# A Shadow By R.K. Narayan Pdf

This is symbolic of the story itself, where truth and lies, like those lights and shadows, are conflated and confused. It is significant that it is when the stranger (later identified as Guru Nayak) lights his cheroot pipe that the astrologer recognises him as the old associate from his past: the light here illuminates his old adversary but Nayak himself remains in the dark concerning the true identity of his interlocutor.

## A Shadow By R.K. Narayan Pdf

We study the influence of parameter \(\alpha \) on the optical features of Schwarzschild-MOG black holes with different thin accretions in scalar-tensor-vector gravity. As \(\alpha \) increases from 0, the radii of the event horizon, photon sphere, and observed shadow increase in comparison with the Schwarzschild black hole. We constrain the parameter \(\alpha \) with the experimental data reported by the Event Horizon Telescope Collaboration for M87\(^*\) and Sagittarius A\(^*\). In the situation of spherical accretions, we unveil that the parameter \(\alpha \) has a positive effect on the shadow size but a negative effect on the observed specific intensities. Considering that the Schwarzschild-MOG black hole is surrounded by an optical and geometrically thin accretion disk, we find that the total observed specific intensities are mainly contributed by the direct emissions, while the photon rings and lensed rings provide small contributions. It is also found that with the increase of \(\alpha \), the black hole shadow expands, the photon rings and lensed rings become larger and thicker. Besides, we emphasize that the boundary of the observed shadow cast by the aim black hole in the disk accretion scenario is determined by the direct emissions rather than the photon ring emissions. Consequently, we unveil that there is a linear relationship involving the critical impact parameter and the starting point of the direct emissions. This finding helps to use the experimental results of the Event Horizon Telescope to infer the critical impact parameter and to test General Relativity.

Black holes are extremely compact celestial bodies predicted by General Relativity, and they have attracted much attention since the first black hole solution was obtained by Karl \(\cdot \) Schwarzschild in 1916 [1]. In recent years, the successful detection of the gravitational-wave signals radiated from binary black hole mergers with Laser-Interferometer Gravitational Wave-Observatory (LIGO) experiments and a wide star-black-hole binary system discovered by radial-velocity measurements strongly confirm the existence of black holes in our universe [2,3,4,5]. While, more intuitive evidences of the existence of black holes are the ultra-high angular resolution images of M87\(^*\) and Sagittarius A\(^*\), which were announced by the Event Horizon Telescope Collaboration (EHT) [6,7,8,9,10,11,12,13,14,15,16,17]. There is a faint region in the center of these images, called the black hole shadow. It is well known that the light ray will deflect in the vicinity of a black hole due to the lensing effect [18,19,20,21], resulting in a deficit of the observed specific intensity inside the sharp-edged boundary, thus a region of brightness depression can be observed on the distant image plane. The black hole shadow is closely related to spacetime geometry. Consequently, it is a robust tool for estimating black hole parameters [22,23,24,25] and testing General Relativity or its alternatives [26,27,28,29,30,31,32].

Spherical accretion is a quietly ideal description of the accretion mechanism in the vicinity of the astrophysical black holes. In fact, the materials in the universe, such as plasma, gas, and dust, will be trapped by a black hole due to the strong gravity and formed a giant disk-shaped accretion flow surrounding the black hole. For a remote observer, the lights emitting from this disk are the main source that illuminates the black hole. Gralla et al. first investigated the shadow cast by a Schwarzschild black hole with an optically and geometrically thin accretion disk, claiming that the features of the black hole shadow are not only photon rings but also lensed rings [62]. However, these two rings are barely visible in the condition of the particular emission profile of the accretion disk because of the deficiency of the angular resolution of the EHT. This elaborate investigation triggers a new era of studying the influence of the thin accretion disk on the observed black hole shadow. By considering a non-rotating black hole in the background of the quintessence dark energy, the authors of [63] studied the effect of the quintessence on the shadow of the black hole with a thin accretion disk and pointed out that the cosmological horizon plays an important role in the shadow images. Peng et al. investigated the optical appearance of the Schwarzschild black hole surrounded by a thin disk accretion with the quantum modification of Einstein gravity and the influence of the quantum corrections on black hole shadows [64]. The effects of a thin accretion disk with different emission profiles on the observed features of shadows cast by the power-Yang-Mills black hole were presented in [65], and the relationship between the power parameter and the shadow size was also studied in this literature. Guo et al. focused on the effect of the magnetic charge on observed shadows of the Hayward black hole with different accretion flows and argued that the luminosities of shadows are affected by the accretion properties and the magnetic charge [66]. In addition, many other studies have focused on the influence of the thin accretion disk on the black hole shadows [67,68,69,70,71,72], and the effect of the thin accretion disk on the optical appearance of wormholes [73, 74] and other compact objects [75] has also received enough interests.

