Revista Brasileira de Física, Vol. 7, N 3, 1977 Causality and Relativity MARIO SCHONBERG Sociedade Brasileira 'de Física, C. P. 20553. São Paulo SP Recebido em 7 de Dezembro de 1976 The causal relation between a pair of physical events is taken as the physical foundation of the Riemannian geometry of the Space-Timeandof the arrow of time, the basic differentiable manifold of the physical continuum being associated to a pre-causal level of the physical theo- ry. The Einstein partial differential equations of General Relativity are associated to a physical field of Causality. The four-dimensionali- ty is related to electromagnetic properties-at the pre-causal level, es- pecially to the field of displacement and the electric current. Gravi- tation is obtained from the physical field of Causality, which may be used also to develop a form of already unified theory. Causality and Determinism are related by a variational principle of the Hilbert type for the fíeld of Causality and matter. Toma-se a relação causal entre um par de eventos físicos como fundamen- to físico da geometria riemanniana & espaço-tempo e da di reçã - tempo- ral, a variedade diferencial básica do continuum físico sendo associa- do a um nível pre-causal da teoria física. Associam-se as equações di- ferenciais parciais, de Einstein, da relatividade geral, a um campo físico de causal idade. Relaciona-se o carater quadridimens ional com as propriedades eletromagnéticas, a um nivel pre-causal , e especialmente com o campo de deslocamentos e com a corrente elétrica. Obtem-se a gra- vitação do campo físico de causal idade, que pode ser tambem uti l izado para se desenvolver uma forma de teoria já unif icada. Re lacionam-se Causal idade e Determi nismo através de um princípio variac ional, do ti- po de Hilbert, para o campo de Causalidade e matéria. 1. PRIMARY PHYSICAL CONTINUUM AND SPACE-TIME In Relativistic Physics, the physical continuum is taken usually as a four dimensional Riemannian manifold endowed with a normal hyperbolic metric defined by a symnetric tensor. with. determinant g(z)#O whose qua- dratic differential form g. .(x) c&' dxJ has signature - 2. That signa- ZJ ture gives a Space-Time structure with.thrse space dimensions and one time dimension. !%e existeme of the normal hyperbotic Riemannian me- Mcof the phys2caZ contimnmi is generally taken as the foundath of the relativistic theozy of Causality. In the mathematical theory of an N-dimensional Riemannian manifold,the Riemannian metric is introduced in a basic differentiable manifold DN by means of a symmetric tensor field g. .(x), with g(x)#O. In Relativi- 23 ty, littlè attention is generally given to the basic differentiable ma- nifold. We s7zuZZ emphasize the role of the basic differentiubte mani- fold as a ftrndwnentat physical structure, distinguishing it carefulZy from the physical strueture given by the noml hyperbolic Riemann&zn Space-Time. In the present paper, we shall take Causal i ty as the physical founda- tion of the Space-Time structure by introducing the metric gij(x) as a tensor associated to the theory of Causal i ty. The signature of g. .(x) L7 dri dizi must obviously have absolute value 2, in order to give satis- factory relations of Causality between physical events, as shown by Special Relativity in a flat Space-Time. The concept of a basic differentLdZe manifold can be introduced inde- pendentZy of CauçaZity. I t wi 11 be related to a pre-causal leve1 of the theory of the physical events in Section 2. The normal hiperbolic Rie- mannian metric of the physical continuum will be introduced in connec- tion with the causal relation between physical events in Section 3. me distinction between the basic differentiabte manifold and the Space -Time gives the possibility of introducing disconnected dmins of Cau- sality in D,, each of them corresponding to a different Space-Time, and thereby to a different universe. The g. .(x) of Causal ity can be taken as variables of a physical field 23 of Causality. This leads to a reinterpretation of the Einstein field of Gravitation of General Relativity, in which the Einstein partia1 differential equations appear as the expression of a physical law re- lating the distribution of energy-momentum-stress of matter to thephy- sical f ield of Causal i ty. Thus, the whole General Rektiuity becomes included in the theory of CausaZity. 