Lie vs Covariant Derivative: Intuitive Understanding

In summary, the Lie derivative and covariant derivative are two different ways of measuring the compatibility of vector fields. The Lie derivative uses the flows of the vector fields, while the covariant derivative uses parallel transport. The Lie derivative is a dynamical invariant, while the covariant derivative is not.
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Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...
 
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The covariant derivative uses a connection, while the Lie derivative doesn't.
 
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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.
 
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FAQ: Lie vs Covariant Derivative: Intuitive Understanding

1. What is the difference between a Lie derivative and a covariant derivative?

A Lie derivative is a measure of how a tensor field changes along the flow of a vector field, while a covariant derivative is a measure of how a tensor field changes along a specified direction in a curved space.

2. How do Lie derivatives and covariant derivatives relate to each other?

Lie derivatives are a special case of covariant derivatives, where the direction of change is given by the vector field generating the flow. In other words, the Lie derivative along a vector field is equivalent to the covariant derivative along that same vector field.

3. Can you give an intuitive explanation of Lie and covariant derivatives?

Imagine a river flowing along a curved path. The Lie derivative measures how a buoy floating in the river moves along with the flow, while the covariant derivative measures how the buoy moves relative to the river bank. Both are measures of how an object changes as it moves along a path.

4. What are some applications of Lie and covariant derivatives?

Lie and covariant derivatives are used extensively in differential geometry, general relativity, and theoretical physics. They are essential tools for understanding the behavior of objects in curved spaces and play a crucial role in the mathematics behind Einstein's theory of gravity.

5. How can one calculate Lie and covariant derivatives?

Lie derivatives can be calculated using the Lie bracket, which describes how two vector fields change with respect to each other. Covariant derivatives, on the other hand, require knowledge of the connection coefficients, which describe how the basis vectors of a curved space change as one moves along a given direction. Both can be calculated using various mathematical techniques, such as the exterior derivative or coordinate transformations.

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