On the nature of space-time: Special Relativity in five dimensions Marcoen J.T.F. Cabbolet∗ Center for Logic and Philosophy of Science, Vrije Universiteit Brussel Paper presented at the Fourth International Conference on the Nature and Ontology of Spacetime, Varna (BG), 2016 Abstract | This paper rigorously introduces a new account of Special Relativity (SR) in a pseudo- Riemannian 5-manifold with one curled-up dimension. This 5D SR consists of four very simple postulates, one of which stating that massive particles and massive antiparticles move in opposite direction through the fifth dimension. A theorem is that all particles move with the speed of light in four-dimensional space. This 5D SR is a theory on its own, but a potential application is discussed in a model of the Elementary Process Theory (EPT). Edit: this 5D approach to a model of the EPT has been abandoned. 1 Introduction Special Relativity (SR), first published in [1], is known for over a century now, and is widely considered to be an untouchable cornerstone of modern physics. This letter introduces a new account of SR in a five-dimensional (5D) space|more precisely, in a pseudo-Riemannian 5-manifold|with one curled-up dimension. The idea of a curled-up dimension is not new: it has been proposed earlier by Klein [2]. Nor is the idea of a 5D account of SR new: earlier 5D accounts of relativity (not necessarily limited to SR) can be found both in the pre-World- War-II literature, e.g. [3] and in the modern literature, e.g. [4, 5, 6, 7]. What is new here is that this 5D SR postulates that massive particles of matter and antimatter move in opposite direction through the curled-up fifth dimension. The outline of this paper is as follows. The next section introduces a pseudo-Riemannian 5-manifold. The section thereafter introduces SR in 5D spacetime; the final section proves that it is empirically equivalent to the `standard' SR in 4D Minkowski spacetime, and states the conclusions. The remainder of this introduction is for some basic mathematical notations and definitions. Notation 1.1 We will use the symbols N; Z; Q; R; C respectively for the sets of natural numbers, integers, rational numbers, real numbers, and complex numbers, and U for the unit interval [0; 1) ⊂ R. The floor function b:c : R ! Z is given by bxc = maxfy 2 Z j y ≤ xg. For any x 2 R and y 2 U, the expression x ≡ y (mod 1) means: x is congruent to y modulo 1, that is, jx − yj 2 N. Definition 1.2 The space (U; +1; ·) consists of the set U together with the binary operations addition +1 : U × U ! U and scalar multiplication · : R × U ! U, which are defined as follows for any a; b; c 2 U and x 2 R: a +1 b = c , (a + b) ≡ c (mod 1) (1) x · a = b , xa ≡ b (mod 1) (2) where the term `xa' in Eq. (2) refers to the product xa 2 R. 1 1 The binary structure (U; +1) is isomorphic to the abelian group (S ; ·), the set S = fz 2 C j jzj = 1g of complex numbers of norm 1 under multiplication. (U; +1) is thus an abelian group; for any a 2 U, we have thus an element `−a' such that a +1 (−a) = 0. Furthermore, 1 · a = a for any a 2 U. Corollary 1.3 The space (U; +1; ·) is not a vector space. Proof A vector space must satisfy the axiom α · (a + b) = α · a + α · b for any scalar α and any vectors a and 3 1 1 3 3 1 3 1 3 3 1 b. However, here we have 2 · ( 2 +1 2 ) = 2 · 0 = 0 but 2 · 2 +1 2 · 2 = 4 +1 4 = 2 . Q.E.D. Definition 1.4 Let (R;T ) be the real line in the standard topology. Let the function % : R ! U be given by % : x 7! y , x ≡ y (mod 1). Then the standard topology on U, notation: TU, is the topology coinduced on U by %. Note that % does not yield a compactification, because % is not injective. We will use the function % throughout this paper. ∗e-mail: [email protected] 1 Corollary 1.5 For any x 2 U, the set Unfxg is open in U. Proof Consider the open subset (x − 1; x) of R. Then by Def. 1.4, %[(x − 1; x)] is open in U. But %[(x − 1; x)] = [0; x) [ (x; 1) = Unfxg as requested. Q.e.d. 1 The space (U;TU) is homeomorphic to the circle S in its standard topology: (U;TU) is thus a path connected, compact Hausdorff space. 