Null string evolution in black hole and cosmological spacetimes

Abstract

We discuss the problem of the motion of classical strings in some black hole and cosmological spacetimes. In particular, the null string limit (zero tension) of tensile strings is considered. We present some new exact string solutions in Reissner-Nordström black hole background as well as in the Einstein Static Universe and in the Einstein-Schwarzschild (a black hole in the Einstein Static Universe) spacetime. These solutions can give some insight into a general nature of propagation of strings (cosmic and fundamental) in curved backgrounds.

Lightlike surfaces of Darboux‐like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3‐space

“…Because v is a nonzero vector, therefore, without loss of generality, we assume v 3,4). 3,4) at s 0 , then one can show that…”

“…From the point of view of physics, Einstein's idea was that the attractive force between massive objects should be viewed as curvature of four-dimensional spacetime, thus, the theory of gravity became a geometrical theory, solutions of Einstein's equations reflect the main relation between the geometry of spacetime and matter distribution. [1][2][3][4] Especially, the lightlike surfaces in Lorentzian-Minkowski space are of importance and are of special interest in relativity theory (particularly in black hole theory) because they are models of different types of horizons. [5][6][7] In the theory of lightlike geometry, lightlike surface is a two-dimensional manifold with a degenerate metric, the normal vector bundle of the lightlike surface intersects with the tangent vector bundle of surface, which is a primary difference with nondegenerate manifolds.…”

Pseudo‐spherical Darboux images and lightcone images of principal‐directional curves of nonlightlike curves in Minkowski 3‐space

“…Lorentz-Minkowski space, as the mathematical setting of Einstein's theory of special relativity, is closely related to physics and provide the supports of theory and methodology for the study of astrophysics and cosmology by considering various of geometric invariants under Lorentzian transformations. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] It is well-known that there exist spacelike curves, timelike curves, and null curves in Lorentz-Minkowski space, which makes the study for them have more abundant contents and more diverse than curves in Euclidean space. The study of problems on curves lying in Lorentz-Minkowski space has received much attention from scholars in the fields of geometry and mathematical physics.…”

“…where k(s) is the curvature function, and (s) is the torsion function, 1 = ⟨B(s), B(s)⟩, 2 = ⟨T(s), T(s)⟩. A curve with given curvature and torsion functions is unique up to isometry of R 3 1 .…”

Lightlike Hypersurfaces and Canal Hypersurfaces of Lorentzian Surfaces

“…In particular the lightlike hypersurfaces, which can be constructed as lightlike ruled hypersurfaces over Lorentzian surfaces in anti-de Sitter space, provide good models for the study of different horizon types of black holes, such as Kerr black hole, Cauchy black hole, and Schwarzschild black hole [1][2][3][4][5][6][7][8]. Hiscock described that the horizon was constituted by lightlike hypersurfaces and lightlike wave front was lightlike hypersurface [6]; Dąbrowski et al have studied the null (lightlike) strings form the photon sphere, moving in the single spacetime of general relativity, including lightlike hypersurfaces [1,3,4]. The authors gave the null string evolution in Schwarzschild spacetime by the solutions of null string equations, which are also the null geodesic equations of general relativity appended by an additional stringy constant [3,4].…”