Null Manifolds Research Articles

On Isometric Immersions of Null Manifolds into Semi-Riemannian Space Forms of Arbitrary Index

A null manifold is a differentiable manifold M endowed with a degenerate metric tensor g. In this work we provide sufficient conditions for a null manifold to be isometrically immersed as a hypersurface into a simple connected semi-Riemannian manifold of constant sectional curvature c and index q

Carroll covariant scalar fields in two dimensions

Conformal Carroll symmetry generically arises on null manifolds and is important for holography of asymptotically flat spacetimes, generic black hole horizons and tensionless strings. In this paper, we focus on two dimensional (2d) null manifolds and hence on the 2d Conformal Carroll or equivalently the 3d Bondi-Metzner-Sachs (BMS) algebra. Using Carroll covariance, we write the most general free massless Carroll scalar field theory and discover three inequivalent actions. Of these, two viz. the time-like and space-like actions, have made their appearance in literature before. We uncover a third that we call the mixed-derivative theory. As expected, all three theories enjoy off-shell BMS invariance. Interestingly, we find that the on-shell symmetry of mixed derivative theory is a single Virasoro algebra instead of the full BMS. We discuss potential applications to tensionless strings and flat holography.

Boosting to BMS

Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, are intrinsically associated to null manifolds and in two dimensions can be obtained as an Inönü-Wigner contraction of the two-dimensional (2d) relativistic conformal algebra. Instead of performing contractions, we demonstrate in this paper how this transmutation of symmetries can be achieved by infinite boosts or degenerate linear transformations on coordinates. Taking explicit cues from the worldsheet theory of null strings, we show boosting the system is equivalent to adding a current-current deformation term to the Hamiltonian. As the strength of this deformation term reaches a critical value, the classical symmetry algebra “flows” from two copies of Virasoro to the BMS algebra. We further explore the situation where the CFT coordinates are asymmetrically transformed, and degenerate limits lead to chiral theories.

A Ricci-type flow on globally null manifolds and its gradient estimates

Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expanding horizons in black hole theory. In the present paper, we introduce a degenerate Ricci-type flow in $M'$ via the intrinsic Ricci tensor of $M$. Several new gradient estimates regarding the flow are proved.

Field theories on null manifolds

We argue that generic field theories defined on null manifolds should have an emergent BMS or conformal Carrollian structure. We then focus on a simple interacting conformal Carrollian theory, viz. Carrollian scalar electrodynamics. We look at weak (on-shell) and strong invariance (off-shell) of its equations of motion under conformal Carrollian symmetries. Helmholtz conditions are necessary and sufficient conditions for a set of equations to arise from a Lagrangian. We investigate whether the equations of motion of Carrollian scalar electrodynamics satisfy these conditions. Then we proposed an action for the electric sector of the theory. This action is the first example for an interacting conformal Carrollian Field Theory. The proposed action respects the finite and infinite conformal Carrollian symmetries in d = 4. We calculate conserved charges corresponding to these finite and infinite symmetries and then rewrite the conserved charges in terms of the canonical variables. We finally compute the Poisson brackets for these charges and confirm that infinite Carrollian conformal algebra is satisfied at the level of charges.

Constant scalar curvature and warped product globally null manifolds

This paper deals with the curvature properties of a class of globally null manifolds ( M, g) which admit a global null vector field and a complete Riemannian hypersurface. Using the warped product technique we study the fundamental problem of finding a warped function such that the degenerate metric g admits a constant scalar curvature on M. Our work has an interplay with the static vacuum solutions of the Einstein equations of general relativity.

On total null manifolds included in a real pseudo-euclidean space of signature (n, n+1)

On total null manifolds included in a real pseudo-euclidean space of signature (n, n+1)

Classical pure states: Information and symmetry in statistical mechanics

The aim of this paper is to find classical counterparts of pure quantum states. It is shown that these are singular probability distributions concentrated on the so-called maximal null manifolds in a phase space. They are equivalent to densities studied by Van Vleck and Schiller and to WKB solutions (cf. Van Vleck, 1928; Schiller, 1962). Properties of such distributions and their relativistic generalisations have been studied in previous papers (Slawianowski, 1971; Slawianowski, 1972). However, it has not been shown there that such distributions arise actually in the limith→0. When working with the standard apparatus of differential geometry we mostly use the language of Kobayashi & Nomizu (1963).

Spectral analysis using ascent, descent, nullity and defect

In this paper the fine structure of the spectrum of a closed linear operator T is studied in terms of the ascent, descent, nullity and defect of the operators )~- T. Several characterizations of poles of the resolvent operator Ra(T) are obtained:and these are used to characterize certain classes of operators, e.g., the class of meromorphic operators. Much of the underlying algebraic theory was developed by A. E. Taylor [17] and M. A. Kaashoek [8]. Their notation will be used throughout this paper. 1. Algebraic Properties of Ascent, Descent, Nullity and Defect Let X be a linear space over the field of complex numbers and let T be a linear operator with domain ~(T) and range ~(T) in X. The ascent of T, ~(T), is the smallest nonnegative integer p such that the null manifolds Jf(T p) and A/,(Tp+ i) are equal. If no such p exists, set ~(T) = ~. The descent of T, 6(T), is the smallest nonnegative integer q such that the range spaces ~(T q) and ~(Tq+ 1) are equal. If no such q exists, set 6(T)= ~. The nullity of T, n(T), is the dimension of A/'(T). The defect of T, d(T), is the dimension of the quotient space X/~t(T). Let r(T)= {2~CI ~(2- T )=6(2- T )=0}. When X is a Banach space and T is a closed linear operator, r(T)= 0(T), the resolvent set of T.