Static Love Numbers of Schwarzschild and Kerr in Four-Dimensional Vacuum GR

The famous zero is a sharply delimited statement: in four-dimensional vacuum general relativity, the conservative linear static tidal Love numbers of an asymptotically flat Schwarzschild or Kerr black hole vanish when the solution is regular on the future horizon and the applied and induced branches are matched in the standard gauge-invariant or worldline scheme. It is not a claim that a horizon cannot absorb, that every dynamic coefficient vanishes, or that an arbitrary finite coefficient of a bulk master field is zero.

Required background. Presymplectic Potentials and Currents supplies the role of boundary regularity and physical versus gauge variations.

Helpful background. Black-Hole Geometry and Mechanics supplies the Schwarzschild and Kerr horizons; PDEs and Green Functions for Gravitation supplies the two radial branches and boundary-value logic; Formulations of General Relativity supplies gauge-invariant metric perturbations; Relativistic Two-Body Dynamics supplies STF tidal and multipole moments; and Black-Hole Perturbation Theory supplies the common mode normalization.

Conservative static tidal response

Theorem used on this page. Let $g_{ab}$ be an asymptotically flat Schwarzschild or subextremal Kerr metric of mass $M$ in four-dimensional vacuum general relativity. Consider a linear spin-2 perturbation with a fixed integer multipole $\ell\geq2$, stationary external tidal data, and a solution smooth in coordinates regular on the future event horizon. Define the electric and magnetic induced moments by buffer-zone matching to normalized STF tidal moments, equivalently by the corresponding local worldline Wilson coefficients in the prescription specified below. Then the conservative static electric and magnetic response maps vanish. Schwarzschild has coefficients $k^E_\ell=k^B_\ell=0$; for Kerr the statement applies to the symmetric conservative map, not the full complex $m$-resolved response.

Asymptotic flatness describes the background. A stationary external tide grows in the buffer solution and is matched to distant sources; it is not an isolated asymptotically flat perturbation extended unchanged to infinity.

The response convention is

$$Q_L(\omega)=-\lambda^E_\ell(\omega)\mathcal E_L(\omega), \qquad S_L(\omega)=-\lambda^B_\ell(\omega)\mathcal B_L(\omega),$$

Here the scalar notation refers to a spherical or specified diagonal angular sector. For Schwarzschild’s conservative static response the electric normalization is $\lambda^E_\ell(0)=2(\ell-2)!k^E_\ell R_0^{2\ell+1}/[G(2\ell-1)!!]$, with fixed reference radius $R_0\gt0$. The result is static and conservative, in four dimensions, for a gravitational field. The applied branch is the normalized growing solution and the induced branch is the matched falling solution. Schwarzschild may be computed in horizon-regular light-cone gauge; Kerr may be computed from a Teukolsky curvature variable and then matched to the same worldline operator basis. The order is: fix the background, $m$, gauge-invariant observable, and subtraction scheme; impose the horizon condition; match in the overlap region; then take the conservative $\omega\to0$ coefficient. The extremal limit is not taken inside this theorem.

These hypotheses and the Schwarzschild construction are explicit in Binnington and Poisson 2009, Eqs. (1.5)–(1.7), (3.12)–(3.22), and Table I, pp. 3–4 and 10–11 (PDF). The Kerr effective-theory statement, including separation of conservative and dissipative terms, is given in Charalambous, Dubovsky, and Ivanov 2021, §§ 5.1–5.3 and 6.1, pp. 35–39 (PDF).

Schwarzschild metric invariants

Write $\mu=GM$, $z=2\mu/r$ and $f=1-z$. The background chart

$$ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega^2$$

is regular at the future horizon. Freeze one tidal harmonic $\mathcal E_m$ with integer $\ell\ge2$. Using the spherical metric harmonics, the even light-cone representative has $h_{vr}=h_{rr}=j_r=0$ and

$$\begin{aligned} p_{vv}&=h_{vv}Y,&p_{vA}&=j_vY_A,\\ p_{AB}&=r^2(\mathsf K\Omega_{AB}Y+\mathsf GY_{AB}),\\ h_{vv}&=-\frac{2r^\ell}{(\ell-1)\ell}e_1\mathcal E_m,\\ j_v&=-\frac{2r^{\ell+1}}{(\ell-1)\ell(\ell+1)}e_4\mathcal E_m,\\ \mathsf G&=-\frac{4r^\ell}{(\ell-1)\ell^2(\ell+1)}e_7\mathcal E_m. \end{aligned}$$

