On the global Gaussian bending measure and its applications in stationary spacetimes

TL;DR

This paper introduces a global, coordinate-free Gaussian bending measure $\mathring{\alpha}_{M}$ defined over singular spacetime regions by removing singular subregions and applying a global Gauss-Bonnet framework. The core result is a bending relation that expressively links boundary angles, topology (Euler characteristic) and total Gaussian curvature, enabling the bending of both massless and massive messengers to be probed in stationary spacetimes, including Kerr black holes. The authors derive explicit forms and provide Kerr-specific expressions, showing that inner singular boundaries do not contribute to the absolute bending measure, while the outer boundary determines the observable bending, especially for geodesic polygonal patches. They further discuss experimental prospects, contrasting with traditional lensing measures, and demonstrate strong-field effects via numerical Kerr analyses, illustrating how this global theory could help distinguish gravity theories by enabling local strong-field tests and measurements of black-hole parameters through bending of light and other messengers.

Abstract

Modified gravity theories have been suggested to address the limitations of general relativity, each exhibiting differences, particularly in their strong-field limits. Nonetheless, there lacks effective means to distinguish or test these theories through local strong-field measurements. In this work, we define a global Gaussian bending measure over singular spacetime regions, establish a corresponding global theory, and demonstrate its applications in a general stationary spacetime. The global theory is based on differential geometry, rather than on specific gravity theories, allowing it to depict various physics within general relativity and beyond. For example, it can be applied to describe the gravitational bending of massless or massive messengers, such as photons, neutrinos, cosmic rays, and possibly massive gravitational waves predicted in certain theories of gravity. Besides, the global theory is applicable to any stationary spacetime regions outside a rotating black hole. As an instance of its direct applications, we investigate the highly-curved spacetime effects of the black hole in its immediate surrounding regions and design local strong-field experiments involving different shapes of singular lensing patches. New means can be therefore anticipated to be developed according to the global theory to differentiate between different gravity theories and test them in their strong-field regions.

Figures (4)

• Figure 1: Illustration of global Gaussian bending over the lensing patch $D\subset\Sigma$, featuring a singularity located at the origin ${\rm O}$ on the physical surface $\Sigma$. The sub-region ${ D_{\rm c}}\subset{D}$ containing the singularity can be removed along a closed curve, denoted as $\partial{D}_{\rm c}$. The region that remains, $\mathring{D}$, is free of singularities, but it has a geometric "hole", with its boundaries $\partial{\mathring D}$ made up of geodesic line segments. As illustrated in the figure, four vertices (or points), $\gamma_{i}, \gamma_{j}, \gamma_{l},~{\rm and}~ \gamma_{m}$, can be chosen in such a way that the region $\mathring{D}$ can be cut into two parts, namely ${ D_{\rm u}}$ and ${ D_{\rm d}}$, along the line segments $C_{il}=\overline{\gamma_{i}\gamma_{l}}$ and $C_{jm}=\overline{\gamma_{j}\gamma_{m}}$. Here, $D=D_{\rm u}\bigcup D_{\rm d}\bigcup D_{\rm c}$. Note that the boundaries, ${\partial D_{\rm u}}$ and ${\partial D_{\rm d}}$, of the two parts can be jointly described by a parametrisation $\gamma:\,[0,l]\to{\partial D_{\rm u}}+{\partial D_{\rm d}}~(\supsetneq\partial{\mathring D})$, forming a closed curve, where $l$ is the total arc length of the closed curve $\gamma$. Assume that $\gamma$ is parametrised by arc length $\lambda$ in the right-handed direction, as marked by the brown arrows, whereas it is parametrised by arc length $\bar{\lambda}=l-\lambda$ in the left-handed direction, as indicated by the cyan arrows. Let $\gamma\left(\lambda_{k}\right)=\gamma_{k}, {\rm for}~k=0, ... , \aleph$, be the vertices of $\gamma$, with $\lambda_{k}$ denoting the value of $\lambda$ at the $k$-th vertex. Suppose that the source and observer are located at points ${\cal S}=\gamma(\lambda=0)=\gamma(\lambda=l)$ and ${\cal O}=\gamma\left(\lambda=\lambda_{0}\right)$, respectively. All the symbols are detailed in Table \ref{['tab:symbol']}. For a black hole, the singularity is hidden behind the event horizon located at a radius of $r=r_{\rm H}$, and thus, $\partial{D}_{\rm c}$ can be chosen as the intrinsic boundary at the event horizon. If $D$ is singularity-free, $D_{\rm c}$ is left empty, and $\partial{\mathring D}=\partial{D}$ is a simple closed curve.
• Figure 2: Geodesic digon and monogon: the geometry of global Gaussian bending from a side-on view. The left panel shows that the light rays emitted from the source at the point $\mathcal{S}$ propagate along the geodesic curves $L$ and $\bar{L}$, respectively, pass by a black hole, and ultimately reach the observer at the point $\mathcal{O}$, forming a geodesic digon. The right panel illustrates a geodesic monogon, along the outer boundary of which the light rays originating from the point $\mathcal{O}$ propagate, are then bent by a black hole, finally arrive back at the point $\mathcal{O}$, i.e., $\mathcal{S}=\mathcal{O}$. As marked by the red circle, there exists a self-intersection at the vertex of the geodesic monogon.
• Figure 3: Geodesic triangle. At the three vertices $\gamma_{k},~k=1,2,3$, there are three devices (or mirrors) that function as both detectors and emitters of photons. Here, $d$ denotes the distance of each vertex to the center of mass of the black hole. The beamed light rays from these devices may travel along the outer boundary of the geodesic triangle, establishing a physically simple, closed loop of photons.
• Figure 4: The excess of $2\pi$ over the Gaussian deflection angle, i.e., $2\pi\!-\!\mathring{\alpha}_{M}$, is shown as functions of distance $d$ and spin $a$ in panels (a) and (b), respectively. Here, $d$ represents the distance of each vertex of a regular geodesic polygon to its center, which is exactly the same as Figure \ref{['fig:triangle']}. Results from the lensing patches in the shapes of a regular digon and triangle are colored red and black, respectively.