On Application of Algebra of Quaternions II

International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 341 ISSN 2250-3153

On application of algebra of quaternions II

Emran Sasi Althabit*, Ismail.M.Mohamed **

* Higher Institute for water Affairs ** Academic of Engineering and Medical Sciences

DOI: 10.29322/IJSRP.11.02.2021.p11041 http://dx.doi.org/10.29322/IJSRP.11.02.2021.p11041

Abstract- We introduced algebraic formulations of time as vector producing the modern description of spacetime, as a four- quaternions, Clifford Algebras , we deduce the Lorentz transform- vector ation from the Minkowski Space. 푋 = [푡, 푥⃗] (1.3) Defining a involution 푋∗ = [푡, −푥⃗] we then Index Terms- Clifford Algebras, K-algebra , multivector, Affine produce the invariant distance vector space , Minkowski Space. 푋푋∗ = [푡, 푥⃗][푡, −푥⃗] = 푡2 − 푥2 (1.4)

Comparing these two descriptions we can see that they I. ALGEBRAIC FORMULATIONS OF TIME provide some significant differences. To begin with, Hamilton’ s n order to proceed with our approach, we now wish to description views time as an intrinsic part of the description of I algebraically model three- dimensional physical space. The three-dimensional space. On the other hand, with the Minkowski distinct approaches are commonly utilized to model various formulation, the immed-iate implication is that time is an physical systems such as three-vectors,four-vectors, matrices, additional Euclidean-type dimension. Although this assu-mption complex numbers, quaternions and complex spinors, depending is qualified by the fact that time contributes an opposite sign to the on the specific application. This fact is indeed quite surprising, in metric distance and so distinct from a regular four dimensional that, one of the specific goals of nineteenth century science was to Cartesian vector. Note that the octonions, being the generalization find the correct algebra for physical space [6]. The objective was of quaternions, have also been considered as an expanded arena initially led by Hamiltonn who produced the quaternion algebra for spacetime. through generalizing the two-dimensional complex numbers to It is an historical fact that the vector quaternions were found three dimensions. Indeed, due to the general success of complex difficult to work with and not suitable to describe Cartesian numbers in algebraically describing the properties of the plane vectors and were replaced by the Gibbs vector system in use today Hamilton reasoned that quaternions should therefore properly [6]. The reason for Hamilton’s failed attempt to algebraically describe the algebra of three-dimensional space. This then lead to describe three-dimensional space, is that by generalizing the the first attempt at a rigorous mathematical definition of a unified complex numbers of the plane to three dimensions he actually only produced the rotational algebra for three dimensions. space and time.Hamilton wrote the quaternion as 푞 = 푡 + 푥1푖 + 3 To properly describe three-dimensional space we further 푥2푗 + 푥3푘 (1.1) where 푥1 , 푥2 , 푥3 ∈ ℝ and the three basis vectors are subject to the well known quaternionic relations 푖2 = need to generalize the quaternions to include a true Cartesian 푗2 = 푘2 = −1 vector component [13]. This generalization of quaternions was in fact achieved by Clifford with the eight dimensional Clifford Hamilton’s quaternions form a 4-D associative normed division 3 algebra over the real numbers represented by ℍ. Hamilton defined algebra over three dimensions of 퐶ℓ(ℝ )[14]

푥⃗ = 푥1푖 + 푥2푗 + 푥3푘 as a vector quaternion to take the role of Cartesian vectors in order to describe spatial vectors, and then also 2. Clifford Algebras Action the group on quaternions(2-1) proposed, around 50 years before Minkowski, that if the scalar “푡” We define an action of the group of unit quaternions, 푆푈(2), on was identified with time then the quaternion can be used as a 3 3 representation for a unified four-dimensional spacetime. Hamilton ℝ . For this, we use the fact that ℝ can be identified with the pure quaternions in ℍ, namely, the quaternions of the form 푥⃗ = 푥1푖 + stating “Time is said to have only one dimension, and space to 3 have three dimensions. ... 푥2푗 + 푥3푘 , where (푥1, 푥2, 푥3) ∈ ℝ . Then, we define the action 3 −1 ∗ The mathematical quaternion partakes of both these of 푆푈(2) over ℝ by 푍 . 푋 = 푍푋푍 = 푍푋푍 (2.1) elements; it may be said to be ‘time plus space’, or ‘space plus where 푍 ∈ 푆푈(2) and 푋 is any pure quaternion. [9] time’: and in this sense it has,or at least involves a reference to, Now, it turns out that the map 휌푍 where −1 four dimensions”. Indeed squaring the quaternion we find 휌푧(푋) = 푍푋푍 is indeed a rotation, and that the map 휌: 푍 ↦ 휌푧is 푞2 = 푡2 − 푥2 + 2푡푥 (1.2) the scalar component 푡2 − 푥2 thus a surjective homomorphism, 휌 ∶ 푆푈(2) → 푆푂(3), whose kernel producing the invariant spacetime distance. is {−1,1}, where 1 denotes the multiplicative unit quaternion. Minkowski, after indeed considering the quaternions, but We can also define an action of the group 푆푈(2) × 푆푈(2) over 4 4 viewing them as too restrictive for describing spacetime, chose ℝ by identifying ℝ with the quaternions. In this case, rather to extend the Gibbs-Heaviside thre-vector system with the (푌, 푍 ). 푋 = 푌 푋 푍∗ (2.2) addition of a time coordinate to create a four-component

