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Register a new userSome globally conserved currents from generalized Killing vectors and scalar test fields
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Justin C. Feng
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In this article, I discuss the construction of some globally conserved currents that one can construct in the absence of a Killing vector. One is based on the Komar current, which is constructed from an arbitrary vector field and has an identically vanishing divergence. I obtain some expressions for Komar currents constructed from some generalizations of Killing vectors which may in principle be constructed in a generic spacetime. I then present an explicit example for an outgoing Vaidya spacetime which demonstrates that the resulting Komar currents can yield conserved quantities that behave in a manner expected for the energy contained in the outgoing radiation. Finally, I describe a method for constructing another class of (non-Komar) globally conserved currents using a scalar test field that satisfies an inhomogeneous wave equation, and discuss two examples; the first example may provide a useful framework for examining the arrow of time and its relationship to energy conditions, and the second yields (with appropriate initial conditions) a globally conserved energy- and momentumlike quantity that measures the degree to which a given spacetime deviates from symmetry.
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Based on an examination of the solutions to the Killing Vector equations for the FLRW-metric in co moving coordinates , it is conjectured and proved that the components(in these coordinates) of Killing Vectors, when suitably scaled by functions, are emph{zero modes} of the corresponding emph{scalar} Laplacian. The complete such set of zero modes(infinitely many) are explicitly constructed for the two-sphere. They are parametrised by an integer n. For $n,ge,2$, all the solutions are emph{irregular} (in the sense that they are neither well defined everywhere nor are emph{square-integrable}). The associated 2-d vectors are also emph{not normalisable}. The $n=0$ solutions being constants (these correspond to the zero angular momentum solutions) are regular and normalizable. Not all of the $n=1$ solutions are regular but the associated vectors are normalizable. Of course, the action of scalar Laplacian coordinate independent significance only when acting on scalars. However, our conclusions have an unambiguous meaning as long as one works in this coordinate system. As an intermediate step, the covariant Laplacians(vector Laplacians) of Killing vectors are worked out for four-manifolds in two different ways, both of which have the novelty of not explicitly needing the connections. It is further shown that for certain maximally symmetric sub-manifolds(hypersurfaces of one or more constant comoving coordinates) of the FLRW-spaces also, the scaled Killing vector components are zero modes of their corresponding scalar Laplacians. The Killing vectors for the maximally symmetric four-manifolds are worked out using the elegant embedding formalism originally due to Schrodinger . Some consequences of our results are worked out. Relevance to some very recent works on zero modes in AdS/CFT correspondences , as well as on braneworld scenarios is briefly commented upon.
Numerical simulations are performed of a test scalar field in a spacetime undergoing gravitational collapse. The behavior of the scalar field near the singularity is examined and implications for generic singularities are discussed. In particular, our example is the first confirmation of the BKL conjecture for an asymptotically flat spacetime.
In this paper we consider homothetic Killing vectors in the class of stationary axisymmetric vacuum (SAV) spacetimes, where the components of the vectors are functions of the time and radial coordinates. In this case the component of any homothetic Killing vector along the $z$ direction must be constant. Firstly, it is shown that either the component along the radial direction is constant or we have the proportionality $g_{phiphi}propto g_{rhorho}$, where $g_{phiphi}>0$. In both cases, complete analyses are carried out and the general forms of the homothetic Killing vectors are determined. The associated conformal factors are also obtained. The case of vanishing twist in the metric, i.e., $omega= 0$ is considered and the complete forms of the homothetic Killing vectors are determined, as well as the associated conformal factors.
Some recent results obtained by the author and collaborators about QFT in asymptotically flat spacetimes at null infinity are summarized and reviewed. In particular it is focused on the physical properties of ground states in the bulk induced by the BMS-invariant state defined at null infinity.
In this paper, we investigate conformal Killings vectors (CKVs) admitted by some plane symmetric spacetimes. Ten conformal Killings equations and their general forms of CKVs are derived along with their conformal factor. The existence of conformal Killings symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. Considering the cases of time-like and inheriting CKVs, we obtain spacetimes admitting plane conformal symmetry. Integrability conditions are solved completely for some known non-conformally flat and conformally flat classes of plane symmetric spacetimes. A special vacuum plane symmetric spacetime is obtained, and it is shown that for such a metric CKVs are just the homothetic vectors (HVs). Among all the examples considered, there exists only one case with a six dimensional algebra of special CKVs admitting one proper CKV. In all other examples of non-conformally flat metrics, no proper CKV is found and CKVs are either HVs or Killings vectors (KVs). In each of the three cases of conformally flat metrics, a fifteen dimensional algebra of CKVs is obtained of which eight are proper CKVs.
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