Price-Law Tails in Schwarzschild

Quasinormal ringing is not the final signal of a massless scalar on Schwarzschild. Waves scattered by the long-range curvature potential leave an inverse-power tail. For a fixed spherical multipole with generic smooth, compactly supported initial data, its leading powers are $t^{-(2\ell+3)}$ at fixed radius and $u^{-(\ell+2)}$ for the radiation field at null infinity. This page explains the centrifugal cancellation behind those powers and why changing the initial data can remove the leading tail. The calculation is linear, on a fixed four-dimensional Schwarzschild exterior.

Required background. PDEs and Green Functions for Gravitation supplies retarded solutions, spectral transforms, and asymptotic estimates.

Helpful background. Black-Hole Geometry and Mechanics supplies the Schwarzschild exterior; Formulations of General Relativity distinguishes scalar and metric master variables; Causality and Global Structure supplies horizons and null infinity; Black-Hole Perturbation Theory places the tail after prompt response and quasinormal ringing.

The scalar equation and its observation limits

Let $M\gt0$ be the physical mass and $\mu=GM$. In the exterior $r\gt2\mu$, set

$$f(r)=1-\frac{2\mu}{r}, \qquad x=r_*=r+2\mu\log\left(\frac{r}{2\mu}-1\right), \qquad \frac{dx}{dr}=f^{-1}.$$

For $\Box_g\Phi=0$, use orthonormal spherical harmonics and write

$$\Phi=\sum_{\ell m}\Phi_{\ell m}(t,r)Y_{\ell m}(\theta,\phi), \qquad \Phi_{\ell m}=\frac{\psi_{\ell m}}{r}.$$

Spherical symmetry preserves each $(\ell,m)$ independently. Substitution into the wave equation gives

$$\begin{aligned} (\partial_t^2+A_\ell)\psi_{\ell m}&=0, & A_\ell&=-\partial_x^2+V_\ell,\\ V_\ell(r)&=f(r)\left[\frac{\ell(\ell+1)}{r^2} +\frac{2\mu}{r^3}\right]. \end{aligned}$$

Fix one multipole and take its initial displacement $h(x)=\psi_{\ell m}(0,x)$ and velocity $g(x)=\partial_t\psi_{\ell m}(0,x)$ to be smooth and compactly supported in $x$. Thus their support is separated from both the horizon and spatial infinity on this static slice. Compact support alone does not guarantee a nonzero leading coefficient.

The three observations use different restrictions of the solution. Define $v=t+x$, $u=t-x$, and the radiation field $\mathcal F_{\ell m}(u)=\lim_{r\to\infty}r\Phi_{\ell m}(u,r)$. The horizon field is the regular limit at $r=2\mu$ with $v$ fixed.

Observation	Quantity and limiting order	Generic leading power
Fixed exterior radius $r$	$\Phi_{\ell m}(t,r)$, then $t\to\infty$	$t^{-(2\ell+3)}$
Future event horizon	$\Phi_{\ell m}^{\mathcal H}(v)$, then $v\to\infty$	$v^{-(2\ell+3)}$
Future null infinity	First form $\mathcal F_{\ell m}(u)$ at fixed $u$, then let $u\to\infty$	$u^{-(\ell+2)}$

“Generic” means that the relevant leading data functional is nonzero. The fixed-radius and horizon statements include differentiated remainder control in regular coordinates; they are not estimates uniform in a simultaneous large-radius limit. A rigorous compact-data result for every fixed $\ell$ is given by Hintz 2022, Theorem 5.1 and Corollary 5.4, pp. 533 and 538, PDF. The radiation-field leading coefficients are described by Angelopoulos, Aretakis, and Gajic 2023, Theorem 2.3, preprint PDF p. 12, Eq. 2.5 for $\ell\ge1$, and Angelopoulos, Aretakis, and Gajic 2018, Theorem 1 and Remark 1.4, preprint PDF pp. 8–10 for the monopole.

To locate these observations geometrically, inspect the right exterior, future horizon and future null infinity in the existing Schwarzschild conformal diagram. A static observer stays in the exterior; the radiation field is defined on its null boundary. Replacing $t$ by $u$ in a fixed-radius formula does not perform that boundary limit.

