Lie derivative

From Wikipedia, the free encyclopedia

In differential geometry, the Lie derivative /ˈliː/, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), 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 ${\displaystyle {\mathcal {L}}_{X}(T)}$. The differential operator ${\displaystyle T\mapsto {\mathcal {L}}_{X}(T)}$ 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 ${\displaystyle {\mathcal {L}}_{X}(Y)}$. 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

${\displaystyle {\mathcal {L}}_{[X,Y]}T={\mathcal {L}}_{X}{\mathcal {L}}_{Y}T-{\mathcal {L}}_{Y}{\mathcal {L}}_{X}T,}$

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.

Motivation

A "naive" attempt to define the derivative of a tensor field with respect to a vector field would be to take the directional derivative of the components of the tensor field with respect to the vector field. However, this definition is undesirable because it is not invariant under coordinate transformations, and is thus meaningless when considered on an abstract manifold. In differential geometry, there are two main notions of differentiation (of arbitrary tensor fields) that are invariant under coordinate transformations: Lie derivatives, and derivatives with respect to connections. The main difference between these is that taking a derivative with respect to a connection requires an additional geometric structure (e.g. a Riemannian metric or just an abstract connection) on the manifold, but the derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field; by contrast, when taking a Lie derivative, no additional information about the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X at a point p depends on the value of X in a neighborhood of p, not just at p itself.

Definition

The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.

The (Lie) derivative of a function

The problem with generalizing the derivative of a function when we consider functions over manifolds is that the usual difference quotient requires we define addition on the function's inputs but it is meaningless to add points on a manifold which is not a vector space. But the critical issue is to consider how the function changes relative to smooth displacements of the points. The Lie derivative of a scalar function can be thought of as a definition of the derivative where we are using the flows defined by vector fields to displace the points:

The Lie derivative of a function f with respect to a vector field X at a point p of the manifold M is the value

${\displaystyle {\mathcal {L}}_{X}f(p)=\lim _{t\to 0}{\frac {f(p'(t))-f(p)}{t}}}$

where ${\displaystyle p'(t)}$ is the point to which the flow defined by ${\displaystyle Xt}$ maps the point ${\displaystyle p}$. We typically express the action of the flow in terms of coordinates ${\displaystyle X=X^{\mu }(x)\partial _{\mu }}$ for the coordinate map ${\displaystyle x\mapsto p}$. We can then identify the Lie derivative of a function at p with the directional derivative:

${\displaystyle {\mathcal {L}}_{X}f(p)=\nabla _{X}f(p(x))=X^{\mu }{\frac {\partial }{\partial x^{\mu }}}f(p(x))}$

The Lie derivative of a vector field

If X and Y are both vector fields, then the Lie derivative of Y with respect to X is also known as the Lie bracket of X and Y, and is sometimes denoted ${\displaystyle [X,Y]}$. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:

• The Lie bracket of X and Y at p is given in local coordinates by the formula

${\displaystyle {\mathcal {L}}_{X}Y(p)=[X,Y](p)=\partial _{X}Y(p)-\partial _{Y}X(p),}$

where ${\displaystyle \partial _{X}}$ and ${\displaystyle \partial _{Y}}$ denote the operations of taking the directional derivatives with respect to X and Y, respectively. Here we are treating a vector in n-dimensional space as an n-tuple, so that its directional derivative is simply the tuple consisting of the directional derivatives of its coordinates. Note that although the final expression ${\displaystyle \partial _{X}Y(p)-\partial _{Y}X(p)}$ appearing in this definition does not depend on the choice of local coordinates, the individual terms ${\displaystyle \partial _{X}Y(p)}$ and ${\displaystyle \partial _{Y}X(p)}$ do depend on the choice of coordinates.

• If X and Y are vector fields on a manifold M according to the second definition, then the operator ${\displaystyle {\mathcal {L}}_{X}Y=[X,Y]}$ defined by the formula

${\displaystyle [X,Y]:C^{\infty }(M)\rightarrow C^{\infty }(M)}$

${\displaystyle [X,Y](f)=X(Y(f))-Y(X(f))}$

is a derivation of order zero of the algebra of smooth functions of M, i.e. this operator is a vector field according to the second definition.

The Lie derivative of a tensor field

More generally, if we have a differentiable tensor field T of rank ${\displaystyle (q,r)}$ and a differentiable vector field Y (i.e. a differentiable section of the tangent bundle TM), then we can define the Lie derivative of T along Y. Let, for some open interval I around 0, φ:M×I → M be the one-parameter semigroup of local diffeomorphisms of M induced by the vector flow of Y and denote φ_t(p) := φ(p, t). For each sufficiently small t, φ_t is a diffeomorphism from a neighborhood in M to another neighborhood in M, and φ₀ is the identity diffeomorphism. The Lie derivative of T is defined at a point p by

${\displaystyle ({\mathcal {L}}_{Y}T)_{p}=\left.{\frac {d}{dt}}\right|_{t=0}\left((\varphi _{-t})_{*}T_{\varphi _{t}(p)}\right)=\left.{\frac {d}{dt}}\right|_{t=0}\left((\varphi _{t})^{*}T_{p}\right).}$

where ${\displaystyle (\varphi _{t})_{*}}$ is the pushforward along the diffeomorphism and ${\displaystyle (\varphi _{t})^{*}}$ is the pullback along the diffeomorphism. Intuitively, if you have a tensor field ${\displaystyle T}$ and a vector field Y, then ${\displaystyle {\mathcal {L}}_{Y}T}$ is the infinitesimal change you would see when you flow ${\displaystyle T}$ using the vector field −Y, which is the same thing as the infinitesimal change you would see in ${\displaystyle T}$ if you yourself flowed along the vector field Y.
We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms:

