Double Conformal Space-Time Algebra for General Quadric Surfaces in Space-Time

Robert Benjamin Easter Email: [email protected]

Abstract The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric G4;8 Algebra that extends the concepts introduced with the Double Conformal / Darboux Cyclide G8;2 Geometric Algebra (DCGA) with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA). Two Conformal G1;3 G2;4 Space-Time subalgebras (CSTA) provide spacetime entities for points, hypercones, hyperplanes, hyperpseudospheres (and their intersections) and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reection, and (pseudo)spherical inversion. DCSTA is a doubling product of two orthog- G4;8 onal CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds G2;4 new 2-vector entities for general (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The pseudo surface entities are spacetime surface entities that use the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in hyperpseudospheres. An operation for the directed non-uniform scaling (anisotropic dilation) of the 2-vector general quadric entities is dened using the boost operator and a spatial projection. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the Thomas-Wigner rotation and length contraction of special relativity. DCSTA is an algebra for computing with general quadrics and their inversive geometry in spacetime. For applications or testing, DCSTA can be computed using various software packages, such as G4;8 the symbolic computer algebra system SymPy with the Algebra module. G Keywords: Cliord algebra; conformal geometric algebra; space-time algebra A.M.S. subject classication: 15A66, 51N25, 51B20, 83A05

1 Introduction This is an extended paper1 on the Double Conformal Space-Time Algebra [10] (DCSTA) [7]. DCSTA G4;8 is a high-dimensional 12D Geometric Algebra [13][15][3] over the twelve-dimensional (12D) vector G4;8 space R4;8 that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geo- metric Algebra (DCGA) 8;2 [4][5][6][8][9] with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadricsG, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) [12]. G1;3 Two orthogonal, and isomorphic, Conformal Space-Time subalgebras (CSTA) [2] provide G(1+1);(3+1) spacetime entities for points, hypercones, hyperplanes, and hyperpseudospheres (in their intersections) and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reection, and (pseudo)spherical inversion. The double CSTA (DCSTA) 4;8 is a doubling product of two orthogonal CSTA subalgebras 2;4 that inherits doubled CSTA entitiGes and versors from CSTA and adds new bivector entities for genGeral (pseudo)quadrics and Darboux (pseudo)cyclides in space-time that are also transformed by the doubled versors. The pseudo surface entities are spacetime surface entities that use the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in hyperpseudospheres. DCSTA allows general quadric surfaces to be transformed in spacetime by a complete set of doubled CSTA versor (i.e., DCSTA versor) operations. General quadric surface entities can be boosted into moving surfaces with constant velocities that display the Thomas-Wigner rotation and length contraction of special relativity. DCSTA also denes an operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities using the boost operator followed by a spatial projection.

1. First preprint version v1 (viXra), 22 Aug 2017. This paper was written as an invited extended paper, extending on the published 10-page conference proceedings paper Double Conformal Geometric Algebra (DOI:10.1063/1.4972658), for possible later submission to the journal Mathematics in Engineering, Science and Aerospace (MESA). If later submitted, the paper may then include a certain corresponding coauthor that helps handle the entire publication process, in which the paper then also undergoes further collaborative revision.

1 2 Section 2

As will be explained further in more detail, the new DCSTA bivector entities for quadrics and Darboux cyclides are formed by extracting polynomial terms from the coecients on the basis 2-blade terms of the DCSTA 2-blade point entity using reciprocal (or pseudoinverse) basis 2-blades as extraction operators. The reciprocal basis 2-blades that extract the same polynomial term s from the DCSTA point entity are added and averaged as the DCSTA 2-vector extraction operator for value s. The DCSTA has a basis of twelve orthonormal vector elements e , 1 i 12, with metric G4;8 D i (squares or signatures) m : D

m = m = diag(1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1) = [mij] (1) D ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ = diag(m 1; m 2) = diag(1; m 1; 1; m 2) (2) C C CS CS = diag(m 1; 1; 1; m 2; 1; 1) = diag(1; m 1; 1; 1; 1; m 2; 1; 1) (3) M ¡ M ¡ S ¡ S ¡ m = diag(m 1; m 2); mij = ei ej: (4) DS CS CS The above metric also includes the metrics of the subalgebras: 1 2 2;4 CSTA1 : m 1 2;4 CSTA2 : m 2 G C C G C C 1 2 1;4 Conformal SA1 (CSA1) : m 1 1;4 CSA2 : m 2 G CS CS G CS CS 1 2 1;3 STA1 : m 1 1;3 STA2 : m 2 G M M G M M 1 2 0;3 Space Algebra 1 (SA1) : m 1 0;3 SA2 : m 2 G S S G S S 2;8 Double Conformal SA (DCSA) : m G DS DS

2 Notation of Space-Time Algebra (STA)

1 The basis of the space-time algebra 1;3 STA = 1;3 STA1 is 0; 1; 2; 3 = e1; e2; e3; e4 , and G M G 2 M f g f g for the second copy of the space-time algebra 1;3 STA2 we have the basis 0; 1; 2; 3 = e7; e8; e ; e . The space algebra SA basis, inGcluded in thMe space-time algebra, fis ; ; g. Thfe STA 9 10g G0;3 S f 1 2 3g unit four-dimensional pseudoscalar is I = 0 1 2 3, and for SA the unit three-dimensional pseudoscalar M is I = 1 2 3. Moreover, STA denes a symbolic space-time test position with symbolic coordinates (w =S ct; x; y; z) by the four-dimensional (4D) vector

t = t = w 0 + x 1 + y 2 + z 3 = w 0 + t ; (5) M S a specic space-time position with specic coordinates (pw; px; py; pz) by the 4D vector

p = p = pw 0 + px 1 + py 2 + pz 3 = pw 0 + p ; (6) M S and a 4D space-time velocity

v = v = c 0 + vx 1 + vy 2 + vz 3 = c 0 + v ; (7) M S with 4D STA vectors in bold italic, and 3D SA spatial vx 1 + vy 2 + vz 3 vectors v = v in bold. An S algebra symbol as a subscript of an element A indicates that A , and similarly for the other S S S 2 S algebra symbols. The geometric product of vectors u and v is uv = u v + u v; (8) ^ where the inner product is the symmetric product

u v = (uv + vu)/2 (9) and the outer product is the anti-symmetric product

u v = (uv vu)/2: (10) ^ ¡ We will make frequent use of the square of t , known as the space-time interval, M t2 = w2 x2 y2 z2: (11) M ¡ ¡ ¡ Notation of Space-Time Algebra (STA) 3

A vector t is time-like for t2 > 0, space-like for t2 < 0, and null (or light-like) for t2 = 0. The interval M a2 is often used to avoid imMaginary numbers. InM(137), the interval a2 is the squaMre of a hyperbolic radius r2 = a2. A null radius r = 0 is associated with a null hypercone. A real radius r > 0, for a2 > 0, is associated with a (hyper)hyperboloid of one sheet. An imaginary r = r p 1, or a2 < 0, is associated with a (hyper)hyperboloid of two sheets. The (hyper)hyperboloids are cjirjcu¡lar and are also called real or imaginary hyperpseudospheres, respectively. A null vector a2 = (s + t)2 = 0 is the sum of space-like s2 < 0 and time-like t2 > 0 vectors that are orthogonal 2s t = st + ts = 0 and of equal magnitude t = s (31), where a2 = s2 + t2 = t 2 s 2 = 0. 2 j j j j 1 2 j j ¡ j j A non-null vector a has a non-zero interval a =/ 0 and has an inverse a¡ = a/a . v v v v The vector projections [13] of any vector u = uk + u? parallel uk and perpendicular u? to any non-null vector v is dened by

1 1 1 v v u = (uv)v = (u v)v + (u v)v = uk + u = (u) + ?(u): (12) ¡ ¡ ^ ¡ ? Pv Pv The untranslated (at origin) observer worldline, in the rest frame of the observer, is

ot = ct 0 (13) with proper time (coordinate time) t and light speed c. See also, the CSTA line entity L of (150). The SA spatial dualization of SA spatial vector n is C 1 n = n I¡ : (14) S ¡ S S For an SA spatial unit vector rotation axis n^, the unit directional 2-blade n^ of the rotation plane is 2 isomorphic to a pure unit quaternion, where (n^) = 1. Using (14), the correspondence with unit ¡ quaternions is i; j; k = 1; 2; 3 . The STA spface-timg e dfualizationgof STA space-time vector v is M 1 v = v I¡ : (15) M M M The vector (1-blade) conjugate ay [16] of any vector a, written using Einstein notation as

p p+q a = aie = aie + aie 1 (16) i 0 i i 12 Gp;q i=1 i=p+1 X X @ A on the standard orthonormal basis of vectors e : 1 i p + q having pseudo-Euclidean sig- f i g nature (p; q) with Euclidean signatures e : e2 = 1; 1 i p and anti-Euclidean signatures i i e : e2 = 1; p + 1 i p + q , is i i ¡ p p+q a = aie aie ; (17) y i ¡ i i=1 i=p+1 X X such that all of the anti-Euclidean basis vector terms are multiplied by 1. For any STA vector (1- 1 ¡ blade) a = aw 0 + a = aw 0 + ax 1 + ay 2 + az 3 1;3, its conjugate is (changing the sign of the spatial component) 2 G

ay = a = a a = a a a a : (18) 0 0 w 0 ¡ w 0 ¡ x 1 ¡ y 2 ¡ z 3 A k-blade A k , of grade k denoted by subscript k [16], is the outer product of k vectors ai, h i h i k

A k = ai = ai = a1 a2 ::: ak: (19) h i ^ ^ ^ i=1 ^ ^ A scalar a is also called a 0-blade. A k-vector A k , often denoted Ak, is a sum of k-blades. A multivector h i A is a sum of k-vectors of various grades k. The reverse A of any multivector A reverses the products of all vectors in A (e.g., I = 3 2 1 0). The reverse of a k-blade A k is M h i k(k 1)/2 Ak = ( 1) ¡ A k = ak ak 1 ::: a1: (20) h i ¡ h i ^ ¡ ^ ^

The k-blade conjugate Ayk [16] of any k-blade A k is (n.b. the reverse order) h i h i k

Ayk = aky+1 i = aky aky 1 ::: a1y: (21) h i ¡ ^ ¡ ^ ^ i=1 ^ 4 Section 2

k For any STA k-blade A k = a1 a2 ::: ak 1;3, 1 k 4, its conjugate is (a composition of reversion h i ^ ^ ^ 2 G Ak with sandwiching between 0 factors) h i

Ayk = 0Ak 0 = aky aky 1 ::: a1y: (22) h i h i ^ ¡ ^ ^ The Euclidean norm (l -norm [14]) a of any STA vector a is 2 k k2

a = pa ay 0; (23) k k2 which is positive or zero. Similarly, the Euclidean norm A k 2 of a k-blade A k is k h ik h i

A k 2 = A k Ayk 0: (24) k h ik h i h i q The Euclidean normalization of any (but restricting to null) STA vector a is the Euclidean normalized vector a^ = a/ a ; (25) k k2 where a^ 2 = 1 is unit Euclidean norm. Although the Euclidean normalization (25) is dened for any STA vekctokr a, in this paper we restrict the Euclidean normalization to STA null vectors a, where a2 = 0. For any STA non-null vector a, we dene a^ by the pseudo-Euclidean normalization (30) as a unit vector, where a^2 = 1. A null vector (1-blade) a has a zero interval a2 = 0 and no inverse, but has a pseudoinverse [16]