This publication is licensed under Creative Commons Attribution CC BY. http://dx.doi.org/10.29322/IJSRP.11.02.2021.p110XX www.ijsrp.org International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 342 ISSN 2250-3153 where (푌, 푍 ) ∈ 푆푈(2) × 푆푈(2) and 푋 ∈ ℍ is any quaternion. identity element, 1퐼퐴so that 퐼퐴 ∙ 푎 = 푎 ∙ 퐼퐴 = 푎, for all 푎 ∈ 퐴. ∗ Then, the map 휌푌,푍∗is a rotation where 휌푌,푍∗(푋) = 푌 푋 푍 ,and the Given two 퐾-algebras 퐴 and 퐵, a 퐾-algebra homomorphism, ℎ ∶ map 휌: (푌, 푍 ) ↦ 휌푌,푍∗is a surjective homomorphis,휌 ∶ 푆푈(2) × 퐴 → 퐵, is a linear map that is also a ring homomorphism, with 푆푈(2) → 푆푂(4), whose kernel is {(1,1), (−1. −1)}. ℎ(퐼퐴) = 퐼퐵. For example, the ring, 푀푛푛(퐾) , of all 푛 × 푛 matrices over a field, Thus, we observe that for 푛 = 2, 3, 4, the rotations in 푆푂(푛) 퐾, is a 퐾-algebra. can be realized via the linear action of some group (the case n = 1 Tensor products( 2-3) is trivial, since 푆푂(1) = {1, −1}. It is also the case that the action Let 퐸, 퐹 be two vector spaces over the field K (퐾 = ℝ, ℂ). of each group can be somehow be described in terms of Consider the (infinite) direct sum multiplication in some larger algebra “containing” the original 푇(퐸, 퐹) =⊕(푒,푓)∈퐸×퐹 퐾 (2.4) vector space ℝ푛 (ℂ for 푛 = 2, ℍ for 푛 = 3, 4). However, these i.e., A tensor product of 퐸 and 퐹 is a pair, groups appear to have been discovered in an ad hoc fashion, and (퐸 ⊗ 퐹,⊗), where 퐸 ⊗ 퐹 is a 퐾-vector space and ⊗ : 퐸 × 퐹 → there does not appear to be any universal way to define the action 퐸 ⊗ 퐹 is a bilinear map. of these groups on ℝ푛 . It would certainly be nice if the action was Proposition.1. For any bilinear map 휙 ∶ 퐸 × 퐹 → 퐺 there exists always of the form 푍 . 푋 = 푍푋푍−1(= 푍푋푍∗) a unique linear A systematic way of constructing groups realizing rotations map 훷 ∶ 퐸 ⊗ 퐹 → 퐺 such that the diagram below is in terms of linear action, using a uniform notion of action, does commutative. [15] exist. Such groups are the spinor groups, to be described in the following paragraphs. We explained in Paragraph.1how the rotations in 푆푂(3) can be realized by the linear action of the group of unit quaternions, 푆푈(2), on ℝ3, and how the rotations in 푆푂(4) can be realized by 4 the linear action of the group 푆푈(2) × 푆푈(2) on ℝ .The main reasons why the rotations in SO(3) can be represented by unit quaternions are the following: The vector space 퐸 ⊗ 퐹 is defined up to isomorphism. The vectors 3 (1) For every nonzero vector 푢⃗⃗⃗⃗ ∈ ℝ , the reflection 푠푢⃗⃗⃗about the 푢⃗⃗ ⊗ 푣⃗, where 푢⃗⃗ ∈ 퐸 and 푣⃗ ∈ 퐹 , generate 퐸 ⊗ 퐹 . hyperplane Perpendi-cular to 푢⃗⃗ is represented by the map 푣 ↦ CLIFFORD’S DESCRIPTION OF SPACE (2-4) −푢푣푢−1 (2.3) here 푢⃗⃗ and 푣⃗ are viewed as pure quaternions in ℍ A Clifford geometric algebra 퐶ℓ(ℝ푛) defines an associative real 3 (i.e., if 푢⃗⃗ = (푢1, 푢2, 푢3) , then view algebra over 푛 dimensions and in three dimensions 퐶ℓ(ℝ )is 푢⃗⃗, as 푢1푖 + 푢2푗 + 푢3푘, and similarly for 