Axiom 1. The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula

${\displaystyle {\mathcal {L}}_{Y}f=Y(f)}$

Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields S and T, we have

${\displaystyle {\mathcal {L}}_{Y}(S\otimes T)=({\mathcal {L}}_{Y}S)\otimes T+S\otimes ({\mathcal {L}}_{Y}T).}$

Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction:

${\displaystyle {\mathcal {L}}_{X}(T(Y_{1},\ldots ,Y_{n}))=({\mathcal {L}}_{X}T)(Y_{1},\ldots ,Y_{n})+T(({\mathcal {L}}_{X}Y_{1}),\ldots ,Y_{n})+\cdots +T(Y_{1},\ldots ,({\mathcal {L}}_{X}Y_{n}))}$

Axiom 4. The Lie derivative commutes with exterior derivative on functions:

${\displaystyle [{\mathcal {L}}_{X},d]=0}$

If these axioms hold, then applying the Lie derivative ${\displaystyle {\mathcal {L}}_{X}}$ to the relation ${\displaystyle df(Y)=Y(f)}$ shows that

${\displaystyle {\mathcal {L}}_{X}Y(f)=X(Y(f))-Y(X(f)),}$

which is one of the standard definitions for the Lie bracket.

The Lie derivative of a differential form is the anticommutator of the interior product with the exterior derivative. So if α is a differential form,

${\displaystyle {\mathcal {L}}_{Y}\alpha =i_{Y}d\alpha +di_{Y}\alpha .}$

This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions.

Explicitly, let T be a tensor field of type (p,q). Consider T to be a differentiable multilinear map of smooth sections α¹, α², ..., α^q of the cotangent bundle T*M and of sections X₁, X₂, ... X_p of the tangent bundle TM, written T(α¹, α², ..., X₁, X₂, ...) into R. Define the Lie derivative of T along Y by the formula

${\displaystyle ({\mathcal {L}}_{Y}T)(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )=Y(T(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots ))}$

${\displaystyle -T({\mathcal {L}}_{Y}\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )-T(\alpha _{1},{\mathcal {L}}_{Y}\alpha _{2},\ldots ,X_{1},X_{2},\ldots )-\ldots }$

${\displaystyle -T(\alpha _{1},\alpha _{2},\ldots ,{\mathcal {L}}_{Y}X_{1},X_{2},\ldots )-T(\alpha _{1},\alpha _{2},\ldots ,X_{1},{\mathcal {L}}_{Y}X_{2},\ldots )-\ldots }$

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the Leibniz rule for differentiation. Note also that the Lie derivative commutes with the contraction.

The Lie derivative of a differential form

A particularly important class of tensor fields is the class of differential forms. The restriction of the Lie derivative to the space of differential forms is closely related to the exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an interior product, after which the relationships falls out as an identity known as Cartan's formula. Note that Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.

Let M be a manifold and X a vector field on M. Let ${\displaystyle \omega \in \Lambda ^{k+1}(M)}$ be a (k + 1)-form, i.e. for each ${\displaystyle p\in M}$, ${\displaystyle \omega (p)}$ is an alternating multilinear map from ${\displaystyle (T_{p}M)^{k+1}}$ to the real numbers. The interior product of X and ω is the k-form ${\displaystyle i_{X}\omega }$ defined as

${\displaystyle (i_{X}\omega )(X_{1},\ldots ,X_{k})=\omega (X,X_{1},\ldots ,X_{k})\,}$

The differential form ${\displaystyle i_{X}\omega }$ is also called the contraction of ω with X. Note that

${\displaystyle i_{X}:\Lambda ^{k+1}(M)\rightarrow \Lambda ^{k}(M)}$

and that ${\displaystyle i_{X}}$ is a ${\displaystyle \wedge }$-antiderivation. That is, ${\displaystyle i_{X}}$ is R-linear, and

${\displaystyle i_{X}(\omega \wedge \eta )=(i_{X}\omega )\wedge \eta +(-1)^{k}\omega \wedge (i_{X}\eta )}$

for ${\displaystyle \omega \in \Lambda ^{k}(M)}$ and η another differential form. Also, for a function ${\displaystyle f\in \Lambda ^{0}(M)}$, that is, a real- or complex-valued function on M, one has

${\displaystyle i_{fX}\omega =f\,i_{X}\omega }$

where ${\displaystyle fX}$ denotes the product of f and X. The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function f with respect to a vector field X is the same as the directional derivative X(f), it is also the same as the contraction of the exterior derivative of f with X:

${\displaystyle {\mathcal {L}}_{X}f=i_{X}\,df}$

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:

${\displaystyle {\mathcal {L}}_{X}\omega =i_{X}d\omega +d(i_{X}\omega ).}$

This identity is known variously as Cartan formula, Cartan homotopy formula or Cartan's magic formula. See interior product for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that

${\displaystyle d{\mathcal {L}}_{X}\omega ={\mathcal {L}}_{X}(d\omega ).}$

The Lie derivative also satisfies the relation

${\displaystyle {\mathcal {L}}_{fX}\omega =f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega .}$

Coordinate expressions

In local coordinate notation, for a type (r,s) tensor field ${\displaystyle T}$, the Lie derivative along ${\displaystyle X}$ is

