Killing vector field

A killing vector field (named after the German mathematician Wilhelm Killing ) is a vector field on a Riemannian manifold that receives the metric . Killing vector fields are the infinitesimal generators of isometries (see also Lie group ).

The same applies to pseudo-Riemannian manifolds , for example in general relativity .

Definition and characteristics

A vector field is a killing vector field if the Lie derivative of the metric with respect to : ${\displaystyle X}$ ${\displaystyle g}$ ${\displaystyle X}$

${\displaystyle {\mathcal {L}}_{X}g=0.\,}$

With the help of the Levi-Civita context , this means point by point

${\displaystyle g(\nabla _{Y}X,Z)+g(Y,\nabla _{Z}X)=0}$

for all vectors and , respectively, that there is an endomorphism that is skew-symmetrical with respect to the tangent space . ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle \nabla _{\bullet }X}$ ${\displaystyle g}$

In local coordinates, this leads to the so-called Killing equation

${\displaystyle \nabla _{i}X_{j}+\nabla _{j}X_{i}=0.}$

A killing field is uniquely determined on the whole manifold by a vector at a point and the covariant derivatives of the vector at that point.

The lie bracket of two killing fields is again a killing field. The Killing Fields of diversity thus form a Lie algebra on . This is the Lie algebra of the isometric group of the manifold (if is complete). ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

A vector field is a killing vector field if and only if it is a Jacobi vector field along every geodetic .

Conservation quantities

Since killing vector fields generate isometrics, there is a conservation quantity of the corresponding spacetime in physics for every killing vector field . In general relativity theory , Killing vector fields are therefore of great importance when characterizing solutions to Einstein's field equations . The conservation quantity for a killing vector field is calculated as ${\displaystyle Q_{X}}$ ${\displaystyle X}$

${\displaystyle \mathrm {Q} _{X}=\int \mathrm {d} ^{3}x{\sqrt {|g|}}\,T_{0\mu }X^{\mu }}$,

where is the energy-momentum tensor and the magnitude of the 4x4 determinant of the metric tensor. Einstein's sums convention was used in the formula . ${\displaystyle T}$ ${\displaystyle |g|}$

Space-time itself is a four-dimensional pseudo-Riemannian manifold with one time coordinate ("upper indices") and three space coordinates , and , with a mixed signature , for example according to the scheme (-, +, +, +). The killing vector field also has four components; the g-matrix (“4x4”), for example, has one negative and three positive eigenvalues. The Lorentz transformations in the flat pseudo-Riemannian Minkowski space can be understood as pseudo-rotations and have the value one as a determinant. However, the results also apply in non-flat rooms. ${\displaystyle x^{0}}$ ${\displaystyle x^{1}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}}$

Integration areas and causality

The question of the range of integration in formulas of the above kind is u. a. therefore difficult - it is no coincidence that precise information is missing above - because i. A. take into account the limitation of the spatial areas in question (see cause and effect or causal structure ) as well as the lead time ("retardation", from lat. Retardare " to delay") of the causes and for all sizes i. A. Must explicitly state the respective arguments and the summation ranges. That is also deliberately not the case above.

In fact, with the above formula, the integration area of ​​the spatial coordinates is the full one, provided that cause and effect are infinitely far apart in time. Instead, you can choose any three-dimensional hypersurface that is causally similarly structured. This also means that the formula does not apply to black holes . ${\displaystyle \mathbb {R} ^{3},}$ ${\displaystyle \mathbb {R} ^{3}}$

Examples

Exactly when the coefficients of the metric in the base are independent of a local coordinate , is a killing vector field. In these same local coordinates it is then , where is the Kronecker delta . ${\displaystyle g_{\mu \nu }}$ ${\displaystyle g}$ ${\displaystyle dx^{\mu }\otimes dx^{\nu }}$ ${\displaystyle x^{k}}$ ${\displaystyle X={\tfrac {\partial }{\partial x^{k}}}}$ ${\displaystyle X_{\mu }=\delta _{\mu }^{k}}$ ${\displaystyle \delta _{\mu }^{k}}$

A set of independent killing vector fields of the unit sphere with the induced metric in spherical coordinates are: ${\displaystyle S^{2}}$ ${\displaystyle \mathrm {d} s^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}(\theta )\mathrm {d} \phi ^{2}}$

${\displaystyle K^{(x)}=-z\partial _{y}+y\partial _{z}=-\sin(\phi )\partial _{\theta }-\cot(\theta )\cos(\phi )\partial _{\phi }}$

${\displaystyle K^{(y)}=-x\partial _{z}+z\partial _{x}=\cos(\phi )\partial _{\theta }-\cot(\theta )\sin(\phi )\partial _{\phi }}$

${\displaystyle K^{(z)}=-y\partial _{x}+x\partial _{y}=\partial _{\phi }}$

This corresponds to the rotations around the - or - or - axis and in quantum mechanics, apart from one factor , the components of the angular momentum operators . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle \hbar /i}$

All linear combinations of these vector fields again represent Killing vector fields. The induced isometries are exactly the elements of the rotating group ${\displaystyle \mathrm {SO} (3)}$ . The corresponding conservation law is the angular momentum law .

literature

• Steven Weinberg: Gravitation and Cosmology . John Wiley & sons, New York 1972, ISBN 0-471-92567-5 .
• Jürgen Jost: Riemannian Geometry and Geometric Analysis . Springer Verlag, Berlin 2002, ISBN 3-540-42627-2 .
• Adler, Ronald; Bazin, Maurice & Schiffer, Menahem: Introduction to General Relativity . 2nd Edition. McGraw-Hill, New York 1975, ISBN 0-07-000423-4 (see Chapters 2 and 9).

Individual evidence

1. ↑ Misner, Thorne, Wheeler: Gravitation . WH Freeman and Company, 1973, ISBN 0-7167-0344-0 .