What part of the potential produces a tail?

The flat-space centrifugal term $\ell(\ell+1)/r^2$ is not itself evidence for a tail. In four-dimensional Minkowski spacetime, a massless scalar with compact data obeys Huygens propagation: after the pulse has passed a fixed observation point, there is no continuing interior signal. Curvature changes the asymptotic radial scattering problem.

Write $L_\ell=\ell(\ell+1)$. Inverting the tortoise relation at large positive $x$ and substituting into the exact potential yields

$$\begin{aligned} V_\ell(x) ={}&\frac{L_\ell}{x^2} +\frac{4\mu L_\ell\log(x/(2\mu))+2\mu(1-L_\ell)}{x^3}\\ &+O\left(\frac{\mu^2\log^2(x/(2\mu))}{x^4}\right). \end{aligned}$$

This is an expansion at fixed $\mu\gt0$ and fixed finite $\ell$; each term has units of inverse length squared. The logarithm comes partly from expressing the centrifugal term in the wave-propagation coordinate $x$, rather than the areal radius $r$.

An outgoing pulse encounters weak scattering at arbitrarily large radius. A signal scattered farther away can return later. The potential becomes exponentially small toward the horizon, but its far end decreases only algebraically. In frequency space this far end produces a zero-frequency branch point in the outgoing solution. Isolated damped poles give exponentials; the branch endpoint can give a power that eventually outlives each such exponential when its coefficient is nonzero. This does not assert a universal crossover time or an expansion consisting only of poles.

Why the spatial logarithm does not give a leading logarithmic tail

There is a nontrivial cancellation hidden in a rule such as “an inverse cubic potential gives Price’s law.” Treat the centrifugal equation exactly and the weak far-zone correction to first Born order, meaning first order in scattering by that correction. For a power family $L^{-2}(x/L)^{-\alpha}$ with a fixed length $L\gt0$ and $\alpha\gt2$, the time-domain contribution has the structure $B_\ell(\alpha)C_\ell(\alpha)T^{-(2\ell+\alpha)}$, where $T=t/L$. The factor $B_\ell$ contains the normalization and the matching to the interior. Near $\alpha=3$ it is regular and nonzero for the scalar problem. The centrifugal factor is

$$C_\ell(\alpha) =\prod_{j=0}^{\ell-1}\frac{\alpha-2j-3}{\alpha+2j+1}, \qquad C_0(\alpha)=1.$$

The Hankel-function scattering calculation producing this factor and the cut power $\sigma^{2\ell+\alpha-1}$ is developed in Ching, Leung, Suen, and Young 1995, § VI, preprint PDF pp. 21–22, Eqs. 6.11 and 6.15–6.18. Its use here explains the far-zone exponent; it is not a first-Born approximation to the strong potential everywhere outside the horizon.

For $\ell\ge1$, the numerator with $j=0$ vanishes at $\alpha=3$. Thus a pure $x^{-3}$ correction has no first-Born $t^{-(2\ell+3)}$ coefficient in these sectors. Schwarzschild also has $\log(x/(2\mu))/x^3$. Generate that term by differentiating the power family with $-\partial_\alpha$. The same differentiation of the response gives

$$-\partial_\alpha\left[ B_\ell C_\ell T^{-(2\ell+\alpha)} \right] = \left[B_\ell C_\ell\log T -\partial_\alpha(B_\ell C_\ell)\right] T^{-(2\ell+\alpha)}.$$

At $\alpha=3$ the coefficient of $\log T$ vanishes, whereas

$$C_\ell'(3)=\frac{(-1)^{\ell-1}}{2\ell(\ell+1)} \ne0, \qquad \ell\ge1.$$

The leading result is therefore a pure $t^{-(2\ell+3)}$ power. For $\ell=0$ there is no centrifugal cancellation, but the logarithmic term in the displayed Schwarzschild potential vanishes; its $2\mu/x^3$ term directly gives $t^{-3}$. Neither argument excludes logarithms at subleading orders. The exact low-frequency calculation does contain them.