${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}=&X^{c}(\partial _{c}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}})\\&-(\partial _{c}X^{a_{1}})T^{ca_{2}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}-\ldots -(\partial _{c}X^{a_{r}})T^{a_{1}\ldots a_{r-1}c}{}_{b_{1}\ldots b_{s}}\\&+(\partial _{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{cb_{2}\ldots b_{s}}+\ldots +(\partial _{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}c}\end{aligned}}}$

here, the notation ${\displaystyle \partial _{a}={\frac {\partial }{\partial x^{a}}}}$ means taking the partial derivative with respect to the coordinate ${\displaystyle x^{a}}$. Alternatively, if we are using a torsion-free connection (e.g., the Levi Civita connection), then the partial derivative ${\displaystyle \partial _{a}}$ can be replaced with the covariant derivative ${\displaystyle \nabla _{a}}$. The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor

${\displaystyle ({\mathcal {L}}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}\partial _{a_{1}}\otimes \cdots \otimes \partial _{a_{r}}\otimes dx^{b_{1}}\otimes \cdots \otimes dx^{b_{s}}}$

which is independent of any coordinate system.

The definition can be extended further to tensor densities of weight w for any real w. If T is such a tensor density, then its Lie derivative is a tensor density of the same type and weight.

${\displaystyle ({\mathcal {L}}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}=X^{c}(\partial _{c}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}})-(\partial _{c}X^{a_{1}})T^{ca_{2}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}-\ldots -(\partial _{c}X^{a_{r}})T^{a_{1}\ldots a_{r-1}c}{}_{b_{1}\ldots b_{s}}+}$

${\displaystyle +(\partial _{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{cb_{2}\ldots b_{s}}+\ldots +(\partial _{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}c}+w(\partial _{c}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}}$

Notice the new term at the end of the expression.

For a linear connection ${\displaystyle \Gamma =(\Gamma _{jk}^{i})}$, the Lie derivative along ${\displaystyle X}$ is^[3]

${\displaystyle ({\mathcal {L}}_{X}\Gamma )_{jk}^{i}=X^{p}\partial _{p}\Gamma _{jk}^{i}+\partial _{j}\partial _{k}X^{i}-\Gamma _{jk}^{p}\partial _{p}X^{i}+\Gamma _{pk}^{i}\partial _{j}X^{p}+\Gamma _{jp}^{i}\partial _{k}X^{p}}$

Examples

For clarity we now show the following examples in local coordinate notation.

For a scalar field ${\displaystyle \phi (x^{c})\in {\mathcal {F}}(M)}$ we have:

${\displaystyle ({\mathcal {L}}_{X}\phi )=X^{a}\partial _{a}\phi }$

So less abstractly, consider the scalar field ${\displaystyle \phi (x,y)=x^{2}-\sin(y)}$ and the vector field ${\displaystyle X=\sin(x)\partial _{y}-y^{2}\partial _{x}}$. The corresponding Lie derivative evaluates as

$${\displaystyle {\begin{alignedat}{4}{\mathcal {L}}_{X}\phi &={\mathcal {L}}_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(x^{2}-\sin(y))\\&=d(i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(x^{2}-\sin(y)))+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(d(x^{2}-\sin(y)))\\&=i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(2xdx-\cos(y)dy)\\&=-2xy^{2}-\sin(x)\cos(y)\end{alignedat}}}$$

For a covector field, i.e., a differential form, ${\displaystyle A=A_{a}(x^{b})dx^{a}}$ we have:

${\displaystyle ({\mathcal {L}}_{X}A)_{a}=X^{b}\partial _{b}A_{a}+A_{b}\partial _{a}(X^{b})}$

Concretely, consider the 2-form ${\displaystyle \omega =(x^{2}+y^{2})dx\wedge dz}$ and the vector field ${\displaystyle X}$ from the previous example. Then,

$${\displaystyle {\begin{aligned}{\mathcal {L}}_{X}\omega &=d(i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}((x^{2}+y^{2})dx\wedge dz))+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(d((x^{2}+y^{2})dx\wedge dz))\\&=d(-y^{2}(x^{2}+y^{2})dz)+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(2ydy\wedge dx\wedge dz)\\&=(-2xy^{2}+2y\sin(x))dx\wedge dz+(-2yx^{2}-2y^{3})dy\wedge dz\end{aligned}}}$$

For a covariant symmetric tensor field ${\displaystyle g=g_{ab}(x^{c})dx^{a}\otimes dx^{b}}$ we have:

${\displaystyle ({\mathcal {L}}_{X}g)_{ab}=X^{c}\partial _{c}g_{ab}+g_{cb}\partial _{a}X^{c}+g_{ca}\partial _{b}X^{c}}$

Properties

The Lie derivative has a number of properties. Let ${\displaystyle {\mathcal {F}}(M)}$ be the algebra of functions defined on the manifold M. Then

${\displaystyle {\mathcal {L}}_{X}:{\mathcal {F}}(M)\rightarrow {\mathcal {F}}(M)}$

is a derivation on the algebra ${\displaystyle {\mathcal {F}}(M)}$. That is, ${\displaystyle {\mathcal {L}}_{X}}$ is R-linear and

${\displaystyle {\mathcal {L}}_{X}(fg)=({\mathcal {L}}_{X}f)g+f{\mathcal {L}}_{X}g.}$

Similarly, it is a derivation on ${\displaystyle {\mathcal {F}}(M)\times {\mathcal {X}}(M)}$ where ${\displaystyle {\mathcal {X}}(M)}$ is the set of vector fields on M:

${\displaystyle {\mathcal {L}}_{X}(fY)=({\mathcal {L}}_{X}f)Y+f{\mathcal {L}}_{X}Y}$

which may also be written in the equivalent notation

${\displaystyle {\mathcal {L}}_{X}(f\otimes Y)=({\mathcal {L}}_{X}f)\otimes Y+f\otimes {\mathcal {L}}_{X}Y}$

where the tensor product symbol ${\displaystyle \otimes }$ is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.

