Tidal Deformability and Love Response

A material star placed in a slowly varying quadrupolar tidal field develops an induced mass quadrupole. In general relativity that response is determined by a regular interior perturbation and its matching to the growing and decaying pieces of the exterior metric. The result is a stellar Love number $k_2$, a dimensionful deformability $\lambda$, and a dimensionless deformability $\Lambda$. Their definitions—and especially the powers of $G$—must be fixed before values are compared.

Helpful background. Stress–Energy and Conservation supplies material perturbations; the Einstein equations supply the interior and exterior fields; Relativistic Two-Body Dynamics supplies the applied tidal field; Relativistic Matter and Compact Objects supplies TOV backgrounds; Gravitational Radiation and Waveforms supplies radiative observables; and Computational Relativity supplies numerical sequence checks.

Response convention and dimensions

In the weak-field buffer region outside the star but well inside the binary separation, choose local Cartesian axes and expand the external Newtonian potential as $\Phi_{\rm ext}=\Phi_0+a_i x^i+\tfrac12{\cal E}_{ij}x^ix^j+\cdots$. The trace-free Hessian ${\cal E}_{ij}=\partial_i\partial_j\Phi_{\rm ext}$ defines the tidal sign. For a companion along the $z$ axis, ${\cal E}_{zz}=-2GM'/D^3$: nearby freely falling particles separate along that axis. The induced trace-free mass quadrupole $Q_{ij}$ is defined by

$$Q_{ij}=-\lambda\,{\cal E}_{ij}.$$

Thus $\lambda\gt0$ gives $Q_{zz}\gt0$, an elongation toward the companion. The relativistic definition identifies the corresponding growing and decaying metric coefficients in that buffer region; it does not require the whole tidally perturbed binary to be static. For a nonrotating star of areal radius $R$,

$$\lambda=\frac{2}{3G}k_2 R^5, \qquad C=\frac{GM}{R}, \qquad \Lambda=\frac{\lambda}{G^4M^5} =\frac{2}{3}k_2 C^{-5}.$$

Here $M$ is physical gravitational mass. Equivalently, if geometric multipoles are used, $\lambda_{\rm geom}=G\lambda$ and $\Lambda=\lambda_{\rm geom}/(GM)^5$. This translation prevents a common normalization error. With $c=1$, ${\cal E}_{ij}$ has dimension $L^{-2}$, $GQ_{ij}$ has dimension $L^3$, and $\lambda_{\rm geom}$ has dimension $L^5$.

This page treats the static, linear, even-parity response of a nonrotating material star in four-dimensional general relativity. Spin–tide couplings, dynamical tides, and nonlinear response are separate. Vacuum black-hole response and subtraction conventions are treated in Relativistic Love Numbers and Gauge Invariance; inserting the stellar response into orbital Hamiltonians and phasing is treated in Adiabatic Tidal Dynamics.

Interior perturbation from the center

Begin with a TOV solution $m(r)$, pressure $p(r)$, energy density $e(r)$, and sound speed $c_s^2=dp/de$. A static quadrupolar even-parity perturbation can be represented by a radial metric function $H(r)$. Defining

$$y(r)=\frac{rH'(r)}{H(r)}$$

turns the second-order equation into a Riccati equation

$$r y'+y^2+F(r)y+r^2Q(r)=0,$$

For $A(r)=1-2Gm(r)/r$, the complete coefficients are

$$F=\frac{1-4\pi Gr^2(e-p)}{A},$$ $$Q=\frac{4\pi G}{A}\left(5e+9p+\frac{e+p}{c_s^2}\right) -\frac{6}{r^2A} -\frac{4G^2\bigl(m+4\pi r^3p\bigr)^2}{r^4A^2}.$$

Here $m$ is physical enclosed mass and $c_s^2=dp/de$ belongs to the static barotropic EOS. The term $Q(r)$ is an ODE coefficient, not the quadrupole tensor $Q_{ij}$. Regularity gives $H\propto r^2$ and hence $y(0)=2$. Indeed, $F\to1$ and $r^2Q\to-6$ at the center, so the local roots are $y=2$ and $y=-3$; the latter is singular. These coefficients follow by substituting $H'/H=y/r$ into Hinderer 2008, §4.2, Eq. (15), with the corrected Eqs. (20) and (23) in the 2009 erratum.