The font distinguishes the metric amplitude $\mathsf G$ from Newton’s constant. In vacuum choose $\mathsf K=0$. The remaining one-parameter gauge transformation shifts the component functions:

$$\begin{aligned} \delta e_1&=-\ell a_1z^{\ell+2},\\ \delta e_4&=a_1[(\ell-1)(\ell+2)+2z]z^{\ell+1},\\ \delta e_7&=2\ell a_1z^{\ell+1}. \end{aligned}$$

Thus $e_1,e_4,e_7$ are not individually gauge invariant, even when their normalization at infinity is fixed. The general even metric invariants are

$$\begin{aligned} \epsilon_a&=j_a-\frac{r^2}{2}\nabla_a\mathsf G,\\ \widetilde h_{ab}&=h_{ab}-2\nabla_{(a}\epsilon_{b)},\\ \widetilde K&=\mathsf K+\frac{\ell(\ell+1)}2\mathsf G -\frac2r r^a\epsilon_a. \end{aligned}$$

Here $a,b$ are orbit-space indices. In the static representative, $\epsilon_v=j_v$ and $\epsilon_r=-r^2\mathsf G'/2$, so

$$\widetilde K=\mathsf K-\frac2rj_v +\frac{\ell(\ell+1)}2\mathsf G+rf\mathsf G'.$$

The $\mathsf K$ term is essential in a general gauge. BP’s displayed formula omits it because $\mathsf K=0$ has already been chosen, not because the trace amplitude never transforms. Define $\widetilde K=-2r^\ell e_K\mathcal E_m/[(\ell-1)\ell]$. Then in that representative

$$e_K=-\frac2{\ell+1}e_4 +\frac{\ell+3-2z}{\ell+1}e_7 +\frac{2rf}{\ell(\ell+1)}e_7'.$$

Substituting the three gauge shifts cancels $a_1$, using $r(d/dr)z^n=-nz^n$. These definitions, gauge conditions and radial invariants follow Binnington and Poisson 2009, Eqs. (3.2)–(3.13), pp. 8–10 (PDF).

Electric horizon regularity

The two vacuum radial solution branches are a primary input from BP’s projected Einstein equations. In its minimally gauged representative,

$$\begin{aligned} e_1&=A_1+2k^E_\ell(R_0/r)^{2\ell+1}B_1,\\ e_4&=A_4-\frac{2(\ell+1)}{\ell}k^E_\ell(R_0/r)^{2\ell+1}B_4,\\ e_7&=A_7+2k^E_\ell(R_0/r)^{2\ell+1}B_7. \end{aligned}$$

The middle coefficient is different. All six radial functions are normalized to one at infinity. Substitution into the invariant yields

$$e_K=A_7+2k^E_\ell(R_0/r)^{2\ell+1}B_7,$$

with

$$\begin{aligned} A_7={}&\frac{\ell+1}{\ell-1}{}_2F_1(-\ell,-\ell;-2\ell;z)\\ &-\frac2{\ell-1}{}_2F_1(-\ell,-\ell-1;-2\ell;z),\\ B_7={}&\frac{\ell}{\ell+2}{}_2F_1(\ell+1,\ell+1;2\ell+2;z)\\ &+\frac2{\ell+2}{}_2F_1(\ell,\ell+1;2\ell+2;z). \end{aligned}$$

$A_7$ is a terminating polynomial. The first hypergeometric function in $B_7$ is zero-balanced. For $a=\ell+1\gt0$ its Euler integral is

$${}_2F_1(a,a;2a;z) =\frac{\Gamma(2a)}{\Gamma(a)^2} \int_0^1\frac{t^{a-1}(1-t)^{a-1}}{(1-zt)^a}\,dt.$$