푣⃗). eight-dimensional [14]. In this case we can adopt the three (2) The group SO(3) is generated by the reflection. quantities 푒1, 푒2, 푒3for basis vectors that are defined to The group of unit quaternions 푆푈(2) turns out to be anticommute in the same way as Hamilton’s quaternions, but isomorphic to the spinor group unlike the quaternions these quantities square to positive one, that 3 2 2 2 푆푝푖푛(3). Because Spin(3) acts directly on ℝ , is 푒1 = 푒2 = 푒3 = 1. Also, similar to quaternions, we the representation of rotations in 푆푂(3)by elements of 푆푝푖푛(3) can combine scalars and the various algebraic components into a may be viewed as more natural than the representation by unit single number, called in this case a multivector quaternions. The group S푈(2) × 푆푈(2) turns out to be 푋 = 푡 + 푥1푒1 + 푥2푒2 + 푥3푒3 + 푛1푒2푒3 + 푛2푒3푒1 + 푛3푒1푒2 isomorphic to the spinor group Spin(4), but this isomorphism is + 푏푒1푒2푒3 (2.5) less obvious. In summary, we are going to define a group 푆푝푖푛(푛) where 푡, 푥1, 푥2, 푥3, 푛1, 푛2, 푛3, 푏 ∈ ℝ. Now define 푗 = 푒1푒2푒3we representing the rotations in 푆푂(푛), for any ≥ 1 , in the sense that find the relation 푗푒1 = 푒2푒3, 푛 there is a linear action of 푆푝푖푛(푛) on ℝ which induces a 푗푒2 = 푒3푒1 and 푗푒3 = 푒1푒2, which allows us to write surjective homomorphism, 휌: Spin(푛) → 푆푂(푛), whose kernel 푋 = 푡 + 푥⃗ + 푛⃗⃗푗 + 푏푗 (2.6) 푛 is {– 1,1}1g. Furthermore, the action of Spin(푛) on ℝ is given in With vectors 푥⃗ = 푥1푒1 + 푥2푒2 + 푥3푒3 and 푛⃗⃗ = 푛1푒1 + 푛2푒2 + terms of multiplication in an algebra, 푙푛 , containing Spin(푛), and 푛3푒3 in which ℝ푛 is also embedded. It turns out that as a bonus, for 푛 ≥ It can be shown that quaternions are isomorphic to the even 3, the group Spin(푛) is topologically simpler than 푆푂(푛), since subalgebra of the multivector, with the mapping 푖 ↔ 푒2푒3, Spin(푛)is simply connected, but 푆푂(푛) is not. By being astute, we 푗 ↔ 푒1푒3, 푘 ↔ 푒1푒2 and the Gibbs vector can be replaced by the can also construct a group, Pin(푛), and a linear action of Pin(푛) vector component of the multivector. Thus, within Clifford’s on ℝ푛that induces a surjective homomorphism, 휌 ∶ Pin(푛) → system we can absorb the quaternion as 푂(푛) , whose kernel is {– 1,1}. The difficulty here is the presence 푞 = 푡 + 푗푛⃗⃗ and a Gibbs vector 푥⃗ into of the negative sign in (2). We will see how Atiyah, Bott and a unified system. Shapiro circumvent this problem by using a “twisted adjoint We define Clifford conjugation on a multivector 푋 as action,” as opposed to the usual adjoint action (where 푣 ↦ 푋̅ = 푡 − 푥⃗ − 푛⃗⃗푗 + 푏푗 (2.7) 푢푣푢−1). We define the amplitude squared of a multivector 푋 as |푋|2 = 푋푋̅ Definition (2-2) Given a field, 퐾, a 퐾-algebra is a 퐾-vector that gives space, 퐴, together with a bilinear operation, ∙ : 퐴 × 퐴 → 퐴, |푋|2 = 푡2 − 푥2 + 푛2 − 푏2 + 2푗(푡푏 − 푥⃗ ∙ 푛⃗⃗) called multiplication, which makes 퐴 into a ring with unity, 퐼퐴 . (2.8) This means that is associative and that there is a multiplicative

