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Derivation from Landau and Liffshitz, vol 6

  1. Oct 20, 2015 #1

    Geofleur

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    I have starting working through section 134 of Landau and Lifshitz, vol 6, and it seems I have entered some kind of twilight zone where all my math/physics skills have left me :cry:

    The derivation starts with the energy-momentum tensor for an ideal fluid:

    ## T^{ik} = wu^i u^k - p g^{ik} ##,

    where the Latin indices range from 0 to 3 (Greek indices would range from 1 to 3), ## w ## is the enthalpy, ## u^i ## is component i of the four-velocity, ## p ## is the pressure, and ## g^{ik} ## is the component ik of the Minkowski metric (with signature ## g^{00} = 1 ##). The derivation also employs the equation for conservation of particle number:

    ## \frac{\partial}{\partial x^i} \left( nu^i \right) = 0 ##,

    where ## n ## is the proper number density of the particles. We lower the first upper index of ## T^{ik} ## using the metric tensor as

    ## T_{i}^{\ k} = g_{im}T^{mk} = wg_{im}u^m u^k - p g_{im} g^{mk} = wu_i u^k -p \delta_i^k. ##

    Now we take the four divergence and set it equal to zero,

    ## \frac{\partial T_i^{\ k}}{\partial x^k} = \frac{\partial}{\partial x^k} \left[ wu_i u^k \right] - \frac{\partial p}{\partial x^i} = u_i \frac{\partial}{\partial x^k}\left[ w u^k \right] + w u^k \frac{\partial u_i}{\partial x^k} - \frac{\partial p}{\partial x^i} = 0 ##.

    And here is where the trouble starts, because Landafshitz has the above equation with a plus sign next to the pressure term, not a minus. But it gets worse! In the next step, they say that ## u_i u^i = -1 ##. Now I must be really confused, because I thought that ## (u^i ) = \gamma (1,\mathbf{v}) ##, so that

    ## u_i u^i = u_0 u^0 + u_\alpha u^{\alpha} = \gamma^2 (1 - v^2) = 1 ##,

    where ## \gamma ## is the Lorentz factor, and the speed of light has been set to unity.

    Can anyone out there help me get this mess straightened out?
     
    Last edited by a moderator: Oct 21, 2015
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  3. Oct 20, 2015 #2

    Orodruin

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    What? 5 dimensional space time? :wink:

    Some authors use different conventions for the metric (+---) or (-+++). The difference is the appearance of some signs here and there. You should check what convention is being adopted in each text you are dealing with.
     
  4. Oct 20, 2015 #3

    Geofleur

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    Ah yes, thanks, I made the appropriate edit.

    I did check this - in a footnote at the beginning of the chapter, the say that the metric has diagonal (1,-1,-1,-1), which is what I'm used to. At any rate, I can't see how the magnitude of the four velocity could end up negative o_O
     
  5. Oct 20, 2015 #4

    Orodruin

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    In the (-+++) convention, the norm squared of all time-like vectors are negative. Admittedly, I do not like this convention and usually use (+---), but it is good to know about it.
     
  6. Oct 20, 2015 #5

    Geofleur

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    I see - and when I worked it out just now it did come out that way! I knew that the Minkowski norm is not positive definite, but a negative magnitude of the four velocity is just weird. Also, it isn't consistent with their footnote...

    I am not sure how that would help the issue with the pressure derivative having the wrong sign.
     
    Last edited: Oct 20, 2015
  7. Oct 20, 2015 #6

    Geofleur

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    OK, this is unbelievable. Out of all 10 volumes, I happen to also have the version of 6 that is in Russian. When I turn to the same page where the problems occur, the signs all make sense!! In the Russian edition, ## u^i u_i = 1 ## and there is a negative sign in front of the pressure term. Здорово!
     
  8. Oct 20, 2015 #7

    PAllen

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    I also prefer the convention (+---), but books are all over the place and you must be very careful about the convention used and the signs. No less that Gerard 't Hooft has argued that only the (-+++) is worthwhile, and that (+---) is idiotic and leads only to sign errors (see, for example, the intro to http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf). Of course, this being purely a convention, neither I nor you have to follow t'Hooft's opinion.
     
  9. Oct 21, 2015 #8

    Orodruin

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    Personally, I would do many more sign errors with the (-+++) convention ... There is just something with the four-momentum squared being minus the mass squared which is unappealing to me.
     
  10. Oct 21, 2015 #9

    martinbn

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    Sorry for the off topic but since the question is resolved and it seems to be just a typo, I don't feel too guilty. I find it interesting that people here prefer (+---). I thought, for some reason, that relativists in general prefer (-+++) and the other convention is preferred by 'particle theorists who don't understand relativity'.
     
  11. Oct 21, 2015 #10

    Orodruin

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    Well, in the same sense as the (-+++) convention is preferred by relativists who don't understand particle theory.
     
  12. Oct 21, 2015 #11
    I think it also has to do with which convention you're first introduced to. I started off SR with Schutz, and after going through Sean Carroll's notes on GR, I just can't get out of the (-+++) habit. Usually when I see vectors with positive norms in relativity I first think "so we're dealing with spacelike vectors huh", and it often takes me a while to realize that the (+---) convention is being used instead.
     
  13. Oct 21, 2015 #12

    martinbn

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    :) Yes, but they do relativity, not understanding other areas is fine.

    I don't know. I used to like the (+---) but then I realized I was wrong and the (-+++) is the 'right' way to go.
     
  14. Oct 21, 2015 #13

    Orodruin

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    To throw in a famous quote: This is not even wrong.
    For things which are conventions there can be no wrong or right. They are physically indistinguishable and only a matter of philosophical debate and personal taste - much like debating QM interpretations, which I also find utterly repetitive and not bringing any new actual scientific value.
     
  15. Oct 21, 2015 #14

    martinbn

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    I guess the intended tone was not clear, it was a joke, I should have put some of those silly smilies not just the quotes' '
     
  16. Oct 21, 2015 #15

    dextercioby

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    Most of the quantum field theory texts use the +--- convention. It has been with us since Schweber and Bjorken/Drell. However, this mostly minus one has the disadvantage that you cannot go to 5,6,.. space-time dimensions. That is why if helps to use +++- throughout.
     
  17. Oct 23, 2015 #16

    Ben Niehoff

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    The (-+++) convention is infinitely better if you work in a variable number of dimensions. It minimizes the number of ##(-1)^d## you have to write and keep track of.
     
  18. Oct 23, 2015 #17

    Dale

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    In my personal study I tend to write the interval as ##ds## when I am using the (-+++) convention and ##d\tau## when I am using (+---). That helps me keep things straight in my mind. I haven't seen anyone else do that, so there is probably a problem with it.
     
  19. Oct 23, 2015 #18

    PAllen

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    I do the same thing, and have seen no problems with that convention.
     
  20. Oct 23, 2015 #19
    I'm pretty sure Susskind does that in his lectures.
     
  21. Oct 23, 2015 #20

    Dale

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    That may be where I picked it up. I have seen those lectures several years ago.
     
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