Killing vectors on a 3D Euclidean space

[Magister]

I have read that the Killing vectors in a 3D euclidean space are the 3 components of the ordinary divergence plus the 3 components of the ordinary rotational. I have being trying to find a derivation of this but it isn´t being easy.
I really apreciates any clues.
Thanks

 

[pervect]

I'm not quite sure what you're trying to prove.

Every one parameter group of isometries is associated with a Killing vector - the Killing vector is the infinitesimal "generator" of said one parameter group.

Now space translations are isometries (distance preserving isomorphisms) in Euclidiean space, therefore space translations generate a Killing vector field in Euclidean space.

This Killing vector field is just the generator of the translation group, i.e. a unit vector field pointing everywhere in the same direction.

Similarly, rotations are also isometries, so they also generate a Killing vector field. So they also generate Killing vectors. If you want the Killing vector field itself, you need to find the "generator" of the rotation group.

 

[Chris Hillman]

The Lie algebra of Killing vector fields in euclidean three space?

I have read that the Killing vectors in a 3D euclidean space are the 3 components of the ordinary divergence plus the 3 components of the ordinary rotational. I have being trying to find a derivation of this but it isn´t being easy.
I really apreciates any clues.

I suspect you misunderstood something you read (if you disagree, can you cite your source and provide an exact quotation?), but more to the point, I think I know why you have been having a hard time finding answers: you should have phrased the question as "how can I compute the Lie algebra of Killing vector fields in euclidean three space $E^3$?"

The answer to this question is found in almost any good gtr textbook: write down and solve the Killing equation. In this case this turns out to be particularly easy, and we find a six dimensional Lie algebra generated by three translations and three rotations. That is, the isometry group of $E^3$ is the Lie group $E(3) = R^3 \, | \! \! \times O(3)$, the semidirect product of the translation group (normal) and the usual rotation group (non-normal; a congugate copy sits over every point in the base manifold), which has Lie algebra $e(3)$, which is generated as stated.

While in principle that provides the answer you seek, this is a good opportunity for me to experiment with providing a little Maple code. I'll assume you have Maple running with an X Windows interface; under unix this is called as xmaple), and I'll assume you have GrTensorII installed (with a working path to a directory holding your frame field definitions).

None of this is necessary to perform the computation, of course! I am simply illustrating something which works to greater effect when finding the isometry group of more interesting manifolds, such as a vacuum gravitational plane wave solution with some extra symmetries.

Use your favorite text editor to create a file called E3_cart.mpl in the proper directory:

Ndim_ := 3:
x1_ := x:
x2_ := y:
x3_ := z:
eta11_ :=  1:
eta22_ :=  1:
eta33_ :=  1:
bd11_ :=  1:
bd22_ :=  1:
bd33_ :=  1:
Info_ := `Coordinate frame in Cartesian chart for E^3`:
# Chart covers -infty < x,y,z < infty
# Beltrami-Laplace equation
#  *d*dh = h_(xx) + h_(yy) + h_(zz) = 0

This file defines $E^3$ a la Cartan, in terms of the coframe field
$$\sigma^1 = dx, \; \; \sigma^2 = dy, \; \; \sigma^3 = dz$$
which has the dual frame field
$$\vec{e}_1 = \partial_x, \; \; \vec{e}_2 = \partial_y, \; \; \vec{e}_3 = \partial_z$$
The metric tensor is then given as
$$g = \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3$$
which expresses the "orthonormal" nature of the coframe. Multiplying out, you immediately recover the usual line element
$$ds^2 = dx^2 + dy^2 + dz^2$$
(The hash marks give an example of how you can comment your GrTensorII files.)

Note that you can replace the three given covector fields [itex]\sigma^1, \, \sigma^2, \, \sigma^3[/tex] with more interesting ones, and then declaring these to form a coframe field will usually give some curved Riemannian three-manifold. Since vector and covector fields are coordinate-free notions, giving a coframe or frame field provides a coordinate-free description of any Riemannian (or Lorentzian) manifold.

