What is Lie derivative: Definition and 60 Discussions
In differential geometry, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted
L
X
(
T
)
{\displaystyle {\mathcal {L}}_{X}(T)}
. The differential operator
T
↦
L
X
(
T
)
{\displaystyle T\mapsto {\mathcal {L}}_{X}(T)}
is a derivation of the algebra of tensor fields of the underlying manifold.
The Lie derivative commutes with contraction and the exterior derivative on differential forms.
Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.
The Lie derivative of a vector field Y with respect to another vector field X is known as the "Lie bracket" of X and Y, and is often denoted [X,Y] instead of
L
X
(
Y
)
{\displaystyle {\mathcal {L}}_{X}(Y)}
. The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity
L
[
X
,
Y
]
T
=
L
X
L
Y
T
−
L
Y
L
X
T
,
{\displaystyle {\mathcal {L}}_{[X,Y]}T={\mathcal {L}}_{X}{\mathcal {L}}_{Y}T-{\mathcal {L}}_{Y}{\mathcal {L}}_{X}T,}
valid for any vector fields X and Y and any tensor field T.
Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.
Generalisations exist for spinor fields, fibre bundles with connection and vector-valued differential forms.
Hi,
starting from a recent thread in this section, I decided to start a new thread about the following:
Take a generic irrotational/zero vorticity timelike congruence. Do the 4-velocity and the direction of proper acceleration (i.e. the vector in that direction at each point with norm 1)...
Hi, a doubt related to the calculation done in this old thread.
$$\left(L_{\mathbf{X}} \dfrac{\partial}{\partial x^i} \right)^j = -\dfrac{\partial X^j}{\partial x^i}$$
$$L_{\mathbf{X}} {T^a}_b = {(L_{\mathbf{X}} \mathbf{T})^a}_b + {T^{i}}_b \langle L_{\mathbf{X}} \mathbf{e}^a, \mathbf{e}_i...
We had a thread long time ago concerning the Lie dragging of a vector field ##X## along a given vector field ##V## compared to the Fermi-Walker transport of ##X## along a curve ##C## through a point ##P## that is the integral curve of the vector field ##V## passing through that point.
We said...
I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
Consider a fluid flow with density ##\rho=\rho(t,x)## and velocity vector ##v=v(t,x)##. Assume it satisfies the continuity equation
$$
\partial_t \rho + \nabla \cdot (\rho v) = 0.
$$
We now that, by Reynolds Transport Theorem (RTT), this implies that the total mass is conserved
$$...
Hi,
reading Carrol chapter 5 (More Geometry), he claims that a maximal symmetric space such as Minkowski spacetime has got ##4(4+1)/2 = 10## indipendent Killing Vector Fields (KVFs). Indeed we can just count the isometries of such spacetime in terms of translations (4) and rotations (6).
By...
I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative
1) for scalar function;
$$\nabla_a\phi \equiv \partial_a\phi~~(1)$$
2) for vectors;
$$\nabla_bV^a = \partial_bV^a + \Gamma^a_{bc}V^c~~(2)$$
$$\nabla_cV_a = \partial_cV_a -...
The first two parts I think were fine, I expressed the tensors in coordinate basis and wrote for the first part$$
\begin{align*}
\mathcal{L}_X \omega = \mathcal{L}_X(\omega_{\nu} dx^{\nu} ) &= (\mathcal{L}_X \omega_{\nu}) dx^{\nu} + \omega_{\nu} (\mathcal{L}_X dx^{\nu}) \\
&= X^{\sigma}...
a) I found this part to be quite straight forward. From the Parallel transport equation we obtain the differential equations for the different components of ##X^\mu##:
$$
\begin{align*}
\frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\
\frac{\partial...
I'm going through the "Advanced Lectures on General Relativity" by G. Compère and got stuck with solving one set of conditions on the subject of asymptotic flatness. Let ##(M,g)## be ##4##-dimensional spacetime and ##(u,r,x^A)## be a chart such that the coordinate expression of ##g## is in Bondi...
Hello PF, here’s the setup: we have a geodesic congruence (not necessarily hypersurface orthogonal), and two sets of coordinates. One set, ##x^\alpha##, is just any arbitrary set of coordinates. The other set, ##(\tau,y^a)##, is defined such that ##\tau## labels each hypersurface (and...
