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Lowering tensor indices

  1. Jan 27, 2013 #1
    Hi
    have i got this corresct :
    [itex]g_{\mu\nu}g_{\mu\nu}T^{\mu\nu} = T g_{\mu\nu} = T_{\mu\nu}[/itex]
    and is:
    [itex]g_{\mu\nu}g_{\mu\nu}u^{\mu} = g_{\mu\nu} u_{\nu} = u_{\mu}[/itex]
     
  2. jcsd
  3. Jan 27, 2013 #2

    Mentz114

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    No, it should be

    gapgqbTpq = Tab


    gapvp = va
     
  4. Jan 27, 2013 #3
    how would i go about going from [itex] T^{ab}[/itex] to [itex] T_{ab}[/itex] i.e with the same indices that is mainly why i am confused it needs to be the same indices not different ones.
     
  5. Jan 27, 2013 #4

    WannabeNewton

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    [itex]g_{\gamma \mu }g_{\delta \nu }T^{\mu \nu } = T_{\gamma \delta }[/itex] and then you are free to relabel the indices back to mu and nu. What you wrote is incorrect because you have 2 mu's on the bottom and a mu on the top which isn't how einstein summation works.
     
  6. Jan 27, 2013 #5
    ok i think i understand so to lower:
    [itex]T^{\mu\nu} = u^{\nu}u^{\mu}[/itex]

    would one multiply by [itex]g_{\gamma \mu}g_{\delta \nu}[/itex] and relable T as instructed, but how would the u's become:
    [itex] u_{\mu}u_{\nu} [/itex]

    I get the relabelling thing but wouldn't they have three lower indices i.e:
    [itex]u_{\mu \gamma \delta} u_{\nu \gamma \delta}[/itex]
     
  7. Jan 27, 2013 #6

    WannabeNewton

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    [itex]g_{\gamma \mu }g_{\delta \nu }T^{\mu \nu } = g_{\gamma \mu }g_{\delta \nu }u^{\mu }u^{\nu }[/itex] so [itex]T_{\gamma \delta } = u_{\gamma }u_{\delta }[/itex] and since these are free indices I can relabel them accordingly that is [itex]\gamma \rightarrow \mu , \delta \rightarrow \nu [/itex] provided I do so on both sides; this gets the desired result.
     
  8. Jan 27, 2013 #7
    thanks you have been really helpful one last question, is [itex]T_{ab} g^{ab}= T^{a}_{b}[/itex] if not how can we get there becuase the metric always has two indices?
     
  9. Jan 27, 2013 #8

    WannabeNewton

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    [itex]T_{ab} g^{ab} = T_{a}^{a}[/itex] because the metric tensor will first raise the lower index b on [itex]T_{ab}[/itex] up to an a and now you just have [itex]T_{a}^{a} = T[/itex] where T is just the trace. Note that here it is arbitrary whether you choose to raise a or b because in the end you end up summing over the same index on the bottom and on the top and you can relabel that to w\e you want; that is [itex]T_{a}^{a} = T_{b}^{b}[/itex] because you end up summing over all components regardless.
     
  10. Jan 27, 2013 #9
  11. Jan 27, 2013 #10

    WannabeNewton

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    Yep anytime and to answer your final question you could just use [itex]T_{ab}g^{bc} = T_{a}^{c}[/itex]
     
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