Additional properties are consistent with that of the Lie bracket. Thus, for example, considered as a derivation on a vector field,

${\displaystyle {\mathcal {L}}_{X}[Y,Z]=[{\mathcal {L}}_{X}Y,Z]+[Y,{\mathcal {L}}_{X}Z]}$

one finds the above to be just the Jacobi identity. Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.

The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M, and let X and Y be two vector fields. Then

• ${\displaystyle {\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )}$
• ${\displaystyle [{\mathcal {L}}_{X},{\mathcal {L}}_{Y}]\alpha :={\mathcal {L}}_{X}{\mathcal {L}}_{Y}\alpha -{\mathcal {L}}_{Y}{\mathcal {L}}_{X}\alpha ={\mathcal {L}}_{[X,Y]}\alpha }$
• ${\displaystyle [{\mathcal {L}}_{X},i_{Y}]\alpha =[i_{X},{\mathcal {L}}_{Y}]\alpha =i_{[X,Y]}\alpha ,}$ where i denotes interior product defined above and it's clear whether [·,·] denotes the commutator or the Lie bracket of vector fields.

Generalizations

Various generalizations of the Lie derivative play an important role in differential geometry.

The Lie derivative of a spinor field

A definition for Lie derivatives of spinors along generic spacetime vector fields, not necessarily Killing ones, on a general (pseudo) Riemannian manifold was already proposed in 1972 by Yvette Kosmann.^[4] Later, it was provided a geometric framework which justifies her ad hoc prescription within the general framework of Lie derivatives on fiber bundles^[5] in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.^[6]

In a given spin manifold, that is in a Riemannian manifold ${\displaystyle (M,g)}$ admitting a spin structure, the Lie derivative of a spinor field ${\displaystyle \psi }$ can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the André Lichnerowicz's local expression given in 1963:^[7]

${\displaystyle {\mathcal {L}}_{X}\psi :=X^{a}\nabla _{a}\psi -{\frac {1}{4}}\nabla _{a}X_{b}\gamma ^{a}\,\gamma ^{b}\psi \,,}$

where ${\displaystyle \nabla _{a}X_{b}=\nabla _{[a}X_{b]}}$, as ${\displaystyle X=X^{a}\partial _{a}}$ is assumed to be a Killing vector field, and ${\displaystyle \gamma ^{a}}$ are Dirac matrices.

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a generic vector field ${\displaystyle X}$, but explicitly taking the antisymmetric part of ${\displaystyle \nabla _{a}X_{b}}$ only.^[4] More explicitly, Kosmann's local expression given in 1972 is:^[4]

${\displaystyle {\mathcal {L}}_{X}\psi :=X^{a}\nabla _{a}\psi -{\frac {1}{8}}\nabla _{[a}X_{b]}[\gamma ^{a},\gamma ^{b}]\psi \,=\nabla _{X}\psi -{\frac {1}{4}}(dX^{\flat })\cdot \psi \,,}$

where ${\displaystyle [\gamma ^{a},\gamma ^{b}]=\gamma ^{a}\gamma ^{b}-\gamma ^{b}\gamma ^{a}}$ is the commutator, ${\displaystyle d}$ is exterior derivative, ${\displaystyle X^{\flat }=g(X,-)}$ is the dual 1 form corresponding to ${\displaystyle X}$ under the metric (i.e. with lowered indices) and ${\displaystyle \cdot }$ is Clifford multiplication. It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on the spinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.

To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article,^[8][9] where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the Kosmann lift.

Covariant Lie derivative

If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. it has horizontal and vertical components), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.

Now, if we're given a vector field Y over M (but not the principal bundle) but we also have a connection over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matches Y and its vertical component agrees with the connection. This is the covariant Lie derivative.

Nijenhuis–Lie derivative

Another generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ω^k(M, TM) of differential forms with values in the tangent bundle. If K ∈ Ω^k(M, TM) and α is a differential p-form, then it is possible to define the interior product i_Kα of K and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:

${\displaystyle {\mathcal {L}}_{K}\alpha =[d,i_{K}]\alpha =di_{K}\alpha -(-1)^{k-1}i_{K}\,d\alpha .}$

History

In 1931, Władysław Ślebodziński introduced a new differential operator, later called by David van Dantzig that of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms.

The Lie derivatives of general geometric objects (i.e., sections of natural fiber bundles) were studied by A. Nijenhuis, Y. Tashiro and K. Yano.

For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940, Léon Rosenfeld^[10]—and before him Wolfgang Pauli^[11]—introduced what he called a ‘local variation’ ${\displaystyle \delta ^{\ast }A}$ of a geometric object ${\displaystyle A\,}$ induced by an infinitesimal transformation of coordinates generated by a vector field ${\displaystyle X\,}$. One can easily prove that his ${\displaystyle \delta ^{\ast }A}$ is ${\displaystyle -{\mathcal {L}}_{X}(A)\,}$.