Clifford conjugation that produces the multivector amplitude turns 3. The Minkowski spacetime out to be the only viable definition for the metric as it is the only A few years later to Einstein’s seminal paper of 1905 option that produces a commuting resultant and is thus an element introduced a revolutionary new description of space and time , of the center of the algebra. This is essential as we require the Minkowski showed that these phenomena could be simply metric distances to be isotropic in space in order to be generally interpreted as the geometrical properties of a four-dimensional consistent with the principles of relativity. Once again we can space-time continuum subject to the metric observe the required invariant distance 푑푠2 = 푐2푑푡2 − 푑x2 2 2 2 2 2 2 2 푡 − 푥 appearing in the metric. = 푐 푑푡 − 푑푥1 − 푑푥2 − 푑푥3 (3.1) An important point to note for the multivector, is that in order Where 2 2 2 2 to produce a meaningful metric, which consists of a combination 푑x = 푑x ∙ 푑x = 푑푥1 + 푑푥2 + 푑푥3 of its various elements, then all of these components must be is the contribution from three spatial dimensions and 푡 is the measured in the same units, as shown in Equation (8). time in this reference frame, where we have assumed units such Now, beginning from 1983, the General Conference on that the speed of light 푐 = 1. Weights and Measures (CGPM) decided that the speed of light should be assumed constant and that a meter was then the distance In the Minkowski formulation we can write a space-time event as traveled by light in a specified time interval equal to 1/c s [15].The the four vector 푋 = [푡, v⃗⃗]푇, where the corresponding event for CGPM defines the speed of light as a universal constant 푐 = other inertial frames is found through the Lorentz transformation 299792458 m/s ≈ 3 × 108m/s. Hence both time and distance 푋′ = Λ푋 where are now measured in units of seconds, and it is therefore natural to Λ is a 4 ×4 real matrix, which preserves the metric distance in Eq. adopt these units for all components of the multivector. Distances (3.1). We reiterate these elementary results to illustrate firstly the typically measured in meters therefore appear in the multivector unified nature of space and time, and secondly, that the metric has with the conversion 푥⃗ → 푥⃗/푐 and so are in units of seconds. a mixed signature, as shown in Eq. (3.1). Interestingly, from the perspective of the multivector, c is Now, due to the fundamental importance of Minkowski simply taking the role of a units conversion factor, and so therefore spacetime, a variety of arguments have been proposed over the the value of 푐 is obviously invariant between observers. Hence one years seeking to derive this spacetime structure from more confusion regarding time could arise due to the poor selection of elementary considerations, such as the consistency of Newton’s units, as in order to properly relate space and time they should be First Law, spacetime isotropy with non-instantaneous interaction measured in the same units, as now carried out by CGPM. propagation [31,20], a probability distribution resulting from The role of the various terms in the metric can be understood quantum fluctuations of the spacetime metric, the requirements for from several perspectives. Geometrically they refer to the four the existence of observed elementary particles and dynamics in the geometric elements of three-dimensional space (points, lines, context of basic quantum mechanical considerations [5, 33, 34], areas and volumes) combined in a natural way to form an invariant implications of electric/magnetic reciprocity [25], and the distance. These four geometric elements described algebraically effective ill-posed nature of the Einstein field equation unless the by the scalar, vectors, bivectors and trivectors of Clifford Minkowski metric holds [35]. geometric algebra. In terms of physical quantities they are We will adopt the formalism of Clifford geometric algebra as commonlyreferred to as scalars, vectors, pseudo vectors and it has been used previously by several authors to duplicate the pseudo scalars. Now, scalars and vectors are well understood Minkowski spacetime structure. Two well-known physically as quantities such as energy and momentum implementations are respectively. Pseudo vectors, also called axial vectors, can be (i) spacetime algebra (STA) [22] and understood physically as rotational quantities, such as angular (ii) the algebra of physical space (APS) [2]. The formalism of APS momentum, torque and the magnetic field. Pseudo scalars are less will be shown to be a special case of our approach detailed shortly, commonly understood and describe the nature of magnetic whereas in the case of STA the Minkowski metric is assumed monopoles or helical motion. axiomatically rather than deduced from the algebra. The multivector generalization of the quaternions now allows We now begin with the Clifford algebra of three dimensions a full algebraic description of three-dimensional space, as the and produce the Minkowski metric as an emergent property of this scalar, vector, bivectors and trivectors components now algebraic structure. While recovering standard results we correspond directly with the geometrical quantities of points, lines, nevertheless generalize the representation of spacetime events as areas and volumes found in three dimensions.Additionally these well as the group of Lorentz transformations. The metric is found four quantities describe the range of physical quantities described to produce a suitable Lagrangian for relativistic processes that as scalars, vectors, pseudo vectors (or axial vectors) and pseudo indicates possible extensions to the electromagnetic interaction. scalars [13]. Now, similar to quaternions, as we also identify time As we see before the Clifford’s geometric algebra (GA) over three as the scalar part of space, time is now imputed the geometrical dimensions as an appropriate algebraic representation. This can be meaning and topology of a scalar point-like quantity. expressed using an object called a Multivector This can be contrasted with the linear description of space as 푀 = 푡 + 푥⃗ + 푗푛⃗⃗ + 푗 (3.2) vectors implied by Minkowski and so we can see now a sharp where 푥⃗ = 푥1푒1 + 푥2푒2 + 푥푛3푒3 a vector 푗푛⃗⃗ = 푛1푒2푒3 + geometrical distinction between space and time when described 푛2푒1푒3 + 푛3푒1푒2 a bivector, and 푗 = 푒1푒2푒3 trivector with within the multivector. We now wish to show briefly that the 푡, 푏, 푥1, 푥2, 푥3, 푛1, 푛2, 푛3 real scalars multivector provides a viable formalism to describe both classical and relativistic dynamics from which a definition of time can

