The concept of simultaneous clicks of all the cameras rely on the the existence of a clock within each camera and depends on how all the clocks can be synchronized. Einstein's theory of Special Relativity (which deals with the electro-magnetic properties of matter, in particular with the light- and radio-rays) grows out from the principle that nothing moves faster than light, in particular the best (the fastest) way for synchronizing clocks is by using light signals. More precisely the first Einstein's principle states that all the physical laws are the same in all inertial frames of reference, and the second Einstein's principle states that the speed of light is a constant which does not even depend on the speed of its source (neither it depends on the inertial frame of reference in which it is measured). A frame of reference is a 3-dimensional Cartesian coordinate system endowed with a clock at each point, and all the clocks are synchronized. With respect to a given frame of reference, an event takes place at a certain space-point at a certain moment, so it is described by its space coordinates (x,y,z) and a time coordinate t, i.e. by a point (x,y,z,t) in the 4-dimensional Minkowski space-time. In the plane we can sketch a restricted picture of Minkowski space-time by measuring only (x,t), see the figure below.
/\ A B \ FUTURE ; / t | : : T s / | F F H TIME i E | O O G CONE x N s | : : I a O i | E E L ; C x (U,V,W,S) | N N SPACE \;/ a SPACE | I I CONE e/ \ (X,Y,Z,T) CONE | L L c /; \ | : : a T e E | D D p H m PAST N | L L s G i O | R R I t TIME C -(0,0)-O----O---------------L-----;-----CONE----------\----> | W W / ; \ x | : : / ; \
Each object in the Universe has its world line in the Minkowski space-time: if the object A stays at rest then only its time coordinate changes and the world line is a straight line parallel to the time-axis (read upward along the lines); if the object B moves with a constant velocity then its space coordinates change proportionally to its time coordinate, so its world line is again a straight line but non-parallel to the time axis; if an object accelerates then its world line is curved. A light signal from the point (X,Y,Z,T) can be considered as consisting of some light- particles - photons - which spread in all directions in the space and, because the speed of light is constant, the world lines of these photons form a cone in the Minkowski space-time called the future light cone at (X,Y,Z,T). The past light cone at (X,Y,Z,T) consists of all points whose future light cone contains (X,Y,Z,T); it is symmetrical to the future light cone at (X,Y,Z,T). The future time cone at (X,Y,Z,T) consists of all points which can be reached by a signal from (X,Y,Z,T) of speed less than the speed of light. The points which can reach (X,Y,Z,T) by a signal of speed less than the speed of light form the past time cone at (X,Y,Z,T). The time cone at (X,Y,Z,T) is the union of the future time and past time cones at (X,Y,Z,T); the light cone at (X,Y,Z,T) is the union of the future light and past light cones at (X,Y,Z,T). Because nothing moves faster than light there are points which neither can be reached by any signal from (X,Y,Z,T), nor they can reach (X,Y,Z,T) by any signal; these points and (X,Y,Z,T) form the space cone at (X,Y,Z,T). (The light cones are 3-dimensional subsets of Minkowski space-time, and the time and space cones are 4-dimensional.)
The following example is considered in the Special Relativity: Let someone travels in a space-ship with a constant velocity with respect to our frame of reference - in which we stay at rest and measure (x,y,z,t); let the traveller measures primed (x',y',z',t') in his or her frame of reference in which the space-ship stays at rest. At the moment R' at the middle of the space-ship the traveller sends a light signal which later at the moment T' reaches both the back side and the front side of the space-ship. >From our point of view at the moment R at the middle of the space-ship the traveller sends a light signal which later at the moment T reaches the back side of the space-ship, but meanwhile the front side has gone ahead, so the light reaches the front side later at the moment S > T. So, the two events (that the light reaches the back side and that it reaches the front side) are simultaneous from traveller's point of view but are not from ours. One might think this is not possible and suspect that the second Einstein's principle is wrong, but an analysis shows there is not any logical inconsistency, and moreover Einstein's theory has the power to explain and to predict physical laws, so this theory is accepted in the contemporary physics, see [1]. Some future development may only refine Einstein's theory - as Einstein's theory does not completely reject Newton's theory but only refines it (and with respect to "small velocities" both theories practically coincide). Concerning the above example let (X,Y,Z,T) and (U,V,W,S) denote the two events from our point of view, and (X',Y',Z',T') and (U',V',W',S') denote the two events from traveller's point of view (T < S and T'= S'). Although T < S, we cannot say that the event (X,Y,Z,T) precedes (U,V,W,S), because no signal from (X,Y,Z,T) of speed less than (or even equal to) the speed of light can reach (U,V,W,S) (such a signal cannot overtake the reflection of the light signal from the back side at (X,Y,Z,T) but the reflection reaches the front side after the moment S). Of course we cannot also say that (X,Y,Z,T) and (U,V,W,S) are simultaneous, we just say that (U,V,W,S) belongs to the space cone at (X,Y,Z,T) or, equivalently, that the vector (U-X,V-Y,W-Z,S-T) is space-like. This is not relative: if (X",Y",Z",T") and (U",V",W",S") denote the two events with respect to any other inertial frame of reference then (U",V",W",S") belongs to the space cone at (X",Y",Z",T") (and possibly T"= S", or T"> S", or T"< S"). Let M denotes the space-time which corresponds to our frame of reference. Any 3-dimensional affine subspace of M (the image of a 3-dimensional linear subspace under a translation) which is contained in the space cone at some point in M is called a space axis; all the events of a space axis are simultaneous with respect to a suitable frame of reference. Any straight line in M which is contained in the time cone at some point in M is called a time axis; all the events of a time axis take the same space-place with respect to a suitable frame of reference.