Singularity Chat with Vernor Vinge and Ray Kurzweil

June 13, 2002 by Vernor Vinge, Ray Kurzweil
Original link:  http://www.kurzweilai.net/singularity-chat-with-vernor-vinge-and-ray-kurzweil
Originally posted June 11, 2002 on SCIFI.COM. Published June 13, 2002 on KurzweilAI.net.

Vernor Vinge (screen name “vv”) and Ray Kurzweil (screen name “RayKurzweil”) recently discussed The Singularity — their idea that superhuman machine intelligence will soon exceed human intelligence — in an online chat room co-produced by Analog Science Fiction and Fact and Asimov’s Science Fiction magazine on SCIFI.COM. Vinge, a noted science fiction writer, is the author of the seminal paper, “The Technological Singularity.” Kurzweil’s The Singularity Is Near book is due out in early 2003 and is previewed in “The Law of Accelerating Returns.” (Note: typos corrected and comments aggregated for readability.)

ChatMod: Hi everyone, thanks for joining us here. I’m Patrizia DiLucchio for SCIFI. Tonight we’re pleased to welcome science fiction writer Vernor Vinge, and author, innovator, and inventor Ray Kurzweil — the founder of Kurzweil Technologies. Tonight’s topic is Singularity. The term "singularity" refers to the geometric rate of the growth of technology and the idea that this growth will lead to a superhuman machine that will far exceed the human intellect.

ChatMod: This evening’s chat is co-produced by Analog Science Fiction and Fact and Asimov’s Science Fiction (www.asimovs.com), the leading publications in cutting edge science fiction. Our host tonight is Asimov’s editor Gardner Dozois.

ChatMod: Brief word about the drill. This is a moderated chat — please send your questions for our guest to ChatMod, as private messages. (To send a private message, either double-click on ChatMod or type "/msg ChatMod" on the command line – only without the quotes.)…Then hit Enter (or Return on a Mac.)

ChatMod: Gardner, I will leave things to you :)

Gardner: So, I think that Vernor Vinge needs no introduction to this audience. Mr. Kurzweil, would you care to introduce yourself?

RayKurzweil: I consider myself an inventor, entrepreneur, and author.

Gardner: What have you invented? What’s your latest book?

RayKurzweil: My inventions are in the area of pattern recognition, which is part of AI, and the part that ultimately will play a crucial role because the vast majority of human intelligence is based on recognizing patterns. I’ve worked in the areas of optical character recognition, music synthesis, speech synthesis and recognition. I’m now working on an automated stock market fund based on pattern recognition.

As for books, there is one coming out next week "Are We Spiritual Machines? Ray Kurzweil vs. the Critics of Strong AI" from the Discovery Institute Press.

My next trade book will be the Singularity is Near (Viking) expected early next year (not the Singularity, but the book, that is).

Gardner: We recognize them even when they don’t exist, in fact. Like the Face On Mars.

RayKurzweil: The face on Mars shows our power of anthropomorphization.

Gardner: So how far ARE we from the Singularity, then? Guesses?

RayKurzweil: I think it would first make sense to discuss what the Singularity is. The definition offered at the beginning of this chat is one view, but there are others.

Gardner: Go for it.

RayKurzweil: I think that once a nonbiological intelligence (i.e., a machine) reaches human intelligence in its diverse dimensions, it will necessarily soar past it because (i) computational and communication power will continue to grow exponentially, (ii) machines can master information with far greater capacity and accuracy already and most importantly, machines can share their knowledge. We don’t have quick downloading ports on our neurotransmitter concentration patterns, or interneuronal connection patterns. Machines will.

We have hundreds of examples of "narrow AI" today, and I believe we’ll have "strong AI" (capable of passing the Turing test) and thereby soaring past human intelligence for the reasons I stated above by 2029. But that’s not the Singularity. This is "merely" the means by which technology will continue to grow exponentially in its power.

Gardner: Vernor? That fit your ideas?

vv: I agree that there are lots of different takes.

RayKurzweil: If we can combine strong AI, nanotechnology and other exponential trends, technology will appear to tear the fabric of human understanding by around the mid 2040s by my estimation. However, the event horizon of the Singularity can be compared to the concept of Singularity in physics. As one gets near a black hole, what appears to be an event horizon from outside the black hole appears differently from inside. The same will be true of this historical Singularity.

Once we get there, if one is not crushed by it (which will require merging with the technology), then it will not appear to be a rip in the fabric; one will be able to keep up with it.

Gardner: What will it look like from the inside?

vv: That depends on who the observer is. Hans Moravec once pointed out that if “you” are riding the curve of intellect improvement, then no singularity is visible. But if “you” have only human intellect, then the transition could be pretty unknowable. In that latter case, all we have are analogies.

RayKurzweil: We’re thinking alike here. We can only answer that by analog or metaphor. By definition, we cannot describe processes whose intelligence exceeds our own. But we can compare lesser animals to humans, and then make the analogy to our intellectual descendents.

vv: Yes, :-)

Gardner: Seems unlikely to me that EVERYONE will have an equal capacity for keeping up with it. There are people today who have trouble keeping up even with the 20th Century, like the Amish.

RayKurzweil: The Amish seem to fit in well. I could think of other examples of people who would like to turn the clock back.

Gardner: Many. So won’t the same be true during the Singularity?

RayKurzweil: But in terms of opportunity, this is the have-have not issue. Keep in mind that because of what I call the "law of accelerating returns," technology starts out unaffordable, becomes merely expensive, then inexpensive, then free.

vv: True, but the better analogy is across the entire kingdom of life

Gardner: How do you mean that, Vernor?

