---
title: Centroid Angular Deviation in Lensing
url: https://www.emergentmind.com/topics/centroid-angular-deviation-cad
type: topic
---

# Centroid Angular Deviation in Lensing

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 [1506.04825].

## 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
$$
\mu_i = \left[\frac{\sin \beta}{\sin \theta_i} \frac{d\beta}{d\theta}\right]^{-1}_{\theta = \theta_i}.
$$
The total absolute magnification is
$$
\mu_{\rm tot} \equiv \sum_{i=1}^{4} |\mu_i|.
$$

## 2. Analytic and Numerical Framework

CAD analysis for strongly naked singularities relies on the Virbhadra–Ellis lens equation (in the small-angle approximation):
$$
\tan \beta = \tan \theta - \frac{D_{ds}}{D_s} \left[ \tan \theta + \tan \left( \hat{\alpha}(\theta) - \theta \right) \right],
$$
where $D_s$, $D_{ds}$, and $D_d$ are the observer–source, lens–source, and observer–lens distances, respectively, and $\hat{\alpha}(\theta)$ is the total bending angle:

For the JNW metric (characterized by mass $M$, massless scalar charge $q$, parameter $\nu = 2M/b$, and $b = 2\sqrt{M^2 + q^2}$), the bending angle is evaluated by:
$$
\hat{\alpha}(\rho_0) = 2\int_{\rho_0}^{\infty} \frac{d\rho}{\rho \sqrt{
(1-1/\rho)^{1-2\nu}
(1-1/\rho_0)^{2\nu-1}
\left( \frac{\rho}{\rho_0} \right)^2 - 1
}} - \pi,
$$
with $\rho \equiv r/b$, $\rho_0 = r_0/b$.

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

## 3. Computational Methodology

The practical procedure for evaluating CAD is as follows:
- For a chosen source position $\beta$, solve the lens equation numerically (typically via root-finding algorithms in systems such as Mathematica) for the four real image angles $\theta_i$.
- Numerically perform the JNW deflection integral to compute $\hat{\alpha}(\theta)$ at each image position.
- Compute the signed image magnifications $\mu_i$ as per the analytic formula.
- Evaluate the magnification centroid $\theta_c$ and subsequently $\Delta\theta$.
No closed analytic forms exist for $\theta_i$ or $\mu_i$ 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 $\beta$ exhibits the following qualitative and quantitative features:
- At perfect alignment ($\beta = 0$), $\Delta\theta$ vanishes, and two concentric Einstein rings are formed.
- As $\beta$ increases from zero, $\Delta\theta$ rises rapidly, reaching a maximum ($\Delta\theta_{\rm max}$) at an intermediate value of $\beta$.
- For example, in the case of a SNS with $\nu = 0.01$, the maximum value is $\Delta\theta_{\rm max} \approx 0.491^{\prime\prime}$ at $\beta \approx 2^{\prime\prime}$.
- Beyond this peak, $\Delta\theta$ decays monotonically toward zero as $\beta \to \infty$, with the shift falling approximately as $1/\beta$ and becoming observationally negligible for large $\beta$.

Representative numerical results for a SNS with $\nu = 0.01$ are summarized:

| $\beta$ $(^{\prime\prime})$ | $\Delta\theta$ $(^{\prime\prime})$ |
|-----------------------------|-------------------------------------|
| 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 $D_{ds}/D_s = 1/2$, the peak value is $\Delta\theta_{\rm SBH} \approx 0.35^{\prime\prime}$ at $\beta \approx 2^{\prime\prime}$. In contrast, the SNS with $\nu = 0.01$ yields a maximum shift of $\sim 0.49^{\prime\prime}$—a $\sim 40\%$ enhancement relative to the SBH case. This suggests that the presence of a scalar charge, as parameterized by $\nu$, significantly amplifies lensing asymmetry and thus the centroid shift [1506.04825]. 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 $\sim 100$ 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 $\beta$ may thus serve as an empirical test of both general relativity in the strong-field regime and the nature of galactic compact objects [1506.04825].

Source: https://www.emergentmind.com/topics/centroid-angular-deviation-cad