Definition 2.1 (Clifford conjugation). We define Clifford cosh 휙 = 훾 and sinh 휙 = 훾푣. Hence conjugation of a multivector 푀 as 푀̅ = 푡 − 푥⃗ − 푗푛⃗⃗ + 푗푏 (3.3) 푒−휙푣̂ = cosh 휙 − 푣̂ sinh 휙 = 훾(1 − 푣), Definition 2.2 (Multivector amplitude). We define the amplitude where 훾 = 1/√1 − 푣2. squared of a multivector M through Clifford conjugation as In this case the vector v is identified with the relative velocity |푀|2 = 푀푀̅ = 푡2 − 푥2 + 푛2 − 푏2 + 2푗(푡푏 − 푥⃗ ∙ 푛⃗⃗) (3.4) vector between frames whereas for rotations it is identified with forming a complex-like number. So we can define the multivector the rotation axis. These results consistent with the known result amplitude as |푀 | = √|푀|2. that the Lorentz group is a sub-manifold of the Pauli algebra [36]. Note that we have used the conventional dot product 푥⃗ · If we now consider the effect of a conven-tional Lorentz boost on 푛⃗⃗ = ½(푥푛 + 푛푥)which can now be defined in terms of the the generalized eight-dimensional spacetime coordinate algebraic product of Clifford vectors, The group transformations 푀 = 푡 + 푥⃗∥ + 푥⃗⊥ + 푗푛⃗⃗∥ + 푗푛⃗⃗⊥ + 푗푏, Definition 2.4 (Bilinear multivector transformation) where we split the spatial coordinate into We define a general bilinear transformation on a multivector 푀 components perpendicular and parallel to as the boost direction ˆ v. We then find from Eq. (2.11) that ′ −푒̂휙 −푒̂휙 −푒̂휙 −푒̂휙 푀′ = 푌 푀푍, (3.5) 푀 = 푡푒 + 푥⃗∥푒 + 푥⃗⊥ + 푗푛⃗⃗∥푒 + 푗푛⃗⃗⊥ + 푗푏푒 ( 3) where 푀, 푌, 푍 ∈ 퐶ℓ ℝ . = 훾(푡 − 푣푥⃗∥) + 훾(푥⃗∥ + 푣푡) + 푗훾(푛⃗⃗∥ − 푣푏) + 푗훾(푏 − 푣푛⃗⃗∥) We then find the transformed multivector amplitude (3.12), |푀′|2 = 푌 푀푍 푌̅̅̅ 푀푍̅̅̅̅ = 푌 푀푍푍̅푀̅푌̅ which now shows the transformation of the full multivector = |푌 |2|푍|2|푀 |2 (3.6) subject to the conventional Lorentz boost operation. We can see where we have used the anti-involution property of Clifford that the plane 푗푛⃗⃗∥ orthogonal to the boost direction 푣 is expanded conjugation and the commuting property of the amplitude. by the γ factor to 푗훾푛⃗⃗ . 2 2 ∥ Hence, provided we specify a unitary |푌 | |푍| = 1 for these This implies that the bivectors do not refer to quantities such transformations, then the amplitude |푀| will be invariant. Without as areas or angular momentum of extended body as the parallel loss of generality, it is then convenient to impose the condition components are in fact unchanged by such boosts—but must refer 2 2 |푌 | |푍| = ±1 The transformation in Eq. (2.5) is then the most to other axial vector-type quantities such as spin or the magnetic general bilinear transformation that preserves the multi-vector field. In fact, the bivector and trivector components 푗푛⃗⃗ + 푗푏 = amplitude and so produces an invariant distance over the space. 푗 (푏 + 푛⃗⃗) are transformed the same as a four-vector therefore 2 2 We will focus on the special case |푌 | |푍| = +1 that describe have the same transformational properties as the four-spin four- transformations that are continuous with the identity.Now, using vector. We can also see that the conventional Lorentz boost the power series expansion of the exponential function, a transformation splits the multivector space into two four multivector Y can be written in an exponential form [22, 23] dimensional subspaces ℝ ⊕ ℝ3 and ⋀2 ℝ3 ⊕ ⋀3 ℝ3 represented 푐+푝+푗푞+푗푑 푌 = 푒 (3.7) by 푡 + 푥⃗ and 푗 푛⃗⃗ + 푗푏 respectively, where ∧ refers to the exterior provided |푌 | ≠ 0, where 푐, 푑 ∈ ℝ and algebra. The first four-vector 푡 + 푥⃗ can be identified as convent- 3 푝, 푞 ∈ ℜ . Therefore, we find ional spacetime if we identify the scalar a with the time t and the 푐+푝+푗푞+푗푑 푐−푝−푗푞+푗푑 푌푌̅ = 푒 푒 second four-vector 푗 푛⃗⃗ + 푗푏 as four-spin. Thus Eq. (3.1) appears = 푒2푐+2푗푑 (3.8) to describe a unified formulation of spacetime that includes spin. Now writing = 푒푟+푗푠 , with 푟, 푠 ∈ ℝ3, we finally produce the For example, with respect to the metric in Eq. (3.4), by identifying general transformation operation the scalar a with time 푡, and 푛 = 0, 푏 = 0, we obtain the 푀′ = 푒푝+푗푞푀 푒푟+푗푠, (3.9) required invariant spacetime interval 푡2 − 푥2. The fact that the which will leave the multivector amplitude invariant. The four conventional boost operation, shown in Eq. (3.12), effectively three-vectors 푝, 푞, 푟, 푠 illustrate that the transformation is specified splits the multivector into two independent four-vector spaces by twelve real parameters, thus generalizing the conventional six perhaps illustrates why the four-vector is effective, even though a dimensional Lorentz group, consisting of boosts and rotations. unified treatment using the multivector is preferable Hence, we Now, if we represent space by the multivector in Eq. (3.2), then can write a general spacetime event 푋, in differential form, as simple rotations of this space are described by the special case 푑푋 = 푑푡 + 푑푥 + 푗푑푛 + 푗푑푏, (2.13) of Eq. (3.9) providing a generalization to eight-dimensional events, where the 푀′ = 푒−푗휔/2푀 푒푗휔/2, (3.10) special case 푑푋 = 푑푡 + 푑푥 is isomorphic to the conventional which will produce a rotation of 휃 = |휔⃗⃗⃗| radians about the axis four vector 푑푋 = [푑푡, 푑푥]푇. Referring to Eq. (3.4) this therefore 휔⃗⃗⃗. It is convenient to have separate notation for the Pythago-rean has an amplitude |푑푋|2 = 푑푡2 − 푑푥2 + 푑푛2 − 푑푥푏2 + length of a vector 휔⃗⃗⃗, given by the unbolded symbol 2푗 (푑푡푑푏 − 푑푥 · 푑푛) . 2 2 2 휔 = |휔⃗⃗⃗| = √휔1 + 휔2 + 휔3 . (3.14) A further special case of Eq. (2.9) is found by selecting the vector In general, the interval is therefore a complex -like number .Now, exponent, which will correspond to conventional Lorentz boosts. for the special case for particles that have a rest frame, defined as That is 푀′ = 푒−휙푣̂/2푀 푒휙푣̂/2 (3.11) 푑푥 = 0, with a corresponding proper time 푑휏 , we have 2 2 2 2 2 where 휙 is defined through tanh 휙 = 푣 푑휏 = |푑푋0| = 푑푡0 + 푑푛0 − 푑푏0 + 2푗푑푡0푑푏0, where 푣 = |푣⃗| and where 푣 is the relative (2.15) where the zero in the subscripts denotes the rest frame. Now 2 velocity vector between the two observers. as 푑휏 is real we therefore require 푑푡0푑푏0 = 0, and as we assume We can rearrange tanh 휙 = 푣 to give 푑푡0 > 0 then we have 푑푏0 = 0.Hence we have in the rest frame 2 2 2 2 2 2 2 푑휏 = 푑푡0 + 푑푛0 = 푑푡 − 푑푥 + 푑푛 − 푑푏 . This publication is licensed under Creative Commons Attribution CC BY. http://dx.doi.org/10.29322/IJSRP.11.02.2021.p110XX www.ijsrp.org International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 345 ISSN 2250-3153

