Can we consider the structure of space( only space not space-time ) to be that of a vector space? Why can we or why can't. And why cant we give a vector space structure to space-time?
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$\begingroup$ Because the space-time is not flat. In the flat sense you have the minkowski space with corresponding metric. However, in the presence of gravitation, the light has different behaviour. This is proved by time dilation, which means, in the presence of gravitation, the time requiring by light from A to B is different than in the absence of gravitation and this leads to the suggestion that light is taking a longer way in the latter case. Therefore one suggests there is a curvature of space-time. $\endgroup$– quallenjägerCommented Apr 6, 2017 at 14:26
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$\begingroup$ I assumed you are meaning vector space by Euclidean space. In mathematics, vector space is a set over a field. $\endgroup$– quallenjägerCommented Apr 6, 2017 at 14:29
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$\begingroup$ youtube.com/… here they are ... 10 nice lectures about General Relativity from prof L. Susskind. If you don't manage to understand them take the prerequisite course - Special Relativty youtube.com/watch?v=toGH5BdgRZ4&list=PLD9DDFBDC338226CA $\endgroup$– Mihai B.Commented Apr 6, 2017 at 14:52
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$\begingroup$ Related: about vectors, tensors and how they transform physics.stackexchange.com/questions/286457/… $\endgroup$– Mihai B.Commented Apr 6, 2017 at 14:57
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$\begingroup$ For Euclidean space do we adopt a vector space structure? (just conforming) $\endgroup$– quirkyquarkCommented Apr 6, 2017 at 15:15
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A vectors space has a preferred vector, the zero vector. Instead there is no preferred point in classical (flat) physical space. Physical space is better described by a so-called three-dimensional affine space. Metric tools are then represented by a scalar product in the space of translations. This space which describes rigid movements of the points in the affine space is a vector space, but it is not the physical space itself, just describes rigid movements in physical space.
Similarly, the spacetime of special relativity is well described by a four-dimensional affine space whose space of translations is equipped with a Minkowskian scalar product.
answered Apr 6, 2017 at 16:46