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.

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  1. cianfa72

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  2. cianfa72

    I Calculation of Lie derivative - follow up

    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...
  3. cianfa72

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  4. Baela

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  5. S

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  6. cianfa72

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  7. Arman777

    A Calculating Lie Derivatives for Tensors & Vectors

    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 -...
  8. E

    Lie derivative of general differential form

    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}...
  9. M

    Solving the same question two ways: Parallel transport vs. the Lie derivative

    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...
  10. L

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  11. Pencilvester

    I Lie derivative of hypersurface basis vectors along geodesic congruence

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  12. haushofer

    A Issue with the definition of a Lie derivative and its components (Carroll's GR)

    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...
  13. Abhishek11235

    A Calculating Lie Derivative for Case (ii)

    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...
  14. E

    A Lie derivative of vector field defined through integral curv

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  15. K

    I Lie Derivative in Relativity: Examples & Uses

    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...
  16. D

    I Diffeomorphism invariance and contracted Bianchi identity

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  17. binbagsss

    Component of Lie Derivative expression vector field

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  18. F

    I Lie derivative of a metric determinant

    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...
  19. F

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  20. binbagsss

    Lie derivative vector fields, show the Leibniz rule holds

    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)...
  21. binbagsss

    GR Lie Derivative of metric vanish <=> metric is independent

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  22. binbagsss

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  23. TAKEDA Hiroki

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  24. F

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  25. S

    A Definition of the Lie derivative

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  26. D

    I Lie derivative of a differential form

    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...
  27. D

    Lie derivative of tensor field with respect to Lie bracket

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  28. Q

    Lie Derivative of one-form: an identity

    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...
  29. FreeThinking

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  30. B

    Lie derivative of contraction and of differential form

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  31. C

    What are some applications of Lie derivative in physics?

    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
  32. C

    Showing that the Lie derivative of a function is the directional deriv

    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...
  33. Z

    Lie derivative of covariant vector

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  34. S

    Understanding the Lie Derivative of Tensors: A Step-by-Step Approach

    consider t is arbitrary tensor and [x,y] is Lie derivative how can we show that L[x,y]t=Lx Ly t - Ly Lx t
  35. K

    Lie derivative of two left invariant vector fields

    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...
  36. B

    Differential Geometry: Lie derivative of tensor fields.

    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...
  37. Matterwave

    Shutz's derivation of the Lie Derivative of a vector field

    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...
  38. S

    Confusion over the definition of Lie Derivative of a Vector Field

    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...
  39. jfy4

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    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...
  40. jfy4

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  41. W

    Chain rule for commutator (Lie derivative)?

    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...
  42. L

    Understanding Lie Derivative: L_X f^\mu = (\partial_\alpha X^\mu) f^\alpha

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  43. L

    Deriving the Lie Derivative of a Covector: The Leibniz Rule

    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) =...
  44. Q

    Covariant derivative vs. Lie derivative

    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...
  45. 0

    Lie derivative and vector field notion.

    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...
  46. S

    Coordinate independence of Lie derivative

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  47. R

    Lie derivative versus covariant derivative

    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...
  48. F

    Adjoint of the scalar Lie derivative?

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  49. R

    Can a Lie Derivative be Taken in the Direction of a Scalar Function?

    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 () =...
  50. G

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