Near $t=1$, the integrand behaves as $y^{a-1}/(f+y)^a$, where $y=1-t$. The region $f\ll y\ll1$ contributes $-\log f$. The second function in $B_7$ is finite at $z=1$ because $c-a-b=1$. Therefore

$$B_7=-c_\ell\log f+O(1),\qquad c_\ell=\frac{\ell}{\ell+2}\frac{(2\ell+1)!}{(\ell!)^2}\gt0.$$

Smooth metric data in the regular chart give smooth invariant combinations. A nonzero $k^E_\ell$ produces a logarithm in $e_K$ and is excluded. Setting $k^E_\ell=0$ leaves the smooth polynomial metric branch. This establishes both necessity and an allowed solution. The explicit radial inputs and invariant identity are Binnington and Poisson 2009, Eqs. (1.6)–(1.7), (3.13), pp. 3 and 10 (PDF).

Magnetic horizon regularity

For odd parity write $p_{vA}=h_vX_A$, $p_{AB}=h_2X_{AB}$ and $p_{rA}=0$, with

$$h_v=\frac{2r^{\ell+1}}{3(\ell-1)\ell}b_4\mathcal B_m,\qquad h_2=\frac{4r^{\ell+2}}{3(\ell-1)\ell^2}b_7\mathcal B_m.$$

The static invariants and their radial coefficients are

$$\begin{aligned} \widetilde h_v&=h_v,& \widetilde h_r&=\frac{h_2}{r}-\frac12h_2',\\ b_v&=b_4,& b_r&=b_7+\frac r\ell b_7',\\ \widetilde h_r&=-\frac{2r^{\ell+1}}{3(\ell-1)\ell}b_r\mathcal B_m. \end{aligned}$$

The residual shift $b_7\mapsto b_7+\alpha z^\ell$ cancels in $b_r$. The vacuum solution identity is

$$\begin{aligned} b_v=f b_r &=A_4-\frac{2(\ell+1)}{\ell}k^B_\ell(R_0/r)^{2\ell+1}B_4,\\ A_4&={}_2F_1(1-\ell,-\ell-2;-2\ell;z),\\ B_4&={}_2F_1(\ell-1,\ell+2;2\ell+2;z). \end{aligned}$$

The terminating sum gives $A_4(1)=0$, so $A_4/f$ is a regular polynomial. The convergent endpoint sum of the other solution gives

$$B_4(1)=\frac{(2\ell+1)!}{(\ell+2)!(\ell-1)!}\gt0.$$

Consequently a nonzero $k^B_\ell$ makes $b_r$ diverge as $1/f$, and hence makes the invariant covector $\widetilde h_r$ singular in the regular ingoing chart. Thus $k^B_\ell=0$; the remaining minimally gauged metric branch is smooth. Notice that $B_4$ itself has a finite horizon value: the invariant response contains $B_4/f$. The full odd dictionary and solution identity are Binnington and Poisson 2009, Eqs. (3.14)–(3.22), pp. 10–11 (PDF).

This is the core proof architecture: existence of a two-dimensional radial solution space, asymptotic identification of applied and response solutions, and incompatibility of the response solution with horizon regularity. No stellar interior is invoked. A material body instead terminates at $r=R>2GM$ and matches its regular interior to a linear combination of both exterior branches, so its coefficient need not vanish.

The stationary Kerr radial equation

For Kerr use physical mass $M$, $a=J/M$ and $\chi_{\mathrm{BH}}=a/(GM)=J/(GM^2)$, with $|\chi_{\mathrm{BH}}|\lt1$. Set

$$\begin{aligned} r_\pm&=\mu\pm\sqrt{\mu^2-a^2},& d&=r_+-r_-,&r_s&=2\mu,\\ x&=\frac{r-r_+}{d},& \gamma&=\frac ad,&q&=m\gamma. \end{aligned}$$

$R_s(x)$ below is the unrescaled radial Teukolsky field; it is not the reference radius $R_0$. At exactly $\omega=0$, the radial Teukolsky equation becomes

$$\begin{aligned} 0={}&x(1+x)R_s''+(s+1)(2x+1)R_s'\\ &+\left[s(s+1)-\nu(\nu+1) +\frac{q^2+iqs(2x+1)}{x(1+x)}\right]R_s. \end{aligned}$$

Primes now mean $d/dx$. This equation is exact at zero frequency, not an additional near-zone truncation of that stationary equation. The physical angular eigenvalue has $\nu=\ell$. To define the branch split, temporarily continue this radial separation parameter away from integer values; this does not introduce fractional physical spherical harmonics. The equation and horizon branch are Charalambous, Dubovsky, and Ivanov 2021, Eqs. (6.1)–(6.3), p. 39 (PDF).

