[Paper Review] How much of the outgoing radiation can be intercepted by Schwarzschildean black holes?

Edward Malec|ArXiv.org|Sep 11, 2000

Astrophysical Phenomena and Observations3 citations

TL;DR

This paper investigates how much of an outgoing electromagnetic pulse can be backscattered and intercepted by a Schwarzschild black hole due to spacetime curvature. Using a linearized analysis of dipole radiation in Schwarzschild geometry, it derives an upper bound on the fraction of energy lost to backscattering, showing the effect is significant for long wavelengths (near the Schwarzschild radius) but negligible in the short-wavelength limit.

ABSTRACT

The Schwarzschild spacetime is for electromagnetic waves like a nonuniform medium with a varying refraction index. A fraction of an outgoing radiation scatters off the curvature of the geometry and can be intercepted by a gravitational center. The amount of the intercepted energy is bounded above by the backscattered energy of an initially outgoing pulse of electromagnetic radiation, which in turn depends on the initial energy, the Schwarzschild radius and the pulse location. Its magnitude depends on the frequency spectrum: it becomes negligible in the short wave limit but can be significant in the long wave regime.

Motivation & Objective

• To quantify the maximum fraction of outgoing electromagnetic radiation that can be intercepted by a Schwarzschild black hole due to curvature-induced backscattering.
• To investigate the dependence of backscattering on the frequency spectrum of the initial radiation pulse.
• To establish a rigorous upper bound on energy loss due to backscattering in the absence of backreaction on the metric.
• To clarify the conditions under which backscattering becomes non-negligible, particularly for long-wavelength radiation.
• To assess the astrophysical relevance of backscattering for compact objects like black holes and neutron stars.

Proposed method

• Models electromagnetic radiation using the Regge-Wheeler tortoise coordinate and solves the wave equation for dipole (l=1) modes in Schwarzschild spacetime.
• Decomposes the electromagnetic potential into a Minkowski-like solution and a perturbation δ, with initial data assumed purely outgoing (g=0).
• Derives the evolution equation for δ from the Maxwell equations in curved spacetime, incorporating the curvature potential term.
• Defines the inward-directed backscattered intensity h₋ via the radial derivative of δ, and uses energy flux conservation along outgoing null cones.
• Applies energy integral estimates along the initial null cone Cₐ to bound the energy loss δEₐ, leading to an upper bound on δEₐ/Eₐ(0).
• Performs Fourier analysis on initial data to relate pulse width (b−a) to frequency content, showing that narrow pulses correspond to high-frequency radiation.

Experimental results

Research questions

• RQ1 What is the maximum fraction of outgoing electromagnetic energy that can be backscattered and intercepted by a Schwarzschild black hole?
• RQ2 How does the magnitude of backscattering depend on the frequency spectrum of the initial radiation pulse?
• RQ3 Under what conditions does backscattering become significant rather than negligible?
• RQ4 How does the distance from the black hole (a) and the pulse width (b−a) affect the backscattering efficiency?
• RQ5 Can the backscattering effect be quantitatively bounded in a way that is independent of the specific pulse shape?

Key findings

• The fraction of energy intercepted by the black hole is bounded above by an expression proportional to (2m/a)² and a geometric factor depending on a and b, with the bound decreasing for larger a or narrower pulses.
• For a pulse centered at a = 4m with width b−a ≈ 2m (i.e., critical width), the backscattered energy fraction is bounded by δEₐ/Eₐ(0) < 0.37, indicating significant energy loss.
• For shorter-wavelength radiation (e.g., b−a ≈ Rₛ/8 at a = 4m), the bound drops to δEₐ/Eₐ(0) < 0.001, showing that high-frequency radiation is effectively unimpeded.
• The critical frequency ω_c ≈ π/R_S is identified as the threshold below which backscattering becomes significant; most of the energy in a pulse with b−a ≈ 2m lies at or below this frequency.
• For solar-type stars, backscattering is negligible (≤10⁻²⁰), but for white dwarfs it could reach ~10⁻⁵, though sharper estimates would reduce this.
• The results suggest that long-wave echoes from black hole horizons could have amplitudes up to 20% of the incident wave, based on numerical studies of scalar fields.