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Register a new userExtension of photon surfaces and their area: Static and stationary spacetimes
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We propose a new concept, the transversely trapping surface (TTS), as an extension of the static photon surface characterizing the strong gravity region of a static/stationary spacetime in terms of photon behavior. The TTS is defined as a static/stationary timelike surface $S$ whose spatial section is a closed two-surface, such that arbitrary photons emitted tangentially to $S$ from arbitrary points on $S$ propagate on or toward the inside of $S$. We study the properties of TTSs for static spacetimes and axisymmetric stationary spacetimes. In particular, the area $A_0$ of a TTS is proved to be bounded as $A_0le 4pi(3GM)^2$ under certain conditions, where $G$ is the Newton constant and $M$ is the total mass. The connection between the TTS and the loosely trapped surface proposed by us [arXiv:1701.00564] is also examined.
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It is shown that the free motion of massive particles moving in static spacetimes are given by the geodesics of an energy-dependent Riemannian metric on the spatial sections analogous to Jacobis metric in classical dynamics. In the massless limit Jacobis metric coincides with the energy independent Fermat or optical metric. For stationary metrics, it is known that the motion of massless particles is given by the geodesics of an energy independent Finslerian metric of Randers type. The motion of massive particles is governed by neither a Riemannian nor a Finslerian metric. The properies of the Jacobi metric for massive particles moving outside the horizon of a Schwarschild black hole are described. By constrast with the massless case, the Gaussian curvature of the equatorial sections is not always negative.
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In 1981 Wyman classified the solutions of the Einstein--Klein--Gordon equations with static spherically symmetric spacetime metric and vanishing scalar potential. For one of these classes, the scalar field linearly grows with time. We generalize this symmetry noninheriting solution, perturbatively, to a rotating one and extend the static solution exactly to arbitrary spacetime dimensions. Furthermore, we investigate the existence of nonminimally coupled, time-dependent real scalar fields on top of static black holes, and prove a no-hair theorem for stealth scalar fields on the Schwarzschild background.
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The Riemann Hypothesis states that the Riemann zeta function $zeta(z)$ admits a set of non-trivial zeros that are complex numbers supposed to have real part $1/2$. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. We analyze two approaches. In the first approach, suggested by Hilbert and Polya, one has to find a suitable Hermitian or unitary operator whose eigenvalues distribute like the zeros of $zeta(z)$. In the other approach one instead compares the distribution of the zeta zeros and the poles of the scattering matrix $S$ of a system. We apply the infinite-components Majorana equation in a Rindler spacetime to both methods and then focus on the $S$-matrix approach describing the bosonic open string for tachyonic states. In this way we can explain the still unclear point for which the poles and zeros of the $S$-matrix overlaps the zeros of $zeta(z)$ and exist always in pairs and related via complex conjugation. This occurs because of the relationship between the angular momentum and energy/mass eigenvalues of Majorana states and from the analysis of the dynamics of the poles of $S$. As shown in the literature, if this occurs, then the Riemann Hypothesis can in principle be satisfied.
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