How do Chern-Simons and Maxwell couplings affect scalarization of Kerr-Newman black holes?

Two couplings produce two asymmetric threshold branches

The paper studies linear scalar perturbations around Kerr-Newman black holes in the Einstein-Chern-Simons-Maxwell-scalar theory, with a quadratic Chern-Simons coupling gamma (written as gamma-hat = gamma/M^2) and a Maxwell coupling eta. Scalarization begins where the bald black hole becomes marginally unstable, so the quantity of interest is the threshold coupling at which a stationary axisymmetric zero mode first appears.

For pure Maxwell coupling, gamma-hat = 0, the stationary calculation at a/M = 0.9 and Q/M = 0.4 gives two real roots, eta = 20.5987 and eta = -102.8439. These are not symmetric: the negative branch is roughly five times larger in magnitude than the positive one. The paper reports that the roots change by less than 2 x 10^-6 between the two numerical grids used, so the asymmetry is not a resolution artifact.

The two branches are also plotted as global axisymmetric m = 0 curves. The negative branch requires a/M > a_loc/M = 0.6723, and the resolved negative branch covers 0.69 <= a/M <= 0.916, staying above that bound and below the extremal spin a_e/M ~ 0.9165. The positive branch extends to the nonrotating limit, with eta = 32.5549 at a = 0.

Equal couplings lower the onset spin well below the local sign change

When the dimensionless couplings are set equal, gamma-hat = eta = lambda, the threshold problem reduces to an eigenvalue problem in lambda. For lambda = -10 the paper finds a global onset at a_crit/M ~ 0.1558. That is far below the northern-polar local sign-change spin a_loc/M ~ 0.8.

The reason given is that the odd Chern-Simons source permits a negative effective mass squared in the southern exterior at lower spins. This is the paper's clearest statement about mechanism: a local negative-mass region is a necessary-looking ingredient, but it is not sufficient to predict where the global instability actually begins. The stationary calculation retains the full two-dimensional angular dependence, which is what allows the southern contribution to matter.

The paper also reports that for equal couplings both threshold magnitudes decrease as spin increases. So the equal-coupling case is not just a special slice of the pure-Maxwell picture; it changes both the location and the spin dependence of the onset.

Unequal couplings merge the two real branches

For unequal Chern-Simons and Maxwell couplings, the two real threshold branches merge as spin or Chern-Simons coupling increases. At fixed a/M = 0.9, the branch eta_1 crosses eta = 0 at the pure-Chern-Simons threshold gamma-hat = 0.7257, obtained from the stationary equation with eta = 0. Beyond that point both roots are negative and approach each other, merging near (gamma-hat_coll, eta_coll) ~ (1.18, -63.69).

The merger spin depends on the fixed Chern-Simons coupling. The paper reports (a_coll/M, eta_coll) ~ (0.90665, -45.76) for gamma-hat = 1 and (a_coll/M, eta_coll) ~ (0.75714, -638.1) for gamma-hat = 5. A stronger Chern-Simons coupling therefore brings the merger to a smaller spin. The merger spins agree at the quoted precision across the three checked grids, up to (N_s, N_x) = (30, 29).

Beyond the merger the tracked roots form complex-conjugate pairs, shown as gray bands. The paper is explicit that these are coupling eigenvalues of the discretized stationary equation, not mode frequencies, so their complex continuation does not determine stability at real coupling.

Time-domain growth rates are finite-time diagnostics, not thresholds

The second calculation is a hyperboloidal Racz-Toth evolution that places the horizon and future null infinity on a finite computational domain and follows the response to a prescribed Gaussian wave packet. The effective growth or decay rate Gamma is extracted from a fit to the logarithmic scalar amplitude over a finite time window, and the paper reports the dimensionless combinations M*Gamma.

At M = 1, a/M = 0.9, Q/M = 0.4, gamma-hat = 1, and l_init = m = 0, the signals at eta = -20 and -90 grow, while those at eta = -30 and -50 decay and develop late-time tails. The near-threshold signals at eta = -24 and -82.75 have much smaller late-time slopes, with eta = -82.75 changing from early decay to weak growth. The raw and envelope fits agree in sign over 450 <= T/M <= 800.

The paper is careful about what this rate means. A positive or negative Gamma indicates growth or decay of the extracted amplitude over the selected interval only. It need not coincide with the imaginary part of a converged quasinormal-mode frequency, particularly near threshold where transients or late-time tails contribute. Growth or decay is identified only when the raw and envelope fits agree in sign and |Gamma| > delta Gamma, and a negative slope approaching zero is recorded as a tail rather than as evidence for a strictly damped eigenmode.

Local negative mass does not fix the global onset

The paper maps the sign of the effective mass squared in the (a/M, r/M) plane at theta = 0, pi/3, and 0.9pi/2, for M = 1, Q/M = 0.4, gamma-hat = 1, and eta = -70 or -90. Increasing |eta| strengthens a positive Maxwell contribution where F^2 < 0, but a negative one where F^2 > 0 near the polar horizon, which produces the angular contrast between the polar and near-equatorial maps.

The decisive comparison is that both the eta = -20 and eta = -30 maps contain negative regions, yet the corresponding waveforms at a/M = 0.9 grow and decay respectively. A local negative-mass region alone therefore does not determine the time-domain response. The paper states this directly: the maps identify local negative-mass regions, whose presence alone does not determine the time-domain response.