(2.16) (the vectors (푥, 푦, 0, 0)푇). The line may be considered a Euclidean Note that as the imaginary term is now zero in the rest frame and line (the vectors (푥, 1, 0, 0)푇 and finally a point at infinity the interval is invariant, then the imaginary component will also (represented by (1, 0, 0, 0)푇. remain zero in the boosted frame. We can then write for the proper The objects dual to points in ℝℙ3will be planes. Similarly to 푗푢 time 푒 푑휏 = 푑푋0 = 푑푡0 + 푗푑푛0, (3.17) points, planes will also be represented by four-dimensional where 푢⃗⃗ is a three-vector. This then returns |푑푋|2 = vectors. In Euclidean terms we may consider the vector (푒푗푢푑휏)(푒−푗푢푑휏) = 푑휏2, as required. Note that 푒푗푢 = cos 푢 + (푎, 푏, 푐, 푑)푇 as representing the parameters that describe the affine 푗푢̂ sin 푢, where 푢 = √푢2 and 푢̂ = 푢⃗⃗/푢. Thus the proper time plane becomes identified with the even subalgebra of 퐶ℓ(ℝ3), which {(푥, 푦, 푧) ∈ ℝ3| 푎푥 + 푏푦 + 푐푧 + 푑 = 0}. can be shown to be isomorphic to the quaternions. A connection is As usual, nonzero scalar multiples of the vector represent the same now also made to quantum mechanics, as the Pauli spinors are also geometric object.If we interpret the equation in a setup of isomorphic to the quaternions. Additionally, under a relativistic homogeneous coordinates, a point(푥, 푦, 푧, 푤)푇 is incident to a boost we will populate the full eight-dimensional multivector that plane (푎, 푏, 푐, 푑)푇if and only if 푎푥 + 푏푦 + 푐푧 + 푑푤 = 0. has been shown to be isomorphic with the eight-dimensional Dirac As in the two-dimensional setup, incidence is simply expressed by spinor describing spin-½ particles[4]. Comparing a primed rest a scalar product being zero. The plane at infinity has homogeneous frame with the boosted frame we therefore find coordinates (0, 0, 0, 1)푇. Finding a plane that passes through three 푑푡′2 + 푑푛′2 = 푑푡2 − 푑푥2 + 푑푛2 − 푑푏2. given points (2.18) 푝푖 = (푥푖, 푦푖, 푧푖, 푤푖 ), 푖 = 1, 2, 3, thus translates to the task of Now, as we have seen, the two-subspaces of 푑푡 + 푑푥 and 푗푑푛 + solving a linear equation: 푎 푗푑푏 are disjoint 푥1 푦1 푧1 푤1 0 푥 푦 푧 푤 푏 under boosts and so we have [ 2 2 2 2] [푐] = [0]. ′2 2 2 푧 푤 푑푡 = 푑푡 − 푑푥 or 푥3 푦3 3 3 푑 0 푑푡′ = 푑푡√1 – (푑푥/푑푡)2 In the next section we will deal with methods of performing = 푑푡√1 − 푣2 = 푑푡/훾 and so gives the usual time dilation this calculation explicitly. However, before we do so we will consider transformations in this homogeneous projective setup. factor 훾 = 1/√1 − 푣2, where 푣 = 푑푥/푑푡. The situation here is almost completely analogous to the two-D Also, 푑푡′2 > 0 that implies 푑푥2 < 푑푡2 and so we have 푣 < 1, case. A transformation is represented by a simple matrix thus setting an upper limit speed. Hence, despite the fact that time multiplication. This time we need a 4 × 4 matrix. The cases of in the rest frame is quaternionic, nevertheless the amplitude of this usual affine transformations in ℝ3are again covered by special time dτ is real and so we will find that standard results are returned. transformation matrices in which certain entries are zero. The However, this recognition of time as a quaternion is planned to be following matrices represent a linear transformation of ℝ3with further explored in a subsequent paper. matrix A, a pure translation, a General affine transformation, and Affine space of real dimension In the standard embedding we a general projective transformation, respectively: may “imagine” the Euclidean ℝ3embedded in ℝ4as an affine 0 1 0 표 푡푥 • • • • • • • • space parallel to the (푥, 푦, 푧, 0)푇space of ℝ4. 퐴 0 0 1 0 푡푦 [ ] , [ ] , [• • • •] , [• • • •] 0 0 1 푡 • • • • • • • • 0 푧 0 0 0 1 0 0 0 1 0 0 0 1 • • • •