From the zero-frequency radial solution to the decay power

The radial shape of the leading tail is also computable. A static physical scalar multipole $R_\ell(r)$ satisfies

$$\frac{d}{dr}\left[r(r-2\mu)\frac{dR_\ell}{dr}\right] -\ell(\ell+1)R_\ell=0.$$

With $z=r/\mu-1$, this is Legendre’s equation on $z\gt1$. Its solution regular at the future horizon is $P_\ell(z)$; the independent $Q_\ell(z)$ solution is logarithmically singular there. Consequently the zero-frequency master solution normalized to one at the horizon is

$$u_\ell(r)=\frac{r}{2\mu}P_\ell\left(\frac r\mu-1\right), \qquad A_\ell u_\ell=0, \qquad u_\ell(2\mu)=1.$$

For example, $u_0=r/(2\mu)$ and $u_1=r(r/\mu-1)/(2\mu)$. The physical radial factor is $u_\ell/r$, not $r^\ell$ at finite radius. Its large-radius growth is proportional to $r^\ell$; those are different statements. The exact static profile is also the leading profile in Hintz 2022, Remark 5.2 and Eq. 5.8, pp. 533 and 537, PDF.

Let $G_\ell(t;x,x')$ be the retarded kernel of $\partial_t^2+A_\ell$:

$$(\partial_t^2+A_\ell)G_\ell =\delta(t)\delta(x-x'), \qquad G_\ell=0\quad(t\lt0).$$

The frequency-domain solution is ingoing at the horizon and outgoing at infinity. Put the downward cut at $\omega=-i\sigma$, $\sigma\gt0$, and define $\Delta\widetilde G_\ell$ as the east bank minus the west bank. With this orientation,

$$G_\ell^{\mathrm{cut}}(t;x,x') =\int_0^\infty e^{-\sigma t} \frac{\Delta\widetilde G_\ell(-i\sigma;x,x')}{2\pi i}\,d\sigma.$$

The contour normalization and factorized discontinuity are given in Casals and Ottewill 2015, § II A, PDF p. 5, Eqs. 2.19–2.21. Their kernel equation has the opposite wave-operator sign and a source $-\delta$; multiplying both sides by $-1$ gives the positive-wave normalization above without changing the kernel Casals and Ottewill 2015, § II A, PDF p. 3, Eqs. 2.6–2.8. Their small-frequency expansion gives, at fixed exterior $x,x'$,

$$\frac{\Delta\widetilde G_\ell}{2\pi i} =b_\ell u_\ell(r)u_\ell(r')\sigma^{2\ell+2} +O(\sigma^{2\ell+3}), \qquad b_\ell\ne0.$$

Here $b_\ell$ is a background- and normalization-dependent constant, not an initial-data coefficient. The omitted terms have controlled frequency derivatives; at higher orders they include logarithms with dimensionless arguments. The full expansion, not just a guessed spatial power, supplies this statement Casals and Ottewill 2015, §§ VII A–B, PDF pp. 31–32, Eqs. 7.1–7.6.

For $q\gt-1$, setting $y=\sigma t$ gives

$$\int_0^\infty e^{-\sigma t}\sigma^q\,d\sigma =\Gamma(q+1)t^{-q-1}.$$

Apply this to the endpoint with a smooth cutoff equal to one near zero and supported at small frequency; call this piece $G_\ell^{\mathrm{end}}$. The chapter overview spells out sufficient endpoint bounds. At fixed exterior radii this gives

$$G_\ell^{\mathrm{end}}(t;x,x') =\kappa_\ell u_\ell(r)u_\ell(r')t^{-p} +O(t^{-p-1}), \qquad p=2\ell+3,\quad \kappa_\ell=b_\ell\Gamma(2\ell+3).$$

The kernel is dimensionless, so $b_\ell$ and $\kappa_\ell$ have units of length to the power $p$. Their sign is not inferred from the exponent. This endpoint calculation alone does not bound every other contour piece. The full-solution estimates used below require the independent decay theorem. Nor does it permit taking $r\to\infty$ inside a fixed-radius remainder estimate. The radiation-field limit instead yields $u^{-(\ell+2)}$. For the horizon, use a horizon-regular time and the regular extension of the solution, not $t$ at $x=-\infty$.

