Surface temperature of an accretion disk around a wormhole Kerr-mimicker

Technical Summary: Surface Temperature of an Accretion Disk around a Wormhole Kerr-mimicker

Problem Statement
While recent gravitational wave observations and Event Horizon Telescope (EHT) imaging provide strong evidence for Kerr black holes, direct proof of an event horizon remains impossible via electromagnetic observations alone. Consequently, "black hole mimickers"—compact objects lacking an event horizon, such as traversable wormholes—remain viable alternatives. The challenge lies in identifying observables that can distinguish a spinning wormhole from a Kerr black hole, particularly when the wormhole is designed to mimic the Kerr metric's orbital dynamics. Previous studies have explored shadows, gravitational wave echoes, and spectral line profiles, but a robust method to differentiate these objects based on the thermal emission of standard thin accretion disks has been less explored, specifically regarding the impact of the radial metric component ($g_{rr}$).

Methodology
The authors analyze a specific stationary, axisymmetric, asymptotically flat spacetime metric proposed by [27], which describes a Kerr-like wormhole. This metric is identical to the Kerr solution in all components except the radial component, $g_{rr}$. The metric is defined by a function $D = r^2 - 2\tilde{M}r + a^2$, where $M$ is the mass, $a$ is the spin, and $\tilde{M}$ is a parameter determining the deviation from the Kerr solution ($\tilde{M} = M$ recovers the Kerr black hole).

• Traversable Wormhole: Occurs when $\tilde{M} > M$, resulting in $D=0$ at a throat radius $r_W$ without an event horizon.
• Non-traversable Wormhole: Occurs when $\tilde{M} < M$, resulting in an event horizon outside the throat.

The study assumes a geometrically thin, optically thick accretion disk in the equatorial plane. The authors derive the surface temperature profile by:

1. Establishing that the orbital properties (angular velocity $\Omega$, specific energy $E$, specific angular momentum $l$, and Lense-Thirring precession) for circular orbits depend only on $g_{tt}$, $g_{\phi\phi}$, and $g_{t\phi}$. Since these are identical to the Kerr metric, the orbital dynamics are indistinguishable.
2. Calculating the radiative flux $F(r)$ from the disk. The flux is derived from the rate of binding energy release, $dL/dr$, divided by the proper surface area of the disk ring, $4\pi r g_{rr}^{1/2}$.
3. Utilizing the blackbody approximation ($F = \sigma T^4$) to relate the flux to the surface temperature.
4. Deriving the ratio of the wormhole disk temperature ($T$) to the Kerr black hole disk temperature ($\tilde{T}$) as:
   $$\frac{T}{\tilde{T}} = \left( \frac{\tilde{g}_{rr}}{g_{rr}} \right)^{1/8}$$
   This relation demonstrates that the temperature profile is directly sensitive to the modification in the radial metric component.

Key Contributions and Results

• Orbital Indistinguishability: The paper confirms that for any value of the deviation parameter, the orbital properties of test particles in circular equatorial orbits are exactly the same for the Kerr-like wormhole and a Kerr black hole of the same mass and spin. This includes the position of the Innermost Stable Circular Orbit (ISCO), provided the ISCO lies outside the throat.
• Temperature Suppression: The primary finding is that the surface temperature of the accretion disk is suppressed in the wormhole spacetime compared to the Kerr case. Because the proper area of the disk is larger (due to the divergence of $g_{rr}$ near the throat), the same amount of radiated energy is distributed over a larger surface area, resulting in a lower temperature.
• Observable Signatures:
  • Traversable Wormholes ($\tilde{M} > M$): The temperature ratio decreases monotonically as the throat is approached, reaching zero at $r = r_W$ where $g_{rr}$ diverges. This creates a "truncated" appearance. For non-spinning wormholes, a temperature suppression exceeding 10% at the ISCO requires a deviation parameter $\lambda^2 \approx 0.5$ (where $\tilde{M}/M = 1 + \lambda^2$). For rapidly spinning wormholes ($a=0.95M$), a 10% suppression requires a smaller deviation, $\lambda^2 \approx 0.1$.
  • Non-traversable Wormholes ($\tilde{M} < M$): The temperature is slightly higher than the Kerr case (by a few percent) due to a smaller proper area within a given circumference, but the effect is less pronounced.
• Independence from Redshift Factors: The authors note that the gravitational and Doppler shift factors ($\hat{g}$) affecting the observed intensity are formally independent of $g_{rr}$ for photons emitted face-on, meaning the temperature suppression is a direct local effect of the metric geometry rather than an observational artifact.

Significance and Claims
The paper claims to identify a "fairly direct method" of measuring the $g_{rr}$ component of the metric using the thermal emission of accretion disks. The significance lies in the fact that while the wormhole perfectly mimics the Kerr black hole in terms of orbital dynamics (making spin measurements via Lense-Thirring precession or ISCO location inconclusive for distinguishing the two), the surface temperature profile offers a distinct observable signature.

The authors conclude that a traversable wormhole with a sufficiently wide throat would exhibit a visibly suppressed blackbody disk temperature, particularly near the inner edge. This suppression would manifest as a "truncated disk" where the temperature drops to zero at the throat radius, distinct from the behavior of a standard black hole accretion disk. The effect is robust enough to be potentially detectable (e.g., $\ge 10\%$ correction) if the deviation parameter is of order unity for non-spinning objects or $\sim 0.1$ for rapidly spinning objects. The paper does not propose new observational campaigns but highlights that existing X-ray spectral analysis methods could, in principle, test for the presence of such wormholes by looking for this specific temperature suppression pattern.