RayKurzweil: We can imagine some bacteria discussing the pros and cons of evolving into more advanced species like humans. There might be some strong arguments against doing so.

RayKurzweil: If bacteria could talk, of course.

Gardner: From the bacteria’s point of view, they might be right.

vv: When dealing with "superhumans" it is not the same thing as comparing — say — our tech civ with a pretech human civ. The analogies should be with the animal kingdom and even more perhaps with things even further away and more primitive.

RayKurzweil: Bacteria are still doing pretty well, and comprise a larger portion of the biomass than humans. Ants are doing very well also. I agree with your last comment, Vernor.

vv: Yes, there could well be a place for the normal humans — certainly as a safety reservoir in case of tech disasters.

Gardner: So do we end up with civilization split up into several different groups, then, at different levels of evolution?

RayKurzweil: I think that the super intelligences will appear to unenhanced humans (MOSHs – Mostly Original Substrate Humans) to be their transcendent servants. They will be able to meet all the needs of MOSHs with only a trivial portion of their intellectual capacity, and will appreciate and honor their forbears.

vv: I think this is one of the more plausible scenarios (and relatively happy). I thought the movie "Weird Science" was very impressive for that reason (though I’m not sure how many people had that interpretation of the movie. :-)

RayKurzweil: Sorry I missed that movie — most scifi movies are dystopian.

Gardner: If they’re so transcendently superior, though, why should they bother to serve unevolved humans at all? However trivial an effort it would demand on their part?

vv: At Foresight a few years ago, [Eliezer] Yudkowsky put it as simply –"They were designed originally to like us."

RayKurzweil: For a while anyway, they would want to preserve this knowledge, just as we value things from our past today.

Gardner: There are people who don’t like people NOW, though. And people who care nothing about preserving their heritage.

RayKurzweil: Also I see this emerging intelligence as human, as an expression of the human civilization. It’s emerging from within us and from within our civilization. It’s not an alien invasion.

Gardner: Would this change?

vv: I am not convinced that any particular scenario is most likely, though some are much more attractive than others.

ChatMod: Let me jump in for a moment just to let our audience know that we will be using audience questions during our second 30 minutes. We haven’t forgotten you.

RayKurzweil: These developments are more likely to occur from the more thoughtful parts of our civilization. The thoughtless parts won’t have the capacity to do it.

Gardner: Let’s hope.

RayKurzweil: Yes. There are certainly downsides.

Gardner: "The rude mind with difficulty associates the ideas of power and benignity." True. But there have been people with immense power who have not used it at all benignly. Who, in fact, have used it with deliberate malice.

vv: It may be too late to announce a sympathy for animal rights :-)

RayKurzweil: One could argue that network-based communication such as the Internet encourages democracy.

Gardner: Yes, but it also encourages people who deliberately promulgate viruses and try to destroy communications, for no particular reason other than for the pure hell of it.

RayKurzweil: Computer viruses are a good example of a new human-made self-replicating pathogen. It’s a good test case for how well we can deal with new self-replicating dangers.

Gardner: True.

Gardner: Let’s hope our societal immune system is up to the task!

RayKurzweil: It’s certainly an uneasy balance — not one we can get overly comfortable about, but on balance I would say that we’ve kept computer viruses to a nuisance level. If we do half as well with bioengineered viruses or self-replicating nanotechnology entities, we’ll be doing well.

ChatMod: STATION IDENTIFICATION: Just a reminder. We’re chatting with science fiction writer Vernor Vinge, and author, innovator, and inventor Ray Kurzweil — the founder of Kurzweil Technologies. Tonight’s topic is Singularity. Tonight’s chat is co-produced by Analog Science Fiction and Fact and Asimov’s Science Fiction (www.asimovs.com.)

Gardner: Vernor, for the most part, do you think that we’d LIKE living in a Singularity? Do you think positive or negative scenarios are more likely?

vv: I’m inclined toward the optimistic, but very pessimistic things are possible. I don’t see any logical inescapable conclusion. History has had this trajectory toward the good, and there are retrospective explanations for why this is inevitable — but, that may also just be blind luck, the “inevitability” just being the anthropic principle putting rose-colored glasses on our view of the process.

RayKurzweil: My feeling about that is that we would not want to turn back. Consider asking people two centuries ago if they would like living today. Many might say the dangers are too great. But how many people today would opt to go back 200 years to the short (37-year average life span) difficult lives of 200 years ago.

Gardner: It’s interesting that only in the last few years are we seeing stories in SF that share that positive view, instead of the automatic negative default setting. Good point, Ray. I always distrust the longing for The Good Old Days–especially since they sucked, for most people on the Earth.

RayKurzweil: Indeed, most humans lived lives of great hardship, labor, disease-filled, poverty-filled, etc.

Gardner: Audience questions?

ChatMod: to : Hi there. I have a question for Ray Kurzweil: Have the events of Sept. 11 altered your forecast for the evolution of machine intelligence? Could a worldwide nuclear war–heaven forbid–derail the emergence of sentient machines? Or has technology already advanced so far that a singularity is inevitable within the next fifty years?

RayKurzweil: The downsides of technology are not new. There was great destruction in the 20th century — 50 million people died in WW II, made possible by technology. However, these great dislocations in history did nothing to dislocate the advance of technology. I’ve been gathering technology trends over the past century, and they all continued quite smoothly through wars, economic depressions and recessions, and other events such as this. It would take a total disaster to stop this process, which is possible, but I think unlikely.

