Times of arrival (TOA) of signals in the Kerr-MOG black hole

Technical Summary: Times of Arrival (TOA) of Signals in the Kerr-MOG Black Hole

Problem Statement
General Relativity (GR) requires the postulation of dark matter and dark energy to explain galactic rotation curves and cosmological acceleration, respectively. Modified Gravity (MOG) theories, specifically Scalar-Tensor-Vector-Gravity (STVG) developed by Moffat, offer an alternative by introducing a massive vector field and a variable gravitational constant $G = G_N(1 + \alpha)$ without invoking dark matter. The Kerr-MOG black hole (BH) is a spinning solution in this theory, characterized by spin $a$ and the MOG parameter $\alpha$. While the Kerr metric (GR, where $\alpha=0$) has been extensively studied, the specific astrophysical signatures of the Kerr-MOG metric, particularly regarding signal propagation times in binary systems, require further investigation. This paper addresses the need to quantify how the MOG parameter $\alpha$ influences the difference in Times of Arrival (TOA) for signals passing a spinning lens, providing a potential observational test to constrain $\alpha$.

Methodology
The authors employ a theoretical framework combining the thin-lens approximation with a Post-Post-Newtonian (PPN) expansion up to the third order in $(1/r)$. The study focuses on a hypothetical binary system consisting of a variable pulsar (PSR) acting as a source, a spinning Kerr-MOG black hole as the lens, and an observer (Earth).

1. Metric Analysis: The authors first analyze the static-MOG metric ($a=0$) to categorize the geometry based on the value of $\alpha$. They identify three distinct sectors:
   • $\alpha > 0$: Repulsive Yukawa-like force; no central singularity.
   • $-1 < \alpha < 0$: Corresponds to a brane-world black hole with a negative tidal charge.
   • $\alpha < -1$: Corresponds to a naked singularity.
2. Derivation of TOA: Starting from the Kerr-MOG line element, the authors derive the coordinate time difference ($\Delta t$) between two light rays emerging from behind the lens, passing on opposite sides (co-rotating and counter-rotating), and reaching the observer. This difference arises from frame-dragging effects.
3. Approximation and Integration: Using the thin-lens approximation, where light rays travel in straight lines except at the lens, the authors integrate the null geodesic equations. They generalize the zeroth-order Laguna-Wolszczan formula to include finite distances on the optical axis and higher-order PPN terms. The total delay is decomposed into three components: $\Delta t_1$ (first order), $\Delta t_2$ (second order), and $\Delta t_3$ (third order).
4. Numerical Estimation: The derived formulas are applied to two speculative binary systems:
   • PSR-Cygnus X-1: A stellar-mass BH system ($M \approx 14.8 M_\odot$).
   • PSR-SgrA:* A system involving the supermassive BH at the Galactic Center ($M \approx 4.2 \times 10^6 M_\odot$).
     Parameters such as impact parameter $b$, distances $d_{OL}$ and $d_{LS}$, and spin $a$ are chosen to satisfy weak-field conditions ($M/b \ll 1$) and small-angle approximations.

Key Contributions

• Generalization of TOA Formulas: The paper extends the Laguna-Wolszczan formula (originally derived for the Kerr metric) to the Kerr-MOG metric, providing explicit expressions for TOA up to the third PPN order. The results are expressed in terms of parameters $\psi_1, \psi_2, \psi_3$, which depend on $\alpha$.
• Geometric Classification: The work clarifies the geometric nature of the static MOG metric across different $\alpha$ sectors, explicitly linking the $-1 < \alpha < 0$ regime to brane-world tidal charges and identifying the conditions for naked singularities.
• Diagnostic Potential: The study establishes $\Delta t$ as a diagnostic tool to distinguish between GR and MOG. The magnitude and sign of $\alpha$ directly influence the TOA components, offering a method to constrain $\alpha$ if suitable binary systems are identified.

Results
Numerical estimates for the two binary systems reveal the following:

• Magnitude of $\Delta t_1$: The leading-order term $\Delta t_1$ is the dominant component.
  • For PSR-Cygnus X-1, $\Delta t_1$ is on the order of $10^{-2}$ to $10^{-1}$ microseconds ($\mu$s), depending on $\alpha$.
  • For PSR-SgrA*, $\Delta t_1$ is significantly larger, ranging from a few to $\sim 17$ $\mu$s.
• Higher-Order Terms: The second ($\Delta t_2$) and third ($\Delta t_3$) order terms are orders of magnitude smaller (ranging from $10^{-6}$ to $10^{-14}$ $\mu$s) and are deemed currently unmeasurable.
• Dependence on $\alpha$: The TOA values vary linearly with $\alpha$ in the leading order. Negative values of $\alpha$ (brane-world regime) yield negative delays relative to the GR baseline, while positive $\alpha$ yields positive delays. However, the paper notes that the sign is a relative difference and does not preclude measurement.
• Measurability: The authors conclude that while $\Delta t_2$ and $\Delta t_3$ are beyond current technological capabilities, $\Delta t_1$ for the PSR-SgrA* system (few $\mu$s) is potentially measurable with current technology, provided a suitable pulsar orbiting SgrA* is identified. For PSR-Cygnus X-1, the signal is smaller but potentially accessible with future precision (0.1 $\mu$s).

Significance and Claims
The paper claims that the TOA effect is a distinct observable from the Shapiro time delay, as it involves the subtraction of travel times rather than their addition, and it specifically samples the frame-dragging of a spinning lens. The primary significance lies in the potential to use future high-precision timing of pulsars in binary systems to test the validity of MOG theories against GR. The authors modestly state that while no such binary has been conclusively identified yet, the theoretical framework provided allows for the constraint of the MOG parameter $\alpha$ once suitable observational data becomes available. The study does not claim to have detected a deviation from GR but rather provides the necessary theoretical tools and order-of-magnitude estimates to facilitate such a test in the future.