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Question on definition of form-invariant

  1. Jan 3, 2015 #1
    Hello everyone. I'm reading Weinberg's 'Gravitation and Cosmology' and I'm having some problems understanding the definition of a 'form-invariat function'. He says:
    If the previous condition was true doesn't this simply mean that ##g_{\mu\nu}^\prime## is the same function as ##g_{\mu\nu}##? Should we ask for:
    \begin{equation}
    g^\prime_{\mu\nu}(x^\prime)=g_{\mu\nu}(x)
    \end{equation}
    ?

    Thanks a lot!
     
  2. jcsd
  3. Jan 3, 2015 #2
    That is precisely what he means, that they are the same function.
    I think that's why he used the symbol "y" instead of x's, so that it wouldn't be confused as having something to do with the coordinates x and x'.
     
    Last edited: Jan 3, 2015
  4. Jan 3, 2015 #3

    ShayanJ

    User Avatar
    Gold Member

    The problem with [itex] g'_{\mu\nu}(x')=g_{\mu\nu}(x) [/itex] is that it means the two functions have same value in a single point. But because [itex] x' [/itex] associates different numbers to that point compared to [itex] x [/itex], [itex] g'_{\mu\nu}(x')=g_{\mu\nu}(x) [/itex] necessarily means [itex] g [/itex] and [itex] g' [/itex] are different functions and don't have the same form.
    But [itex] g'_{\mu\nu}(y)=g_{\mu\nu}(y) [/itex] means that if we give the two functions, the same numbers, they will give equal results which means they're the same function. Note that y in the right refers to a different point than y in the left because they are same numbers in different coordinate systems.
     
  5. Jan 3, 2015 #4
    Oh I see. That is clear. Thanks!
     
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