The fine topology on M, proposed by Zeeman in [2], is the finest topology on M which induces the 3-dimensional Euclidean topology on every space-axis and the 1-dimensional Euclidean topology on every time-axis. The above consideration shows that equal weight is given to any inertial frame of reference. In order to get able to present main Zeeman result we need just another definition. For a given value V let V^2 denotes the square of V, and let Q denotes the quadratic form
x'= (x-vt) / sqrt x = (x'+vt') / sqrt y'= y y = y' z'= z z = z' t'= (t-(vx)/(c^2)) / sqrt t = (t'+(vx')/(c^2)) / sqrtThe above transformation is exactly the correspondence between the coordinates of events with respect to two frames of reference such that: the corresponding axes have the same direction, the primed frame of reference moves with respect to the unprimed with a constant speed v along only the x-axis, and, (0,0,0,0) and (0',0',0',0') represent the same event. (So, from our point of view the space-ship shorts and the traveller's clock delays by the factor sqrt, but from traveller's point of view our space shorts and our time delays). In general, with a possible reversal of space or time, any Lorentz transformation is exactly the correspondence between the coordinates of events with respect to two suitable frames of reference such that (0,0,0,0) and (0',0',0',0') represent the same event. And vice versa: the correspondence between the coordinates is a Lorentz transformation whenever (0,0,0,0) and (0',0',0',0') represent the same event. (Let us also note that it is customary in physics to describe an event by (x,y,z,ct) so that the product ct measures length of space as well as the other 3 coordinates; and in a coordinate system which measures (x,y,z,ict) where i is the imaginary unit (ii=-1), Lorentz transformations preserve the usual distance and the origin, so in such a system they are compositions of rotations and reflections.)
The main result of Zeeman paper states that the group of all auto-homeomorphisms of the fine topology is generated by the Lorentz group (of all Lorentz transformations), translations and multiplications by scalars. In Zeeman words this result in particular "shows that the fine topology contains intrinsically the linear structure of M. It also contains intrinsically the inhomogeneous Lorentz group and consequently its representations, which are basic invariants in physics". With a little change of notations we also cite: "Another physically interesting property of the fine topology is as follows. We may regard the path of a particle as a continuous map f : I --> M from the unit interval I into M, with the property that if a < b in I then the vector f(b) - f(a) is time like and oriented towards the future. If M is given the 4-dimensional Euclidean topology, then the image of such a map can be very pathological, and everywhere non-analytic (i.e. physically meaningless). But if M is endowed with the fine topology, then the image of such a map is piecewise linear, consisting of a finite number of straight intervals along time axes, exactly like the path of a freely moving particle under a finite number of collisions."
In his paper Zeeman suggested some generalizations: in particular for the curved space-time of General Relativity and for the topology of Lorentz group. Einstein's theory of General Relativity presents the gravitation via a curvature of the space-time: the presence of mass curves the space- time and the bodies move or fall along geodesics, see [3]; locally the curved space-time looks as Minkowski space-time (for this rather geometrical than topological approach see also [4]; see [5] for Kaluza-Klein theories in which the space-time has more than 4 dimensions). Interrelations among different structures on space-time (e.g. Lorentz group, the causal structure, various modifications of Zeeman topology, some groups of transformations which preserve certain subsets of the space-time - for example light cones or time like arcs) were investigated in [6], [7], [8], [9], [10], [11], [12], [13], and [14] where a number of additional references is given.
In mathematics the method for topologizing a set that preserves some predetermined topology on a given family of subsets is basic, see e.g. [15,Ch.VI]. One may check [16], [17], [18], [19], [20] and [21] for some of the mathematical aspects of the above discussed topic.
The author's acquiantance with physics is fairly elementary. The author investigated the finest topology on the n-dimensional Euclidean space which induces the standard topology on each straight line. This topology (for n > 1) is not regular [22] which in a certain manner corresponds with its Baire property [24]. Also, its weight is equal to the cofinality D of the partially ordered set of all functions from \c into \omega , where \c is the cardinal of continuum and \omega is the least infinite ordinal [23]. It is unknown if there is a model of ZFC in which D < exp(\c) . It is known that the cofinality d of the set of all functions from \omega into \omega may vary in different models of ZFC, see [25] and [26]. By definition GCH implies that D = exp(\c) ; by a result of [27] and [28] about the cofinality d_1 of the set of all functions from \omega_1 into \omega , the equality D = exp(\c) follows from CH. A model in which d_1 < exp(\omega_1) is constructed in [29].