Dear all,
I'm having a small issue with the notion of Lie-derivatives after rereading Carroll's notes
https://arxiv.org/abs/gr-qc/9712019
page 135 onward. The Lie derivative of a tensor T w.r.t. a vector field V is defined in eqn.(5.18) via a diffeomorphism ##\phi##. In this definition, both...
I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in picture. The question is,how do I apply ##\phi^*## to given function ##\psi## . I know that...
Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral
curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted
##\phi _ { t } ( p ) .##
Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...
Can someone point me some examples of how the Lie Derivative can be useful in the General theory of Relativity, and if it has some use in Special Relativity.
I'm asking this because I'm studying how it's derived and I don't have any Relativity book in hand so that I can look up its application...
I've been reading Straumann's book "General Relativity & Relativistic Astrophysics". In it, he claims that the twice contracted Bianchi identity: $$\nabla_{\mu}G^{\mu\nu}=0$$ (where ##G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R##) is a consequence of the diffeomorphism (diff) invariance of the...
1. Homework Statement
Hi,
I have done part a) by using the expression given for the lie derivative of a vector field and noting that if ##w## is a vector field then so is ##wf## and that was fine.
In order to do part b) I need to use the expression given in the question but looking at a...
I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is.
Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...
it is often stated in texts on general relativity that the theory is diffeomorphism invariant, i.e. if the universe is represented by a manifold ##\mathcal{M}## with metric ##g_{\mu\nu}## and matter fields ##\psi## and ##\phi:\mathcal{M}\rightarrow\mathcal{M}## is a diffeomorphism, then the sets...
Homework Statement
Homework Equations
##V=V^u \partial_u ##
I am a bit confused with the notation used for the Lie Derivative of a vector field written as the commutator expression:
Not using the commutator expression I have:
## (L_vU)^u = V^u \partial_u U^v - U^u\partial_u V^v## (1)...
Homework Statement
How to show that lie deriviaitve of metric vanish ##(L_v g)_{uv}=0## <=> metric is independent of this coordinate, for example if ##v=\partial_z## then ##g_{uv} ## is independent of ##z## (and vice versa)
2. Relevant equation
I am wanting to show this for the levi-civita...
Homework Statement
Question attached.
Homework Equations 3. The Attempt at a Solution [/B]
I'm not really sure how to work with what is given in the question without introducing my knowledge on lie derivatives.
We have: ##(L_ug)_{uv} =...
In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times...
I've been studying a bit of differential geometry in order to try and gain a deeper understanding of the mathematics of general relativity (GR). As you may guess from this, I am approaching this subject from a physicist's perspective so I apologise in advance for any lack of rigour.
As I...
Consider the Lie derivative of the vector field ##\bf{Y}## with respect to the vector field ##\bf{X}## on manifold ##M^{n}(x)## defined as
##\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}##
Now, I understand that...
Hello,
I have a maybe unusual question. In a paper, I recently found the equation $$\mathcal{L}_v(v_i dx^i) = (v^j \partial_j v_i + v_j \partial_i v^j) dx^i$$
Where v denotes velocity, x spatial coordinates and \mathcal{L}_v the Lie derivative with respect to v. Now I'm an undergraduate who...
I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t
but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...
Homework Statement
I am trying to prove an identity for the Lie derivative of a smooth one-form. The identity is: for X, Y smooth vector fields, alpha a smooth one-form, we have:
$$L_{[X, Y]}\alpha = [L_X, L_Y]\alpha$$ For anyone familiar with the book, this is exercise 5.26 in the first...
Homework Statement
(Self study.)
Several sources give the following for the Riemann Curvature Tensor:
The above is from Wikipedia.
My question is what is \nabla_{[u,v]} ?
Homework Equations
[A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators)...
Hello. I'm learning about Lie derivatives and one of the exercises in the book I use (Isham) is to prove that given vector fields X,Y and one-form ω identity L_X\langle \omega , Y \rangle=\langle L_X \omega, Y \rangle + \langle \omega, L_X Y \rangle holds, where LX means Lie derivative with...