As always, there are many alternative (local) representations of the same manifold; as I have been stressing elsewhere (e.g. in threads of computing the frequency shift observed for an ordered pair of possibly accelerating observers in some spacetime model given by a Lorentzian manifold, perhaps a solution of the EFE or some competing field equation), this multiplicity is often just what we need, but as always in mathematics, the problem of classifying coframe fields up to local equivalance (in this case, local isometry of the Riemannian or Lorentzian manifold they define) arises. I'll say a bit about that below.

(I should add that in this example, the ON coframe just happens to be the coordinate coframe. This is never true except for Cartesian coordinate charts, which exist only in appropriate regions of locally flat manifolds, so this example is in some respects a bit misleading!)

Next, in your maple server window,

qload(E3_cart);
grcalcalter(LieD[kv0,g(dn,dn)],1):grdisplay(_);
[grcalc>X(x,y,z);
[grcalc>Y(x,y,z);
[grcalc>Z(x,y,z);
keqs := [seq(seq(grcomponent(LieD[kv0,g(dn,dn)], [grcomponent(x(up),[j]),
grcomponent(x(up),[k])]),j=1..3),k=1..3)]:
for guy in keqs do value(guy); od;
casesplit(keqs);
op(%)[1]: pdsolve(%): subs(%,[X,Y,Z](x,y,z)): 
sol := subs(_C1=c[1],_C2=c[2],_C3=c[3],_C4=c[4],_C5=c[5],_C6=c[6],%);
subs(c[1]=1,c[2]=0,c[3]=0,c[4]=0,c[5]=0,c[6]=0,sol);
subs(c[1]=0,c[2]=1,c[3]=0,c[4]=0,c[5]=0,c[6]=0,sol);
subs(c[1]=0,c[2]=0,c[3]=1,c[4]=0,c[5]=0,c[6]=0,sol);
subs(c[1]=0,c[2]=0,c[3]=0,c[4]=1,c[5]=0,c[6]=0,sol);
subs(c[1]=0,c[2]=0,c[3]=0,c[4]=0,c[5]=1,c[6]=0,sol);
subs(c[1]=0,c[2]=0,c[3]=0,c[4]=0,c[5]=0,c[6]=1,sol);

Here, note the intermediate result: the Killing equation, written out explicitly for this particular chart on this particular manifold, become
$$X_x = Y_y = Z_z = 0$$
$$Y_x + X_y = Z_x + X_z = Y_x + X_y = 0$$
$$Z_y + Y_z = Z_x + X_z = Z_y + Y_z = 0$$
where I am using the useful notational convention, common to much of the literature dealing with PDEs, in which partials are denoted by subscripts. Here, the undetermined Killing vector field is written
$$\vec{W} = X \, \partial_x + Y \, \partial_y + Z \, \partial_z$$
where $X, \, Y, \, Z$ are three undetermined functions of the three coordinates, namely the components with respect to our frame field of the undetermined Killing vector field. As you can see, the Killing equation implies that for a Killing vector field $\vec{W}$ on $E^3$, the divergence $\vec{\nabla} \vec{W} = X_x + Y_y + Z_z$ and the curl $\vec{\nabla} \times \vec{W}$ both vanish. Maybe what you saw was the claim that every Killing vector field on $E^3$ has vanishing divergence and curl?

You can probably see how to solve this linear system of first order PDEs "by hand"; the utility "casesplit" simply automates the "triangularization" you would probably pursue. It
works by applying Groebner basis methods to differential rings, and draws upon landmark work by Lie, Cartan and Buchberger (see below for some citations), among others. The Buchberger algorithm computes a Groebner basis, and is a kind of common generalization of Gaussian reduction of a system of linear equations, the division algorithm for polynomials (you learned that in grade school, in the disguised form of "long division"), and of Euclid's algorithm in elementary number theory.

From the final result we can read off the six generators
$$\partial_x, \; \partial_y, \; \partial_z$$
$$y \, \partial_x - x \, \partial_y, \; \; z \, \partial_x - x \, \partial_z, \; \; z \, \partial_y - y \, \partial_z$$
I suspect that you saw this result somewhere (correct) and misread it as the "divergence operator plus the curl operator" (incorrect).