+ 2 a = ay/ a ; (26) k k2 where a a+ = 1. For any k-blade A k , its pseudoinverse [16] is h i + 2 A k = Ayk / A k 2 = Ayk /(A k Ayk ); (27) h i h i k h ik h i h i h i + + 1 where A k A k = 1. For any non-null k-blade A k , its pseudoinverse A k equals its inverse A¡k = + h i h i h i h i h i A k , h i 1 2 A¡k = Ak /(A k Ak ) = A k /A k ; (28) h i h i h i h i h i h i which follows from Ayk = Ak when no vector ai in A k is a null vector. h i h i h i The pseudo-Euclidean norm (or seminorm) a of any STA vector a is k k a = a2 = a a 0; (29) k k j j j j and the pseudo-Euclidean normalizationpof any npon-null vector a is the unit vector a^ = a/ a ; (30) k k such that if a is time-like (a2 > 0) then a^2 = 1, and if a is space-like (a2 < 0) then a^2 = 1. For a null ¡ (light-like) vector a, the notation a^ is the Euclidean (l2) normalization a^ = a/ a 2 (25). The pseudo- Euclidean norm a is a seminorm since a = 0 for a null vector a =/ 0 [14]. k k The pseudo-Ekucklidean norm a (29) kis ekquivalent to hyperbolic modulus [20] (or magnitude) k k a = a = a a = a2 (a2 + a2 + a2) ; (31) j j k k j j j w ¡ x y z j q and the non-null unit vector a^ (30) ips also called a unimodular vector. For scalar a, a is the absolute value of a. For SA spatial vector a, the pseudo-Euclidean norm and Euclidean norm jarje equal,

a = a = a = pa ay = p a a 0: (32) k k j j k k2 ¡ Similarly, the pseudo-Euclidean norm A k of any STA k-blade A k is k h ik h i 2 A k = A k = A k = A k A k 0: (33) j h ij k h ik j h ij j h i h ij q For any STA vector (1-blade) a, the unit vector a^ is pdened by

a/ a = a/ a2 : a2 =/ 0 a^ = k k j j (34) 2 ( a/ a 2 = a/pa ay : a = 0: k k p Notation of Space-Time Algebra (STA) 5

For any STA k-blade A k , the unit k-blade A^ k is dened by h i h i

2 2 A k / A k = A k / A k : A k =/ 0 h i k h ik h i j h ij h i A^ k = (35) h i 8 q 2 > A k / A k 2 = A k / A k Ayk : A k = 0: < h i k h ik h i h i h i h i q > 2 In STA, null k-blades exis:t, such as the null 2-blade ( 0 + 1) 2, where (( 0 + 1) 2) = 0. Note that, + 2 the conjugate Ayk , Euclidean norm A k 2, and pseudoinverse A k exist for null k-blades A k = 0 h i k h ik h i h i (including null 1-blade vectors) and are mainly used for the algebra of null k-blades. n The canonical basis of p;q has n = p + q orthonormal basis vector elements ai = ei and a total of 2 basis unit k-blade elementsG

bi b1 b2 bn Ab = ei = e1 e2 :::en ; (36) ^ where the exponents bi are n binary digits of the binary number b = b1b2:::bn, essentially acting as presence bits (generally A0 = 1, A1 = A). A k-blade has a number b with k ones. The basis unit 0-blade (unit scalar) is A0 = 1, and the n-blade unit pseudoscalar is I = A2n 1 = e1e2:::en. However, for the ¡ decimal number dec(b) of b, in general dec(b) =/ i of ei etc., and the binary number b is the subscript on Ab since it intuitively relates to the construction of Ab as the product of canonical basis unit vectors ei in ascending order of subscripts i. The basis k-blade Ab on an arbitrary basis ai, where the ai are not necessarily orthonormal vectors, is

A = abi = ab1 ab2 ::: abn; (37) b i 1 ^ 2 ^ ^ n ^ bi which is not in general equal to the geometric product of the ai , as for orthogonal vectors. On an arbitrary vector basis a , 1 i n, of an n-dimensional algebra , n = p + q, where a = a i Gp;q j is the jth linearly independent basis vector, the pseudoscalar is I = A n = ai (19) (not necessarily a h i unit n-blade) and the reciprocal basis vector aj, to a = a , is [13, page 28] j V j j 1 1 a = ( 1) ¡ (I a )I¡ ; (38) ¡ n j where I a = a (i.e., I without a ) and j is still the index (not exponent), such that n j i=/ j i j V j j 1 1 j 1 1 1 a a = ( 1) ¡ a ((I a ) I¡ ) = ( 1) ¡ (a (I a ))I¡ = II¡ = 1; (39) j ¡ j n j ¡ j ^ n j and a aj = 0 for i =/ j. Using the Kronecker delta i 0 : i =/ j j = (40) i 1 : i = j; the reciprocal basis vectors are often dened by the expression a aj = j. i i The reciprocal basis vector aj is a coecient extraction operator that extracts the (contravariant) j i i j j scalar coecient v on (covariant) basis vector aj in any vector v = v ai = via as v = v a , and j 1 vj = v aj. For basis vector a = aj, its reciprocal basis vector a and its inverse a¡ are not necessarily equal, especially since a null basis vector has no inverse but does have a reciprocal. Similar to the case of vectors (1-blades), it is possible to compute the reciprocal basis k-blade of a canonical (non-null) basis unit k-blade Ab (36) as

b 1 A = s(I A )I¡ ; (41) n b where b is still the index (not exponent), and

1 s = (A A ) I¡ and A = I A where (I = I 1) I = 1; (42) b ^ NOT b NOT b n b ^ n b b + 1 n such that Ab A = 1 and A = Ab = Ab¡ on the canonical basis. The notation NOT b = b XOR (2 1) is the bitwise complement. The formula (41) can also be used to compute the reciprocal basis k-b¡lade b A on an arbitrary basis ai by replacing I with the pseudoscalar A n (19) of the arbitrary basis ai, but b + 1 h i then A = Ab = Ab¡ does not hold in general on an arbitrary basis. + j 1 There are distinctions between pseudoinverse a , reciprocal a , and inverse a¡ vectors (and of k- blades A k ) that can be claried in the context of STA by some further explanation, which follows (until h i (45)). 6 Section 2

In STA , an arbitrary basis can include null basis vectors. There are three Minkowski planes, G1;3 0 1, 0 2, and 0 3, that each span 0, so we have to choose one Minkowski plane for any STA basis. A 2D Minkowski (pseudo-Euclidean) plane can be spanned by two null vectors, e.g., n0 1 = 0 1 ¡ ¡ and n0+1 = 0 + 1 = n0y 1, using (25) for a null unit vector n^0 1 (25), and then the rest of 4D space- ¡ time is spanned by the other two vectors, e.g. 2 and 3. Such a basis is comprised of orthonormal vectors ai n^0 1; n^0+1; 2; 3 , as are the canonical basis vectors ai 0; 1; 2; 3 , so it is not an entirely2afrbit¡rary basis, but ag special basis. The three dierent unit2pfseudoscalars, fgor a dierent orthonormal basis with null vectors, are I = n^0 1 n^0+1 2 3 or I = n^0 2 1 n^0+2 3 or M ¡ ^ ^ ^ M ¡ ^ ^ ^ I = n^0 3 1 2 n^0+3 (19), but they are not actually dierent since all equal I , which is the uMnit pseu¡dos^calarôf thê canonical basis of STA. For the canonical basis unit vectors a M ; ; i 2 f 0; 1 2 3g and the null basis unit vectors n^0 i (25), it can be shown that their pseudoinverse vector (26) and reciprocal basis vector (38) are equal, and we nd it more convenient to use the pseudoinverse to obtain j 1 a reciprocal basis vector, since we are then not concerned with determining the sign ( 1) ¡ in (38). 2 2 ¡ More generally, for any pseudo-Euclidean 0 i-plane, with 0 =1 and i = 1, the canonical orthonormal + ¡ j basis ; and any basis of pseudoinverses a = a a^ = a ; a = a^y/ a = a in the plane, have the f 0 ig f j j j j j g property that their pseudoinverses (26) equal their reciprocals (38). The basis a; a+ includes the null + f g basis n^0 i; n^0+i / or any non-null skew basis a = 0 i; a : =/ around 0 in the f ¡ g ¡ 0 i-plane. The choice of basis for STA, between the canonical basis (of vectors) ai j or a special + + 2 f g basis ai j / 0;i ; a; a , having a particular skew basis a; a for just one of the three particular 2 f 2f g g f g + Minkowski 0 i-planes, is arbitrary since the pseudoinverse basis k-blade Ab (27) provides a uniform b + expression of the reciprocal basis k-blade A = Ab (41) on these special choices of basis. For the general + choice of an arbitrary STA vector basis ai, the pseudoinverse ai (26) is not in general equal to the i + i reciprocal a =/ ai (38), and then the general reciprocal vector a by (38) can always be used instead of the pseudoinverse to obtain the correct reciprocal. If we arbitrarily choose one of the three Minkowski planes 0 i, then the special null basis for STA is an orthonormal basis, aî n^0 1; n^0+1; 2; 3 or aî n^0 2; 1; n^0+2; 3 or aî n^0 3; 1; 2; n^0+3 2 f ¡ g 2 f ¡ g 2 f ¡ g with unit pseudoscalar I , that includes the two null unit vectors n^0 i of the 0 i-plane and the two M other canonical basis vectors j / 0;i . Then, a basis unit k-blade Ab (37) that includes one of the null 2f g 2 b basis unit vectors n^0 i is a null basis unit k-blade Ab = 0, where its reciprocal basis k-blade (41) A and + b + pseudoinverse k-blade (27) Ab are equal A = Ab , but no inverse exists. For a null basis unit k-blade, we nd it again convenient to use its pseudoinverse (27) as its reciprocal (41) and avoid the determination of the sign s in the more general formula for a reciprocal (41). For a canonical (non-null) basis unit k-blade (36) or special null basis unit k-blade Ab, we have

A = a^bi = a^b1 a^b2 ::: a^bn = a^b1a^b2:::a^bn; (43) b i 1 ^ 2 ^ ^ n 1 2 n ^ bi where the basis unit vectors a^j (34) are orthonormal, and where just one of a^i = a^j may be present (bj = 1) that is a special null basis unit vector a^j = n^0 i for a special null basis unit k-blade Ab. Its b + reciprocal basis unit k-blade A (41) equals its pseudoinverse (27) Ab as (n.b. the reverse order) n b + bn+1 1 + bn + bn 1 + b1 + bn + bn 1 + b1 + A = Ab = (a^n+1 ¡i) = (a^n ) (a^n ¡1 ) ::: (a^1 ) = (a^n ) (a^n ¡1 ) :::(a^1 ) : (44) ¡ ¡ ^ ^ ¡ i=1 ^ + This result Ab , on the canonical or any special null basis, is easy to use in the sequel on DCSTA, where + the DCSTA bivector (2-vector) extraction operators Ts are sums of 2-blade extraction operators Ab . + We simply multiply pseudoinverse (reciprocal) CSTA basis vectors aj , as CSTA extraction operators + si + s s2 s1 aj = T i (128), in reverse order to form a DCSTA 2-blade extraction operator Ab = T = T 2T 1 (Table C D C C 1) for polynomial term s = s2s1. In further extensions of CGA beyond doubling, called Extended CGA + (k-CGA), this result Ab is used for dening the k-vector extraction operators Ts of k-CGA, which are + sums of k-blade extraction operators Ab . 1 2 A k-versor Vk is the product of k non-null vectors ai with inverses ai¡ = ai/ai as

