Tidal Disruption Tests of Spacetime

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A star passing close to a massive black hole may be partially stripped, fully disrupted, or captured. Relativistic capture and stream precession depend on spacetime, but the observed transient is separated from that encounter by stellar structure, hydrodynamics, circularization, accretion, winds, and reprocessing. Tidal-disruption events can therefore test threshold and scaling predictions only through a population-aware forward model. The evidence boundary is 2026-08-19.

Helpful background. Weak-Field Gravity supplies the Newtonian tidal and precession limits; Probability and Statistical Inference supplies transient-selection likelihoods; Black-Hole Geometry and Mechanics supplies capture, spin, and horizon scales; and Black-Hole Astrophysics supplies stellar disruptions, stream dynamics, accretion, outflows, and reprocessing.

Disruption and capture scales

For a star of mass $M_*$ and radius $R_*$ approaching a black hole of mass $M$ , the Newtonian tidal radius is

r_t\simeq R_*\left(\frac{M}{M_*}\right)^{1/3}, \qquad \beta=\frac{r_t}{r_p},

where $r_p$ is pericenter and $\beta$ is the penetration factor. Disruption must occur outside the relevant capture boundary to produce returning debris. That boundary depends on orbital energy and angular momentum, black-hole spin and inclination, and the assumed metric; it is not universally the event-horizon radius.

The tidal estimate follows by comparing differential black-hole acceleration across the star, of order $GM R_*/r^3$ , with stellar self-gravity $GM_*/R_*^2$ . Its order-unity coefficient depends on stellar structure and the chosen disruption criterion. Even the simplest relativistic capture example differs from a horizon comparison: a point particle arriving on a parabolic Schwarzschild orbit has specific energy $\mathcal E=1$ and radial equation

\left(\frac{dr}{d\tau}\right)^2 =1-\left(1-\frac{2GM}{r}\right) \left(1+\frac{L^2}{r^2}\right),

where $L$ is specific angular momentum. Requiring a double turning point gives $r=4GM$ and $L=4GM$ , the marginal capture orbit; the horizon is at $2GM$ . This is a point-particle threshold for that energy, not an exact disruption boundary for a finite star. It explains why replacing the capture condition by $r_t\gt2GM$ is inadequate.

A common energy-spread estimate is

\Delta\epsilon\sim\frac{GM R_*}{r_t^2}.

This estimate evaluates the potential difference across the star at $r_t$ ; moving the assumed freeze-in radius to pericenter would change its dependence on $\beta$ and must be physically justified. For debris of negative Newtonian specific orbital energy $\epsilon$ , Kepler’s law gives the return time and its inverse,

t(\epsilon)=\frac{2\pi GM}{(2\lvert\epsilon\rvert)^{3/2}}, \qquad \lvert\epsilon(t)\rvert =\frac12\left(\frac{2\pi GM}{t}\right)^{2/3}.

Thus, for ballistic debris released near pericenter,

\dot M_{\rm fb}(t) =\left.\frac{dM}{d\epsilon}\right|_{\epsilon(t)} \left|\frac{d\epsilon}{dt}\right| =\left.\frac{dM}{d\epsilon}\right|_{\epsilon(t)} \frac{(2\pi GM)^{2/3}}{3t^{5/3}}.

The $t^{-5/3}$ law therefore requires a finite, approximately constant $dM/d\epsilon$ near zero binding energy, as well as nearly Keplerian returns. It does not require prompt conversion of fallback into luminosity. Partial disruptions, stellar density profiles, relativistic energy shifts, surviving cores, and repeated passages can change this distribution. A measured $t^{-5/3}$ light curve additionally needs a radiation model; it is not a direct strong-field signature. The Newtonian energy-to-return map and its relativistic extension are developed in Kesden 2012b, §§II–III.

Relativistic apsidal precession can make outgoing and incoming streams intersect, while spin-induced nodal precession can delay intersection. Yet self-intersection does not uniquely determine circularization or luminosity: shocks, magnetic stress, cooling, viscosity, winds, and optical depth decide how orbital energy becomes radiation. The effect of relativistic tides and capture on disruption rates is calculated in Kesden 2012a, §§II–IV; debris-energy and fallback calculations are a separate result Kesden 2012b, §§III–IV.

From encounter to selected transient

The forward chain should keep its layers visible:

(\text{star},\text{orbit},\text{metric}) \to \frac{dM}{d\epsilon} \to \dot M_{\rm fb} \to \text{stream and disk} \to L_\nu(t) \to \text{survey data}.

