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From elementary general relativity

  1. Sep 20, 2014 #1
    A map h: = T.(M) ---> T.(M) is defined by h(X) = X + g(U,X)U where U ε T.(M) is a fixed vector with g(U,U) = -1.

    i: Give an expression for the components h^i (sub) j (This is "h" with a superscript i and subscript j) of h regarded as a tensor type (1,1)

    ii: Prove that h^2 = h. Interpret h geometrically.

    So I've been playing around with this but I'm getting no where. I could use some guidance on where to really get started. I'm also VERY new to general relativity, so every step/hint/anything would be great
  2. jcsd
  3. Sep 22, 2014 #2


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    So, the first step is probably to figure out what the question is really asking. So we have this map ##h## which takes a vector ##X## and gives you the back the vector ##h(X)=X+g(U,X)U##. Since this mapping is taking a vector into another vector, then we know that ##h## is a type (1,1) tensor which has components ##h^i_{~~j}##. So we know then that in component notation ##h(X)\equiv h^i_{~~j}X^j e_{(i)}## where ##e_{(i)}## are the basis vectors (the parenthesis mean that they are not components of a vector, but each ##i## denotes a different vector). Can you perhaps turn the right hand side of the first equation into component notation to see what it looks like?
  4. Sep 22, 2014 #3


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    Regarding ii, this should be a simple matter of insertion and using the linear property of g. For the interpretation, what happens if you set X = U? What happens when you use X such that g(U,X) = 0? (I suggest not using the coordinate representation for ii as it really is not necessary.)
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