Centroid Angular Deviation in Lensing

• Centroid Angular Deviation (CAD) is a measure of the angular shift between the unlensed source and its magnification centroid, highlighting image asymmetries.
• Its computation involves numerical solutions of the lens equation using the Virbhadra–Ellis framework under the Janis–Newman–Winicour (JNW) metric.
• CAD values offer a practical method to differentiate compact object models, with significant shifts observed in naked singularities compared to Schwarzschild black holes.

Centroid Angular Deviation (CAD) is a quantitative observable in gravitational lensing that characterizes the shift of the image magnification centroid relative to the true, unlensed angular position of a light source. Originally developed for the study of lensing by strongly naked singularities (SNS) within the Janis–Newman–Winicour (JNW) spacetime, CAD serves as a key diagnostic for distinguishing between different types of gravitational lenses, particularly when comparing compact object models such as black holes and naked singularities (DeAndrea et al., 2015).

1. Definitions and Formalism

The Centroid Angular Deviation, denoted $\Delta\theta$, is computed as the difference between the magnification centroid (or centroid angle) $\theta_c$ and the unlensed source position $\beta$:

$\Delta\theta \equiv \theta_c - \beta.$

Here,

• $\beta$ is the angular position of the unlensed source, measured from the optic axis.
• $\theta_i$ ($i = 1,\ldots,4$) are the (lensed) image positions formed by the gravitational lens, as predicted by the lens equation for a SNS.
• $\mu_i$ are the signed magnifications associated with each image.

The magnification centroid is defined as the magnification-weighted mean of the image positions:

$$\theta_c = \frac{\sum_{i=1}^4 \mu_i \theta_i}{\sum_{i=1}^4 \mu_i},$$

with magnifications $\mu_i$ given generally by

$\theta_c$0

The total absolute magnification is

$\theta_c$1

2. Analytic and Numerical Framework

CAD analysis for strongly naked singularities relies on the Virbhadra–Ellis lens equation (in the small-angle approximation):

$\theta_c$2

where $\theta_c$3, $\theta_c$4, and $\theta_c$5 are the observer–source, lens–source, and observer–lens distances, respectively, and $\theta_c$6 is the total bending angle:

For the JNW metric (characterized by mass $\theta_c$7, massless scalar charge $\theta_c$8, parameter $\theta_c$9, and $\beta$0), the bending angle is evaluated by:

$\beta$1

with $\beta$2, $\beta$3.

The lens equation is solved numerically for each $\beta$4 of interest, and the signed magnifications $\beta$5 are computed at each image position.

3. Computational Methodology

The practical procedure for evaluating CAD is as follows:

• For a chosen source position $\beta$6, solve the lens equation numerically (typically via root-finding algorithms in systems such as Mathematica) for the four real image angles $\beta$7.
• Numerically perform the JNW deflection integral to compute $\beta$8 at each image position.
• Compute the signed image magnifications $\beta$9 as per the analytic formula.
• Evaluate the magnification centroid $\Delta\theta \equiv \theta_c - \beta.$0 and subsequently $\Delta\theta \equiv \theta_c - \beta.$1. No closed analytic forms exist for $\Delta\theta \equiv \theta_c - \beta.$2 or $\Delta\theta \equiv \theta_c - \beta.$3 in this context; all results depend on this explicit numerical method.

4. Characteristic Behavior as a Function of Source Position

The CAD as a function of the unlensed source position $\Delta\theta \equiv \theta_c - \beta.$4 exhibits the following qualitative and quantitative features:

• At perfect alignment ($\Delta\theta \equiv \theta_c - \beta.$5), $\Delta\theta \equiv \theta_c - \beta.$6 vanishes, and two concentric Einstein rings are formed.
• As $\Delta\theta \equiv \theta_c - \beta.$7 increases from zero, $\Delta\theta \equiv \theta_c - \beta.$8 rises rapidly, reaching a maximum ($\Delta\theta \equiv \theta_c - \beta.$9) at an intermediate value of $\beta$0.
• For example, in the case of a SNS with $\beta$1, the maximum value is $\beta$2 at $\beta$3.
• Beyond this peak, $\beta$4 decays monotonically toward zero as $\beta$5, with the shift falling approximately as $\beta$6 and becoming observationally negligible for large $\beta$7.

Representative numerical results for a SNS with $\beta$8 are summarized:

$\beta$9 $\theta_i$0	$\theta_i$1 $\theta_i$2
0.1	0.0499
1.0	0.397
2.0	0.491
3.0	0.450
4.0	0.388
5.0	0.334
10.0	0.186

This behavior is quantitatively distinct from Schwarzschild black hole lensing, particularly in the magnitude of the maximum shift.

5. Comparative Analysis and Physical Implications

The CAD is sensitive to the spacetime geometry of the lens. For a Schwarzschild black hole (SBH) of the same mass and $\theta_i$3, the peak value is $\theta_i$4 at $\theta_i$5. In contrast, the SNS with $\theta_i$6 yields a maximum shift of $\theta_i$7—a $\theta_i$8 enhancement relative to the SBH case. This suggests that the presence of a scalar charge, as parameterized by $\theta_i$9, significantly amplifies lensing asymmetry and thus the centroid shift (DeAndrea et al., 2015). A plausible implication is that precision measurement of CAD curves could offer a means of empirically discriminating between black holes and naked singularities or, more generally, probing deviations from standard black hole models.

6. Observational Prospects and Theoretical Significance

Achieving sufficient angular resolution to detect CAD at the predicted levels (sub-arcsecond for SMBH or SNS at galactic-center distances) is feasible in principle with next-generation high-resolution interferometers, such as the proposed NASA MAXIM X-ray interferometer, which targets $i = 1,\ldots,4$0 nanoarcsecond sensitivity. Measurement of a CAD curve significantly above the Schwarzschild prediction would indicate the presence of a naked singularity, thereby challenging the cosmic censorship hypothesis or necessitating consideration of alternative compact object models. The magnitude of the observed centroid shift as a function of $i = 1,\ldots,4$1 may thus serve as an empirical test of both general relativity in the strong-field regime and the nature of galactic compact objects (DeAndrea et al., 2015).