Strong Gravitational Lensing by Compact Object without Cauchy Horizons in Effective Quantum Gravity

This paper theoretically investigates strong gravitational lensing by a compact object in effective quantum gravity that lacks Cauchy horizons, finding that while horizonless wormhole solutions are ruled out by SgrA* observations, they remain consistent with M87* data, and providing estimates for potentially detectable lensing observables.

The Gist

Imagine the universe as a giant, stretchy trampoline. In our everyday understanding of gravity (thanks to Einstein), heavy objects like stars and black holes sit on this trampoline and create deep dips. If you roll a marble (light) across it, the marble follows the curve of the dip. This is gravitational lensing: gravity bending light.

For a long time, we thought we knew exactly how these dips looked. But near the center of a black hole, things get weird. The math breaks down, creating "singularities" (infinite points) that don't make sense. Scientists suspect that Quantum Gravity (the rules of the very small) might fix these glitches, but we don't have a full theory yet.

So, this paper explores a "best guess" model called Effective Quantum Gravity (EQG). The author, Suvankar Paul, asks a simple question: If we tweak the rules of gravity with quantum effects, what happens to the way light bends around a black hole?

Here is the breakdown of the paper's story, using simple analogies:

1. The Two Faces of the Object

The paper studies a specific mathematical solution that can act like two different things, depending on a "dial" called $\zeta$ (zeta). Think of $\zeta$ as a knob you can turn to change the nature of the object.

• The Black Hole (The Deep Pit): If you turn the knob to a low setting, the object acts like a classic black hole. It has an Event Horizon, which is like a one-way door. Once you cross it, you can never get out.
• The Wormhole (The Tunnel): If you turn the knob to a high setting, the "one-way door" disappears. Instead of a pit, the object becomes a Wormhole. Imagine a tunnel connecting two different rooms. Light can go in, hit the narrowest part (the throat), and come back out the other side. There is no "point of no return."

The paper investigates how light behaves when it tries to orbit these objects.

2. The "Photon Sphere" (The Race Track)

Around any heavy object, there is a special zone called the Photon Sphere. Imagine a race track right on the edge of a cliff.

• If a car (light) drives too fast, it flies off into space.
• If it drives too slow, it falls off the cliff.
• But at the perfect speed, it can drive in a circle forever.

In the world of black holes, this track is unstable. A tiny bump, and the car either falls in or flies away. The paper calculates exactly where this track is for their new "Quantum" objects.

3. The Detective Work: SgrA* vs. M87*

The author doesn't just do math; they play detective using real data from the Event Horizon Telescope (EHT). We have pictures of two supermassive black holes:

• SgrA:* The one at the center of our own Milky Way galaxy.
• M87:* The giant one in the M87 galaxy.

The author asks: Do the shadows cast by these objects match our "Quantum" model?

• The Verdict on SgrA:* The data from our galaxy's black hole is very strict. It says, "No wormholes here!" The "knob" ($\zeta$) must be set low. SgrA* looks like a standard black hole with a one-way door.
• The Verdict on M87:* The data from the distant giant is a bit more flexible. It allows for the "knob" to be turned higher. This means M87* could potentially be a wormhole (a tunnel) rather than a black hole, though it could still be a black hole too.

4. The "Strong Lensing" Effect (The Funhouse Mirror)

When light gets very close to these objects, it doesn't just bend once; it can loop around multiple times, creating a stack of images (like a reflection in a funhouse mirror).

• The "Time Delay" Clue: The paper finds a fascinating difference. If you send two light beams around the object, one taking a slightly longer path than the other, they arrive at different times.
  • For Black Holes, this time difference is tiny (minutes for our galaxy).
  • For Wormholes (specifically in the M87* scenario), the time difference could be days.
  • Why this matters: If we could measure light arriving days apart, we could prove we are looking at a wormhole, not a black hole.

5. The Big Picture Conclusion

This paper is like a "field guide" for future astronomers.

• It tells us that Quantum Gravity might change the shape of the "dip" in the trampoline.
• It suggests that while our local black hole (SgrA*) is likely a standard black hole, the giant one in M87* is a candidate for something stranger: a wormhole.
• It gives us specific tools (like measuring the time delay of light) to tell the difference between a black hole and a wormhole in the future.

In short: The universe might have "tunnels" (wormholes) hiding in plain sight, and by watching how light dances around them, we might finally find out if they are real. This paper provides the map for that dance.

Technical Summary

1. Problem Statement

General Relativity (GR) successfully describes gravity in weak-field regimes but faces fundamental challenges in strong-field regimes, specifically regarding spacetime singularities and incompatibility with quantum mechanics. While full quantum gravity theories remain elusive, Effective Quantum Gravity (EQG) offers a phenomenological approach to resolve singularities using semiclassical Hamiltonian constraints.

Previous EQG studies derived two types of static, spherically symmetric solutions (Ref. [23]), which have been extensively analyzed. However, a third type of solution (Ref. [24]) remains relatively unexplored. This specific solution is unique because:

• It does not contain Cauchy horizons (implying stability under perturbations).
• Depending on the theory's parameters, it can describe either a Black Hole or a Horizonless Wormhole.
• The paper aims to investigate the strong gravitational lensing characteristics of this third solution and constrain its parameters using observational data from the Event Horizon Telescope (EHT) regarding SgrA* and M87*.

