Swapping Tensor Indexes

  • #1

Main Question or Discussion Point

What is the difference between ##{T{_{a}}^{b}}## and ##{T{^{a}}_{b}}## ? Both are (1,1) tensors that eat a vector and a dual to produce a scalar. I understand I could act on one with the metric to raise and lower indecies to arrive at the other but is there a geometric difference between the objects to which these are components?
 

Answers and Replies

  • #2
andrewkirk
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I don't think there's a geometric difference. The difference is that, where ##V## and ##V^*## are the underlying vector space and its dual, ##T_a{}^b## is a function from ##V\times V^*## to ##\mathbb R## and ##T^a{}_b## is a function from ##V^*\times V## to ##\mathbb R##. Or if we think of the tensor as being a function that takes two arguments, ##T_a{}^b## takes a vector as its first argument and ##T^a{}_b## takes a dual vector as its first argument.
 
  • #3
Orodruin
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Generally, they are different (1,1) tensors (you could imagine sidestepping #2 by defining the obvious equivalence between functions from ##V\times V^*## and functions from ##V^* \times V##). It is certainly not true that ##T^a_{\phantom ab} \omega_a V^b = T^{\phantom ba}_b \omega_a V^b## except in the case where ##T^a_{\phantom ab} g_{ac}## is symmetric.
 
  • #4
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##T_{\ \ \ \nu}^\mu=g_{\nu\xi}T^{\mu\xi}## and

##T^{\ \nu}_\mu=g_{\mu\xi}T^{\xi\nu}##

are different in general. Obviously when T is symmetric

##T^\mu_{\ \ \ \nu}=T^{\ \nu}_\mu## and they can be denoted as ##T^{\nu}_\mu##.
 
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