Spacelike Killing Vector Research Articles

Twistfree and Papapetrou vacuum spacetimes with a spacelike killing vector

We show that, for the case of vacuum solutions of the Einstein equations with a spacelike hypersurface orthogonal Killing vector ∂/∂ϰ3 and associated metricds2 =e2U (dx3)2 +e−2U γabdxadxb whereU is not a constant, there exists at every point of the quotient 3-space a plane of vectorsKa such that £KRab=0 andKa Rab=0 whereR{inab} is the Ricci tensor formed fromγab. Then in the case whereU{in,a} is a timelike or spacelike vector in the quotient 3-space, Petrov type I solutions of the vacuum field equations are obtained. In the simpler case whereU{in,a} is a null vector in the quotient 3-space, the complete solution of the vacuum field equations is obtained. It is shown that this solution is Petrov type III of Kundt's class. For the case of Papapetrou solutions where there is a twist potentialψ which is a function ofU, solutions corresponding to the twistfree solutions are given.

Killing vectors in conformally flat perfect fluid spacetimes

The existence of Killing vectors in conformally flat perfect fluid spacetimes in general relativity is considered. In particular Killing vectors which are neither orthogonal nor parallel to the fluid velocity vector are considered and stationary fields in which the fluid velocity vector is not parallel to the timelike Killing vector field are shown to exist. This class of solutions is shown to include several stationary (but non-static) axisymmetric fields, thus providing counter-examples to a theorem of Collinson (1976). In the case when the fluid is non-expanding, the number of spacelike Killing vectors is shown to depend on the rank of four functions of time which appear in the metric. Some examples of stationary but non-static fields are presented in closed form.

An asymptotically flat vacuum solution

An asympototically flat algebraically general vacuum metric is obtained. The solution is characterized by two commuting spacelike Killing vectors with flat integral surfaces and depends on one arbitrary function.

Generalised Kerr-Schild transformation from vacuum backgrounds to Einstein-Maxwell spacetimes

The generalised Kerr-Schild transformation on a vacuum background is used to generate exact solutions of Einstein-Maxwell field equations. Two families of algebraically general solutions are obtained. For generic values of a certain parameter a, they consist of the homogeneous non-static metrics given by Datta (1965). For a=1+or-2, there are solutions with only two commuting spacelike Killing vectors.

Studies in space-times admitting two spacelike Killing vectors

Some properties of the space-times admitting two spacelike Killing vectors are studied. In particular, using harmonic maps the degree of freedom on the M′ manifold is exploritd to add scalar and electromagnetic fields to Bonnor’s nonsingular solution. It is also shown that for vacuum space-times the noncommutativity of two spacelike Killing vectors is incompatible with the self-similarity requirement and such a self-similar vacuum space-time has no Taub–NUT equivalent extension.

5D space-time-mass gravity theory with off-diagonal components in the metric tensor

The origin of matter and the nature of diagonalization are studied in the framework of 5D STM gravity theory with off-diagonal components in the metric tensor. The time of change of mass is presented with a simple cosmological solution and the 5D geodesic equations. In the special case, our solution can be compared to Chodos and Detweiler's (1982) procedure assuming that the metric possesses a spacelike Killing vector.

Spacelike conformal Killing vectors and spacelike congruences

Necessary and sufficient conditions are derived for space-time to admit a spacelike conformal motion with symmetry vector parallel to a unit spacelike vector field na. These conditions are expressed in terms of the shear and expansion of the spacelike congruence generated by na and in terms of the four-velocity of the observer employed at any given point of the congruence. It is shown that either the expansion or the rotation of this spacelike congruence must vanish if Dna/dp =0, where p denotes arc length measured along the integral curves of na, and also that there exist no proper spacelike homothetic motions with constant expansion. Propagation equations for the projection tensor and the rotation tensor are derived and it is proved that every isometric spacelike congruence is rigid. Fluid space-times are studied in detail. A relation is established between spacelike conformal motions and material curves in the fluid: if a fluid space-time admits a spacelike conformal Killing vector parallel to na and naua =0, where ua is the fluid four-velocity, then the integral curves of na are material curves in an irrotational fluid, while if the fluid vorticity is nonzero, then the integral curves of na are material curves if and only if they are vortex lines. An alternative derivation, based on the theory of spacelike congruences, of some of the results of Collins [J. Math. Phys. 25, 995 (1984)] on conformal Killing vectors parallel to the local vorticity vector in shear-free perfect fluids with zero magnetic Weyl tensor is given. The necessary and sufficient condition for vortex lines to be material lines is derived and the restriction this places on the flow of a thermodynamical perfect fluid is determined. As an application, a pure magnetic field in a rotational fluid is considered and results similar in nature to Ferraro’s law of isorotation are obtained. Throughout, corresponding results for a timelike conformal motion and for Newtonian theory are given for comparison.

