Observational signatures of hot spots orbiting horizonless objects

Pushed by a number of advances, electromagnetic observatories have now reached the horizon scale of supermassive black holes. The existence and properties of horizons in our universe is one of the outstanding fundamental issues that can now be addressed. Here we investigate the ability to discriminate between black holes and compact, horizonless objects, focusing on the lensing of hot spots around compact objects. We work in particular with boson and Proca stars as central objects, and show that the absence of a horizon gives rise to a characteristic feature—photons that plow through the central object and produce an extra image. This feature should be universal for central objects made of matter weakly coupled to the standard model.

Figure 1

Boson star solutions. Top metric function $g_{tt}$ from Eq. (6) as a function of the normalized radial coordinate $r/M$. Bottom: metric function $g_{rr}$ from Eq. (6) as a function of the normalized radial coordinate $r/M$. The thin black line represents the Schwarzschild solution, i.e., $g_{tt}=g_{rr}^{-1}=1–2M/r$. Finiteness and positiveness of the metric functions guarantee the nonexistence of horizons or singularities in spacetime.

Figure 2

Proca stars. Top: metric function $g_{tt}$ from Eq. (6) as a function of the normalized radial coordinate $r/M$. Bottom: metric function $g_{rr}$ from Eq. (6) as a function of the normalized radial coordinate $r/M$. The thin black line represents the Schwarzschild solution, i.e., $g_{tt}=g_{rr}^{-1}=1–2M/r$. Finiteness and positiveness of the metric functions guarantee the nonexistence of horizons or singularities in spacetime.

Figure 3

Time integrated images for a full orbit, assuming the source to be a spherical luminous spot. The observer inclination is $i=80°$. Different rows show images for different central compact object (Schwarzschild BH, boson star, and Proca star, from top to bottom). The columns portray different orbital radius ($8M,10M,12M,20M$). See text for discussion.

Figure 4

Drawing of the time integrated images depicting the primary and secondary paths for the compact objects as well as the photon ring in the Schwarzschild BH. The primary track is presented as a dashed line, the secondary is a dotted line, the “plunge-through” and photon ring are in full.

Figure 5

Lensed images for different times. Top row: Schwarzschild BH; middle row: boson star; bottom row: Proca star. The inclination is $i=80°$, the orbital radius is $r=8M$. See text for details.

Figure 6

Geodesics connecting the hot spot at $x=8M$ and $z=0$ to the observer at $x=-1000M$ with an observation angle of $i=80°$ for the BSC4, PSC4, and Schwarzschild configurations. The black dashed circle corresponds to the event horizon. The third source does not appear in the lensed images of the Schwarzschild case because the associated geodesic would cross the event horizon.

Figure 7

Temporal magnitude $m_k$ and temporal centroid ${\overrightarrow{c}}_k$ for Schwarzschild and most compact boson and Proca stars (cf. Tables 1 and 2). For the $i=90°$ centroid, the vertical axis is time. See text for details.

Figure 8

Geodesic congruences in the Schwarzschild spacetime (left panel) and in the BSC4 boson star spacetime (right panel). The dashed black circle in the upper panel corresponds to the light-ring. The observer is located towards the right. The position of the equatorial plane is represented in blue with a different hue depending on the inclination of the observer. The entire equatorial plane outside the light-ring is projected onto the observer’s screen for the Schwarzschild case independently of the observation angle. For the BSC4 case, only a certain portion $r>r_c$ of the equatorial plane is projected onto the observer’s screen, where $r_c$ is a critical radius that varies with the inclination angle. In particular, this image shows that for low inclination angles the secondary image is always absent in the BSC4 case.

Figure 9

Scalar field $\gamma$ function as a function of the normalized radial coordinate $r/M$. The exponential decay at large radii confirms the confinement of the boson star in a finite region of space near the origin.

Figure 10

Top: vector field $f$ function as a function of the normalized radial coordinate $r/M$. Bottom: vector field $g$ function as a function of the normalized radial coordinate $r/M$. The exponential decays of both functions at large radii confirm the localization of the Proca star in a finite region of space near the origin.

Figure 11

Orbital velocity $\mathrm{\Omega }$ from Eq. (c1) as function of the radial coordinate $r$ normalized to the total mass $M$. Top: results for the four boson stars considered in Table 1. Bottom: results four Proca stars considered in Table 2. The thin black line represents the orbital velocity of the Schwarzschild solution. An increase in the central density ${\gamma }_0$ and $f_0$ leads to orbital velocities closer to that of a Schwarzschild solution.

Figure 12

Relative error of the orbital period $T=2\pi /\mathrm{\Omega }$ as functions of the radial coordinate $r$ normalized to the total mass $M$. Top: results for the four boson stars considered in Table 1. Bottom: results for the four Proca stars considered in Table 2. The thin black line represents the 1% error level. The relative errors decrease with an increase in the central densities ${\gamma }_0$ and $f_0$.

Figure 13

Minimum orbital radius necessary for a secondary image to appear in the screen of the observer (and, consequently, produce the shifting effect in the centroid) as a function of the observation angle of the observer $\theta$ and the equatorial angle of the source $\varphi$, where $\varphi =0$ was defined as the direction opposite to the observer, for the BSC4 (top panel) and the PSC4 (bottom panel) configurations.

Figure 14

Time integrated images $⟨I{⟩}_{lm}$ for a full orbit assuming a spherical spot. The observer inclination is $i=20°$ (top 3 panels) and $i=50°$ (bottom 3 panels). The rows depict the compact object metrics (Schwarzschild, boson star, and Proca star). The columns portray different orbital radius ($8M,10M,12M,20M$).

Figure 15

Time integrated images $⟨I{⟩}_{lm}$ for an observer inclination of $i=90°$ (see Fig. 14 for details).

Figure 16

Time integrated images $⟨I{⟩}_{lm}$ for an observer inclination of $i=80°$ (see Fig. 14 for details), for boson and Proca stars of increasing compactness (cf. Tables 1 and 2). The orbital radius is $r=8M$. As the compactness increases the lensing is stronger due to a larger mass of the compact objet inside the orbit.

Figure 17

Temporal magnitude $m_k$ and temporal centroid ${\overrightarrow{c}}_k$ for the boson stars (cf. Table 1). For $i=90°$ centroid, the vertical axis is time.

Figure 18

Temporal magnitude $m_k$ and temporal centroid ${\overrightarrow{c}}_k$ for the Proca stars (cf. Table 2). For $i=90°$ centroid, the vertical axis is time.