Brief notes on Killing fields Ivo Terek Notes on Killing fields Ivo Terek* Fix throughout this text a (connected) pseudo-Riemannian manifold (Mn, h·, ·i), and its Levi-Civita connection r. The Riemann curvature tensor of (M, h·, ·i) is the (1, 3)-tensor field defined by . R(X, Y)Z = rX rY Z − rY rX Z − r[X,Y]Z, where [X, Y] denotes the Lie Bracket of X and Y. The Ricci curvature is the (0, 2)-tensor defined by Ric(Y, Z) = trX 7! R(X, Y)Z, and the scalar curvature is the smooth function s = trh·,·i Ric . Proceeding: we can compute the Lie derivative of tensor fields with respect to vector fields. For example, for h·, ·i we have . (LX h·, ·i)(Y, Z) = LX hY, Zi − hLXY, Zi − hY, LX Zi, which can also be rewritten as (LX h·, ·i)(Y, Z) = XhY, Zi − h[X, Y], Zi − hY, [X, Z]i. In terms of the Levi-Civita connection r, we can also write (LX h·, ·i)(Y, Z) = hrY X, Zi + hY, rZXi, in view of the metric compatibility rh·, ·i = 0. In other words, the (0, 2)-tensor LX h·, ·i transformed into an endomorphism with the aid of h·, ·i itself is just rX + (rX)∗. Definition. A vector field x 2 X(M) is a Killing field if Lxh·, ·i = 0. This means that h·, ·i is preserved by the flow of x, which then consists of isome- tries, where defined. Moreover, we see that the covariant differential rx 2 End(X(M)) is skew-adjoint. To give us a first and quick way to identify Killing fields in practice, we ask this first question: when are coordinate vector fields Killing? i Proposition. Let (x ) be a local coordinate system for M. Then the coordinate vector field ¶k is a Killing field if and only if ¶kgij = 0 for all i and j. Proof: By definition, the conclusion holds if and only if for all i and j the equality 0 = ¶kgij − h[¶k, ¶i], ¶ji − h¶i, [¶k, ¶j]i holds, but [¶k, ¶i] = [¶k, ¶j] = 0. *[email protected] Page 1 Brief notes on Killing fields Ivo Terek More generally, we have the “full” coordinate versions of Killing’s equations: Proposition. Let (xi) be a local coordinate system for M. Assume that the field x is written k as x = x ¶k in the domain of the given coordinate system. The following are equivalent: (i) x is Killing; k k k (ii) x ¶kgij + (¶ix )gkj + (¶jx )gik = 0 for all i, j and k; (iii) xi;j + xj;i = 0 for all i and j. Proof: Just compute Lxh·, ·i using the expressions we have seen just now. On one hand we have (Lxh·, ·i)(¶i, ¶j) = x(gij) − h[x, ¶i], ¶ji − h¶i, [x, ¶j]i, which can be rewrit- k k k ten as (Lxh·, ·i)(¶i, ¶j) = x ¶kgij + (¶ix )gkj + (¶jx )gik by using that standard identi- [ ] = [ k ] = −( k) r = k ties x, ¶i x ¶k, ¶i ¶ix ¶k. On the other hand, if we write ¶i x x ;i¶k, then k k k k we have (Lxh·, ·i)(¶i, ¶j) = hx ;i¶k, ¶ji + h¶i, x ;k¶ki = x ;igkj + x ;jgki = xi;j + xj;i. Example. n n (1) Let A = (aij)i,j=1 2 GL(n, R) be a symmetric matrix and consider in R the i j pseudo-Riemannian metric h·, ·iA = aij dx dx . Then all the natural coordinate fields are Killing fields, as all of the aij are constants. This means that translations n are isometries for (R , h·, ·iA). Another class of fields that we can consider are lin- n ear vector fields, X 2 X(R ) for which p 7! X p is a linear map. The Levi-Civita n connection of h·, ·iA is the standard flat connection in R , so that rX is equal to the ∗ total derivative of p ! X p, i.e., X itself. But computing (rX) , in turn, requires considering the matrix A. For instance, we can do this by relating h·, ·iA with the n standard inner product h·, ·iIdn via h·, ·iA = h·, A·iIdn . Thus, given v, w 2 R , regarded as column vectors, we have that: v> AXw = (v> AXw)> = w>X> Av = w> AA−1X> Av, which says that (rX)∗ = A−1X> A. Thus X is Killing if and only if X + A−1X> A = 0, which is equivalent to AX + X> A = 0. In particular, we note that a linear field in Rn is Killing if and only if it is actually skew-symmetric. Of course, this can also be recovered by noting that n > O(R , h·, ·iA) = fT 2 GL(n, R) j T AT = Ag and thus n > so(R , h·, ·iA) = fX 2 gl(n, R) j X A + AX = 0g, by taking derivatives. Page 2 Brief notes on Killing fields Ivo Terek (2) Let R2 be equipped with the standard flat