The EOS must provide thermodynamically consistent $e(p)$ and $de/dp$ over the star. At a sharp first-order transition at $r=r_d$, define the outward density decrease $\Delta e=e(r_d^-)-e(r_d^+)\gt0$. For a static barotropic interface with continuous pressure and no surface stress, the correct jump is

$$y(r_d^+)-y(r_d^-) =-\frac{4\pi r_d^3\Delta e}{m(r_d)+4\pi r_d^3p(r_d)}.$$

To see why pressure belongs in the denominator, retain the singular $(e+p)\,de/dp$ term in $Q$, replace $dr$ using the TOV pressure equation, and integrate across the interface. The factors $G$, $A$, and $e+p$ cancel. Omitting the pressure term is justified only at a zero-pressure surface, not at an interior transition (Takátsy and Kovács 2020, Eqs. (2)–(11)). A tabulated calculation must state interpolation, phase construction, crust join, and derivative policy; smoothing a sharp jump changes the problem.

Independent center checks use a regular series rather than starting exactly at $r=0$. Rescaling $H$ must leave $y$ and $k_2$ unchanged. The background mass and pressure should satisfy TOV residual tolerances, and the response should converge under radial refinement and EOS interpolation changes.

Exterior matching and the Love number

Outside the star, the static vacuum solution is a linear combination of a growing $r^2$ tidal field and a decaying $r^{-3}$ response. Matching $H$ and the appropriate derivative at $r=R$ determines their ratio. The value entering the exterior formula is $y_R=y(R^+)$. If the density tends continuously to zero, it equals the interior value. A self-bound surface of density $e_s$ instead requires $y_R=y(R^-)-4\pi R^3e_s/M$. The quadrupolar result is

$$k_2=\frac{8C^5}{5}(1-2C)^2 \bigl[2+2C(y_R-1)-y_R\bigr]\,D^{-1},$$

with

$$\begin{aligned} D={}&2C\bigl[6-3y_R+3C(5y_R-8)\bigr]\\ &+4C^3\bigl[13-11y_R+C(3y_R-2)+2C^2(1+y_R)\bigr]\\ &+3(1-2C)^2\bigl[2-y_R+2C(y_R-1)\bigr]\ln(1-2C). \end{aligned}$$

The logarithmic expression contains large cancellations at small $C$ and should be evaluated with adequate precision or a controlled series. Binnington and Poisson give a relativistic exterior definition, gauge analysis, and material-body matching in Binnington and Poisson 2009, §§III.A, IV.D, and V.B.

Three checks are especially useful. First, $k_2$ and $\Lambda$ are dimensionless while $\lambda$ carries $R^5/G$. Second, the low-compactness limit should recover the Newtonian Love response of the same density profile, not one universal number for every EOS. Third, direct fitting of exterior growing and decaying coefficients must agree with the closed $C,y_R$ formula. The black-hole limit cannot be inferred by simply extending a material EOS sequence past its domain: its inner boundary condition is different.

An instructive Newtonian limit of the same formula is

$$k_2\longrightarrow\frac{2-y_R}{2(3+y_R)}.$$

For an incompressible Newtonian sphere, $H\propto r^2$ inside gives $y(R^-)=2$, but its finite surface density produces a jump of $-3$. Consequently $y_R=-1$ and $k_2=3/4$. Using the uncorrected interior value would incorrectly give zero: even this simplest benchmark needs surface matching. Incompressibility is a mathematical Newtonian benchmark, not a causal relativistic EOS.

Check the deformability conversion

A stellar model has compactness $C=0.17$ and quadrupolar Love number $k_2=0.080$ in the convention above. Compute $\Lambda$.

Solution

Using $\Lambda=(2/3)k_2C^{-5}$,

$$\Lambda= \frac{2}{3}(0.080)(0.17)^{-5} \simeq 3.76\times10^2 .$$

Thus $\Lambda\simeq376$. The result is dimensionless. Omitting the factor $2/3$ or using $M/R$ instead of $GM/R$ with physical mass would mix conventions and give a different, incorrect value.

References

• Binnington, Taylor, and Eric Poisson. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80 (2009): 084018. DOI.
• Hinderer, Tanja. “Tidal Love Numbers of Neutron Stars.” Astrophysical Journal 677 (2008): 1216–1220; erratum Astrophysical Journal 697 (2009): 964. DOI.
• Takátsy, János, and Péter Kovács. “Comment on ‘Tidal Love Numbers of Neutron and Self-Bound Quark Stars.’” Physical Review D 102 (2020): 028501. DOI; Open PDF.