the correct spacetime distance x 2 {c 2 t 2 , and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i~ffi ffiffiffiffiffiffi ffi {1 p , with the Clifford bivector i~e 1 e 2 for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e 1 and e 2 . We find that with this model of planar spacetime, using a twodimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.

arXiv (Cornell University), 2015

The four dimensional spacetime continuum, as originally conceived by Minkowski, has become the default framework for describing physical laws. Due to its fundamental importance, there have been various attempts to find the origin of this structure from more elementary principles. In this paper, we show how the Minkowski spacetime structure arises naturally from the geometrical properties of three dimensional space when modeled by Clifford geometric algebra of three dimensions Cℓ(ℜ 3). We find that a time-like dimension along with the 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 value of this geometric approach is shown by the emergence of a fixed speed for light, the laws of special relativity and the form of Maxwell's equations, without recourse to any physical arguments.

The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which " 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 he 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 of 1908. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted 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.

2015

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 structur...

2012

Following the development of the special theory of relativity in 1905, Minkowski proposed a unified space and time structure consisting of three space dimensions and one time dimension, with relativistic effects then being natural consequences of this spacetime geometry. In this paper, we illustrate how Clifford's geometric algebra that utilizes multivectors to represent spacetime, provides an elegant mathematical framework for the study of relativistic phenomena. We show, with several examples, how the application of geometric algebra leads to the correct relativistic description of the physical phenomena being considered. This approach not only provides a compact mathematical representation to tackle such phenomena, but also suggests some novel insights into the nature of time.

2015

The Minkowski formulation of special relativity reveals the essential four-dimensional nature of spacetime, consisting of three space and one time dimension. Recognizing its fundamental importance, a variety of arguments have been proposed over the years attempting to derive the Minkowski spacetime structure from fundamental physical principles. In this paper we illustrate how Minkowski spacetime follows naturally from the geometric properties of three dimensional Clifford space modeled with multivectors. This approach also generalizes spacetime to an eight dimensional space as well as doubling the size of the Lorentz group. This description of spacetime also provides a new geometrical interpretation of the nature of time.

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.

International Journal of Theoretical Physics, 2000

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cℓ 1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS) Cℓ 3 . STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA + of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA + are all that is needed to calculate physically measurable quantities (called measurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA + and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out timelike planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to the relative version of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra.