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Intuitively what's the difference between Lie Derivative and Covariant Derivative?

  1. Feb 5, 2007 #1
    Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...
     
  2. jcsd
  3. Feb 5, 2007 #2

    cristo

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  4. Feb 5, 2007 #3

    garrett

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    The covariant derivative uses a connection, while the Lie derivative doesn't.
     
  5. Feb 6, 2007 #4
    Just to spotlight one of these: The Lie derivative L_X(Y) is basically a dynamical invariant. It measures how compatible the flows of the vector fields are, i.e. how much they commute with each other. If X is generated by the flow [itex]\phi_t[/itex] and Y is generated by [itex]\psi_s[/itex], then the Lie derivative [X,Y] at point P is the tangent vector at time 0 of the curve given by: [itex]t \mapsto \psi_{-t} \circ \phi_{-t} \circ \psi_t \circ \phi_t (P)[/itex].

    Berger describes this as moving P forward in time along the X-curves by t, then moving along the Y-curve by t, then moving backward in time along the X-curve and finally moving backward in time along the Y-curve. If you've ended up back at P, then [X,Y]=0 at P.

    In particular, if [itex] \phi_s \circ \psi_t=\psi_t \circ \phi_s [/itex] for all s and t, then [X,Y]=0.

    The covariant derivative acts similarly except instead of pushing Y along the X-curves via X's flow, we are pushing Y along X-curve via parallel transport. This explanation, though, is a bit of circular logic, since one usually uses the specific covariant derivative to generate the parallel transport.
     
    Last edited: Feb 6, 2007
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