Vk = akak 1:::a1: (45) ¡ A unimodular k-versor V^ is the product of k unimodular (non-null) vectors a^2 = 1 as k i V^k = a^ka^k 1:::a^1: (46) ¡ The modulus V of k-versor V is j kj k

Vk = VkVk = VkVk = akak 1:::a1a1:::ak 1ak = ak ak 1 ::: a1 ; (47) j j j j j j j ¡ ¡ j j jj ¡ j j j p p p Notation of Space-Time Algebra (STA) 7

and the unimodular k-versor V^k can be expressed as

1 V^ = V ¡ V = V / V V : (48) k j kj k k j k k j 1 The inverse Vk¡ of k-versor Vk is p 1 1 1 1 Vk¡ = Vk/(VkVk) = a1¡ a2¡ :::ak¡ : (49) 1 1 2 a a The 1-versor V1 = a is an operator for vector reection v0 = V1vV1¡ = ava¡ = a^ a^va^ = vjj v? . The 1 2 a a ¡ negative, ava¡ = a^ a^va^ = v? vjj , reects v in the hyperplane orthogonal to a. The unimodular ¡ ¡ ¡ 1 2 1-versor V^1 = a^ is an operator v0 = V^1vV^1 = a^va^ for reection in the vector a^ = a^¡ when a^ = 1, and for reection in the hyperplane orthogonal to a^ = a^ 1 when a^2 = 1. ¡ ¡ ¡ A unimodular k-versor V^k operates on a vector v using the versor sandwich operation

v0 = V^kvV^k; (50) 1 1 where V^ = V^¡ (use V^¡ when orientation is signicant). The general k-versor V operation is k k k k 1 v 0 = VkvVk¡ : (51) We primarily use (50) since the orientation is usually not signicant for CGA geometric entities. The unimodular k-versor V^k is an operator for successive reections in k vectors or hyperplanes, depending on the particular signatures of the unit vectors a^2 = 1. Reection in two unimodular spatial i vectors a^i is spatial rotation in the plane of the two vectors by twice the angle between them. Reection 2 in two unimodular time-like space-time vectors a^i = 1 in the same space-time v^ 0-plane is hyperbolic rotation (boost in direction v^) by twice the hyperbolic angle ' between them.

By outermorphism [16], the unimodular k-versor V^k (46) transforms the k-blade A k (19) as h i ^ ^ ^ ^ ^ ^ ^ ^ A0k = VkA k Vk = Vka1Vk Vka2Vk ::: VkakVk ; (52) h i h i ^ ^ ^ which rotates or reects the k-blade A¡ k by rotati¡ng or reecting e¡ach vector ai in the k-blade. By the h i linearity of the versor operation (50), as a linear operator, a unimodular k-versor V^k can also transform any k-vector A k or multivector A. h i An STA unimodular 2-versor V^ is the geometric product of two unimodular vectors a^2 = 1, 2 i V^2 = a^2a^1 = a^2 a^1 + a^2 a^1 = exp( A^); (53) ^ which is a scalar a^ a^ plus a 2-blade A = a^ a^ with unit 2-blade A^ by (35) in the direction of the 2 1 2 ^ 1 plane of rotation. A unimodular 2-versor V^2 is a geometric number [20] that is isomorphic to an elliptic a + bi (i = p 1), parabolic a + b" ("2 = 0; " =/ 0), or hyperbolic complex number a + bj (j2 = 1; j =/ 1), ¡ and is generally a rotation operator that preserves the modulus ai (31) of vectors ai (52). The angle of rotation in (53) is given by j j ^ 2 = acos( a^2 a^1) : A = 1 (V2 = elliptic complex number) ¡ ¡ = 1 ^ 2 (54) 8 a^2 a^1 ¡ A 2 : A = 0 (V2 = parabolic complex number) > j j k k ^ 2 < ' = atanh( A /(a^2 a^1)) : A = +1 (V2 = hyperbolic complex number): j j > A unimodul:ar 2-versor V^2 operates on a vector v using the versor sandwich operation

v 0 = V^2vV^2; (55) where the reverse V2 corresponds to the complex conjugate V2¡ = V2. For the unimodular exponential 1 form V^2 = exp( A^), which is our usual preferred form, we have V^2 = V^2¡ exactly. In (55), the vector v is rotated by twice the angle (i.e., 2 ). To rotate by angle , we dene a 2- versor V^ for angle as the square root

V^ = V^ = pa^ a^ = pa^ a^ + a^ a^ = exp( A^ /2): (56) 2 2 1 2 1 2 ^ 1 p We usually assume that our 2-versors are unimodular V2 =V^2 and use V^2 or its square root V^ in operation (55), but for non-unimodular versors Vk =/ V^k the operation (51) could be used. ^ ^ 1 ^ It can be useful to interpret the product of vectors a^2a^1 = b/a^ = ba^¡ as a certain ratio of vectors b 1 1 by a^ that transforms a^ into b^ as (b^/a^)a^ = (b^/a^) 2 a^(b^/a^)¡ 2 , and also transforms other vectors by the same proportion, which is by the same angle in the same plane of rotation. 8 Section 2

2 ^ In (53) and (54) for A = 0, the null unit 2-blade is A = A / A 2 by (35) and V2 = V2V2¡ = 2 1 k k 1 j j j j (a^ a^ ) = a^ a^ by (47), where by (48) V^ = V ¡ V = a^ a^ ¡ V = exp( A^) = (1 + A^ ) j 2 1 j j 2 1j 2 j 2j 2 j 2 1j 2 p (n.b., V = V exp( A^), but in (55) is canceled). Then, V^ is a unimodular 2-versor for a special p 2 j 2j 2 type of translation (rotation around the point at innity), such that the modulus v 0 = v is invariant. General translation does not preserve the modulus. For example, using a^ = + j +j j ajnd a^ = , 2 0 1 2 1 ¡ 2 the product a^ a^ is the ratio a^ / = 1 + ( + ) = 1 + A = exp(A) = V^ , and then the versor 2 1 2 2 2 ^ 0 1 2 V^ = exp(A/2) transforms 2 into a^2 = V^ 2V^ , which is 2 translated by the null vector n = 0 + 1, where 2 + n = 2 = 1. Other vectors are transformed by V^ by the same proportion, which are special translajtions byj vjarioj us null or non-null vectors (not a constant vector) in the plane of A that preserve the modulus. When STA is viewed as the CGA of the -plane (CSA in 2D), then the G1;3 G1;2+1 2 3 translator T = 1 d( 0 + 1)/2 = exp(e d/2) translates CGA entities by d in the 2 3-plane (see CSA in [7] for details)¡. 1 If a is a null vector, then a^ is Euclidean norm by (34), which is not unimodular since the modulus is a^ = 0 =/ 1. The reection v0 = ava of any vector v in any null vector a produces v0 with modulus j j v0 = 0, which is a null vector or the zero vector 0. Therefore, if any vector ai in the k-versor Vk is null j 2 j a =0, then the modulus is V = 0 by (47) and the operation v0 = V vV produces a vector with v0 = 0, i j kj k k j j which is a null vector or the zero vector 0. For a 2-versor V2 = a2a1, if either vector ai is a null vector then the modulus is V = 0 by (47), V has no exponential form (thus, not a proper 2-versor), and all j 2j 2 resulting vectors v0 = V2vV2 are zero modulus v 0 = 0. In general, a null vector ai is not admitted in a proper k-versor V (45). For example, using a =j j + and a = might be interpreted as the ratio k 2 0 1 1 ¡ 2 a2/ 2, but the attempt to transform as a versor operation gives V2 2V2 = 0. The k-versor (46) for even k =2m is a composition of m 2-versors, and the k-versor for odd k =2m+ 1 includes one more 1-versor for a nal reection in a vector or hyperplane. 1 2 An STA 1-versor a is any non-null STA vector a = a with an inverse a¡ = a/a . The reection M p0 of vector p in vector a is given by the versor sandwich operation (or conjugation) 1 a a 1 1 p0 = apa¡ = pjj p? = (p) ?(p) = (p a)a¡ (p a)a¡ : (57) ¡ Pa ¡ Pa ¡ ^ Two successive reections (57), in vector a and then in vector b, forms the 2-versor ba. In general, in the geometric algebra of an nD vector space, k successive reections (57) in 1 k n vectors a i forms a k-versor ak::::a2a1 for an orthogonal transformation (Cartan-Dieudonné theorem). All of the versors in this paper can be derived from successive vector reections (57). The 2-versors are generalized rotation operators with unimodular exponential forms exp(A) = eA. The opposite orientation to (57), 1 p0 = apa¡ , is reection in the hyperplane orthogonal to a. The geometrical distinction between reect¡ion in vectors or in hyperplanes is not very important in this paper since we will only use even k-versors as products of unimodular 2-versors in exponential forms, or transform homogeneous entities in CSTA and DCSTA that are equivalent up to any non-zero scalar multiple, where any changes in orientation ( sign) or scale are usually of little signicance. The STA2-versors include the spatial rotor R and the spacetime hyperbolic rotor (boost) B. The STA 2 2-versor spatial rotor R is dened as (n.b., (n^ ) = 1, see (14)) S ¡ 1 1 n^ R = Rn = (b^ /a^) 2 = exp(n^ /2) = e 2 S = cos(/2) + sin(/2)n^ (58) S S for spatial rotation around the spatial unit vector axis n^ through the origin by the radians angle subtended from a^ to b^ (by right-hand rule) in the ab-planSe orthogonal to n. Two successive reections (57), in vector a then in vector b, rotates by twice the angle = \ab, but R rotates by just . The ratio b^ /a^ is isomorphic to Hamilton's unit quaternion versor [11]. 2 The STA 2-versor space-time hyperbolic rotor (boost) B is dened as (n.b., (v^ 0) = +1)

1 B = Bv = ( v/o) 2 = exp('v^ 0/2) = cosh('/2) + sinh('/2)v^ 0; (59) where three-dimensional spatial speed v in physics is

v = c = v = v2 + v2 + v2; (60) k k x y z light speed is c, natural speed is q 0 ( = = v / o = v/c) 1; (61) v k k k k space-time velocity is by (7)

v = o + v = c 0 + cv^; (62) Notation of Space-Time Algebra (STA) 9 and rapidity (hyperbolic angle in (59)) is

' = 'v = atanh( ): (63) The Lorentz factor = = dt/d = o2 /v2 = o / v = 1/ 1 2 = 1/d (64) v j j j j ¡ v is related to special relativity length cpontraction (from L0 to pL) as L = 1 2L = L / = dL ; (65) ¡ 0 0 0 where = tpv is the proper time of tphe observable with space-time velocity v. The proper time of o is tpo = t, which is also the coordinate time tcv = t of any vector v = o + v directly relative to an observer o. The proper time of vy = o + vy = o v (see (17) and (18)) is t . Relative to o with coordinate time t, ¡ pvy then tpv = tpvy = and vy = v, but these are not equal relative to another observer u =/ o. A space-time velocity u = o + u is a spatial velocity u relative to the spatially stationary observer o. The coordinate time of u is tcu = tpo, and the proper time of u is tpu. The unimodular boost operator B is either an active boost B = Bv by v = o + v into the rest frame 1 of v or an equivalent passive boost B = B¡ relative to vy, boosting o as vy 1 BvoBv¡ = o v = vv = v(o + v) = t(o + v) = vt (66) 1 B¡ oB t = o vyt = tv = t(o vy) = (o vy) = v: (67) vy vy vy vy ¡ ¡ The above active and passive boosts, while algebraically equivalent, have dierent interpretations, espe- 1 cially of the time transformations. For B = B then = t = t , and for B = B¡ the relatively v v v pv po vy corresponding (and numerically equivalent) times are vyt = vytpo = tpvy. We also use the alternative notations

v = 0 v = v = v = o v; (68) where 0 v = v emphasizes the spatial velocity boost by v = o + v from velocity 0 or from an unspecied arbitrary initial velocity, and ' v = o v emphasizes the space-time boost by v and that (o v) o = v and (o v) v = o (and similarly for initial velocity u instead of o). The notation for the act ive boost of u by v is 1 BvuBv¡ = u v = u v(o + u v); (69) where the active dilation factor (including an alternative notation subscript) is u v u v = u v = v 1 =/ ( 0 u v = u v = o u v); (70) ¡ c2 the active time transformation ( = t ) (t = t = t = t ) is pv ! cu po co u v = t; (71) and the spatial velocity addition u v (u boosted by v) is v^ v^ 1 1 vujj + vv + u? v(u v)v¡ + vv + (u v)v¡ (u v) u v = u v = ^ t: (72) u v u v !

