Shadows of quintessence black holes: spherical accretion, photon trajectories, and geodesic observers

Abstract

The presence of a quintessence-like field can influence the black hole shadow through three primary mechanisms: the dynamics of accretion flows, the trajectories of photons, and the motion of observers. Unlike standard shadow analyses that assume a static observer at spatial infinity, the non-asymptotically flat nature of quintessence-corrected spacetimes motivates the consideration of freely falling (geodesic) observers. Using a perturbative approach, we derive analytical expressions for the event-horizon location, photon-sphere radius, innermost stable circular orbit, and critical impact parameter. We compute the observed intensity profiles for both static and infalling spherical accretion flows. We find that, although the photon-sphere radius and the critical impact parameter are invariant properties of the spacetime, the apparent angular size of the shadow depends sensitively on the observer's motion and location. Freely infalling observers systematically measure smaller angular radii than static observers at the same radius, whereas freely outgoing observers measure larger ones, in agreement with relativistic aberration. In contrast to the Schwarzschild case, the impact parameter alone is insufficient to characterize the observed angular structure in non-asymptotically flat spacetimes. Applying our results to the Event Horizon Telescope observation of M87$^\ast$, we show that more negative equations of state lead to stronger constraints on the quintessence parameter, largely independent of the observer prescription. Our analysis highlights the importance of carefully specifying the observer in shadow studies of non-asymptotically flat black-hole spacetimes.

Figures (11)

• Figure 1: Coordinate systems for a photon trajectory (red solid curve) in the equatorial plane ($\theta = \pi/2$). The Euclidean coordinates $(x, y)$ are defined as $x = r \cos\phi$ and $y = r \sin\phi$. The $(x', y')$ coordinates represent the local reference frame of an observer located at radius $r = r_{\rm obs}$. The center of the black hole is located at the origin of the Euclidean coordinate system $(x, y)$.
• Figure 2: Left panel: Angular radius of the photon sphere observed by static observers at different radii for several black-hole spacetimes. For a given value of $\omega$, only the observer located at $r_{\rm m}$ follows a geodesic; all other static observers have nonzero proper acceleration. Note that the curve for Schwarzschild case almost overlaps with the black curve. Right panel: Comparison of the photon-sphere angular radius observed by static observers in quintessence black-hole spacetimes and in the Schwarzschild spacetime. For all quintessence black holes, the normalization constant is fixed to $c=0.001$.
• Figure 3: Difference in the total emission coefficient, $\Delta j_{\rm t} = j_{\rm t} - j_{\rm S}$, between the quintessence–corrected Schwarzschild spacetime and the standard Schwarzschild case. For all choices of the equation-of-state parameter $\omega$, the normalization constant is fixed to $c=0.001$.
• Figure 4: Metric time component $f(r)$ of the quintessence-corrected Schwarzschild black hole for the equation-of-state parameter $\omega=-2/3$ and normalization constant $c=10^{-4}$. The function $f(r)$ attains its maximum at $r_{\rm m}$.
• Figure 5: Difference in the redshift factor $g$ for static spherical emission, $\Delta g = g - g_{\rm S}$, between the quintessence–corrected Schwarzschild spacetime and the standard Schwarzschild case. The static geodesic observer is located at the radius where $f(r)$ attains its maximum value: For $\omega = -1/2$, $r_{\rm obs} \approx 252M$; for $\omega = -2/3$, $r_{\rm obs} \approx 45M$; and for $\omega = -3/4$, $r_{\rm obs} \approx 27M$; for Schwarzchild case, $r_{\rm obs} \to \infty$. For all quintessence–corrected Schwarzschild spacetimes, the normalization constant is fixed to $c=0.001$.