Lie derivative Definition and 63 Threads

For every continuous linear operator A: H \rightarrow H from a Hilbert space H to itself, there is a unique continuous linear operator A^* called its Hermitian adjoint such that \langle Ax, y \rangle = \langle x, A^* y \rangle for all x,y \in H. Given that \mathcal{L}_X: \Omega^0(M)...

I'm working thru Thirring's Classical Mathematical Physics. The lie derivative is defined and used on a vector field. I.e. L(x)f where x is a vector field. () = subscript However, later on, he uses the lie derivative of the hamiltonian, which is a scalar function. I.e. L(H)f () =...

Homework Statement Hi, it's the first time I post here, so apologies if this is not the right place. I'm trying to self-study GR, but I'm stuck with Lie Derivatives. The book I'm using (Ludvigsen - General Relativity. A geometric approach) starts with the usual definitions and then gives...

Homework Statement Calculate the lie derivative of the metric tensor, given the metric, g_{ab}=diag(-(1-\frac{2M}{r}),1-\frac{2M}{r},r^2,R^2sin^2\theta) and coordinates (t,r,theta,phi) given the vector E^i=\delta^t_0 Homework Equations...

Can somone remind me how to see that the Lie derivative of a vector field, defined as (L_XY)_p=\lim_{t\rightarrow 0}\frac{\phi_{-t}_*Y_{\phi_t(p)}-Y_p}{t} is actually equal to [X,Y]_p?

I'm trying to use an example to make sense out of the equation \mathcal{L}_X = d\circ i_X + i_X\circ d. Some simple equations: \omega = \omega^1 dx_1 + \omega^2 dx_2 i_X\omega = X_1\omega^1 + X_2\omega^2 (d\omega)^{11} = (d\omega)^{22} = 0,\quad (d\omega)^{12} =...

Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other...

Suppose we define the Lie derivative on a tensor T at a point p in a manifold by \mathcal{L}_V (T) = \lim_{\epsilon \to 0}\frac{\varphi_{-\epsilon \ast}T(\varphi_\epsilon(p))- T(p)}{\epsilon} where V is the vector field which generates the family of diffeomorphisms \varphi_t. If T is just an...

what is the physical significance of the lie derivative? What is its purpose in tensor analysis?

i was curious as to what exactly this is and more importantly, what motivates it. what are its applications?

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

Let M be a diff. manifold, X a complete vectorfield on M generating the 1-parameter group of diffeomorphisms \phi_t. If I now define the Lie Derivative of a real-valued function f on M by \mathscr{L}_Xf=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f-f}{t}\right)=\frac{d}{dt}\phi_{t}^{*}f |_{t=0}...

I'd like an example of calculating the Lie derivative of a one-form with respect to a vector field, for example, the one-form \omega = 3 dx_1 + 4x dx_2 with the vector field X = 7x \frac{\partial }{\partial x_1} + 2 \frac{\partial }{\partial x_2} Any input would be...