Damour-Solodukhin Wormhole as a Black Hole Mimicker: The Role of Observers' Location

Technical Summary: The Role of Observers' Location in Identifying Damour–Solodukhin Wormholes as Black Hole Mimickers

Problem Statement
In semi-classical gravity, the Damour–Solodukhin wormhole (DSWH) serves as a "black hole mimicker." It is a static, spherically symmetric solution to Einstein's equations characterized by a parameter $\lambda$. When $\lambda = 0$, the metric describes a Schwarzschild black hole (BH) with an event horizon. When $\lambda \neq 0$, the horizon is replaced by a tiny throat, rendering the object a traversable wormhole (WH). Despite topological differences, the DSWH mimics many BH properties (e.g., accretion discs, quasi-normal modes) when $\lambda$ is sufficiently small, making the travel time for signals from the throat to an asymptotic observer extremely large ($\Delta t \sim 2GM \ln(1/\lambda^2)$).

The central problem addressed is the ambiguity in identifying such an object in a single experiment. If observing a BH mimicker is equivalent to observing a BH, how does an observer determine whether the object is truly a BH ($\lambda=0$) or a WH ($\lambda \neq 0$)? Specifically, does the identity of the object depend on the observer's location?

Methodology
The authors adapt a concept introduced by Tangherlini regarding "non-quantum statistical indeterminacy" in ordinary optical media to the gravitational context. The methodology proceeds as follows:

1. Effective Optical Medium: The DSWH spacetime is modeled as an "effective optical medium" with a position-dependent refractive index $n(r)$. This approach relies on the analogy between photon motion in a gravitational field and light propagation in a medium, supported by the Pound–Rebka experiment's confirmation of momentum increase laws ($p' = np$) in gravity.
2. Observer-Dependent Indices: The paper distinguishes between two types of observers:
   • Asymptotic Observers (a.o.): Located far from the source, measuring coordinate quantities ($n_{a.o.}$).
   • Near-Throat Local Observers (l.o.): Located close to the throat, measuring proper quantities ($\tilde{n}_{l.o.}$).
     The refractive indices for these observers differ due to gravitational time dilation and length contraction effects encoded in the metric functions $\Omega(r)$ and $\Phi(r)$.
3. Probabilistic Identification via Fresnel Coefficients: Instead of deterministic geodesic analysis, the authors apply Tangherlini's formulation to derive Fresnel reflection ($R$) and transmission ($T$) coefficients for photons probing the throat.
   • $R$ represents the probability of the photon being reflected (behaving as if hitting a BH horizon).
   • $T$ represents the probability of the photon being transmitted (behaving as if passing through a WH throat).
   • The coefficients are calculated using $R = (n-1)^2/(n+1)^2$ and $T = 4n/(n+1)^2$, where $n$ is the refractive index sensed by the specific observer.
4. Interpretation:
   • $R=1, T=0$ corresponds to a certain identification as a Schwarzschild BH.
   • Deviations from $(1,0)$ imply a probabilistic identification as a WH.
   • The observer with the higher $R$ is more likely to identify the object as a BH; the observer with the higher $T$ is more likely to identify it as a WH.

Key Results
Applying this framework to the DSWH metric yields the following specific results at the throat ($r_{th} = M/2$):

1. Mass Independence: The Fresnel coefficients at the throat are independent of the mass $M$ and depend solely on the deviation parameter $\lambda$.
2. Observer-Dependent Identities:
   • Asymptotic Observer: Measures a refractive index $n_{a.o.} = 4/\lambda$. The resulting coefficients are $R_{a.o.} = \left(\frac{4-\lambda}{4+\lambda}\right)^2$ and $T_{a.o.} = \frac{16\lambda}{(4+\lambda)^2}$. For small $\lambda$, $R_{a.o.}$ is close to 1.
   • Local Observer: Measures a refractive index $\tilde{n}_{l.o.} = 1/\lambda$. The resulting coefficients are $\tilde{R}_{l.o.} = \left(\frac{1-\lambda}{1+\lambda}\right)^2$ and $\tilde{T}_{l.o.} = \frac{4\lambda}{(1+\lambda)^2}$. For small $\lambda$, $\tilde{R}_{l.o.}$ is significantly smaller than $R_{a.o.}$, and $\tilde{T}_{l.o.}$ is larger than $T_{a.o.}$.
3. Divergent Probabilities: For any non-zero $\lambda$ (however small), the asymptotic observer is more likely to identify the object as a BH (higher reflection probability), while the near-throat local observer is more likely to identify it as a WH (higher transmission probability).
4. Limiting Cases:
   • As $\lambda \to 0$, both sets of coefficients converge to $R=1, T=0$, correctly identifying the object as a BH for both observers.
   • As $\lambda \to \infty$, the coefficients also approach BH values, though the physical interpretation involves different time scales and tidal forces.

Significance and Claims
The paper claims to provide a novel approach to the BH/WH distinction problem by introducing observer-dependent probabilistic indeterminacy. The central result is that the identity of a gravitating object (WH or BH) is not absolute but depends on the location of the observer when probing the object with photons.

• Reconciliation with Lemos and Zaslavskii: The authors note that their findings support the conclusion of Lemos and Zaslavskii that DSWHs are "bad mimickers" due to divergent tidal effects as $\lambda \to 0$. However, the current paper offers a distinct perspective: even if tidal effects are managed by considering discrete small $\lambda$, the probabilistic identification remains observer-dependent.
• Ghost Wormholes: The paper suggests a scenario where an asymptotic observer identifies the object as a BH, while a local observer identifies it as a WH. The local observer might reconcile this by concluding the object is a "ghost WH" (a dead or collapsed WH), a phenomenon that does not occur for exact Schwarzschild BHs ($\lambda=0$).
• Observational Prospects: The authors acknowledge that practical astrophysical support is currently lacking. They suggest that future diagnostics should look for exclusive WH signatures in observed BH spectra (e.g., SgrA* or M87) using accretion profiles and strong-field lensing, though they do not propose specific new experiments.

The paper concludes that this "pre-quantal statistical formulation" offers a deeper implication for semi-classical quantum gravity, suggesting that excluding BH solutions does not preclude the probabilistic identification of an object as a BH by certain observers.