2. Methodology

A. Theoretical Framework

The study utilizes a static, spherically symmetric metric derived from EQG with $q=0$ (asymptotically flat case). The line element is:
$$ds^2 = -A^{(q)}(r)dt^2 + \frac{dr^2}{\mu(r)A^{(q)}(r)} + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$
Where the metric functions are:

• $A^{(0)}(r) = 1 - \frac{r^2}{\zeta^2} \arcsin\left(\frac{2M\zeta^2}{r^3}\right)$
• $\mu(r) = 1 - \frac{4M^2\zeta^4}{r^6}$
• $M$: Mass of the object.
• $\zeta$: Free quantum parameter representing deviation from classical GR.
• $q$: Integer parameter (set to 0 for this study).

Classification of Solutions:
The nature of the spacetime depends on the ratio $\zeta/M$:

1. Black Hole ($\zeta/M < 2(\pi/2)^{3/2}$): An event horizon exists ($A(r)=0$) outside the throat radius.
2. Extremal Black Hole ($\zeta/M = 2(\pi/2)^{3/2}$): The horizon and throat coincide.
3. Wormhole ($\zeta/M > 2(\pi/2)^{3/2}$): No event horizon exists; the minimum radius acts as a wormhole throat satisfying the flare-out condition.

B. Geodesic Analysis

The authors analyzed the motion of null geodesics (photons) to determine:

• Effective Potential ($V_{eff}$): Derived to locate the photon sphere ($r_m$).
• Photon Sphere Radius: Found to be independent of $q$, depending only on $\zeta$. The equation is $r_m^6 - 9M^2r_m^4 - 4M^2\zeta^4 = 0$.
• Critical Impact Parameter ($b_m$): Calculated for the photon sphere, which defines the edge of the shadow.

C. Strong Deflection Limit (SDL)

Using the formalism by Bozza and Tsukamoto, the bending angle $\alpha(b)$ near the photon sphere was expanded logarithmically:
$$\alpha(b) = -\bar{a} \log\left(\frac{b}{b_m} - 1\right) + \bar{b} + O((b-b_m)\log(b-b_m))$$
The coefficients $\bar{a}$ and $\bar{b}$ were computed as functions of $\zeta$.

D. Observational Constraints

The study utilized EHT data to constrain $\zeta$:

• M87:* Shadow diameter $d_{sh} = 42 \pm 3 \, \mu\text{as}$.
• SgrA:* Fractional deviation $\delta$ from Schwarzschild shadow size (Keck and VLTI data).
• Theoretical shadow radii ($R_{sh}$) were compared against these observational bounds to determine allowed ranges for $\zeta$.

E. Lensing Observables

The paper calculated key observables for relativistic images formed near the photon sphere:

• Angular position ($\theta_n$) and separation ($s_1$).
• Magnification ($\mu_n$) and relative flux ($R_1$).
• Time Delay ($\Delta T$): The difference in arrival time between the first and second relativistic images.

3. Key Results

A. Parameter Constraints

• SgrA Observations:* The data restricts $\zeta/M$ to $[0, 1.878]$ (VLTI) and $[0, 3.035]$ (Keck). Since the transition to a wormhole occurs at $\zeta/M \approx 3.937$, SgrA observations rule out the wormhole solution*, supporting only the Black Hole scenario.
• M87 Observations:* The data allows $\zeta/M$ up to $5.485$. This range includes both Black Hole and Wormhole solutions, meaning M87* cannot currently distinguish between the two based on shadow size alone.

B. Lensing Coefficients ($\bar{a}$ and $\bar{b}$)

• The coefficient $\bar{a}$ decreases monotonically as $\zeta$ increases.
• The coefficient $\bar{b}$ varies non-trivially.
• For small $\zeta$ (VLTI constraints), the lensing behavior is nearly indistinguishable from the Schwarzschild black hole ($\zeta=0$).

C. Strong Lensing Observables

• Angular Separation ($s_1$): As $\zeta$ increases, the angular separation between the outermost image and the inner images decreases. This makes resolving the first image increasingly difficult.
• Magnification: The magnification of the outermost image decreases significantly with increasing $\zeta$. Wormhole solutions exhibit much lower magnification than black holes.
• Time Delay ($\Delta T$):
  • For SgrA*, the time delay is on the order of minutes ($\sim 0.18$ hours), which is currently difficult to detect.
  • For M87*, the time delay is on the order of days ($\sim 319$ to $348$ hours).
  • Crucial Finding: Unlike previous EQG models where time delay decreased or remained constant with $\zeta$, here time delay increases with $\zeta$. This provides a potential signature to distinguish the wormhole case from the black hole case in future high-precision observations of M87*.

4. Significance and Conclusion

• Phenomenological Viability: The study demonstrates that the third EQG solution is a viable candidate for describing compact objects, as it is consistent with current EHT data for both black holes and (in the case of M87*) horizonless wormholes.
• Discriminating Power: While shadow size alone cannot rule out the wormhole for M87*, strong lensing observables, particularly the time delay, offer a promising avenue for future discrimination. The distinct behavior of time delay (increasing with $\zeta$) contrasts with other modified gravity models.
• Astrophysical Implications: The results suggest that if the central object of M87* is a wormhole described by this EQG model, it would produce a significantly larger time delay between relativistic images compared to a standard black hole.
• Future Prospects: The paper highlights the need for future observations capable of measuring time delays of days (for M87*) to test the existence of horizonless compact objects and constrain the quantum gravity parameter $\zeta$.

In summary, this work bridges the gap between theoretical effective quantum gravity models and observational astrophysics, providing specific, testable predictions for strong gravitational lensing that could validate or falsify the existence of quantum-corrected wormholes.