Selecting the future horizon

For $R_s\sim x^p$, the indicial polynomial is $(p-iq+s)(p+iq)$. The future-ingoing branch has $p=iq-s$. Factor it as

$$\begin{aligned} R_s&=P(x)F(x),& P(x)&=x^{iq-s}(1+x)^{-iq-s},\\ a_h&=-\nu-s,& b_h&=\nu+1-s,& c_h&=1+2iq-s. \end{aligned}$$

Substitution reduces the complete radial equation to

$$x(1+x)F''+[c_h+(a_h+b_h+1)x]F'+a_hb_hF=0.$$

Thus the selected solution is $F={}_2F_1(a_h,b_h;c_h;-x)$, initially at generic parameters. The apparent singular powers in $P$ are not a physical singularity. In ingoing Kerr coordinates $\widetilde\phi=\phi+\gamma\log[x/(1+x)]$, the factor $e^{im\phi}$ contributes $x^{-iq}(1+x)^{iq}$, while the regular-tetrad conversion contributes $[x(1+x)]^s$ up to a smooth nonzero factor. Their product cancels $P$. The remaining $F$ is smooth at $x=0$ when $c_h$ is not a nonpositive integer; the degenerate normalized limit is treated below. This selection does not assert positivity of the stationary energy flux normalized at infinity.

Connecting the two exterior branches

For noninteger exponent difference $2\nu+1$ and generic Gamma parameters, the full connection formula is

$$\begin{aligned} F={}&A x^{-a_h} {}_2F_1(a_h,a_h+1-c_h;a_h+1-b_h;-1/x)\\ &+B x^{-b_h} {}_2F_1(b_h,b_h+1-c_h;b_h+1-a_h;-1/x),\\ A={}&\frac{\Gamma(c_h)\Gamma(b_h-a_h)} {\Gamma(b_h)\Gamma(c_h-a_h)},\\ B={}&\frac{\Gamma(c_h)\Gamma(a_h-b_h)} {\Gamma(a_h)\Gamma(c_h-b_h)}. \end{aligned}$$

Use real $\log x$ for $x\gt0$. Multiplying by $P$ gives applied and response powers $x^{\nu-s}$ and $x^{-\nu-s-1}$, each with its own descendant series. The formula, not just its two leading powers, is needed when descendants overlap at integer $\ell$. It follows from Charalambous, Dubovsky, and Ivanov 2021, Eq. (A.26), p. 48 (PDF).

Normalize the applied coefficient and define the dimensionless response

$$\begin{aligned} \mathcal K_{\nu m}^{(s)} &=\frac BA\left(\frac d{r_s}\right)^{2\nu+1}\\ &=\frac{\Gamma(-2\nu-1)\Gamma(\nu-s+1)\Gamma(\nu+1+2iq)} {\Gamma(-\nu-s)\Gamma(2iq-\nu)\Gamma(2\nu+1)} \left(\frac d{r_s}\right)^{2\nu+1}. \end{aligned}$$

The prescription is to identify these branches at noninteger $\nu$ with the background and $m$ fixed, then take $\nu=\ell+\varepsilon$ to the physical integer. Reading a falling power from a terminating polynomial after setting $\nu=\ell$ first is a different operation: applied descendants can occupy that power. The distinction is explained explicitly in Charalambous, Dubovsky, and Ivanov 2021, Eqs. (3.50)–(3.56), pp. 20–21 (PDF).