v⃗⃗ A “•” stands for an arbitrary number. In every case we must [ 0 , v⃗⃗ ] , assume that the determinant of the matrix is nonzero.Such a transformation T maps a point p to the associated image point. The Fig.1 A one-to-one correspondence between 3 correspond-ing transformation that applies to the homo-geneous pure quaternions and vectors in ℝ [13] coordinates of a plane is (as in the two-dimensional case) given by 3 (푇−1)푇. By this choice incidence of points and planes is preserved by projective transformations: If this is too hard to imagine, then one can also proceed purely 〈푝, ℎ〉 = 0 ⇐⇒ 〈푇 , (푇−1)푇ℎ〉 = 0. formally. In the standard embedding a point (푥, 푦, 푧) ∈ ℝ3is 푝 represented by a four-Dimensional vector (푥, 푦, 푧, 1)푇 ∈ ℝ4. Instead of the transposed inverse (푇−1)푇it is also possible to use Nonzero scalar multiples are identified. The Infinite points are the transposed adjoint (푇∆)푇, since they differ by only a scalar exactly those nonzero vectors of the form (푥, 푦, 푧, 0)푇. They do not factor.Remark 12.1. A word of caution: One should consider the correspond to Euclidean points. Vectors of the form (푥, 푦, 푧, 0)푇 setup of presenting points and planes by four-dimensional vectors may also be interpreted as homogeneous coordinates of the usual and expressing coincidence by the standard scalar product as a projective plane by ignoring the last entry.Thus we can literally kind of interim solution that will be replaced by something more say that ℝℙ3consists of ℝ3 (the homogeneous vectors powerful later on. The problem that we will have to face soon is (푥, 푦, 푧, 1)푇) and a projective plane at infinity (the homogeneous that lines will be represented by six-dimensional vectors and we vectors (푥, 푦, 푧, 0)푇.We may also interpret this process inductively have to create a notational system that handles points, lines, and and consider the projective plane itself as composed of the planes in a unified way. For this we will have to give up the Euclidean plane ℝ2 (the vectors (푥, 푦, 1, 0)푇)and a line at infinity