I would like to express my sincere gratitude to Professor Otto Laback (Technical University of Graz, Austria) who introduced me into the above discussed applications of topology in physics during his visits in Sofia in 1993 and 1995, and to Professor Gary Gruenhage who kindly sent me much information about cofinal subsets, and to the Editors of Topology Atlas for their kind cooperation.
[1] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics, v.1,2,3, Addison-Wesley, 1975.
[2] E.C. Zeeman, The topology of Minkowski space, Topology v.6 (1967), 161-170.
[3] S.W. Hawking and G. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Math. Phys. v.1, 1980.
[4] R. Geroch and G.T. Horowitz, Global structure of space-times, in: General Relativity, ed. S.W. Hawking and W. Israel, Cambridge Univ.Press, 1979, 212-293.
[5] H.C. Lee, ed., An Introduction to Kaluza-Klein Theories, World Scientific, 1984.
[6] E.C. Zeeman, Causality implies the Lorentz group, J.Math.Phys. v.5 (1964), no.4, 490-493.
[7] A.D. Alexandrov, A contribution to chronogeometry, Canad.J.Math. v.19 (1967), 1119-1128.
[8] H.J. Borchers and G.C. Hegerfeldt, The structure of space-time transformations, Commun.Math.Phys. 28 (1972), 259-266.
[9] P.G. Vroegindeweij, The separating topology for the Lorentz group L, J.Math.Phys. v.16 (1975), no.6, 1210-1213.
[10] R. Goebel, Zeeman topologies on space-times of General Relativity theory, Commun.Math.Phys. 46 (1996), 289-307.
[11] S.W. Hawking, A.R. King and P.J. McCarthy, A new topology for curved space-time which incorporates the causal, differential, and conformal structures, J.Math.Phys. 17 (1976), no.2, 174-181.
[12] R. Goebel, The smooth-path topology for curved space- time which incorporates the conformal structure and analytic Feynman tracks, J.Math.Phys. 17 (1976),5, 845-853.
[13] D.B. Malament, The class of continuous timelike curves determines the topology of spacetime, J.Math.Phys.18 (1977), no.7, 1399-1404.
[14] O. Laback, Topologies in General Space-Time, Berichte der Math.-Stat. Sektion im Rechenzentrum Graz, 138 (1980).
[15] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
[16] L.R. Rubin, Line preserving topologies, General Topology and its Applications, 2 (1972), 193-198.
[17] S.P. Franklin and B.V. Smith-Thomas, Topologies determined by paths, Topology Proc. 4 (1979), 371-384.
[18] F.D. Ancel, Collections of paths and rays in the plane which fix its topology, Topol.Appl. 15 (1983), 99-110.
[19] D.W. Curtis and L.R. Rubin, Sectionally continuous injections of Euclidean spaces, Top.Appl.31(1989),159-168.
[20] O. Laback, On the continuity of bijections of R^n , Suppl.Rendic.Circ.Mat.Palermo Ser.II, 18(1988), 337-341.
[21] C.L. Cooper, Determining topologies by collections of proper maps and by weak topologies, Top.Appl. 55 (1994), 203-213.
[22] S.G. Popvassilev, Non-regularity of some topologies on R^n stronger than the standard one, Math.Pannon. 5 (1994), no.1, 105-110.
[23] S.G. Popvassilev, Principle A^tau, Principle A^tau II, Questions Answers Gen.Topol., 13 (1995), no.2, 203-210.
[24] S.G. Popvassilev, Baire property versus non-regularity in some topologies on R^n, Dr. Brechner's teprint service, C.R.Acad.Bulg.Sci. 49 (1996), no.5, to appear.
[25] E.K. van Douwen, The Integers and Topology, in: Handbook of Set-Theoretic Topology, ed. K. Kunnen and J.E. Vaughan, Elsevier Science Publishers, 1984, 111-167.
[26] M.E. Rudin, Martin's Axiom, in: Handbook of Mathematical Logic, ed. J. Barwise, Studies in Logic and Foundations of Mathematics, v.90, North Holland, 1977, 491-501.
[27] J. Steprans, Some Results in Set Theory, Thesis, University of Toronto, Canada, 1982.
[28] T. Jech and K. Prikry, Cofinality of the partial ordering of functions from \omega_1 into \omega under eventual domination,Math.Proc.Camb.Phil.Soc.95(1984),25-32.
[29] R.W. Knight, Dominance in ^\omega_1 \omega, University of Oxford, Oxford, England (submitted for publication).