The active time transformation is a relative time transformation of the proper time tpv of v into the coordinate time tcu of u, which is the proper time tpo = t of o. Then, (u v) = (o + u v)t is a relative transformation of the velocity audition u v back into the frame of o with coordinate time t. In general, u v =/ v u, otherwise the direction of length contraction would be ambiguous. The formulas for u v 1 and u v can be derived and veried by algebraically expanding the boost versor operation BvuBv¡ . 1 For any boost B , we must limit to less than light speed 0 <1 such that 1 < and 1 ¡ >0. v v v v 1 v Note that, in some other literature u v (70) is sometimes dened dierently, as we dene u v (80) for the composition of two boosts (two ), by u and then by v. Our denition of u v is for one boost (one ) of u by v, boosting u into the rest frame of v with new time = tpv, which passively transforms back to t, where u and v are both in the frame of o with coordinate time t. Also note that, due to the anti-Euclidean metric of spatial vectors u and v, the sign on u v is negative compared to some other 1 v¡ literature that uses the positive Euclidean metric for spatial vectors. The expression (u v)= 1 (u v) v¡ may appear more like some other literature. The notation for the passive boost of u relative to v is 1 Bv¡ uBv = u v = u v(o + u v); (73)

10 Section 2 where the passive dilation factor (including an alternative notation subscript) is u v u v = u v = v 1 + =/ ( 0 u v = u v = o u v); (74) c2 the passive time transformation (t = t = t = t ) ( = t ) is cu po co ! pv u vt = ; (75) and the spatial relative velocity u v (u relative to v) is

v^ v^ 1 1 vujj vv + u? v(u v)v¡ vv + (u v)v¡ (u v) u vt = ¡ u vt = ¡ ^ : (76) u v u v !

The passive time transformation is a relative transformation of the coordinate time tcu of u, which is the proper time tpo = t of o, into the proper time tpv = of v. Then, (u v)t = (o + u v) is a relative transformation of the relative velocity u v into (relative to) the rest frame of v , where

(v v)t = (o/ v)t = o and ((o v) v) = o. In the rest frame of v with proper time as coordinate tim e, the worldline o represents ob servable vt that is in the frame of o (i.e., o is the observer space- time velocity within any rest frame). Velocity subtraction (u vy)t = (u v)t in the frame of o uses the spatial velocity addition (72) with a negative velocity vy = v, which is not actually the same as the spatial relative velocity (76) in the frame of v since the fram¡es o and v and their proper times t and are dierent. Our notation u v, which we dene by (72) as u boosted by v, is also found in some other literature, where the notationu v is dened dierently as v boosted by u with a reversed sense of operator and operand (or some other denition). In the expression u v, our operator v is a RHS operator that acts on LHS operand u, while some other literature may dene the LHS operator u that acts on v. By our denitions of RHS unary operators v (72) and v (76), we arguably writemore intuitive expressions such as u v v = (u v) v = u with conventi onal left to right precedence of operations, while in other literature d ening LHS o perators this may be written backwards as v (v u) = u or perhaps as v (v u) = u, which requires the parentheses to order the operations as rightto left. The form of a re lative vector u v, of u relative to v, better agrees with our choice of notational denition, expressing u relative to v ¡as u v (76). However, in other literature using LHS operators this same expression would be written v u or perhaps v u, which is misleading or less intuitive. Some other literature may try to work aro und this notati onal problem by dening our RHS operators with other notations, such as v and v, but whatever the exact dierences may be in our denitions of v (72) and v (76) compared to other literature, we stand by our more general denitions of v (69) and v (73), which can operate not only on vectors u but also on versors and geometric entities. The notations for the relatively equivalent active and passive boosts are

1 1 ¡ ¡ BvuBv = Bv uBvy = u v = u vy = u vy(o + u vy); (77) y with the relatively equivalent active u vtpv = tcu and passive u v tcu = tpv time transformations, y y which are numerically equivalent for tpv = tco since u v = u vy. An active boost can be viewed as the equivalent passive boost, and vice versa, but their interpretat ions are dierent. 1 1 For the composition of active boosts BvBu, as o u v = BvBuoBu¡ Bv¡ , we use the following notations 1 BuoBu¡ = o u = 0 u(o + 0 u) = u(o + u) = u(o + u) = o u(o + u) (78) 1 Bv(o u)Bv¡ = o u v = u u v(o + u v) = u v(o + u v) (79) u v u v = o u v = u u v = u v 1 (80) ¡ c2 u v u v = u vy = u u v = u v 1 + 2 (81) c o u vtpv = o utpu = tco = t: (82) 1 1 For the composition of the relatively equivalent passive boosts B B = B¡ B¡ , we use the following v u vy uy notations 1 B¡ oBu = o uy = o u (o uy) = 0 u (o uy) = u (o uy) = uu (83) uy y y y y 1 ¡ ¡ ¡ ¡ Bv (o uy)Bvy = o uy vy = uy uy vy(o uy vy) = o uy vy(o uy vy) (84) y ¡ ¡ = o u v(o + u v) = o u v tco o u v = tpu u v = tpv : (85) y y y y y y Notation of Space-Time Algebra (STA) 11

1 1 Although B B = B¡ B¡ , their interpretations and time transformations are dierent. v u vy uy For continued compositions of boosts t u ::: v w and t u ::: v w etc., the notations t u ::: v wtpw = t u ::: vtpv = ::: = t utpu = tct and tct t u ::: v w = tpu u ::: v w = ::: = tpv v w = tpw etc. may be most intuitive for the time transformations from frame to frame. For a mixed active/passive composition t u v w = t u v w (as an example), we must have tct t u v w = tpw since the last boost is passive into the frame of w, and then we make an equivalent purely passive factor t u v w = t u v w, where tct t u v w = tpu u v w = tpv v w = tpw, noting that tpv = tpv y y y y y etc. Similarly, a continued boost that ends active t u v w must have t u v wtpw = tct, and then we make an equivalent purely active factor t u v w = t u v w, where t u v wtpw = t u v tpv = y y y t utpu =tct. A continued active boost has a time transformation taking a proper time tpw of w through many proper times to a coordinate time tct of t, and a continued passive boost takes a coordinate time tct through many proper times to the proper time tpw of w. Note that, t can be replaced with o t = o ty, and then the purely active boosts convert time into tco = tpo, and the purely passive boosts convert from tco = tpo = tct, such that only proper times are transformed from frame to frame. For the active boost u v, the spatial velocity u v generally has a natural speed u v = u v /c; (86) k k and for the special case u v of parallel velocities jj u + v u v = : (87) 1 + u v For the passive boost u v, relative to v, the spatial velocity u v generally has a natural speed

u v = u v /c; (88) k k and for the special case u v of parallel velocities jj u v u v = ¡ : (89) 1 ¡ u v For the special case u v of perpendicular velocities ? 2 2 2 u v = u v = (1 v) u + v: (90) ¡ The boost notations and formulas given abovpe are derived directly from the boost operations. The notations can extend to further compositions of boosts.

(1/2)ϕvvˆγ0 1/2 v 1/2 Hyperbolic rotor (boost): B = Bv = e G1 Bv =(γvv/o) =(γv + γvβvˆγ0) 0,3 space ∼ 1 1 Bv† = Bv Bv = exp((1/2)ϕvvˆγ0)= ch ϕv + sh ϕv vˆγ0 2 2 2 (vˆγ0) = 1 2 2 2 γvv = γvβvcvˆ γvv Hyperbolas of constant (rapidity) space-time interval: Observer: Hyperbolic arc length: Boost velocity: v = o + v = cγ0 + βvcvˆ o = cγ0 γvv γvβvcv γvv rise 3 = ˆ ϕv = ath = ath(βv) βv = run Natural speed: 4 2 γv =1/ 1 − βv Lorentz factor: p v = βvcvˆ v = o + v cϕv γvcϕv γv = dtpo/dtpv = dt/dτ ∼ ∼ v = BvovBv = Bv(o/ γv)Bv u ⊕ v ∼ c γvv = BvoBv = γvcγ0 + γvβvcvˆ ϕv γv u = o + u = cγ + βucuˆ Observable velocity: 0 o o o γvo = γvcγ0 G1 γv γu 1,0 time −1 2 βu = u − u 2 γ =1/ 1 β p † Active ⊕ and passive ⊖ boosts: u Observer v or v : ∼ † ∼ † † † v (o ⊕ v)tpv =(BvoBv )tpv = γvvtpv = vtco ov = Bv v Bv = o/ γ −v ∼ ∼ † † † − v v v v (o ⊖ v)tco =(Bv oBv)tco = γo⊖vv tco = v tpv v = o ov = B vB = o/ γ

v 2 Proper times: u||vˆ + 1 − k k u⊥vˆ + v † r c2 − v γv u v γ t = tcv = tcv† = tpo ⊕ = u · v 1 − † c2 τ = tpv = tpv = tpo/ γv

v 2 u||vˆ + 1 − k k u⊥vˆ − v r c2 −γ2v γ2v† u ⊖ v = u v 1+ · u ⊖ v c2

u v u v ∼ dtpo · dtpv · (u ⊕ v)τ =(BvuBv )τ = γu⊕v(o + u ⊕ v)τ =(o + u ⊕ v)t γu⊕v = = γv 1 − 2 , γu⊖v = = γv 1+ 2 dtpv c dtpo c ∼ (u ⊖ v)t =(Bv uBv)t = γu⊖v(o + u ⊖ v)t =(o + u ⊖ v)τ

