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Homework Help: Difference between lorentz invariant and lorentz covariant

  1. Dec 3, 2004 #1
    title says it all. I've heard these two phrases.

    Lorentz invariant: Equation (Lagrangian, or ...?) takes same form under lorentz transforms.

    Lorentz covariant: Equation is in covariant form.

    I'm don't think I know what I mean when I say the latter. Can someone elucidate the difference between these two. Is there some necessary/sufficient condition that relates the two.

    Thanks for your help!
  2. jcsd
  3. Dec 3, 2004 #2


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    By "Lorentz invariant" i understand any expression (that is product) of tensors on the flat manifold [tex] M_{4} [/tex] which has the the same form in every inertial reference frame.So it is a scalar wrt to the Lorentz transformations/group.
    For example,the D'Alembert operator on flat spacetimes is a scalar:[tex] (\partial^{\mu})'(\partial_{\mu})'=\partial^{\mu}\partial_{\mu}[/tex],where,obviously:
    [tex] (\partial^{\mu})'=\Lambda^{\mu}\ _{\nu} \partial^{\nu} [/tex].

    Any expression written correctly wrt to suffices' position and containing a finite tensor product of tensors defined on the same flat manifold (space-time) is a Lorentz covariant.Arbitrary (but finite) rank tensors are Lorentz covariants,including scalars obtained through a finite number of contraction of suffices in a tensor product.
  4. Dec 4, 2004 #3
    I'm sorry I don't understand that explaniation. Can you give me two examples, and say:

    1) Formula , "look at the formula" this is what i mean by lorentz invariant
    2) Another Formula, "this is what i mean by lorentz covariant"

    What I mean is just give an example without so much mathematical jargon
  5. Dec 4, 2004 #4


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    A Lorentz invaraint is a quanirty decsrivbed by a single number and is the same for all inertial observers, an example of this would be mass.

    A Lorentz covaraint is a quantity described by 4^n (n = 0,1,2,3,...) numbers whose componets may change unbder transformation but essientially remain the same quantity (to be non-technical), Lorentz invaraints are alos Lorentz covaraint, any four-vector like four-momentum is Lorentz covariant as indeed is any Lorentz tensor.
  6. Dec 4, 2004 #5


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    The Lorentz invariants are a proper subset of the scalar quantities in physics. The Lorentz invariants are also a proper subset of the Lorentz covariants.
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