Time As a Geometric Property of Space

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.

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: General Physics, 2015

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 an...

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.

Journal of Physics Communications, 2023

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 fourdimensional structure is a natural property of physical three-dimensional space, if modeled with Clifford geometric algebra Cl(R3). 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.

arXiv: History and Philosophy of Physics, 2023

In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space and dynamics, we recall briefly Minkowski's approach in 1910 implementing the nowadays commonly used 4-vector calculus and related tensorial representations as well as Klein's 1910 paper on the geometry of the Lorentz group. To include microscopic representations, we discuss few aspects of Wigner's and Weinberg's 'boost' approach to describe 'any spin' with respect to its reductive Lie algebra and coset theory, and we relate the physical identification to objects in $P^{5}$ based on the case $(1,0)\oplus(0,1)$ of the electromagnetic field. So instead of following this -- in some aspects -- special and misleading 'old' representation theory, based on 4-vector calculus and tensors, we provide and use an alternat...

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.

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

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.