The Schwarzschild-MOG black hole is a static spherically symmetric vacuum solution derived from scalar-tensor-vector gravity (STVG) based on the Einstein-Hilbert action combined with massive vector field and matter action [76,77,78]. The investigation of such black holes is necessary and has profound implications for our understanding of the universe, as STVG not only successfully explains some phenomena in the solar system observation, the dynamics of galactic clusters, and the rotation curves of galaxies [79,80,81,82], but also fits the lensing and Einstein ring in Abell 3827 [83], and the signal of GW150914 [84]. In [85], the author introduced the Hamiltonian formalism for the dynamics of the massive and massless particles in STVG, as well as the post-Newtonian approximation of the equations of motion. The circular motions of neutral test particles on the equatorial plane of a Schwarzschild-MOG black hole were reported in [86], while the dynamics of massive charged particles around this black hole with an external magnetic field were investigated in [87]. In addition, the light deflections and lensing effects in this gravitational field were solved in [88,89,90]. The shadow, which is the fingerprint of the black holes, was first investigated by Moffat [91], and late studied by the authors of [92]. However, these studies only focus on the size and silhouette of the Schwarzschild-MOG black hole shadow, while the influences of the accretion flow on the shadow of this object are still unclear. Since the accretion flow plays a pivotal role in black hole shadow observations, here we concentrate on the brightness depression and optical images of this black hole illuminated by different accretions, which is the main purpose of this paper.

The remainder of this paper is organized as follows. In Sect. 2, we study the photon deflections and shadow radii of Schwarzschild-MOG black holes with the help of Lagrangian formalism. The constraints of the parameter \(\alpha \) with the EHT data are also presented. In Sect. 3, we investigate the specific intensities and optical appearances of Schwarzschild-MOG black holes with static and infalling spherical accretion. The effects of an optically and geometrically thin accretion disk on the observed features of this black hole are investigated in Sect. 4. Finally, the conclusions and discussions are concluded in Sect. 5.

The shadow is the fingerprint of the spacetime geometry, and the parameter \(\alpha \) of the Schwarzschild-MOG black hole can be inferred by the observed shadow. For a distant observer, the black hole shadow is always measured by angular diameter \(\varOmega \) [30]

By solving Eqs. (14) and (19), we study the dependence of angular diameter \(\varOmega \) of the black hole shadow on different values of the parameter \(\alpha \), as shown in Fig. 3. Here we have \(\gamma =4.14 \times 10^6\) and distance \(D=8.127\) kpc for the red line, \(\gamma =6.2 \times 10^9\) and distance \(D=16.8\) Mpc for the black line. It is found that \(\varOmega \) grows linearly with \(\alpha \) in the range of \(0\le \alpha \le 1\). The green and blue regions are the shadow diameters of M87\(^*\) (\(42\pm 3\) \(\mu \)as) and Sagittarius A\(^*\) (\(51.8\pm 2.3\) \(\mu \)as) estimated with the EHT observations [6, 12]. Consequently, we constrain the parameter \(\alpha \) of the Schwarzschild-MOG black hole as \(0.040

There is an ocean of free-moving matter in the universe that can be accreted by a black hole. The light rays radiated by these substances are the light source that illuminates the black hole. In this section, we investigate the shadows and photon rings of Schwarzschild-MOG black holes with static and infalling spherical accretions.

The relationship between the angular diameter \(\varOmega \) of the observed shadow and the parameter \(\alpha \). The green and blue regions are the experimental data of M87\(^*\) (\(42\pm 3\) \(\mu \)as) and Sagittarius A\(^*\) (\(51.8\pm 2.3\) \(\mu \)as) reported by the EHT, respectively. Thus, the parameter \(\alpha \) can be constrained as \(0.040

The observed specific intensity \(I_\text obs\) is not only a function of impact parameter b but also affected by the parameter \(\alpha \). Figure 4 depicts the \(I_\text obs\) radiated by a static spherical accretion flow surrounding Schwarzschild-MOG black holes, while \(I_\text obs\) corresponding to negative b is obtained by the symmetry of spacetime. In the region of positive b, we find that no matter \(\alpha \) takes, the specific intensity \(I_\text obs\) increases with the increase of b and reaches a peak quickly, then drops asymptotically to a small value with increasing b. However, the parameter \(\alpha \) significantly harm the specific intensity, which means that the brightness around the Schwarzschild-MOG black hole is lower than that of the Schwarzschild case, and this evidence allows us to distinguish the former from the latter. Besides, the impact parameter corresponding to the maximum specific intensity is the critical impact parameter \(b_p\), which is the radius of the black hole shadow. Clearly, larger \(\alpha \) correspond to a larger shadow diameter.