Gardner: A worldwide effective plague, like the Black Death, might be even more effective in setting back the clock a bit. In fact, World War II accelerated the rate of technological progress. Wars tend to do that.

RayKurzweil: A bioengineered pathogen that was unstoppable is a grave danger. We’ll need the antiviral technologies in time.

ChatMod: Next Question

ChatMod: to : Question for Vernor: Saw a reference to a possible film version of "True Names," Vinge’s "Singularity" novella on "Coming Attractions" http://www.corona.bc.ca/films/filmlistingsFramed.html. What is happening with that?

vv: Aha — It is under option. True Names was recently pitched to one of the heads of the scifi network. I’ve heard that it was very well received and that they’re thinking about it for a TV series. It would be an action series focused on things like anonymity but leading ultimately to singularity issues.

ChatMod: Next Question

ChatMod: to : How does Mr. Kurzweil reconcile his statement that superintelligent AI can be expected in 2029, but the Singularity will not begin to "tear the fabric of human understanding" until 2040?

RayKurzweil: The threshold of a machine being able to pass the Turing test will not immediately tear the fabric of human history. But when you then factor in continuing exponential growth, creation of many more such intelligences, the evolution of these machines into true super intelligences, combining with nanotech, etc., and the maturing of all this, well that will take a bit longer.

Gardner: If it’s the AIs who are becoming superintelligent, maybe it’s them who’ll have the Singularity, not the rest of us.

ChatMod: Next Question

ChatMod: to : For both guests. So far, with humans, evolution seems to have favored intelligence as a survival trait. Once machines can think in this "Singularity" stage, do you think intelligence will still be favored? Or will stronger, healthier bodies be preferred for the intelligent tech we’ll all be carrying like symbiotes?

RayKurzweil: The us-them issue is very important. Again, I don’t see this as an alien invasion. It’s emerging from our brains, and one of the primary applications, at least initially will be to literally expand our brains from inside our brains.

RayKurzweil: Intelligence will always win out. It’s clever enough to circumvent lesser distinctions.

vv: I think the intelligence will be even more strongly favored (as the facilitator of all these other characteristics).

ChatMod: Next Question

ChatMod: to : This question for Vinge and Kurzweil. What other paths are there to Singularity and which is more likely to lead to singularity?

RayKurzweil: Eric Drexler asks this question: Will we first have strong AI, which will be intelligent enough to create full nanotechnology (i.e., an assembler that can create anything). OR will we first have nanotechnology which can be used to reverse-engineer the human brain and create strong AI. The two paths are progressing together, it’s hard to say which scenario is more plausible. They may come together.

Gardner: Speaking of us-them issues, will the Singularity affect dirt farmers in China and Mexico? Will it spread eventually to include everyone? Or will their lives remain essentially unchanged, no matter what’s happening elsewhere?

RayKurzweil: Dirt farmers will soon be using cell phones if they’re not using them already. Some Asian countries have skipped industrialization and gone directly to an information economy. Many individuals do that as well. Ultimately these technologies become extremely inexpensive. How long ago was it that when you saw someone in a movie use a cell phone that indicated that this person was a member of the power elite. That was only a few years ago.

ChatMod: Next Question:

ChatMod: to : Do you have suggestions for how we can participate in research leading to the singularity. I am a programmer for example. Are there projects someone like me can help with?

RayKurzweil: The Singularity emerges from many thousands of smaller advances on many fronts: three-dimensional molecular computing, brain research (brain reverse engineering), software techniques, communication technologies, and many others. There’s no single Singularity study. It’s the result of many intersecting revolutions.

vv: I think there are a number of paths and they intertwine: Pure AI, IA (growing out of human-computer interface work whereby our automation becomes an amplification of ourselves, perhaps becoming what David Brin calls our “neo-neocortex”), the Internet and large scale collaboration, and some improvements in humans themselves (e.g., improved natural memory would make early IA interfaces much easier to set up, as in my story “Fast Times at Fairmont High”).

ChatMod: to : For both guests — will we notice the Singularity when it happens, or will life seem routine until we suddenly wake up and realize we’re on the other side?

Gardner: Or WILL we realize that?

RayKurzweil: Life will appear progressively more exciting. Doesn’t it seem that way already. I mean we’re just getting started, but there’s already a palpable acceleration.

vv: Yes, I know some people who half-seriously suggest it has already happened.

ChatMod: Gardner, maybe you and VV can answer this one — in fiction when did the first realistic discussion of Singularity appear?

Gardner: I think that Vernor was the first person to come up with a term for it.

vv: I think that depends on how constrained the definition is. In my 1993 NASA essay, I tried to research some history though that wasn’t related to fiction so much: I first used the term/idea at an AAAI meeting in 1982, and then in OMNI in 1983.

Gardner: Although the idea that technology would in the future give people the power of gods and make them incomprehensible to us normal humans, goes way back. At least as far as Wells.

vv: Before that, I didn’t find a use of Singularity that was in the sense I use it (but von Neumann apparently used the term for something very similar).

RayKurzweil: Von Neumann said that technology appeared to be accelerating such that it would reach a point of almost infinite power at some point in the future.

vv: Gardner: yes, apocalyptic optimism has always been a staple of our genre!

Gardner: Some writers have pictured the far-future techno-gods as being indifferent to us, though. Much as we usually are to ants (unless we want to build a highway bypass where their nest is!)

RayKurzweil: We’re further away from ants, evolutionarily speaking, than the machines we’re building will be from us.

vv: The early machines anyway.

RayKurzweil: Good point.

Gardner: What about from the machines the machines we build, though?