This is why the stationary zero-mode problem is solved on the full (r, theta) domain with m = 0, regularity at the horizon and on the axes, and decay at infinity. The resulting curves are global axisymmetric marginal-stability curves and are not obtained by interpolating the finite-time waveforms. The paper also notes that m = 0 denotes axisymmetry; it does not impose a pure l = 0 harmonic or reduce the calculation to the local slice theta = 0.

Spin and charge change the response without defining a scaling law

The charge and spin scans show why the local effective mass alone is not enough. For the negative-coupling samples at Q/M = 0.1 and a/M = 0.9, growth occurs at eta = -4860, whereas the fitted slope at eta = -4800 changes sign with the fitting window. The positive-coupling sample eta = 90 also grows. The factor Q^2 in the Maxwell invariant explains the weaker Maxwell source, but changing Q/M also changes the distance from extremality, so the paper does not treat these samples as defining a charge-scaling law.

At a/M = 0.5 and Q/M = 0.4, all three sampled scalar perturbations decay through T/M = 850. The spin changes both the source profiles and the wave operator, so these time-domain samples are used as dynamical observations rather than as global linear instability thresholds.

At fixed a/M = 0.9, Q/M = 0.4, and eta = -50, increasing gamma-hat also changes the response. The signals for gamma-hat = 0, 0.5, and 1 decrease through T/M = 400, whereas the signal for gamma-hat = 1.25 grows. Because only the Chern-Simons contribution to the effective mass is varied here, the paper uses this as an isolation of its role in the competition with the Maxwell term.

Mode mixing leads to late-time monopole dominance

The background conserves the azimuthal number m but not the angular multipole l: the rotating wave operator and the angular dependence of the effective mass squared mix harmonics during evolution. The paper tests this with wave packets proportional to Y_{l_init 0} for l_init = 0, 1, 2, 3 at a/M = 0.9, Q/M = 0.4, and (gamma-hat, eta) = (1, -20).

All four initial multipoles yield M*Gamma ~ 5.55 x 10^-3 over 250 <= T/M <= 380, and the normalized monopole power fraction P_0 at T/M = 400 exceeds 0.99. The late-time growth is therefore dominated by the monopole component with nearly the same rate for all four initial multipoles, although their transient responses differ. The paper explains that the rotating wave operator and the Maxwell source preserve equatorial parity, whereas the odd Chern-Simons source couples even and odd harmonics within the same m sector, so an initially odd wave packet can also excite the l = 0 component.

Two boundaries matter here. The monopole dominance refers to the evolved field, while the stationary zero-mode calculation retains the full angular dependence. And the normalized multipolar power fraction P_l is computed with the denominator retaining the five harmonics 0 <= l <= 4, so it is a normalized multipolar power fraction at infinity, not an energy fraction of the full perturbation. Higher harmonics and the convergence of the fractions with angular resolution have not been tested.

Azimuthal sectors and initial data limit what the rates show

For m = 0, 1, 2 with the lowest allowed initial multipole l_init = |m|, corresponding to (l_init, m) = (0,0), (1,1), (2,2), at a/M = 0.9, Q/M = 0.4, and (gamma-hat, eta) = (1, -20), the m = 0 signal grows while the m = 1 and m = 2 signals decrease through T/M = 400. Extending the m = 1, 2 runs at eta = -20, -50, -90 to T/M = 850 gives decaying tails in all six cases.

The paper states that this decay over a finite time does not establish stability of the m = 1, 2 sectors, which requires separate stationary or spectral threshold calculations. The analysis is restricted to linear perturbations, and no nonlinear scalarized solutions are constructed.

Initial data also matter near marginality. For eta = -24 and -82.75, the late-time slopes for the original Gaussian initial data remain positive under the tested changes of fitting window, resolution, time step, and observation angle. But changing the radial centre or packet width delays the visible growth, so the modified packets initially give decaying fits and show growth only in longer evolutions, with different fitted rates. The paper concludes that the critical coupling is determined from the stationary equation, while the time-domain evolution tests the associated dynamical response.

What to watch in this threshold map

The most useful way to read this paper is as a comparison between two kinds of statement. The stationary zero-mode calculation gives global axisymmetric marginal-stability curves with the full angular dependence retained. The time-domain evolution gives finite-time growth or decay rates at representative points, and those rates depend on the initial data and the fitting window near marginality. The paper's own checks, such as the equal-coupling case lambda = -10 where the fitted rates at a/M = 0.15 and 0.16 are negative and positive respectively, support the correspondence between the two, but the rates are not themselves the thresholds.

The paper-specific priority is therefore the branch structure rather than any single number. For pure Maxwell coupling at Q/M = 0.4, the two branches are asymmetric, with eta = 20.5987 and eta = -102.8439 at a/M = 0.9. For equal couplings, the onset moves to a_crit/M ~ 0.1558 at lambda = -10, well below the local sign-change spin a_loc/M ~ 0.8. For unequal couplings, the two real branches merge, and a stronger fixed Chern-Simons coupling brings the merger to a smaller spin: a_coll/M ~ 0.90665 at gamma-hat = 1 versus a_coll/M ~ 0.75714 at gamma-hat = 5.

The practical threshold to carry away is that a negative effective-mass region in a polar or near-equatorial map is not a scalarization criterion. The onset is set by the global axisymmetric zero mode, and the Chern-Simons and Maxwell couplings can move it, split it into two branches, or merge those branches depending on their relative size and on the spin.

Sources used in this answer