The integer-multipole limit

Two elementary Gamma identities finish the calculation:

$$\begin{aligned} \lim_{\varepsilon\to0} \frac{\Gamma(-2\ell-1-2\varepsilon)} {\Gamma(-\ell-s-\varepsilon)} &=\frac{(-1)^{\ell+s+1}(\ell+s)!}{2(2\ell+1)!},\\ \frac{\Gamma(\ell+1+2iq)}{\Gamma(2iq-\ell)} &=\prod_{j=-\ell}^{\ell}(2iq+j)\\ &=2iq(-1)^\ell\prod_{j=1}^{\ell}(j^2+4q^2). \end{aligned}$$

In the first line the numerator pole is displaced by $2\varepsilon$, not $\varepsilon$: omitting that factor doubles the response. Combining these identities with $d/r_s=\sqrt{1-\chi_{\mathrm{BH}}^2}$ yields

$$\begin{aligned} \mathcal K_{\ell m}^{(s)}={}&(-1)^{s+1}\frac{i m\chi_{\mathrm{BH}}}{2} \frac{(\ell+s)!(\ell-s)!}{(2\ell)!(2\ell+1)!}\\ &\times\prod_{j=1}^{\ell} \left[j^2(1-\chi_{\mathrm{BH}}^2)+m^2\chi_{\mathrm{BH}}^2\right]. \end{aligned}$$

This preserves the product of Charalambous, Dubovsky, and Ivanov 2021, Eq. (6.5), p. 39, and Eq. (A.21), p. 47 (PDF); the notation $\mathcal K$ distinguishes it from BP’s dimensionless $k^E$.

Converting to physical response

For the gravitational case take $s=2$ and $\ell\ge2$. In the electric moment convention $Q=-\lambda_E\mathcal E$, the conversion is

$$\begin{aligned} \lambda^E_{\ell m} &=\frac{(\ell-2)!\,r_s^{2\ell+1}} {G(2\ell-1)!!}\mathcal K^{(2)}_{\ell m} =-\frac{\lambda^{\mathrm{CDI}}_{\ell m}}{G},\\ r_s^{2\ell+1}\mathcal K^{(2)}_{\ell m} &=2R_0^{2\ell+1}\widehat k^{E,\mathrm{BP}}_{\ell m}. \end{aligned}$$

The hat denotes the full complex response in BP’s moment/radius normalization, not the conservative Love number. In this diagonal Kerr basis the latter is

$$k^{E,\mathrm{Love}}_{\ell m} =\operatorname{Re}\widehat k^{E,\mathrm{BP}}_{\ell m} =\frac{r_s^{2\ell+1}}{2R_0^{2\ell+1}} \operatorname{Re}\mathcal K^{(2)}_{\ell m}.$$

CDI’s mass-moment response uses the opposite sign and geometric mass units. Its electric curvature matching fixes the same physical multipole as the buffer calculation; see Charalambous, Dubovsky, and Ivanov 2021, Eqs. (2.12)–(2.14), footnote 6 and Eqs. (5.4)–(5.23), pp. 7–8 and 35–37 (PDF).

Every factor after $i$ in the product is real, so $\operatorname{Re}\mathcal K_{\ell m}^{(2)}=0$. Moreover $\mathcal K_{\ell,-m}^{(2)}=\overline{\mathcal K_{\ell m}^{(2)}}$. On real STF components the imaginary, $m$-odd coefficient represents an antisymmetric response map. Its contraction with two copies of the same real tidal tensor is zero; it cannot supply a nonzero symmetric conservative $\mathcal E^2$ coefficient. For example,

$$\mathcal K_{22}^{(2)}\big|_{\chi_{\mathrm{BH}}=3/5} =-\frac{26i}{625},$$

not zero as a complex number. The same gravitational curvature solution encodes electric and magnetic data with the common response factor (Charalambous, Dubovsky, and Ivanov 2021, Eqs. (5.7)–(5.15), pp. 35–36 (PDF)). The zero conservative map survives a fixed nonsingular translation of tidal and induced-moment normalization in either parity. The nonzero electric conversion above must not be used as a formula for a physical magnetic current coefficient; that requires its own complete dictionary.