As pervect mentioned in his reply, these six generators "exponentiate" to give six unidimensional subgroups of the Lie group of isometries. The rotations generate the (proper) isotropy group $SO(3)$. The group of (proper) isometries SE(3) is a fiber bundle over $R^3$ with fibers diffeomorphic to $SO(3)$.

For some insight into why it is so advantageous to think about vector fields as linear first order partial differential operators, particularly in the context of Lie theory, see (in order):

Roger Howe, "Very basic Lie theory", American Math. Monthly Nov. 1983: 600-623

Roger Carter, Graeme Segal, and Ian MacDonald, Lectures on Lie Groups and Lie Algebras, LMS Student Texts 32, Cambridge University Press, 1995.

Peter J. Olver, Applications of Lie Groups to Differential Equations, GTM 107, second edition, Springer, 1993.

Peter J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995.

The second book by Olver listed above is devoted to explaining Cartan's formulation and solution of a huge class of "equivalence problems", including a very important one for readers of this newsgroup, the problem of equivalence up to local isometry of Riemannian manifolds (the analogous problem for Lorentzian manifolds also yields to Cartan's methods, although this gets a bit trickier due to the noncompactness of the Lorentz group, and important questions about how much work is required in the Lorentzian case appear to remain open).

By the way, the best way to understand the interaction here of ideas from the theory of homogeneous spaces (Kleinian geometries, if you like) and more general Lorentzian or Riemannian manifolds is probably the theory of Cartanian manifolds; see the recent paper by John Baez's student Derek Wise (see http://math.ucr.edu/home/baez/this.week.html and then http://www.arxiv.org/abs/gr-qc/0611154) .

Last but not least, for a superb introduction to Groebner bases in algebraic geometry (which ultimately is concerned with systems of polynomial equations, the simplest generalization of the systems of linear equations solved algorithmically by Gauss), see
David Cox, John Little, Donal O'Shea, Ideals, Varieties, and Algorithms, Springer, 1992.
This undergraduate textbook, known as IVA, is to computational algebraic geometry what MTW is to gravitation. It is a much more important, readable, and useful book than others which are unfortunately more widely read, such as popular physics books.

If what I said above didn't sufficiently whet the appetites of PF members to run out and purchase their own copy (trust me, you should own your own copy!), I might mention that in additition to applications in algebraic geometry, invariant theory, robotic motion, and automated proof verification (all discussed in the book), this stuff has important and practical applications in differential equations (as hinted above), enumerative combinatorics, game theory (want to find a Nash equalibrium?), statistics (want to analyze a graphical model of statistical independence relations, in the sense of Lauritzen?), and so on; see for example

Bern Sturmfels, Solving Systems of Polynomial Equations, AMS, 2002.

 

[Magister]

First of all, sorry for not replaying early.
Thanks a lot for your answers. Chris Hillman, you are totaly rigth, I misunderstood the solution I saw. But I still don t know how to " compute the Lie algebra of Killing vector fields in euclidean three space" by hand. I have searched in some books of GR, from Ray D'Inverno (where I have saw the solution) to the Misner, Thorne and Wheeler, and didnt found anything which could lead me to the result. I believe that I am missing some basic stuff which don't permit me to go further.
Thanks to any replies.

 

[BrunoSantos]

Can you explain better how to compute the (d_x,d_y,d_z) generators for euclidean space?
I don't understand really well how you can you do it just from the fact rot(W)=0 e div(W)=0
Regards

 

[salparadise]

I was wondering why in 3D Euclidean space, the transformation of axis inversion, which is a length preservation one, doesn't show up in the Killing vector field. Where only translations and rotations are present.

My guess is that this is because, axis inversion is a discrete symmetry, and Killing vector fields are always associated with a Lie group which is by definition a continuous one.

Could someone please explain better the link between: Continuous/discrete transformation metric preserving transformations; Killing vectors; Lie group.

Thanks