Which initial data excite the leading tail?

The initial-value solution is

$$\psi_{\ell m}(t,x) =\int_{\mathbb R} \left[G_\ell(t;x,x')g(x') +\partial_tG_\ell(t;x,x')h(x')\right]dx'.$$

The two terms do different jobs: $G_\ell$ propagates initial velocity, whereas its time derivative propagates initial displacement. The endpoint expansion identifies the leading coefficient after integrating over the compact data support. Independently controlling the full solution gives the following statement at fixed exterior radius; we use a weaker half-power remainder than the detailed endpoint expansion:

$$\begin{aligned} J_g&=\int_{\mathbb R}u_\ell(r(x'))g(x')\,dx',\\ \Phi_{\ell m}(t,r) &=\frac{\kappa_\ell u_\ell(r)}{r}J_g\,t^{-p} +O(t^{-p-1/2}). \end{aligned}$$

This makes “generic” concrete on the chosen static initial slice: $J_g\ne0$. It is a weighted velocity moment, not the unweighted integral of the field. The static solution is positive outside the horizon, so one-signed nonzero real velocity data cannot cancel this moment; sign-changing profiles can.

In the physical-space treatment the corresponding leading functional is expressed using a time-inverted Newman–Penrose charge: an asymptotic conserved quantity of a solution whose time derivative is the original field. Its precise normalization depends on the harmonic and field conventions. We do not identify it numerically with $J_g$ without that translation. The compact-slice velocity pairing and the full-solution remainder follow independently from the static profile, resolvent estimate and forcing calculation in Hintz 2022, § 5, pp. 537–538, Eq. 5.11, PDF.

For momentarily stationary compact data, $g=0$. If $J_h=\int u_\ell h\,dx'$ is nonzero, differentiating the endpoint term identifies the coefficient. Using differentiated full-solution remainder control gives

$$\Phi_{\ell m}(t,r) =-\frac{p\kappa_\ell u_\ell(r)}{r}J_h\,t^{-p-1} +O(t^{-p-3/2}).$$

The extra inverse-time power is not an assumption about a fit: the velocity term is absent and the displacement term differentiates the tail. The corresponding generically sharp bound is $t^{-(2\ell+4)}$ on fixed bounded radial regions, including the horizon in regular time Hintz 2022, Corollary 5.4 and proof, p. 538, PDF. If $J_g=0$ for sign-changing velocity data but $g$ is not identically zero, subleading velocity terms also contribute. A vanishing leading moment alone does not determine the next nonzero coefficient or power.

Compact momentarily stationary data are not a noncompact initially static exterior multipole. The latter already extends to arbitrarily large radius and has different low-frequency content; compact-support convolution and remainder arguments cannot simply be reused. Likewise, scalar results do not directly give a metric component, a normalized Teukolsky variable, nonlinear stability, or detector strain.

Interpreting a measured power

A numerical evolution should test the declared observation limit: compare two fixed radii, use regular advanced time on the horizon, and extract the radiation field along outgoing null hypersurfaces with retarded time held fixed. Refine the spatial grid, extraction procedure, and time window. Keep the outer boundary causally out of the fitted interval; a reflected pulse can imitate or obscure a tail. These are requirements for a future evolution test, not a claim that this page has run one.

A local logarithmic slope is useful away from zeros, but a short plateau can be misleading. For the diagnostic signal as a function of dimensionless time, $X(T)=A T^{-p}+B T^{-p-1}$, with $T=t/L\gt0$, one finds

$$p_{\mathrm{loc}} =-\frac{d\log(|X|/X_0)}{d\log T} =p+\frac{B}{AT+B}, \qquad X_0\gt0,\quad AT+B\ne0.$$

Here $A,B$ are real, and $X_0$, $A$ and $B$ have the signal’s units. For complex amplitudes the correction is instead $\operatorname{Re}[B/(AT+B)]$. If $A$ is small, the signal can resemble the faster subleading power until very large $T$. If $A=0$ and $B\ne0$, that faster power really is the leading one. This elementary two-term diagnostic is not a Schwarzschild evolution.