The normalized Schwarzschild check

The product vanishes for $\chi_{\mathrm{BH}}=0$ or $m=0$. At $q=0$, however, the unnormalized $s=2$ function has $c_h=-1$ and cannot simply be evaluated as an ordinary hypergeometric series. For $\ell=s=2$, normalize the growing coefficient $A=24/(c_h)_4$ first. Then

$$\frac FA=\frac1{24} \sum_{n=0}^4(-4)_n(c_h+n)_{4-n}(-x)^n \ \longrightarrow\ x^2(1+x)^2.$$

Here $(a)_n=\Gamma(a+n)/\Gamma(a)$ is the rising factorial. The usual $n!$ cancels because $b_h=1$. Multiplication by $P$ at $q=0$ gives $R_2=1$, which directly solves the stationary Schwarzschild quadrupolar equation. This finite limit checks the normalization; the all-$\ell$ metric regularity argument above establishes the Schwarzschild statement.

There is an essential qualification for $m\neq0$. A perturbation with $\omega=0$ relative to infinity has $k_H=\omega-m\Omega_H=-m\Omega_H$ and is not stationary relative to the horizon generator. Its response can therefore contain an imaginary, dissipative part even though the conservative static Love number is zero. For $m=0$, that frame-dragging channel is absent. Thus “vanishing Love number” means the conservative matched coefficient vanishes; it does not mean the full complex response function is identically zero. Since the external tide has $\dot{\mathcal E}=0$, it does no stationary work through $Q:\dot{\mathcal E}$ at infinity. Angular-momentum transfer and positive horizon-generator energy, hence area growth, may nevertheless remain.

Scope, order of limits, and counterexamples

The conclusion only establishes the stated four-dimensional asymptotically flat vacuum result. It does not imply any of the following:

• zero absorption or zero tidal heating;
• zero finite-frequency local coefficients, logarithmic running, or nonlocal retarded response;
• zero response for a material star, a charged or hairy object, a black hole in higher dimensions or with a cosmological constant, or a solution of modified gravity;
• zero for every coefficient obtained before gauge-invariant matching; or
• commutation of $\omega\to0$, $a\to GM$, the near-zone expansion, and the asymptotic limit.

A material star is a direct discriminating counterexample: its surface lies outside the would-be horizon, so the exterior response branch is allowed and is fixed by the interior equation of state. Higher-dimensional black holes also need not share the four-dimensional integer-multipole cancellation. These cases do not contradict the theorem because they violate its hypotheses.

Exercises

Consider the $\ell=2$ even-parity solution above. Show directly that a future-horizon-regular solution has zero matched response, and check that this does not forbid absorption.

Solution

For $\ell=2$, the invariant electric solution has

$$A_7=1-\frac{z^2}{2},\qquad z^5B_7=-30A_7\log(1-z)-5z(6+3z-z^2).$$

These are BP’s Table I functions. At $z=1$, $A_7=1/2$ is finite, whereas

$$B_7\sim-15\log(1-z)+O(1).$$

The coefficient of $B_7$ in the invariant $e_K$ is $2k^E_2(R_0/r)^5$. Since $(R_0/r)^5$ is finite and nonzero at the horizon, regularity forces $k^E_2=0$. At $z\to0$, $A_7=1-z^2/2$ is the normalized polynomial applied solution and the separately normalized $B_7$ branch is the one that would carry the $r^{-5}$ response relative to the applied tidal power. Hence the matched conservative coefficient is zero.

The check concerns the exactly static, time-even radial solution. At $\omega\neq0$, the ingoing solution has a phase and a nonzero horizon flux; its imaginary response begins at a positive power of $GM\omega$. Nothing in the regularity argument sets that flux to zero. For Kerr with $m\neq0$, even $\omega=0$ at infinity leaves $k_H=-m\Omega_H$, so dissipation can remain.

The complementary calculation is the positive-area flux and possible energy extraction discussed under horizon absorption and tidal heating.

References

• Binnington, Taylor, and Eric Poisson. 2009. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80, 084018. DOI.
• Charalambous, Panagiotis, Sergei Dubovsky, and Mikhail M. Ivanov. 2021. “On the Vanishing of Love Numbers for Kerr Black Holes.” Journal of High Energy Physics 05, 038. DOI.

Further reading

• Chia, Horng Sheng. 2021. “Tidal Deformation and Dissipation of Rotating Black Holes.” Physical Review D 104, 024013. DOI.