Figure 1. Space-time diagram of space-time boost B operations. 12 Section 2

Figure 1 shows a space-time diagram of space-time boost B operations on space-time velocities as 1 hyperbolic rotations. The orientation places the pseudospatial time axis 0 1;0 horizontal and the anti- Euclidean spatial axis v^ 1 vertical. The slope of a space-time velocit2yGv = o + v = c + cv^ is the 2 G0;3 0 natural speed = = v / o = v /c. The Lorentz factor of v is = = o / v = c/ c2 2c2 = v k k k k v j j j j ¡ v 1/ 1 2 . In close analogy to elliptic trigonometry in a Euclidean plane where x =r cos(), y = r sin(), ¡ v p and = atan(y / x), in the Minkowski space-time plane of Figure 1, we have hyperbolic trigonometry p where x = r cosh('), y = r sinh('), and ' = atanh(y /x) = atanh( ). Here, r is the constant hyperbolic radius under hyperbolic rotations (r2 is the constant interval), and ' is the hyperbolic angle (rapidity). The stationary observer has space-time velocity o = r 0 = c 0. The observer worldline is ot. The hyperbolic rotation u0 = u v = B uB of a space-time velocity u by angle ' = atanh( ) is an v v v v active boost by spatial velocity v = cv^ that transforms the slope u of the space-time velocity u 2 2 into u0 = u v of u0 = u v, while holding the interval u = u0 constant. In the gure, the active j j j j boost of o by v is o v = B oB = v, where the new time = t is the proper time of v, where v v v pv v = t and vv = vt = (o + v)t is in the frame of o. Note that, no boost B can ever boost a speed to exactly 0 = 1 since the hyperbolic rotation can only asymptotically approach, but never reach, the direction of a light-speed null vector c 0 cv^ on the light-like null hypercone. The time-like hyperbolic 2 2 2 (pseudo-Euclidean) length vt = (o + v)t = (c 0 + cv^) t = 1 ct = ct/ = c = ctpv gives the proper time = t of v. Pjropj erjtime is tjhe pseudo-Euclidean lengt¡h of the worldline when using only pv p p natural speeds with c = 1. For = 0, the observable vt coincides with the observer ot, and the observer measures time t = L . For 0 1, the observer computes the time t = L = 1 2L = L / , which 0 pv ¡ 0 0 is the special relativity length contraction formula (65) for length (and time, or space-time) contraction p in direction v^ as experienced by the observable v relative to o. In eect, vt = (v o)t, relative to ot, experiences contraction o = ot/ = (v v)t. v A composition of two successive unimodular boosts BvBu = B^vBû = B^wR^n^, by velocities u then v, is equivalent to a Thomas-Wigner rotation R^n^ that is followed by a single resultant boost B^w by a velocity w = 0 u v = u v. The Thomas-Wigner rotation R ^ is a spatial rotation in the plane of u and v n (with normal n = n^) by an angle . However, the factoring of BvBu can be done in two dierent ways ^ ^ ^ ^ ^ ^ as BvBu = Bw2Rn = RnBw1, where Bw2 = Bw. The product of two unimodular boosts BvBu expands as 1 1 B B = exp ' v^ exp ' u^ (91) v u 2 v 0 2 u 0 1 1 1 1 = cosh ' cosh ' + sinh ' sinh ' ( vû^) + (92) 2 v 2 u 2 v 2 u ¡ 1 1 1 1 cosh ' sinh ' u^ + sinh ' cosh ' v^ 2 v 2 u 0 2 v 2 u 0 = h + h ( vû^) + (h u^ + h v^) : (93) cc ss ¡ cs sc 0 The part of the expanded product BvBu that is purely a spatial rotation is

R = h + h ( v^u^) = h + h ( v^ u^ v^ u^) = h + h cos() + sin()u^ v^ u^ (94) n cc ss ¡ cc ss ¡ ¡ ^ cc ss ^ ? 1 1 1 1 R^n = Rn ¡ Rn = exp n^ = cos + sin n^ = R^n^; (95) j j 2 S 2 2 S where the angle 0 between u^ and v^ is = acos( v^ u^); (96) ¡ 2 2 u^ the modulus of Rn, as an elliptic complex number with (n^ ) = u^ v^? = 1, is S ^ ¡ 2 2 R = R¡R = pRR = h + h + 2h h cos(); (97) j nj n n n n cc ss cc ss and the spatial Thomas-Wigpner rotation angle opf R^n is

Rn Rn h sin() = 2 atan j ¡ j = 2 atan ss : (98) R + R h + h cos() n n cc ss Using trigonometric identities, the Thomas-Wigner rotation angle is also expressed as [19]

2 (1 + u v + u + v) 1 + cos() = > 0: (99) (1 + u v)(1 + u)(1 + v) Notation of Space-Time Algebra (STA) 13

The axis n^ of the Thomas-Wigner rotation R^n = R^n^ is (by the undual operation), S

Rn Rn v^ u^ n^ = (n^ )I = ¡ I = ¡ ^ I ; (100) S ¡ S S ¡ R R S ¡ sin(acos( v^ u^)) S j n ¡ n j ¡ where the rotation is from u^ toward v^ by angle . With the anti-Euclidean metric of SA, the rotation direction is the reverse of the direction of a similar rotor using Euclidean metric. Using the pure rotor Rn, the composition of boosts BvBu factorizes two dierent ways as 1 BvBu = Rn(1 + Rn¡ (hcsu^ + hscv^) 0) = RnBw1 (101) 1 = (1 + (hcsu^ + hscv^)Rn¡ 0)Rn = Bw2Rn (102) 1 1 1 1 B^ = B ¡ B = exp ' w^ = cosh ' + sinh ' w^ (103) w1 j w1j w1 2 w1 1 0 2 w1 2 w1 1 0 1 1 1 B^ = B 1B = exp ' w^ = cosh ' + sinh ' w^ ; (104) w2 j w2j¡ w2 2 w2 2 0 2 w2 2 w2 2 0 where the inverse of spatial rotor Rn is

1 Rn hcc + hss( u^v^) Rn R^n Rn¡ = = 2¡ = 2 = ; (105) R Rn Rn Rn Rn n j j j j j j 1 ^ ^ 2 and the modulus of Bw = exp 2 'ww 0 , as a hyperbolic complex number with (w 0) = 1, is

¡ 2 B = B¡B = BB = B = : (106) j wj j w wj j w wj j wj j j We now have the unimodular fapctorings p p ^ ^ ^ ^ ^ ^ BvBu = BvBu = Bw2Rn = RnBw1: (107)

The boosts Bw1 and Bw2 can also be written as R^(h u^ + h v^) 1 B = 1 + n cs sc = 1 + w^ = 1 + tanh ' w^ (108) w1 R 0 1 1 0 2 w1 1 0 j nj R^ (h u^ + h v^) 1 B = 1 + n cs sc = 1 + w^ = 1 + tanh ' w^ ; (109) w2 R 0 2 2 0 2 w2 2 0 j nj and we see that w^ = w^ = . Therefore, k 1 1k k 2 2k hcsu^ + hscv^ 'w = 'w = 'w = 2 atanh( ) = 2 atanh k k (110) 1 2 R j nj w = tanh('w) = w1 = w2: (111) ^ ^ ^ In the composition BvBu = Bw2Rn, the purely spatial rotation Rn is applied rst to a purely spatial ^ point (or other geometric entity) with zero velocity, and then the space-time boost Bw2 is applied second (subscript 2). Therefore, the velocity addition direction u v is w^ = w^ = u v = R^ (h u^ + h v^); (112) 2 n cs sc and the unimodular boost B^w = B^w = B^u v can be expressed as 2

B^w = B^u v = B^vB^uR^n^; (113) with boost space-time velocity w = o + w = o + u v = c + cw^ : (114) 0 w ^ ^ ^ A triad of boosts BwyBvBu that returns to zero boost velocity is a Thomas-Wigner rotation

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ Bw BvBu = Bu v BvBu = Rn^Bu Bv BvBu = B0Rn^ = Rn^: (115) y ¡ ^ ^ ¡^ ^ ^ ^ ¡ However, a quad of boosts BuyBvyBvBu = 1 = B0R0 that returns to zero boost velocity also returns to zero rotation. ^ ^ In the composition BvBu = Rn^Bw1, the space-time boost Bw1 is applied rst (subscript 1), and then the spatial rotation R^n^ is applied second, where R^n^ rotates the boost velocity direction w^ 1 as

^ ^ ^ ^ ^ ^ Rn^w^ 1Rn^ = Rn^Rn^(hcsu^ + hscv^)Rn^ = Rn^(hcsu^ + hscv^) = w^ 2 = w^ : (116) 14 Section 3

Applying the rotation R^n^ second has the eect of making a trajectory correction by the rotation R^n^w^ 1R^n^ = w^ 2 of the boost velocity direction w^ 1 into w^ 2 = w^ , as well as spatially rotating any spatial entity. Spatial rotation of a boosted point or entity also rotates its velocity direction. To apply only an arbitrary spatial rotation Rn to an entity with space-time velocity u=o+u, without changing the velocity u or changing from the frame of u, we use the boosted rotor

R u = B R B; (117) n u n u where Rn u relative to u is the spatial rotation (Rn u) u = Rn in the frame of u. Applying Rn directly to an entity with velocity u will also rotate the velocity into u0 = RnuRn , which rotates the entity as stationary in the frame of u0 = o + u0. Similarly, to apply an arbitrary boost Bv to an entity in the frame of u, without changing from the frame of u, we use the boosted boost

B u = B B B; (118) v u v u where Bv u relative to u is the boost (Bv u) u = Bv in the frame of u. For a stationary entity in the frame of u, then Bv u is eectively equal to applying BuBv (=/ BvBu) to the entity when stationary in the frame of o. Note that, BuBv boosts into the frame of u, while BvBu boosts into the frame of v. 2 As we will show, the boost eects of BvBu, including length contraction L= 1 wL0 and Thomas- ^ ¡ Wigner rotation Rn^, are easy to demonstrate when boosting the DCSTA 2-vectopr quadric surface entities (see Figure 4).

3 Notation of Conformal STA (CSTA)

1 The basis of CSTA 2;4 , isomorphic to (=) CSTA1 2;4 (index = 1), is G C G C 0; 1; 2; 3; e+; e = e1; e2; e3; e4; e5; e6 ; (119) f ¡g f g and for the second copy CSTA2 2 (index = 2), G2;4 C 0; 1; 2; 3; e+; e = e7; e8; e9; e10; e11; e12 : (120) f ¡g f g The six-dimensional CSTA unit pseudoscalar is

I = 0 1 2 3e+e : (121) C ¡ The 1;4 Conformal Space Algebra (CSA), subalgebra of 2;4 CSTA, omits the time-like basis vector 0 and tGhe time coordinate w = ct = 0, and then has only spaGtial entities and operations that are similar to those of 4;1 CGA. CSTA denes three geometric inner product null space (GIPNS) [16] 1-blade entities, as followGs. The CSTA GIPNS 1-blade null hypercone entity K (growing sphere in time from a point), equal to the null point embedding P , is C C 1 2 2 K^ = P^ = (p ) = p + p e + eo ; P = 0; (122) C C C M M 2 M 1 C centered at vertex p with null innity point (representing the point at innity) M 2 e = e+ + e ; e = 0; (123) 1 ¡ 1 and null origin point (representing the point at the origin)