The massless flat-space limit is another check on interpretation: as $\mu\to0$, the curvature-induced cut disappears in a consistently specified flat-limit problem. One cannot keep a nonzero Schwarzschild tail amplitude while deleting the curvature. Nor should a fixed-black-hole late-time expansion be assumed uniform in that limit.

Exercises

A spatial logarithm and a temporal logarithm. For $\ell=1$, compute $C_1(3)$ and $C_1'(3)$. Compare the first-Born leading terms generated by $x^{-3}$ and by $x^{-3}\log(x/L)$.

Solution

Here $C_1(\alpha)=(\alpha-3)/(\alpha+1)$, so $C_1(3)=0$ and $C_1'(3)=1/4$. The pure inverse-cubic correction has a vanishing first-Born $t^{-5}$ term. For the logarithmic correction, differentiating with $-\partial_\alpha$ leaves $-B_1(3)T^{-5}/4$; its potential’s overall coefficient must also be included. The $T^{-5}\log T$ term vanishes. This does not say that a pure inverse-cubic potential has no tail at any higher scattering order.

Cancel the leading tail without time symmetry. Let $g_1$ and $g_2$ be two nonnegative smooth compact velocity profiles with disjoint supports and nonzero moments $J_1,J_2$ against $u_\ell$. Construct nonzero velocity data with zero leading tail moment. What can be concluded about its decay?

Solution

Take

$$g=g_1-\frac{J_1}{J_2}g_2.$$

Its moment is $J_g=J_1-(J_1/J_2)J_2=0$, while disjoint supports ensure $g$ is not identically zero. The $t^{-(2\ell+3)}$ coefficient disappears. The next coefficient depends on subleading velocity contributions and on the initial displacement; its power cannot be determined from $J_g=0$ alone. These data are not momentarily stationary.

Identify a false plateau. In the diagnostic above let $p=3$, $A,B\gt0$, and $B/A=100$. At $T=100$, what is the local power? How large must $T$ be for the relative error from $3$ to be at most one percent? Finally, give the generic fixed-radius and radiation-field powers for a scalar quadrupole.

Solution

The local power is $3+100/(T+100)$, hence $3.5$ at $T=100$. The one-percent condition is

$$\frac{100}{3(T+100)}\le0.01 \quad\Longleftrightarrow\quad T\ge\frac{9700}{3}.$$

A fit much earlier than this can overestimate the asymptotic power. For $\ell=2$ the generic fixed-radius field decays as $t^{-7}$, whereas the radiation field decays as $u^{-4}$. The second is not obtained by substituting $t=u$ in the first.

Continue to Branch-Cut Contributions to Retarded Response for the contour calculation, or Kerr Mode Coupling and Tail Exponents to see why spherical multipoles no longer evolve independently.

References

• Angelopoulos, Yannis, Stefanos Aretakis, and Dejan Gajic. “Late-Time Asymptotics for the Wave Equation on Spherically Symmetric, Stationary Spacetimes.” Advances in Mathematics 323 (2018): 529–621. DOI. Open PDF.
• Angelopoulos, Yannis, Stefanos Aretakis, and Dejan Gajic. “Price’s Law and Precise Late-Time Asymptotics for Subextremal Reissner–Nordström Black Holes.” Annales Henri Poincaré 24 (2023): 3215–3287. DOI. Open PDF.
• Casals, Marc, and Adrian C. Ottewill. “High-Order Tail in Schwarzschild Space-Time.” Physical Review D 92 (2015): 124055. DOI. Open PDF.
• Ching, E. S. C., P. T. Leung, W. M. Suen, and K. Young. “Wave Propagation in Gravitational Systems: Late Time Behavior.” Physical Review D 52 (1995): 2118–2132. DOI. Open PDF.
• Hintz, Peter. “A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes.” Communications in Mathematical Physics 389 (2022): 491–542. DOI. Open PDF.