Physical approach to cosmological homogeneity.

This paper shows that in general relativity space-time will admit three linearly independent spacelike Killing vectors if the cosmic matter is a perfect fluid with a functional relationship between pressure and density and the spacelike eigenvectors of the shear tensor are parallel transported along the world lines of the fluid. In the Newtonian case it is proved that for an irrotational motion of a fluid with pressure and density functionally related spatial uniformity of the matter distribution leads to the homogeneity of the velocity field.

Positivity of energy in five-dimensional classical unified field theories

Five-dimensional classical unified field theories as well as in Yang-Mills theory with gauge group U(1), are described in terms of a Lorentzian five-dimensional space V5 with metric tensor y;; which admits a space-like Killing vector ξα. It is assumed that: (1) V5 has the topology of V4×S1, S1 is a circle and V4 is a four-dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γαβ of V5 satisfies \(\Gamma _{\alpha \beta } u^\alpha v^\beta \leqslant 0\), where uα and vβ are future oriented time-like vectors with \(\gamma _{\alpha \beta } \upsilon ^\alpha \xi ^\beta = 0\). The spinor approach of Witten, Nester, and Moreschi and Sparling is used to show that the conserved five-dimensional energy momentum vector P; is nonspace-like. If P;=Γαβ=0 then V5 must admit a time-like Killing vector. Lichnerowicz's results then imply that V5 must be flat. A lower bound for P4 (the mass) similar to that found by Gibbons and Hull is obtained.

Positivity of energy in five dimensional classical unified field theories, II

Five dimensional classical unified field theories as well as Yang-Mills theory with gauge group U (1), are described in terms of a Lorentzian five dimensional space V 5 with metric tensor γαβ which admits a space-like Killing vector ζ α. It is assumed that: (1) V 5 has the topology of V 4 x S 1, S 1 is a circle and V 4 is a four dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γ αβ of V 5 satisfies Γ αβ U α υ β ⩽ 0 where U α and υ α are future oriented time-like vectors with γ αβ υ α ζ β = 0. The spinor approach of Witten [4], Nester [3] and Moreschi and Sparling [5] is used to show that the conserved five dimensional energymomentum vector P α = ifΓ αβ = 0 then V 5 must admit a time-like Killing vector. Lichnerowicz's results [1] then imply that V 5 must be flat. A lower bound for P 4 (the mass) similar to that found by Gibbons and Hull [6] is obtained.

On the Nutku—Halil solution for colliding impulsive gravitational waves

The equations appropriate for space-times with two space-like Killing-vectors are set up, ab initio , and explicit expressions for the components of the Riemann, the Ricci, and the Einstein tensors in a suitable tetrad-frame are written. The equations for the vacuum are reduced to a single equation of the Ernst type. It is then shown that the simplest linear solution of the Ernst equation leads directly to the Nutku-Halil solution for two colliding impulsive gravitational waves with uncorrelated polarizations. Thus, in some sense, the Nutku-Halil solution occupies the same place in space-times with two space-like Killing-vectors as the Kerr solution does in space—times with one time-like and one space-like Killing-vector. The Nutku-Halil solution is further described in a Newman-Penrose formalism; and the expressions for the Weyl scalars, in particular, make the development of curvature singularities manifest. Finally, a theorem analogous to Robinson’s theorem (but much less strong) is proved for space-times with two space-like Killing-vectors.

Rotating charged dust as a source for cylindrically symmetric electrovacs

Exact solutions for stationary rotating charged dust with cylindrical symmetry, for which one of the spacelike Killing vectors is hypersurface orthogonal, are considered. The general solution for the case of rigid rotation and constant charge to mass ratio is given. It is proved that all rigidly rotating solutions can be joined smoothly to the exterior electrovac solutions involving third Painlevé transcendents. Also the solutions for differentially rotating charged dust, which have a force-free electromagnetic field can be matched to a particular class of the electrovac metrics.

