Does Special Relativity assume an 'Inclined Space' at heart - for space enclosed in moving objects? Otherwise in the famous Train Experiment how would the Observer on Platform see the light moving in inclined direction? Is there any link where I can find more information on this?
Relativity assumes that coordinate systems are a human invention overlaid on reality for the purpose of quantifying nature. If you overlay a coordinate scheme such that the ends of the train are at constant spatial coordinates and the time axis is perpendicular to the spatial axes, you get one set of coordinates for events. Use a coordinate system in which the sleepers of the track are fixed but the train is moving and you get a different set of values for the same events. It is the axes that are inclined to each other, relativity does not reify space.
No, special relativity uses flat geometry where four-vector dot product is taken with the help of Minkowski metric diag(1,1,1-1). First three positive entries are for the space part and the fourth one is for the time part. For example ds2=dx2+dy2+dx2-dt2, in a curved space time the dot product will be of the form ds2=gμν dxμdxν where the entries in metric gμν will vary from point to point.
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