2 eo = (e e+)/2; eo = 0; eo e = 1; e+e = eo e : (124) ¡ ¡ 1 ¡ ¡ ^ 1

A normalized point entity P^ has unit scale on the homogeneous term eo as C

P^ = P /( P e ): (125) C C ¡ C 1 The vector p and its embedding P^ = (p ) represent a specic position point (pw; px; py; pz). M C C M The symbolic vector t and its embedding T^ = (t ) represent the symbolic variable test point (w = ct; x; y; z). M C C M The projection (inverse of embedding) of a point P^ = (p ) to its embedded STA vector is C C M 1 1 p = ¡ (P ) = P^ I I¡ = P^ e+ e (e+ e ); (126) M C C C M M C ^ ^ ¡ ^ ¡ ¡ ¡ Notation of Conformal STA (CSTA) 15

which is geometrically projection onto I or rejection from e+e = e+ e = e eo . M ¡ ^ ¡ 1 ^ The inner product of two normalized points P^ = (p ) and Q^ = (q ) is C C M C M 1 1 1 P^ Q^ = p q p2 q2 = (p q )2; (127) C C M M ¡ 2 M ¡ 2 M ¡2 M ¡ M which is 1/2 the space-time interval (p q )2 between p and q . ¡ M ¡ M M M The reciprocals of 0; 1; 2; 3; e+; e are 0; 1; 2; 3; e+; e , and the reciprocals of f ¡g f ¡ ¡ ¡ ¡ ¡g eo ; e are e ; eo . Using these reciprocals, we dene the CSTA extraction operators f 1 g f¡ 1 ¡ g s w t x y z 1 t2 T T = T ; T ; T ; T ; T ; T ; T M = 0; 0/c; 1; 2; 3; e ; 2eo ; (128) C 2 C C C C C C C C f ¡ ¡ ¡ ¡ 1 ¡ g which are for the extractions of the correspon ding coecients s w = ct; t; x; y; z; 1; t2 from any point 2 f Mg T^ = (t ) as the inner product s = T^ T s. Linear combinations of the extraction operators T s T (eCxtraCctinMg values s) can form the CSTACG IPCNS 1-blade entities for hypercones K , hyperplanes EC ,2andC C C hyperpseudospheres in terms of their algebraic polynomial implicit surface functions F (w; x; y; z) = 0 C in space-time. The inner product of the symbolic test point T^ = (t ) with point P^ is C C M C 1 1 1 T^ P^ = t p t2 p2 = (t p )2 (129) C C M M ¡ 2 M ¡ 2 M ¡2 M ¡ M 1 = ((w p )2 (x p )2 (y p )2 (z p )2); (130) ¡2 ¡ w ¡ ¡ x ¡ ¡ y ¡ ¡ z which represents the implicit surface function F (w; x; y; z) = T^ P^ of a hypercone F = 0 with vertex C C p . A point T is on the hypercone surface presented by P i (if and only if) T P = 0. As an IPNS M C C C C entity, the conformal point embedding P^ = (p ) represents the hypercone implicit surface function (130) for a hypercone with vertex p , notCjustCtheMembedded point p . However T P = 0 i T P , M M C ^ C C ' C and we call P a geometric outer product null space (GOPNS) point entity. The relation denotes the C equality, up to a non-zero scalar multiple, of homogeneous entities representing the same'geometry. The IPNS hypercone entity P = K can be written in terms of the CSTA extraction operators T s as C C C ^ t2 w x y z 2 2 2 2 1 2K = T M 2pwT + 2pxT + 2pyT + 2pzT + (pw px py pz)T (131) C C C C C ¡ C ¡ 2 C ¡ ¡ ¡ = 2eo 2p p e : (132) ¡ ¡ M ¡ M 1 It can then be veried that T^ K^ = F (w; x; y; z) of (130). Linear combinations of the extraction operators T s construct IPNS enCtitiesCthat directly correspond to certain polynomial functions F (w; x; y; z) that canCbe formed as linear combinations of the available terms s. While the T s represent symbolic variables and constants of a polynomial function F (w; x; y; z), their linear combinCations form specic elements of CSTA, which we call geometric inner product null space (GIPNS) entities. If points p are restricted to spatial points p = p with no time (w = ct = 0), the conformal M M S embedding is the CSA null 1-blade spatial point entity P^ = (p ) of the CSA subalgebra of CSTA . Both the IPNS and OPNS of the CSA spatial point CPS CrepSresent only theCSpoint p when tested C CS S against the CSA test point T^ = (t ) (i.e., T P = 0 i T P ). CS C S CS CS CS ' CS The CSTA GIPNS 1-blade space-time hyperplane (3D subspace of 4D ST) entity E is C

E = n + (p n )e ; (133) C M M M 1 which represents the 3D subspace orthogonal to n and passing through space-time position p . A M M normalized hyperplane E^ has n = n^ by (34). When n is a null vector (n2 = 0), then E = L is the CSA GIPNS 1-blade Cnull lineM(lightM-line) entity L thrMough the point p inMthe null direcCtion nC C M M that includes point e on the null line. 1 The inner product of the symbolic test point T^ = (t ) with E is C C M C T^ E = t n p n = (t p ) n (134) C C M M ¡ M M M ¡ M M = (w pw)nw (x px)nx (y py)ny (z pz)nz; (135) ¡ ¡ ¡ ¡ ¡ ¡ ¡ which represents the implicit surface function F (w; x; y; z) = T^ E of a hyperplane F = 0 orthogonal to n and passing through p (by translation). The hyperplaCne enCtity E can be written in terms of M M C the CSTA extraction operators T s as C w x y z 1 E = nwT nxT nyT nzT (p n )T : (136) C C ¡ C ¡ C ¡ C ¡ M M C 16 Section 3

For position p =p and normal n =n restricted to spatial vectors, the entity E is the CSA GIPNS M S M S C 1-blade spatial plane entity = E (i.e., holding w = ct = 0 removes the 0 dimension), where the CS C CSA point T^ = (t ) is on the plane i T = 0. In CSTA, is a purely spatial plane CS C S CS CS CS CS entity at zero velocity (no time t dependency), which can be boosted Bv Bv into a velocity v. CS The CSTA GIPNS 1-blade hyperpseudosphere entity centered at point P^ = (p ) with initial C C C M radius r0, or through point Q^ = (q ), is C C M ^ ^ 2 ^ ^ ^ = P + (1/2)r0e = P + P Q e ; (137) C C 1 C C C 1 where the initial radius r0 can be real or imaginary. For¡spatial points P^ = (p ) and Q^ = (q ), r0 is the initial radius of a spatial sphere that grows with time in space-time, aCndCtheSn is aC(hyCperS)hyper- 2 2 D boloid of one sheet in space-time. More generally r0 = (p q ) (cf.(127)), and for a space-like 2 ¡ M ¡ M interval (p q ) <0 then r0 is real and is a (hyper)hyperboloid of one sheet, for a time-like interval M2 ¡ M C (p q ) > 0 then r0 is imaginary and is a (hyper)hyperboloid of two sheets, and when P^ = Q^ M ¡ M C C C then r0 = 0 and ^ = P^ is a null hypercone. The inner product of the symbolic test point T^ = (t ) C C C C M with ^ is C ^ ^ 1 2 1 2 2T = 2 (t p ) r0 (138) ¡ C C ¡ ¡2 M ¡ M ¡ 2 = r2 + (w p )2 (x p )2 (y p )2 (z p )2; (139) 0 ¡ w ¡ ¡ x ¡ ¡ y ¡ ¡ z which represents the implicit surface function F (w; x; y; z) of a pseudosphere (space-time circular hyperboloid) in the pseudospatial time w dimension with any two space dimensions, and a hyperpseudosphere F = 0 in all four STA dimensions. For P^ = P^ = (p ) restricted to a spatial center point p C CS C S S with w = ct = 0, the hyperpseudosphere ^ is the CSA GIPNS 1-blade spatial sphere entity S^ = ^ C CS C with radius r = r0, where the CSA null 1-blade spatial point T^ = (t ) is on the spatial sphere S i T S =0. The hyperpseudosphere entity can be written iCnSterCms Sof the CSTA extraction operaCStors T sCSas CS C C ^ 2 1 t2 w x y z 2 1 2 = r0T + T M 2pwT + 2pxT + 2pyT + 2pzT + p T : (140) ¡ C C C ¡ C C C C M C The quasi-sphere implicit surface is dened by at2 + bn t + c = 0, which can be represented as the linear combination a + bE + cT 1. The quaMsi-spherMe is Ma hyperpseudosphere for a =/ 0, and is a hyperplane for a = 0 and b=/C 0. TheC quasiC-sphere generalizes the hyperpseudosphere and hyperplane. The hyperpseudosphere entity ^ / r0 through point q with center p = q + r0 n^ becomes, in the limit C j j M M M j j r0 , the normalized hyperplane E^ with normal n^ through q , j j ! 1 C M ^ ^ 1 ^ 2 ^ 2 q + r0 n + 2 ((q + r0 n) ( r0 n) )e + eo E^ = lim C = lim M j j M j j ¡ j j 1 (141) C r0 r0 r0 r0 j j!1 j j j j!1 j j = n^ + (q n^)e : (142) M 1 Thus, the hyperpseudosphere entity generalizes to the hyperplane entity, like the quasi-sphere. The extraction operators (128) can also form the following CSTA dierential operators

1 w 1 Dw = T (T )¡ = e 0 = 0 e (143) C C 1 1 1 t 1 ¡ ^ Dt = T (T )¡ = ce 0 = c 0 e (144) C C 1 1 1 x 1 ¡ ^ Dx = T (T )¡ = e 1 = 1 e (145) C C 1 1 1 y 1 ¡ ^ Dy = T (T )¡ = e 2 = 2 e (146) C C 1 1 1 z 1 ¡ ^ Dz = T (T )¡ = e 3 = 3 e : (147) C C ¡ 1 ^ 1 The CSTA dierential elements are free vectors [3], which represent directions without locations, and are invariant by the translation operator (158). The n-directional ( n 2 = pn ny = 1) derivative of any CSTA GIPNS 1-blade entity A is, by the commutator product (k176k) with a dierential operator, @ A = (n D + n D + n D + n D ) A: (148) n w w x x y y z z The outer product of two to six of the above three CSTA GIPNS 1-blade entities (null-hypercones K , hyperplanes E , hyperpseudospheres ) forms, by intersection, more CSTA GIPNS entities of higheCr grades. C C The CSTA GIPNS 2-blade space-time plane entity

= D (p D )e ; (149) C M ¡ M M 1 Notation of Conformal STA (CSTA) 17 in direction of 2-blade D through point p , is the intersection (wedge) of two space-time hyperplanes M M (133). A normalized plane ^ has a unit 2-blade direction D =D^ by (35). Note that, the CSA GIPNS C M M 2-blade spatial line entity L = d^S p d^S e , viewed as a CSTA entity that is tested against CS ¡ 1 the CSTA point T = (t ) instead of the CSA point T = (t ), gains the span of the pseudospatial C C M ¡ CS C S direction 0 and is the CSTA plane with 2-blade direction D = d 0 through point p = p . The CSTA GIPNS 3-blade space-tiCme line entity M ^ M S

L = d + (p d ) e ; (150) C M M M ^ 1 in the direction d through point p = pw 0 + p , is the intersection (wedge) of three space-time M M S hyperplanes (133). A normalized line L^ has a unit direction d =d^ by (34). If the line direction vector d is a null vector, then the line entiCty L is a null 3-bladeMrepreMsenting a null line (light-line). The M C point at innity e is a point of any line L . The line L can represent the worldline of an observable 1 C C with STA velocity v = d = c 0 + cv^ with initial spatial position p = p0. The initial position p0 can also be found as the CSTMA GOPNS 2-blade at point2 [3] position M

1 (p0) e ( 0 L )I¡ (151) C ^ 1 ' ^ C C at t =0, where E = 0 is the t =0 hyperplane (133) and 0 L =P is a CSTA GIPNS 4-blade at point C ^ C C 1 entity P , which is the intersection of four hyperplanes (133). By CSTA dualization (153), ( 0 L )I¡ C ^ C C is a CSTA GOPNS 2-blade at point entity P = P e . The point P^ = (p ) of a CSTA GOPNS C C ^ 1 C C M 2-blade at point P is projected as C (eo e ) (eo P) p = ^ 1 ^ C : (152) M (eo e ) P ¡ ^ 1 C The boost B, and the other CSTA versors, can operate on the line L to implement space-time transfor- C mations of a worldline representation. Intersecting L with the time t hyperplane E = 0 + cte nds the spatial position at coordinate time t in the currenCt frame as the resulting CSTACGIPNS 4-b1lade at point P = L E . A passive boost changes the coordinate time t to be the proper time in the new frame. C C ^ C CSTA dualization of a CSTA GIPNS k-blade entity X gives its dual CSTA geometric outer product null space (GOPNS) [16] (6 k)-blade entity C ¡ 1 X = X I¡ : (153) C C C A CSTA point P is on CSTA GIPNS entity X i C C P X = 0: (154) C C A CSTA point P is on the corresponding dual CSTA GOPNS entity X i C C P X = 0: (155) C ^ C The outer product of up to six well-chosen CSTA points P produces various CSTA GOPNS (1:::6)-blade Ci space-time surface entities X = P i for surfaces that the points span as surface points. The CSTA GOPNS null 1-blade point (emC beddinCg) P equals the CSTA GIPNS null 1-blade hypercone P = K . CSTA inherits the STA 2-versVor spatialC rotor C C