Static conformally invariant scalar field

For a massless conformally invariant scalar field, a class of solutions is obtained to the Einstein equations for which the geometry of the space-time admits one timelike and two spacelike Killing vectors. The class of solutions admit plane-symmetric and conformally flat solutions as special cases. The metrics can be interpreted as static cylindrically symmetric solutions if one of the spacelike Killing vectors is associated with the rotational motion.

Static perfect fluid in Brans-Dicke theory

The static perfect fluid in Brans-Dicke theory with spherical symmetry and conformal flatness leads to a differential equation in terms of the scalar field only. We obtain a unique exact solution for the casep=ɛρ, but density and pressure are singular at the center. We further consider the metric corresponding to a static nonrotating space-time with two mutually orthogonal spacelike Killing vectors in Brans-Dicke theory. We obtain a differential equation involving only the scalar field for the equation of statep=ɛρ The general solution is found as a transcendental function. Finally, we generalize a theorem given by Bronnikov and Kovalchuk (1979) for perfect fluid in Einstein's theory.

Stiff magnetic field lines. I. A geometrical foundation

It is shown that the foliation of a space-time manifold of codimension 2 provides a basis for the study of the deformation of magnetic field lines. It is found that the fluid flow vector and the curvature vector of a nongeodesic “stiff” magnetic field line are always orthogonal. Further, it is shown that the metric tensor of the 2-space orthogonal to the “Maxwellian string” is Lie-transported along the magnetic field lines when the magnetic field lines are “stiff.” If there exists a spacelike Killing vector field parallel to the magnetic field, then the magnetic field lines must be “stiff.”

An extension of the Bonnor transformation

The Bonnor transformation, which maps stationary, axially symmetric vacuum fields into static, axially symmetric Einstein–Maxwell fields is extended to spacetimes possessing only one spacelike Killing vector. The transformation generates two distinct Einstein–Maxwell solutions from any vacuum solution.

Spacelike motions admitted by the magnetofluid space-times

In this paper we have obtained a set of necessary and sufficient conditions for the existence of a spacelike Killing vector collinear to the magnetic-field vector. Using these conditions, we have shown that the rotation of the congruence of magnetic-field lines is Lie invariant (or «frozen-in») for the rotating «stiff» magnetic-field lines. Furthermore, we have investigated a set of necessary and sufficient conditions for the spacelike homothetic Killing vector collinear to the magnetic-field vector. It is observed that the spacelike proper homothetic motion along the congruence of magnetic-field lines is not possible in a space of constant curvature filled with a self-gravitating magnetofluid.

Analogs in general relativity of Ferraro's law of isorotation

Two relativistic analogs of Ferraro's law of isorotation for a fluid with infinite electrical conductivity are given. In the first case it is assumed that the flow is shear-free withHa=0 and it is shown that £(H) ωa=0. In the second case it is assumed that there exists a space-like Killing vector field parallel toHa, and it is proved thatω,aHa=0.

Unified geometry of internal space with space-time

The unified theory of gravitation with gauge fields proposed earlier is shown to be a ($4+n$)-dimensional Einstein theory with $n$ linearly independent complete spacelike Killing vector fields which span an $n$-dimensional metric submanifold and form a closed Lie algebra. In this picture the internal space is combined with space-time and the internal symmetry becomes a symmetry of the metric of the ($4+n$)-dimensional combined space. The 4-dimensional space-time appears as the quotient space of the ($4+n$)-dimensional enlarged space by the equivalence relation of the group of motions of the $n$ Killing vector fields.

Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions

Spacetimes with closed spacelike hypersurfaces and spacelike two-parameter isometry groups are investigated to determine their possible global structures. It is shown that the two spacelike Killing vectors always commute with each other. Connected group-invariant spacelike hypersurfaces must be homeomorphic to S 1 ⊗ S 1 ⊗ S 1 (three-torus), S 1 ⊗ S 2 (three-handle), S 3 (three-sphere), or to a manifold which is covered by one of these. The spacetime metric and Einstein equations are simplified in the absence of nongravitational sources and used to establish the impossibility of topology change as well as other limitations on global structure. Regularity conditions for spacetimes of this type are derived and shown to be compatible with Einstein's equations.