Multi-band light curves, temperatures, line profiles, X-ray behavior, polarization, and radio outflows probe different downstream stages. Host-galaxy photometry and spectroscopy infer black-hole mass only through scaling relations with intrinsic scatter. Extinction, host subtraction, bolometric corrections, cadence, and classification affect the measured population.

A versioned dataset records alert and forced-photometry release, filters, calibration, difference-imaging response, cadence and seasonal gaps, host reference, spectra, X-ray and radio products, quality cuts, and classification rules. The Zwicky Transient Facility sample construction illustrates how alert selection, host association, spectroscopy, and completeness enter a population statement van Velzen et al. 2021, §§2–5. A bright, optically selected sample does not directly constrain all disruptions, especially obscured, X-ray-selected, jetted, faint, or directly captured encounters.

For population parameters $\Lambda$ ,

p(\{d_i\}\mid\Lambda,\mathrm{det}) \propto \prod_i\frac{\int p(d_i\mid\lambda_i) p(\lambda_i\mid\Lambda)d\lambda_i} {\alpha(\Lambda)},

where $\alpha$ is the survey detection fraction and $\lambda_i$ contains stellar type, orbit, black-hole mass and spin, hydrodynamics, reprocessing, extinction, and host properties. A rate-aware likelihood also includes the expected count. Priors on the stellar mass function, nuclear dynamics, occupation fraction, black-hole mass function, and emission model can dominate a proposed capture cutoff.

Reproduction requires transient identifiers, calibrated photometry and spectra, selection code and injections, host and mass-estimator versions, simulation initial data, stellar model, relativistic dynamics, resolution and convergence, radiation or reprocessing prescription, priors, and posterior samples. This page performs no new hydrodynamic calculation.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
Multi-band transients with nuclear locations and evolving spectra	Support a population of stellar-disruption candidates	Classification, host subtraction, extinction, and emission mechanism remain uncertain
Fallback-like decline or stream-collision signatures	Connect some light curves to disruption and circularization models	Stellar structure, partial disruption, shocks, viscosity, and reprocessing alter the mapping
Mass-dependent event rates or high-mass cutoff	Could probe the competition between disruption and relativistic capture	Black-hole masses, spin inclinations, stellar populations, survey selection, and obscuration are degenerate

The conservative position is that tidal events test coupled encounter-and-radiation models. A competing interpretation may identify a light-curve break, delayed flare, or population cutoff directly with precession or a horizon, but hydrodynamic and selection alternatives can produce similar patterns. Absence of luminous high-mass events is negative evidence only after occupation, feeding, extinction, and detection efficiency are controlled. No current transient alone establishes a horizon or unique metric; the conclusion remains unresolved and model-dependent.

The adversarial check simulates encounters spanning full disruption, partial disruption, and direct capture, then passes independent hydrodynamic and reprocessing outputs through the survey cadence and selection. An invariant threshold or precession scaling should be recovered across resolutions and stellar models and should predict held-out spectral or timing behavior. Agreement from codes sharing the same freeze-in and radiation assumptions is not independent.

What could change the conclusion. The interpretation could change after a superseding transient catalog or calibration, a validated selection function across optical and X-ray samples, or converged relativistic simulations show that capture or precession signatures survive independent hydrodynamic and reprocessing prescriptions.

Exercise

At fixed stellar mass and radius, how does $r_t/(GM)$ scale with black-hole mass, and why does that scaling motivate a high-mass suppression of luminous disruptions?

Solution

The tidal radius scales as $M^{1/3}$ , whereas the gravitational radius scales as $M$ . Their ratio therefore scales as $M^{-2/3}$ . As black-hole mass increases, disruption occurs progressively closer in gravitational units and can fall inside the orbit-dependent capture boundary, leaving no returning debris to power a conventional flare. The population cutoff is not universal because spin, inclination, stellar structure, nuclear feeding, emission efficiency, and survey selection all shift the observed suppression.

References

Kesden, M. (2012a), “Tidal-Disruption Rate of Stars by Spinning Supermassive Black Holes,” Physical Review D 85, 024037. DOI.
Kesden, M. (2012b), “Black-Hole Spin Dependence in the Light Curves of Tidal Disruption Events,” Physical Review D 86, 064026. DOI. Open PDF.
van Velzen, S., et al. (2021), “Seventeen Tidal Disruption Events from the First Half of ZTF Survey Observations,” Astrophysical Journal 908, article 4. DOI.