Hello everybody, I am an undergrad physics student and I'm taking some "Geometry and Topology for physicist" course. We saw Lie Derivative some time ago and I still don't know how can I use it on physics, can anyone give me some examples? thanks
Hi!
To boost my understanding of the mathematics in relation to general relativity, I'm reading about Lie derivatives in Sean Carroll's "Spacetime and geometry". Here he defines the Lie derivative of a (k,l) tensor at the point p along the vectorfield V as
$$\mathcal{L}_V T^{\mu_1 \cdots...
Homework Statement
Derive L_v(u_a)=v^b \partial_b u_a + u_b \partial_a v^b
Homework Equations
L_v(w^a)=v^b \partial_b w^a - w^b \partial_b v^a
L_v(f)=v^a \partial_a f where f is a scalar.
The Attempt at a Solution
In the end I get stuck with something like this,
L_v(u_a)w^a=v^b...
Hi all,
I was following Nakahara's book and I really got my mind stuck with something. I would appreciate if anybody could help with this.
The Lie derivative of a vector field Y along the flow \sigma_t of another vector field X is defined as
L_X...
Homework Statement
Let M be a differentiable manifold. Let X and Y be two vector fields on M, and let t be a tensor field on M. Prove
\mathcal{L}_{[X,Y]}t = \mathcal{L}_X\mathcal{L}_Yt -\mathcal{L}_Y\mathcal{L}_Xt
Homework Equations
All is fair game, though presumably a coordinate-free...
I have a question about Bernard Shutz's derivation of the Lie derivative of a vector field in his book Geometrical Methods for Mathematical Physics.
I will try to reproduce part of his argument here.
Essentially, we have 2 vector fields V and U which are represented by \frac{d}{d\lambda} and...
Hello all, I was hoping someone would be able to clarify this issue I am having with the Lie Derivative of a vector field.
We define the lie derivative of a vector field Y with respect to a vector field X to be
L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y), where \phi_t is the...
Hi,
I have been looking around, and I can't seem to find a slightly different version of the lie derivative where the lie derivative is taken with respect to a tensor field, rather than a vector field. That is, a quantity which measures the change in a vector field, along the "flow" of a...
Hi,
This thread looks like GR/SR, but it has grounds QM and maybe only stays in that realm, which is what I'm asking
I was looking at some everyday non-relativistic quantum mechanics and I spotted something I thought was interesting. Consider the time evolution of an observable
\frac{d...
I'm curious if there's a chain rule for the commutator (I'll explain what I mean) just like there's a product rule ([AB,C]).
So, say you have an operator, which can be expressed in terms of another operator, and we know the commutation relationship between x and another operator, y. I'll call...
I'm trying to show that L_X f^\mu = ( \partial_\alpha X^\mu) f^\alpha where f^\mu is a basis for the cotangent space T_p^*(M)
The answer says
L_X dx^\mu = dL_X x^\mu (ive already shown this)
=dX(x^\mu) by properties of lie derivative on a function
=dx^\mu (dX) using X(f)=df(X)...
Use the Leibniz rule to derive the formula for the Lie derivative of a covector \omega valid in any coordinate basis:
(L_X \omega)_\mu = X^\nu \partial_\nu \omega_\mu + \omega_\nu \partial_\mu X^\nu
(Hint: consider (L_X \omega)(Y) for a vector fi eld Y).
Well I have the formula L_X(Y) =...
Hey there,
For quite some time I've been wondering now whether there's a well-understandable difference between the Lie and the covariant derivative. Although they're defined in fundamentally different ways, they're both (in a special case, at least) standing for the directional derivative of...
Here is an approach for lie derivative. And i would like to know how wrong is it.
Assuming lie derivative of a vector field measures change of a vector field along a vector field, take a coordinate system, xi , and the vector field fi along which Ti is being changed. I go this way, i take the...
Hello Forum,
since my GR tutor can't help me with some issues arising I thought it is time to register here.
I am very confused about the phrase "coordinate independence". Especially regarding the Lie Derivative and the Commutator of two vector fields.
1)
The Lie Derivative is said...
When calculating the derivative of a vector field X at a point p of a smooth manifold M, one uses the Lie derivative, which gives the derivative of X in the direction of another vector field Y at the same point p of the manifold.
If the manifold is a Riemannian manifold (that is, equipped...
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...