concept of indexing a vector entry with the position where it is [2] [2]http://en.wikipedia.org/wiki/Minkowski-space placed in a vector. [3] [3]James M.Chappel1* and others ,Time As Geometric Property of Space, Dumitru Baleanu, University of Craiova, Romania , 17 November 2016 [4] [4] Harvey Reall, Part 3 General Relativity,4/11/2013 II. CONCLUSION [5] [5] Sean M. Carroll, Lecture Notes on General Relativity, nstitute for Theoretical Physics University of California Santa Barbara, CA 93106 Minkowski space is a vector space or affine space of real December 1997 dimension non which there is an inner product or pseudo- [6] [6] David Hestenes, Spacetime Physics with Geometric Algebra , Department Riemannian metric of signature (n−1,1), i.e., in the above of Physics and Astronomy Arizona State University, Tempe, Arizona 85287- terminology, n−1 "pluses" and one "minus".The Poincaré groupis 1504 June 2003. the group of all isometries of Minkowski spacetime including [7] [7] James M. Chappell and others, Exploring the origin of Minkowski boosts, rotations, and translations. The Lorentz groupis the Spacetime,arXiv:1501.04857v4[physics.gen-ph] , 6 Aug 2016 subgroup of iso-Metries which leave the origin fixed and includes [8] [9] [9] Jean Gallier , Clifford Algebras, Clifford Groups, and a Generalization of the boosts and rotations; members of this subgroup are called the Quaternions:The Pin and Spin Groups, Department of Computer and Lorentz transfor-mations. Among the simplest Information Science University of Pennsylvania Philadelphia, PA 19104, Lorentztransformations is a Lorentz boost. USAJanuary 31, 2012 [10] [10] Bernhard Rothenstein1(Politehnica University of Timisoara, Physics The archetypal Lorentz boost is[2] Department)and Stefan Popescu2(Siemens AG, Erlangen, Germany) Lorentz ′ transformations: Einstein’s derivation simplified, without date 푀0 훾 −훽훾 0 0 푀0 ′ [11] [11] J¨urgen Richter-Gebert, Perspectives on Projective Geometry , (Draft 푀1 −훽훾 훾 0 0 푀1 Version)October 3, 2010 Springer ′ = [ ] [ ] 푀2 0 0 1 0 푀2 [12] [12] Darryl D Holm , Geometric MechanicsPart II: Rotating, Translating and ′ Rolling, Mathematics Department Imperial College London 2nd Edition 23 [푀3] 0 0 0 1 푀3 1 푣 Sept 2011 where 훾 = is the Lorentz factor, and 훽 = . 푣2 푐 [13] [13] R.Mukunda , Quaternions: From Classical Mechanics to Computer √1− 푐2 Graphics, and Beyond, University of Canterbury Christchurch, New Zealand, Conference in Mathematics 2002. Invited Paper. All four-vectorsin Minkowski space transform according to the same formula under Lorentz transformations.Minkowski diagramsillustrate Lorentz transformations.[2] AUTHORS First Author – Emran Sasi Althabit*, PH.D of mathematics, SUDAN UNIVERISITY SEEINCE AND TECHNOLOGY (Khartoum), +218926083695 [email protected] Second Author – Ismail Mustafa Mohamed**, PH.D in mathematics, AL NEELAN UNIVERISITY (Khartoum), +249124699207 [email protected]

Fig .2

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