2. PHYSICAL CONTINUUM AND PHYSICAL EVENTS We shall now use the concept of physical event for the discussion of the primary physical continuum, by means of the following physical law: Law ofthe PhysicaZ Events: A physical event is associated to a point of the basic differentiable manifold D, of the physical continuum. The point x(E), associated to the physical event E, wi l l be cal led the position of E in the D, of the physical continuum. D, is the setofthe possibte positwns of events, and endowed Lnth a different2abZe struc- twle of the class C', tnth r > 1, taken suffzciently Zarge to alhthe constructwn of the physical theory. The differentiable structure of D, allows for the existence of relati- ve tensors of all orders, with all the values of the weight, as well as that of the relative pseudo-tensors. The condition r > 1 aZhs ah the existeme of affinities. The level of the theory of the physical events based only on the above law will be called pre-causal. At this level, there is not yet the dis- tínguishable system of lines of the physical continuum allowing forthe introduction of Causality relations between events at different points. We have the following theorem: meorem of the üirections: At the pre-causal level of the theory ofthe 373 physicai events there is still no distinction of different kinds of linear directions at any point x of the physical continuum. At the points r, of an N-dimensional differentiable manifold DN' with N > 1, the linear directions constitute an irnportant projective space i (x) whose "poi nts" have as homogeneous coord i nates the X compo- 'N- I , nents 'of the contravariant vectors X#O at x. The straight 1 i nes of (x) describe the planar directions at the point x of DN. We have p~-1 the following theorem: IPheorem of P,(x): The dimensionality N=4, of the physical continuum, is distinguished by special properties of the planar directions at its points z, corresponding to those of the straight i ines of P3(x), given by the self-duality of the straight line in the thre-diniensional pro- jectiye geometry. The special properties of the planar directions at the points of Dsare related to the existence of the. Levi .Civita . basic relative tensoreGhk' The homogeneous coordinates xZ3 = -SZof a planar direction at x sa- tisfy the quadratic condition E xij $k = Q-hk ,, , (2-1) which is the Plücker condition for the homgeneous Plücker coordinates X" X" of the straight 1ines of P,(x). In the pre-causal leve1 of the theory of the physical events of the present Section, the a priori assumption of the four-dimensionality of the physical continuum is made, since it is based on the law of the physi cal events. fiereby, in the above pre-causal ZeveZ, the fow - di- mensionatity directty as an experimental fact is asswned, but not the Space-Time stmccture. 3. THE CAUSAL RELATION OF PHYSICAL EVENTS We shall now introduce the concept of a causal relation between two 3 74 physical events, as a nonsymmetric type of relation, in which one of the events plays the role of cause, and the other event that of effect. The asymetry of the causal relation will be asswned as the fowzdation of the amou of time. We shall also introduce the concept of the possibilitv of a causal re- lation between two events, associated to their positions in the physi- cal continuum, without further specification of their special nature . In a similar way, we shall also introduce the concept of the impossi- bility of a causal relation between two events specified only by their positions. In this approach to the possibility íimpossibility) of a causa2 retation between two events, we shaZl not distinguish the "cause" and the "effect" , unless it becomes necessary in connection with the arrow of time. In both the Special and the General theories of Relativity, Causality is based on the previous introduction of a normal hyperbolic Riemannian metric in the physical continuum. The Riemannian metric, gij(x), allows us to formulate purely positional conditions for the possibility of cau- sal relations between physical events. In our approach, we shallintro- duce the tensor g . (x) as a symmetric tensor of CausaZity, Znstead of 23 taking it as a geometric tensor, by means of a physicaz Zaw giving the positionaz conditions for the possibitity of causa2 relations between physicaZ events. The quadratic differential form, gij(x) &cz &, of Causality will be taken wi th the signature (-2), as the quadratic differential form of the Riemannian metric in General Relativi ty. The signature (-2) is related to the role played by that qwzdratic differential form in the defini- tion of the lines of Causatity, the tines in which gij(x) &' dxi > O for any infinitesiml displacement dx on them, at any of their points. We shall now introduce the following physical law: Lm of Causa2 PossibiZity: There is the possi bi 1i ty of a causal rela- tion between two physical events at different points of D,, if and only if they can be connected by a line of. Causality . of the tensor field g..(x) of Causal ity, having d3cZ & with signature (-2).