A fine causal topology

eng Ansari Education and Research Society Journal of Ultra Scientist of Physical Sciences August 2010 22 2 511 516 jusps-A 999 article A fine causal topology Ujagar Patel 1 Somnath Choudhury 1 Satish Kumar Patel 2 Department of Mathematics, Govt. Autonomous College, Bhawanipatna, Dist, Kalahndi, Orissa - 766001 (INDIA) Department of Physics, G.M. Autonomous College, Sambalpur, Orissa - 768004 (INDIA In the paper `Fine topologies for Minkowski space', Williams1 has suggested a fine topology for M, the four-dimensional flat space-time, with the following properties: (i) the induced topology on each time like line and light like line is Euclidean and (ii) the group of C1 -homeomorphisms of this topology is G. While suggesting this topology, Williams has argued that defining a topology in terms of lines rather than in terms of timelike lines and spacelike hyperplanes2,3,4 has certain advantages. For example, this procedure lends itself to possible generalization to curved space-times where curves are significant, whereas, spacelike hypersurfaces are of little physical significance. However, William's topology has an unsatisfactory feature. If we imagine the path of a particle as the continuous image of I, the closed unit interval, then such a path in Williams' topology is a finite connected union of timelike intervals. Since photons travel along lightlike lines, it follows that photons are excluded from the category of particles whose paths are intuitively thought of as continuous images of I. In this paper we show that one can define a fine topology (that is, a topology which is finer than the Euclidean topology) on Minkowski space in terms of lines such that (1) the C1-group of homeomorphisms is G and (ii) any continuous image of I is a connected union of a finite number of timelike and (or) lightlike intervals. Thus, we are able to remove the unsatisfactory features of Williams' topology while retaining its defining features. One of the unsatisfactory features of this topology (as well as of Williams' topology) is that we have to make the C1 assumption while deriving the group of homeomorphisms. It is reasonable to conjecture that without the C1 hypothesis, the group of homeomorphisms will be too wild. https://www.ultrascientist.org/paper/999/ A fine causal topology