R = R = exp(n^ /2) = cos(/2) + sin(/2)n^ ; (156) C S S and STA 2-versor space-time hyperbolic rotor (boost)

B = B = exp('v^ 0 /2) = cosh('/2) + sinh('/2)v^ 0: (157) C Compositions of rotor and boost, such as the boosted rotor and boosted boost are also inherited. CSTA introduces the CSTA 2-versor space-time translator

T = exp(e d /2) = 1 + e d /2; (158) C 1 M 1 M which translates by d . As versor compositions, CSTA also introduces the following three translated 2- versors. The translatoMr T with (e d )2 =0 is a geometric number form of unimodular T =1 parabolic complex number (dual nuCmber) a1+ bM" with "2 = 0. j C j The CSTA 2-versor spatial translated-rotor is

1 L = T R T ¡ = exp 0 L^ /2 = cos(/2) + sin(/2)L^ ; (159) C C C C ¡ C CS

2. Flat point P e in [3] is called h¡omogeneous point p e + eo e in [16]. C ^ 1 M ^ 1 ^ 1 18 Section 4

which rotates by angle anticlockwise (by right-hand rule) around the spatial CSA line L^ = n^ CS S ¡ (d n^ )e through point d in the rotor axis direction n^ . ST heS C1STA 2-versor translaSted-boost is S d B = exp('(v^ 0 (d (v^ 0))e )/2) = exp '^ /2 = cosh('/2) + sinh('/2)^ ; (160) C ¡ M 1 C C ¡ centered on point d and with plane direction D^ = (v^ 0)I . The CSTA 2-verMsor translated-isotropic dilatorMis M

D = exp ln(d)P^ e /2 = cosh(ln(d)/2) + sinh(ln(d)/2)P^ e (161) C C ^ 1 C ^ 1 for isotropic dilation b¡y factor d > 0 relative to normalized center point P^ , i.e. P^ e = 1. By versor outermorphism [16], all CSTA versors correctly transform all CSTA GICPNS anCd d1ual C¡STA GOPNS entities.

4 Construction of Double CSTA (DCSTA) In double conformal space-time algebra (DCSTA) , CSTA1 1 and CSTA2 2 are orthogonal subalge- D C C bras and their geometric or outer product is a doubling extension. Any CSTA1 entity or versor A 1 and its C double A 2 in CSTA2 (with the same scalar coecients on corresponding basis blades) can be multiplied C to form the corresponding DCSTA entity or versor A = A 1A 2 = A 1 A 2. By versor outermorphism, the DCSTA versors operate correctly on all DCSTA DentitieCs. C C ^ C The DCSTA null 2-blade point

1 2 T = (t ) = T 1 T 2 = (t 1) (t 2) (162) D D M C ^ C C M ^ C M is an extended, doubled form of the CSTA point embedding T = (t ) (122). Note that, as in CSTA, the DCSTA point is a geometric OPNS (GOPNS) null point, bCut Ca GMIPNS null hypercone. The construction method is further extensible to an Extended CGA (k-CGA) , which is using not K just a double k = 2, but some k corresponding orthogonal CGAi i of a vector space Rp;q i, Gp+1;q+1 C V 1 i k, where the k-CGA entities or versors are A = A 1A 2:::A k = A 1 A 2 ::: A k, and points 1 2 k K C C C C ^ C ^ ^ C are T = (t ) = T 1T 2:::T k = (t 1) (t 2)::: (t k). SiKmilaKr toVthe DCCSCTA nuCll 2-CbladVe pCointVentitCy, oVther standard doubled 2-blade entities are formed as the product of corresponding CSTA1 and CSTA2 GIPNS 1-blade entities, which include the DCSTA GIPNS 2-blade hyperplane E = E 1E 2, the DCSTA GIPNS 2-blade hyperpseudosphere = 1 2, D C C D C C and the DCSTA GIPNS null 2-blade hypercone K = K 1K 2 = T 1T 2. The CSTA GIPNS intersection entities of grades 2, 3, 4, and 5 can also be doubledD into CtheiCr correCspoCnding standard DCSTA GIPNS entities of even grades 4, 6, 8, or 10, respectively. The same holds that, the CSTA GOPNS entities can be doubled into DCSTA GOPNS entities, or obtained from DCSTA GIPNS entities by using the DCSTA dualization operation (167). The doublings of the CSTA versors include the DCSTA 4-versor translator T = T 1T 2, the DCSTA d d d D C C 4-versor rotor R =R 1R 1 and its translated form R =R 1R 2 =L 1L 2, and the DCSTA 4-versor boost D C C d d d D C C C C B =B 1B 2 and its translated form B =B 1B 2. The DCSTA GIPNS 2-blade hyperplane E =E 1E 2 isDalso tChe DC CSTA 2-versor reector inDthe hCypeCrplane. The DCSTA GIPNS 2-blade hyperpsDeudosCpherCe = 1 2 is also the DCSTA 2-versor inversor in the hyperpseudosphere. When time is xed as t =D0, DCCSTCA eectively becomes the DCSA subalgebra, where the DCSA null point T = T represents onlyDthe point by both IPNS and OPNDSS(i.e., P T = 0 i P = T ), the DCSAD2S-bladDe sphere S = is the DCSA 2-versor inversor in the sphDeSre,DaSnd the DCDSA 2-DblSade plane = E is the DCDSSA 2-vDersor reector in the plane. DS D From a DCSTA point T , certain scalar polynomial terms, or values s, in variables x, y, z, and D w = ct can be extracted from the basis 2-blade coecients in T by inner products s = T Ts with D D certain corresponding value s extraction operators Ts (see Table 1), which are each a certain bivector that is an averaged sum of up to k = 2 reciprocal (pseudoinverse) basis 2-blades that extract the same s coecient value s. A DCSTA 2-blade extraction operator T i, 1 i k, for value s is the product s s2 s1 s2 s1 D s2 s1 T i = T 2T 1 = T 2 T 1 of CSTA2 and CSTA1 extraction operators T 2 and T 1 (128), respectively, such D C C C ^ C s C C that s = s2s1. Note that, the reciprocal 2-blade T i is formed by using the reverse order of multiplication of CGAi elements, as compared to the order thaDt forms a point T . In Extended CGA (k-CGA) , up s sk s2 s1 D K to k reciprocal basis k-blades T i = T k:::T 2T 1, 1 i k, extract the same coecient value s = sk:::s2s1 from a k-CGA point T . K C C C K Construction of Double CSTA (DCSTA) 19

For example, in DCSTA , which is a 2-CGA, we can form a DCSTA 2-blade extraction operator x D x 1 x x x 1 2 1 2 1 T i for value x in two ways, as T 1 = T T and T 2 = T T . Then, the DCSTA bivector (2-vector) D D C C D C C x 1 x x 1 extraction operator T = Tx for value x is Tx = (T + T ) = (e 2 e2 + e8 e 1), which is the D 2 D1 D2 2 1 ^ ^ 1 average of the reciprocal 2-blades T x and T x . The value x is then extracted from a DCSTA point T D1 D2 D as x = T Tx = Tx T . D D

1 1 1 Tx = (e 2 e2 + e8 e 1) Ty = (e 2 e3 + e9 e 1) Tz = (e 2 e4 + e10 e 1) 2 1 ^ ^ 1 2 1 ^ ^ 1 2 1 ^ ^ 1 Tx2 = e8 e2 Ty2 = e9 e3 Tz2 = e10 e4 1 ^ 1 ^ 1 ^ T = (e e + e e ) T = (e e + e e ) T = (e e + e e ) xy 2 9 ^ 2 8 ^ 3 yz 2 10 ^ 3 9 ^ 4 zx 2 8 ^ 4 10 ^ 2 Txt2 = eo2 e2 + e8 eo1 Tyt2 = eo2 e3 + e9 eo1 Tzt2 = eo2 e4 + e10 eo1 M ^ ^ M ^ ^ M ^ ^ T1 = e = e 1 e 2 Tt2 = eo2 e 1 + e 2 eo1 Tt4 = 4eo = 4eo1 eo2 1 1 1 1 1 ¡1 ¡ ^ M ^ ^ M ¡ ¡ ^ Tw = (e1 e 2 + e 1 e7) Tw2 = e7 e1 Twt2 = e1 eo2 + eo1 e7 2 1 1 1 ^ ^ 1 ^ M 1 ^ ^ Twx = 2 (e1 e8 + e2 e7) Twy = 2 (e1 e9 + e3 e7) Twz = 2 (e1 e10 + e4 e7) 1 ^ ^ 1 ^ ^ 1 ^ ^ 2 2 2 2 Tt = Tw Tt = 2 Tw Ttt = Twt c c M c M T = 1 T T = 1 T T = 1 T tx c wx ty c wy tz c wz

Table 1. DCSTA bivector extraction elements Ts.

Table 1 gives all 27 of the DCSTA bivector extraction operators (or elements) Ts for the extractions of scalar values s (indicated by the indices x; :::; tz) from any DCSTA point T = (t ) by the inner D D M products

s = Ts T : (163) D Note that, in Table 1, the scalar time t = w/c is not the vector t (in bold italic), where t = t is the M STA symbolic test position vector (5).

A linear combination of the DCSTA extraction operators Ts forms a DCSTA GIPNS bivector geometric entity that represents a polynomial function F (w; x; y; z), which in turn represents a Darboux cyclide implicit surface F (w; x; y; z) = 0 in space-time, where

= sTs (164) s and X

F (w; x; y; z) = T = ss; (165) D s X with real scalar coecients s. For w = ct = 0, then t = t and T = T = (t ) is a DCSA spatial point, and the rst ve rows M S D D D S in Table 1 are the DCSA extraction operators T for spatial Darboux cyclide surfaces in the G2(1);2(3+1) s anti-Euclidean space R0;3. DCSA is similar to the Double Conformal / Darboux Cyclide Geometric G2;8 Algebra (DCGA) with opposite signature. G8;2 Darboux cyclides are quartic (polynomial degree 4) surfaces that include quartic Dupin cyclides (including tori), quartic Blum cyclides, cubic (polynomial degree 3) parabolic cyclides, and general quadric (polynomial degree 2) surfaces. In Extended CGA (k-CGA), linear combinations of the k-vector extraction operators T s form k-vector entities that represent a further generalization of the Darboux cyclide polynomial funcKtion F that includes general degree k implicit surfaces and certain other specic implicit surfaces of degrees k < l 2k of inversive geometry. The DCSTA GIPNS bivector entities for quadrics and cyclides can be directly written as linear combinations of the extraction operators Ts. For example, an ellipsoid (centered at the origin, aligned along the SA axes 1; 2; 3) is