metric dx2 + dy2. Restricting it to the suitable domain R2 n f(x, 0) 2 R2 j x < 0g, we may rewrite it using polar coordi- 2 2 2 2 2 nates (r, q), and dx + dy = dr + r dq . So ¶q is a Killing field, and the isometries it generates are the rotations around the origin, given by the matrices cos q − sin q , sin q cos q as q ranges over [0, 2p[. (3) Let I ⊆ R be an interval and f : I ! R>0 be a smooth function. Any surface of revolution in R3 (generated by a rotation around the z axis, say) is modeled, with 1 2 2 2 a suitable f , by a warped product I × f S , with metric h·, ·i = ds + f (s) dq , where s is the natural coordinate in I and dq is the (non-exact) angle-form in S1. We see that the field ¶q is a Killing field. The isometries generated by ¶q are further revolutions around the original axis of rotation. The field ¶s, in turn, is not Killing. (4) Let M > 0 and h: R>0 ! R be given by h(r) = 1 − 2M/r. Consider the 2 Schwarzschild half-plane PI = f(t, r) 2 R j r > 2Mg, equipped with the Lo- 2 −1 2 2 rentzian metric −h(r) dt + h(r) dr , and let the warped product PI ×r S be the full Schwarzschild space, with metric h·, ·i = −h(r)2 dt2 + h(r)−1 dr2 + r2 dq2 + r2 sin2 q dj2, 2 where (q, j) are spherical coordinates in the S factor. Then ¶t is a Killing field, and the isometries generated by it are translations in the time coordinate. The field 2 ¶j is also Killing, and the flow generated by it consists on rotations in the S factor. The fields ¶r and ¶q are not Killing. (5) Focusing on PI only, we can make a more detailed analysis. One may compute that the Gaussian curvature is just K(t, r) = 2M/r3. Since isometries preserve K, any isometry F : PI ! PI necessarily has the form F(t, r) = ( f (t, r), r). So in particular we have that ¶ f ¶ f dF(¶ ) = ¶ and dF(¶ ) = ¶ + ¶ , t ¶t t r ¶r t r whence dF(¶t) being unit timelike forces ¶ f /¶t = ±1, and thus dF(¶t) and dF(¶r) being orthogonal gives that ¶ f /¶r = 0. This means that f (t, r) = ±t + c for some constant c 2 R, and so F(t, r) = (±t + c, r). That being understood, we know that every Killing field arises as the vector field associated to a 1-parameter family of isometries. Since our only choices are Fs(t, r) = (±t + c(s), r), for any functions c of the flow parameter s, we compute d 0 Fs(t, r) = c (0)¶t. ds s=0 0 And c (0) can be any real number. The conclusion is that all Killing fields in PI are multiples of ¶t. Page 3 Brief notes on Killing fields Ivo Terek (6) Let (G, h·, ·i) be a Lie group equipped with a left-invariant pseudo-Riemannian metric. So right-invariant vector fields are Killing fields, because their flows consist R of left translations: if X 2 X (G), then FX (t, g) = FX (t, e)g. Similarly, if the metric is right-invariant, then left-invariant vector fields are Killing fields. Proposition. Let x0 and x00 be two Killing fields. Then [x0, x00] is also a Killing field. Thus the space of Killing fields is a Lie algebra, denoted by iso(M, h·, ·i). In particular, we see that the dimension of iso(M, h·, ·i) is at most1 n(n + 1)/2. Proof: Just directly compute L[x0,x00]h·, ·i = Lx0 (Lx00 h·, ·i) − Lx00 (Lx0 h·, ·i) = Lx0 (0) − Lx00 (0) = 0. Example. (1) Consider the usual Euclidean space (R3, dx2 + dy2 + dz2). This turns out to have the maximum 3(3 + 1)/2 = 6 of linearly independent Killing fields. Let’s exhibit them all, divided in two types: • Ts(x, y, z) = (x + s, y, z) is a 1-parameter family of isometries, since we have that DTs(x, y, z) = IdR3 . Thus d d Ts(x, y, z) = (x + s, y, z) = ¶x ds s=0 ds s=0 is a Killing field (we already knew that). Similarly we recover that ¶y and ¶z are Killing fields. • We have three 1-parameter families of rotations, around each axis, and the associated Killing fields are given by: 0 1 0 1 0 1 0 1 1 0 0 x 0 0 0 x d @0 cos q − sin qA @yA = @0 0 −1A @yA = −z¶y + y¶z, dq q=0 0 sin q cos q z 0 1 0 z and similarly: 0 1 0 1 0 1 0 1 cos q 0 − sin q x 0 0 −1 x d @ 0 1 0 A @yA = @0 0 0 A @yA = −z¶x + x¶z dq q=0 sin q 0 cos q z 1 0 0 z 0 1 0 1 0 1 0 1 cos q − sin q 0 x 0 −1 0 x d @sin q cos q 0A @yA = @1 0 0A @yA = −y¶x + x¶y.