RayKurzweil: That was Vernor’s point. However, I still maintain that we will evolve into the machines. We will start by merging with them (by, for example, placing billions of nanobots in the capillaries of our brain), and then evolving into them.

ChatMod: Let me jump in and tell our audience that we’ll do about ten more minutes of questions, then open the floor

Gardner: Like the Tin Man paradox from THE WIZARD OF OZ.

ChatMod: to : I’d like to know what the guests think of the idea that perhaps everyone will be left behind after the Singularity, that humans will/may have to deal with the idea that most meaningful human endeavors will be better performed by AIs. Or at least by enhanced humans.

vv: I think this will be the case for standard humans. Also I think Greg Bear’s
fast progress scenario is possible, so that the second and third generation stuff happens in a matter of weeks.

Gardner: I think it makes a big difference whether the AIs are enhanced humans or pure machines. I’m hoping for the former. If it’s the later, we may have problems.

RayKurzweil: The problem stems more from self-replication of entities that are not all that bright — basically the gray goo scenario.

vv: Actually, an argument can be made for the reverse , Gardner. We have so much bloody baggage that I’m not sure who to trust as the early "god like" human :-)

ChatMod: ‘Nother Question:

ChatMod: to : to ‘vv’ and ‘RayKurzweil’: When will our mode of economics change relative to the singularity? Will capitalism "survive" the singularity?

Gardner: Are you saying that machines would be kinder to us than humans are likely to be?

vv: I think there will continue to be scarce resources, though "mere humans" may not really perceive things that way.

RayKurzweil: Capitalism, to the extent that it creates a Darwinian environment, fuels the ongoing acceleration of technology. It’s the "Brave New World" scenario, in which technology is controlled by a totalitarian Government that slows things down. The scarce resources will all be knowledge-based.

vv: Hmm, they might be matter-based in the sense that some form of matter is needed to make computing devices (Hans’ [Moravec] notion of the universe converted into thinking matter).

RayKurzweil: Very little matter is needed. Fredkin shows no lower bound to the energy and matter resources needed for computation and communication. I do think that the entire Universe will be changed from "dumb matter" to smart matter, which is the ultimate answer to the cosmological questions.

ChatMod: We’ll make this next one the final audience question

ChatMod: to : To both authors: If you knew that the first project team to create an enhanced human or build a self-improving AI were hanging out in this chatroom, what would you most want to say to them? What kind of moral imperatives are involved with implementing or entering a Singularity?

RayKurzweil: Although this begs the question, I don’t think the threshold is that clear cut. We already have enhanced humans, and there are self-improving AIs within narrow constraints. Over time, these narrow constraints will get less narrow. But putting that aside, how about "Respect your elders"?

vv: I find it very hard to think of surefire safety advice, but I bet that time-constrained arms races would be most likely to end in hellish disasters.

Gardner: If there were Singularities elsewhere, what signs would they leave behind that we could recognize? Wouldn’t the universe already have been converted to "smart matter" if other races had hit this point?

RayKurzweil: Yes, indeed. That is why I believe that we are first. That’s why we don’t notice any other intelligences out there and why SETI will fail. They’re not there. That may seem unlikely, but so is the existence of our universe with its marvelously precise rules to allow matter to evolve, etc. But here we are. So by the anthropic principle, we’re here to talk about it.

Gardner: You feel that way too, Vernor?

vv: I’m not that confident of any particular explanation for Fermi’s paradox (which this question is surely a part of). From SF we have lots of possibilities: maybe we are the first or we are Not At All the first — in which latter case you get something like Robert Charles Wilson’s Darwinia or Hans Moravec’s “Pigs in Cyberspace.” Or maybe there are big disasters that can happen and we have been living in anthropic-principle bliss of the dangers. However, being the first would be nice, if only now we can pull it off and transcend!

RayKurzweil: The only other possibility is that it is impossible to overcome the speed of light as a limitation and that there are other superintelligences beyond our light sphere. The explanation that a superintelligence may have destroyed itself is plausible for one or a few such civilizations, but according to the SETI assumptions, there should be millions or billions of such civilizations. It’s not plausible that they all destroyed themselves, or that they all decided to remain hidden from us.

Gardner: Well, before we open the floor, let’s get plugs. We already know that Ray has two books out or coming out (repeat the titles, please!). Vernor, what new projects do you have coming?

vv: Well, I have the story collection from Tor out late last year. I’m waiting on the TV True Names stuff from SCIFI channel that I mentioned earlier. I’ve got some near-future stuff based on "Fast times at Fairmont High" (about wearable computers and fine-grained ubiquitous networking).

RayKurzweil: With regard to plugs, I would suggest people check out KurzweilAI.net. We have about 70 big thinkers (like Vernor, Hans Moravec, and many others) with about 400 articles, about 1000 news items each year, a free e-newsletter you can sign up for, lots of discussion on our MindX boards on these issues. I’m also working on a book called "A Short Guide to a Long Life" with coauthor Terry Grossman, MD, so that we can all live to see the Singularity. One doctor has already cured type I Diabetes in rats with a nanotechnology-based module in the bloodstream. There are dozens of such ideas in progress.

ChatMod: Our hour is long gone. Thanks, gentlemen for a great chat. Tonight’s chat is co-produced by Analog Science Fiction and Fact and Asimov’s Science Fiction (www.asimovs.com.) Just a reminder…Join us again on June 25th when we’ll be chatting with Asimov artists Michael Carroll, Kelly and Laura Freas, and Wolf Read. Good night everybody.