2 2 2 E = T 2 /a + T 2 /b + T 2 /c T ; (166) x y z ¡ 1 and a general point P is on it i P E = 0. The DCSTA dualization of the bivector E, D D 1 1 ED = EI¡ = E(I 1I 2)¡ ; (167) D C C 20 Section 4

is a valid GOPNS 10-vector entity where P is on it i P ED = 0. If time is always xed as t = 0, D D ^ then the DCSTA GIPNS bivector entities formed from the Ts correspond to entities of 8;2 DCGA [6], up to some sign dierences in some scalar expressions, due to the dierent choice of signGature.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 2. DCSA 2-vector quadrics Q and their inversions = SQS in sphere S = S . DS Figure 2 shows various DCSA 2-vector quadrics (and tori) Q and their inversions in a DCSA 2-blade sphere S = S (S = holding t = 0). Figure 2(a) shows the inversion of a quartic torus, which is a quartic DupinDSring cyclDide. Figure 2(b) shows the inversion of a quadric cylinder, which is a quartic Dupin needle cyclide. Figure 2(c) shows the inversion of a quadric cone, which is a Dupin horned cyclide. Figures 2(d,e,f,g,h) show the inversions of various other quadrics (ellipsoid, one sheet hyperboloid, two sheets hyperboloid, paraboloid, and hyperbolic paraboloid, resp.), which are various quartic Darboux cyclides. Figure 2(i) shows the inversion of a torus in a sphere that is centered on a surface point of the torus, which is a cubic parabolic cyclide. The DCSTA point at innity e = e 1e 2 is an outlier surface 11 1 1 point of the quadric ellipsoid entity Q = E of Figure 2(d), where Se S¡ = P^ = (p ) is the center 1 DS D S point of sphere S and an outlier surface point (not visible in the gure) of the Darboux cyclide =SES. Any DCSTA GIPNS bivector spatial quadric surface entity Q, formed as a linear combination of the Ts from the rst three rows in Table 1, has no time t dependency and appears to have zero velocity in space-time. The entity Q is also a purely spatial entity in the 2;8 DCSA subalgebra. The spatial quadric entity Q can be actively boosted into a spatial velocity vG= cv^ using the DCSTA 4-versor boost operator B = B 1B 2 (157). In physics, the speed 0 v < c of massive bodies can only approach D C C light speed c, and the natural speed = v/c is then limited to 0 < 1. If Q has been translated by d from the origin (perhaps by using a DCSTA translator T = T 1T 2) and is centered at spatial position d d d D C C p0 = d, then the translated boost operator B = B 1B 2 (160) can be used on Q. The boosted quadric d d D C C entity Q = B QB has center position pt = p0 + vt at time t, and has a geometrical length contraction D D (directed scaling) of the surface in the direction v^ by factor d = 1 2, which is consistent with ¡ special relativity length contraction. While Q (in bold) is a spatial entity with no time t dependency, p the boosted entity Q (in bold italic) is a space-time entity with time dependent position according to a constant velocity v of the boost, and also a contraction eect at all times. At t = 0, the contraction eect, which is a geometrical dilation, is present, and projecting Q on the DCSA subalgebra, eectively setting t = 0, produces a purely spatial entity Q0 = (Q) (168) at t = 0, centered at p = d, that retains P 0 the geometrical dilation or length contraction. The result Q0 is a directed scaling operation, but (so far) limited to a scaling factor 0 < d 1. The boost natural speed for a length contraction factor d is by (16) = p1 d2. Admitting the ¡ imaginary scalar p 1, the boost of a quadric by an imaginary dilates by d > 1, and then the result can be projected to¡the spatial subalgebra DCSA to discard time components and achieve directed G2;8 spatial scaling in the direction v^ of the boost velocity v. Construction of Double CSTA (DCSTA) 21

z 3 Ellipsoid: d d Boost direction: E0 = (B EB ) 1 P D D v^ = ( 1 + 2 + 3) 15 d = 5( + + ) p3 1 2 3 = p1 d2 ¡ E0 E d = 3 d x 1 5

15 S

E00

Darboux cyclide: Time: E00 = SE0S y 2 w = ct = 0

Figure 3. Spherical ellipsoid E(r = 5) dilated by factor d = 3 in direction v^ as ellipsoid E0 and then reected in sphere S as Darboux cyclide E00.

Figure 3 visualizes [17] a DCSTA GIPNS bivector spherical ellipsoid E dilated in situ by factor d = 3 d in the direction v^ as E0 using a translated-boost operator B centered on the center position p0 = d of D E and E0. The DCSA projection is G2;8 1 (A) = (A I )I¡ ; (168) P DS DS where the DCSA unit pseudoscalar is

I = I 1e5e6I 2e11e12: (169) DS S S

E0 is reected in a DCSTA GIPNS 2-blade (hyperpseudo)sphere S = (t = 0; r0 = 15) = 1 2 as E00, which is a Darboux cyclide. The sphere S, initially centered on the origin, was translated usiCng aCDCSTA 1 1 4-versor translator T by a displacement vector d + (5 + 15)Rd^R, using R = exp ( 2 1)S , D 2 2 p2 ¡ to bring the sphere into a tangential position to E0. All are at time t = 0.

∼ ∼ ∼ BvEBv BvBuEBu Bv BvBu = BwRǫnˆ v yγ2 ∼ = βvcγ2 BwEBw w = u ⊕ v βv =6/10 2 2 2 (u⊥v) ∴ βw =p (1 − βv)βu + βv t = 10 t = 10 y =6 wt

Thomas-Wigner rotation: ∼ ∼ ∼ ∼ t =0 Bw†(BvBuEBu Bv )Bw† = RǫnˆERǫnˆ ◦ ǫ = 23.602 nˆ = γ3

ǫ xγ1

∼ BuEBu x =9 u = βucγ1 x = 10 βu =9/10 t = 10 Ellipsoid E(a =4,b = c = 2) Using c =1

Figure 4. Boosts of ellipsoid E, showing length contractions and Thomas-Wigner rotation. 22 Section 4

Figure 4 shows the DCSA 2-vector ellipsoid entity E (166) with a= 4 (x-diameter 2a =8) and b =c =2 (y-diameter 2b = 4), centered on the origin with zero initial velocity, which is then boosted by various velocities u, v, and w = u v. In the gure, natural speeds are used with c = 1, and all of the boost B and rotation R versors are assumed to be their unimodular DCSTA doubled forms (e.g., Bu = B^ 1uB^ 2u). C C The boosted ellipsoid BuEBu =E u (extending the notation of (69)), by u= uc 1 =(9/10) 1 and with 1 2 center position ut =9 1 at t =10, has length contraction factor u¡ = 1 u 0.436 and the contracted 1 ¡ x-diameter is approximately u¡ 2a 3.487. The boosted ellipsoid pBvEBv = E v, by v = vc 2 = 1 2 (6/10) 2 with center position vt = 6 2 at t = 10, has length contraction factor v¡ = 1 v = 0.8 and 1 ¡ the contracted y-diameter is v¡ 2b = 3.2. The boosted ellipsoid BvBuEBuBv = BpwRnÊRn^Bw (by (107)), with a resulting velocity w = cw^ 0.9372c(0.7682 + 0.6402 ) (by (111) and (112)) and w 1 2 center position wt = 7.2 1 + 6 2 at t = 10, has a more complicated contraction due to the composition of boosts; however, when boosted back to zero velocity as Bwy(BwRnÊRn^Bw)Bwy = RnÊRn^, then it is only the Thomas-Wigner rotation Rn^ (95) of the ellipsoid E. In this example, u and v are perpendicular 2 2 2 (u v), so we can also obtain w by (90) as w = u v = (1 v) u + v. The boosted ellipsoid ? ¡ B EB has the same velocity w as B B EB B , but is a much dierent result: it is boosted into the w w v u u v p frame of w = o + w, not into the frame of u then into the frame of v; it does not include the Thomas- 2 Wigner rotation Rn^; and, it has a simple contraction by the factor 1 w 0.3487 in only the direction w^ . ¡ p

Σ w

E+ Ω y

(a) (b) (c)

+ + Figure 5. Inversion = E of pseudoquadric E in pseudosphere . D D D

+ Figure 5 shows the inversion = E of a DCSTA 2-vector pseudoquadric ellipsoid D D + 2 2 2 E = T 2 /a + T 2 /b + T 2 /c T (170) x y w ¡ 1 in a DCSTA 2-blade hyperpseudosphere , which is a pseudosphere (circular space-time hyperboloid) D in the three dimensional space-time of the two spatial dimensions x and y with the pseudospatial dimension w, holding z = 0. A DCSTA 2-vector space-time pseudoquadric (pseudospatial quadric) Q+ is formed from a DCSA 2-vector spatial quadric Q by replacing one of the coordinates x, y, or z with the pseudospatial coordinate w. The inversion of the corresponding spatial quadric ellipsoid = E D D viewed in the same three dimensions x, y, and w sees the spatial ellipsoid as a circle (or ellipse) in the xy-plane and as a cylinder in xyw-spacetime (i.e., the same xy-plane circle for all time w = ct), and therefore its inversion appears quite dierent than the inversion of the corresponding pseudoquadric. The DCSTA dierential elements are

1 Dw = 2TwTw¡2 (171) 1 Dx = 2TxTx¡2 (172) 1 Dy = 2TyTy¡2 (173) 1 Dz = 2TzTz¡2 (174) 1 Dt = 2TtTt¡2 (175) and the commutator product of multivectors A and B is A B = (AB BA)/2 = B A: (176) ¡ ¡ Bibliography 23

Using the commutator product, the DCSTA differential elements are differential operators on any bivector surface entity that is formed as a linear combination of the DCSTA extraction elements Ts. The time t derivative of is @ _ = @ = = D : (177) t @t t

For direction n with unit magnitude n = pn ny = 1, the n-directional derivative operator is k k2 @ @ = = D =(n D + n D + n D + n D ) (178) n @n n w w x x y y z z and the n-directional derivative of any bivector entity is @ = D : (179) n n The entity represents an implicit surface function F (w; x; y; z), and its n-directional derivative @n represents the derivative implicit surface function @nF . Mixed partial derivatives are obtained by taking successive derivatives in any order.

5 Conclusion

4;8 DCSTA extends 2;8 Double Conformal Space Algebra (DCSA), which is dierent in space signature G G from the DCGA 8;2 of [8], into a high-dimensional 12D embedding of Space-Time Algebra 1;3 that has general quadric sGurface entities with a complete set of space-time transformation operationGs as versors and projections. The DCSTA 2-vector general quadric surface entities provide an accurate representation of quadric surfaces in space-time. As discussed, boosts of the DCSTA quadric surface entities are moving surfaces that include the special relativity eects of length contraction and Thomas-Wigner rotation. In geometry and physics, the DCSTA 2-vector surface entities, including general quadrics and their inversions in hyperpseudospheres, may nd uses in education and applications for modeling inversive geometry and surfaces in the space-time of special relativity. DCSTA is an algebra for computing with general quadric surfaces and their inversions in hyperpseudospheres in space-time. For applications, testing, or education, DCSTA 4;8 can be computed using various software packages. During the research and writing of this paper, thGe author used the free symbolic computer algebra system Sympy [18] with the Algebra [1] module. All gures were rendered G using Mayavi [17] and annotated with mathematical text using TEXMACS [21]. As discussed in the paper, but not elaborated in full detail, not only is it possible to construct a doubling of CGAs as for DCSTA, but it is also possible (in theory) to extend to any number k of orthogonal CGAs . This CGA extension theory is to be called Extended CGA or k-CGA. Gk(p+1);k(q+1) In k-CGA, there are k-vector entities (linear combinations of k-blade extraction operators) that represent general degree k surfaces and also certain other surfaces of degrees l, k < l 2k, representing all possible inversions (and compositions of inversions) of the general degree k surfaces in hyperpseudospheres.

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