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A Continuous map from one manifold to another

  1. May 15, 2016 #1
    I need a reminder. What numbers or functions characterize a map from one manifold to another? More specifically, is there a continuous function that goes from one manifold to another to another to another is some parameterized way? What is that called?

    I'm thinking of a manifold of spacetime. And with acceleration one goes from the initial frame of reference (the first manifold) to a frame of reference traveling with respect to the first frame. This can be viewed as a different manifold but shares some points with the first. Can this acceleration from one frame (manifold) to another frame (manifold) be characterized by numbers that tell us something about the mass or energy need to go from one frame to another? I hope this is enough information to generate answers. Thanks you.
     
  2. jcsd
  3. May 15, 2016 #2

    stevendaryl

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    A change from one frame of reference to another does not change the manifold. It only changes the coordinates used.

    Now, you can think of a "coordinate system" as a map between manifolds: Spacetime is one manifold, [itex]\mathcal{M}[/itex] and [itex]R^4[/itex] is another manifold (the set of all possible 4-tuples [itex](x,y,z,t)[/itex] of real numbers). A coordinate system [itex]\mathcal{C}[/itex] is a map from [itex]\mathcal{M}[/itex] to [itex]R^4[/itex] with the restriction that the map must be continuous and must have an inverse. (More generally, a coordinate system might be just a map from a region in [itex]\mathcal{M}[/itex] to [itex]R^4[/itex], rather than from all of [itex]\mathcal{M}[/itex].) A coordinate change can be thought of as a map between two different copies of [itex]R^4[/itex].

    But as far as the physics is concerned, there is no mass or energy associated with a change of coordinate system. A more meaningful question might be: How much energy does it take to accelerate a given mass to a given final velocity?
     
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