Lie vs Covariant Derivative: Intuitive Understanding

[sit.think.solve]

Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...

 

[cristo]

There was a recent discussion of the relationship between the Lie and covariant derivatives, which may be useful to you: https://www.physicsforums.com/showthread.php?t=150200.

After reading this, do you ahave any other specific questions?

 

[garrett]

The covariant derivative uses a connection, while the Lie derivative doesn't.

 

[Doodle Bob]

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 \phi_t and Y is generated by \psi_s, then the Lie derivative [X,Y] at point P is the tangent vector at time 0 of the curve given by: t \mapsto \psi_{-t} \circ \phi_{-t} \circ \psi_t \circ \phi_t (P).

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 \phi_s \circ \psi_t=\psi_t \circ \phi_s 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.