2 Definition of a pseudo-Riemannian 5-manifold Definition 2.1 The pseudo-Riemannian 5-manifold (M;TM;AM;T M; g) consists of the set 4 M = R × U, the standard product topology TM, the atlas AM, the tangent bundle T M, and the metric tensor field g, such that (i) the atlas AM = f(Ut; φt) j t 2 Qg contains for every t 2 Q a chart (Ut; φt) given by 3 Ut = (t; t + 1) × R × Unf%(t)g (x0; x1; x2; x3; x4) 7! (x0; x1; x2; x3; bt + 1c + x4) if x4 2 [0;%(t)) (3) φ : U ! 5 ; φ : t t R t (x0; x1; x2; x3; x4) 7! (x0; x1; x2; x3; btc + x4) if x4 2 (%(t); 1) where % is the function from Def. 1.4, and [0;%(t)) = ; for t 2 Z. See Fig. 1 for an illustration. S 5 (ii) the tangent bundle T M is the sum set fTP (M) j P 2 Mg = M × R , where TP (M) is the tangent space at the point P 2 M given by 5 0 1 2 3 4 0 1 2 3 4 j 10 TP (M) = fP g × R = f(p ; p ; p ; p ; p ; x ; x ; x ; x ; x ) j x 2 Rg ⊂ R (4) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 If we denote (p ; p ; p ; p ; p ; x ; x ; x ; x ; x ) 2 TP (M) as ~xP , with ~xP = (x ; x ; x ; x ; x )P , and if we endow TP (M) with the operations addition and scalar multiplication given by 0 0 1 1 2 2 3 3 4 4 ~xP + ~yP = (x + y ; x + y ; x + y ; x + y ; x + y )P (5) 0 1 2 3 4 α · ~xP = (αx ; αx ; αx ; αx ; αx )P (6) 5 then (TP (M); +; ·) is isomorphic to the standard five-dimensional vector space (R ; +; ·). (iii) the tensor field g adds to every point P 2 M a metric tensor gP : TP (M) × TP (M) ! R such that for any two vectors ~xP ; ~yP 2 TP (M) 0 0 1 1 2 2 3 3 4 4 (5) i j gP (~xP ; ~yP ) = −x y + x y + x y + x y + x y = ηij x y (7) i j where x ; y are the coordinates with respect to a standard basis of TP (M). Definition 2.2 The binary operation scalar multiplication · : R × M ! M is for any number a 2 R and any X = (x0; x1; x2; x3; x4) 2 M given by a · (x0; x1; x2; x3; x4) = (ax0; ax1; ax2; ax3; a · x4) (8) 4 where the product a · x is given by Def. 1.2. Definition 2.3 The O-group M (M; +; O) consists of the set M, given in Def. 2.1, the binary operations addition + : M × M ! M and the set of operators O, such that (i) the sum of any two elements X = (x0; x1; x2; x3; x4) and Y = (y0; y1; y2; y3; y4; y5) of M is given by 0 1 2 3 4 0 1 2 3 4 0 0 1 1 2 2 3 3 4 4 (x ; x ; x ; x ; x ) + (y ; y ; y ; y ; y ) = (x + y ; x + y ; x + y ; x + y ; x +1 y ) (9) 4 4 where the sum x +1 y is given by Def. 1.2. Note that (M; +) is thus an abelian group. 2 3 Figure 1: Illustration of the chart (Ut; φt) in the atlas AM. Suppressing R , the square to the left represents 3 Ut = (t; t + 1) × R × Unf%(t)g; the red part is (t; t + 1) × [0;%(t)), the green part is (t; t + 1) × (%(t); 1). The blue line is the line ` in M with parametrization (u; 0; 0; 0;%(u)). The square to the right represents φt[Ut], the image of Ut under φt. The red part, the green part and the blue line in the right square are the images of the corresponding items in the left square. The vertical axis of the right figure shows the total distance traveled in the curled-up dimension since t = 0 by an observer moving on ` (see Sect. 3); if t = 0 coincides with the beginning of the universe, then an integer value is the degree of evolution of the EPT that the observer is at (see Rem.4.7). 0 1 λ00 λ01 λ02 λ03 B λ10 λ11 λ12 λ13 C (ii) for every Lorentz transformation Λ with matrix B C on Minkowski space with @ λ20 λ21 λ22 λ23 A λ30 λ31 λ32 λ33 0 1 λ00 λ01 λ02 λ03 0 B λ10 λ11 λ12 λ13 0 C B C signature (−; +; +; +) there is an operator λ 2 O with matrix B λ20 λ21 λ22 λ23 0 C so that B C @ λ30 λ31 λ32 λ33 0 A 0 0 0 0 1 0 0 1 0 1 0 0 1 y λ00 λ01 λ02 λ03 0 x 1 1 B y C B λ10 λ11 λ12 λ13 0 C B x C B 2 C B C B 2 C λ(X) = Y , B y C = B λ20 λ21 λ22 λ23 0 C B x C (10) B 3 C B C B 3 C @ y A @ λ30 λ31 λ32 λ33 0 A @ x A y4 0 0 0 0 1 x4 Remark 2.4 The space (M; + ; · ) is not a vector space, for the same reason as mentioned in the proof of Cor.