Classical Log Soft Graviton Theorem

• The classical log soft graviton theorem is a modification of the four-dimensional soft graviton expansion incorporating non-analytic logarithmic terms in the low-frequency regime.
• It connects the appearance of logarithmic corrections to observable phenomena such as gravitational memory steps and waveform tails, with explicit scaling relations in time and frequency domains.
• The theorem also constrains the asymptotic structure of spacetime at null infinity, offering insights into the deviation from smooth peeling and the universal properties of gravitational scattering.

The classical log soft graviton theorem is the four-dimensional modification of the classical soft graviton expansion in which the low-frequency radiative field is no longer a pure Laurent series in the soft frequency $\omega$. Instead, beyond the leading Weinberg pole $1/\omega$, the expansion develops non-analytic logarithmic terms such as $\log \omega$ and $\omega(\log\omega)^2$. In the time domain these terms control universal late-time and early-time tails of the gravitational waveform, including the $1/u$ correction to ordinary memory and the $u^{-2}\ln|u|$ tail. The theorem is classical in origin: it is tied to the long-range $1/r$ gravitational field in four spacetime dimensions, to logarithmic deviations of asymptotic worldlines, and to backscattering of the soft graviton itself in the background gravitational field (Laddha et al., 2018, Sahoo et al., 2018, Saha et al., 2019).

1. Four-dimensional origin and classical content

The defining feature of the theorem is that it is specific to $D=4$. In higher dimensions, asymptotic forces decay fast enough that the standard soft expansion remains polynomial in $\omega$. In four dimensions, by contrast, long-range Coulomb/Newton fields modify asymptotic trajectories by logarithmic-in-time terms, so the orbital angular momentum entering the subleading soft factor becomes logarithmically divergent. This is the classical reason the naive subleading soft graviton theorem fails as an ordinary power series and must be replaced by a logarithmically corrected expansion (Sahoo et al., 2018).

A standard asymptotic form is

$$r_a^\mu(\sigma)=\eta_a \frac{p_a^\mu}{m_a}\sigma + c_a^\mu \ln |\sigma|+\cdots ,$$

so that

$1/\omega$0

The replacement $1/\omega$1 then produces the classical logarithmic soft term (Sahoo et al., 2018).

This logarithmic structure is not intrinsically a quantum loop effect. The relevant mechanisms are classical: outgoing massive particles continue to accelerate in the long-range gravitational field of the remnant or of the other asymptotic particles, and the soft graviton experiences a propagation phase or drag in that same background field (Laddha et al., 2018). A direct proof from the classical Einstein equations coupled to matter was given for general four-dimensional scattering or explosion processes, thereby replacing the earlier infrared prescription by an explicit classical derivation (Saha et al., 2019).

2. Low-frequency expansion and theorem statements

A convenient radiative variable is the trace-reversed perturbation

$1/\omega$2

whose far-zone Fourier mode admits the generic low-frequency expansion

$1/\omega$3

In this form, $1/\omega$4 is the leading soft coefficient responsible for memory, while $1/\omega$5 is the first logarithmic coefficient responsible for the $1/\omega$6 tail (Das, 2023).

The same structure can be written as an all-order logarithmic tower,

$1/\omega$7

which makes explicit that the four-dimensional classical soft expansion is best viewed as an expansion in powers of $1/\omega$8 dressed by powers of $1/\omega$9 (Alessio et al., 2024).

Relative to the usual tree-level or $\log \omega$0 hierarchy,

$\log \omega$1

the four-dimensional theorem replaces the naive subleading and higher coefficients by logarithmically corrected quantities. A useful formulation of the first logarithmic coefficient is

$\log \omega$2

with

$\log \omega$3

Here the first term is the drag contribution from the outgoing soft null ray, and the second is the divergent-angular-momentum contribution from the hard particles (Boschetti et al., 25 Aug 2025). In the asymptotic radiative frame with $\log \omega$4, this becomes

$\log \omega$5

which is the clean radiative-frame form of the classical log soft factor (Boschetti et al., 10 Mar 2026).

3. Frequency-space logarithms, memory, and waveform tails

The observational content of the theorem is the Fourier correspondence between soft non-analyticities and long-time waveform behavior. For retarded time $\log \omega$6, the basic dictionary is

$\log \omega$7

This is the core reason the logarithmic soft theorem has an immediate waveform interpretation (Laddha et al., 2018, Sahoo, 2020).

For the first logarithmic correction, the late-time waveform behaves as

$\log \omega$8

with the retarded time shifted by a Shapiro-type delay,

$\log \omega$9

The leading coefficient $\omega(\log\omega)^2$0 is the ordinary linear memory term, while $\omega(\log\omega)^2$1 is the tail coefficient fixed by the logarithmic soft factor (Laddha et al., 2018).

At the next order, the universal sub-subleading logarithmic theorem gives

$\omega(\log\omega)^2$2

more precisely a $\omega(\log\omega)^2$3 tail at late and early retarded times. More generally, the leading non-analytic contribution to the $\omega(\log\omega)^2$4-leading classical soft graviton theorem is conjectured to scale as

$\omega(\log\omega)^2$5

This pattern was established explicitly for the sub-subleading case and then conjectured at all orders (Sahoo, 2020).

A compact summary of the large-$\omega(\log\omega)^2$6 behavior is

$\omega(\log\omega)^2$7

with the coefficients determined solely by asymptotic scattering data (Sen, 2024).

4. Null infinity, asymptotic geometry, and peeling

The classical log soft graviton theorem is not only a statement about waveforms; it also constrains the differential structure of future null infinity. If the radiative field has the low-frequency expansion

$\omega(\log\omega)^2$8

then the inverse Fourier transform implies

$\omega(\log\omega)^2$9

at large $1/u$0. This tail induces non-artifact $1/u$1 terms in the asymptotic metric and thereby obstructs smooth peeling behavior at $1/u$2 (Das, 2023).

In Bondi gauge the resulting metric expansion takes the form

$1/u$3

and the appearance of the $1/u$4 term is interpreted as physical rather than gauge removable. In that sense the logarithmic soft theorem provides a direct route from soft radiation to failure of smoothness at null infinity, yielding a weakened partial peeling property rather than standard Penrose peeling (Das, 2023).

A later asymptotic proof recast the log theorem entirely within the framework of timelike, spatial, and null infinity. In that approach the low-frequency waveform

$1/u$5

is derived from Einstein equations near $1/u$6, $1/u$7, and $1/u$8, together with matching across their boundaries. The standard asymmetry between future and past hard contributions is then traced to a discontinuity at spatial infinity,

$1/u$9

which obstructs imposing independent radiative log frames at $u^{-2}\ln|u|$0 and $u^{-2}\ln|u|$1 (Boschetti et al., 10 Mar 2026).

5. Higher logarithmic orders, towers, and structural constraints

Beyond the first logarithmic term, several works organize a full tower of classical logarithmic soft structures. The first rigorous step was the sub-subleading classical theorem, whose universal non-analytic part is of order $u^{-2}\ln|u|$2, together with mixed $u^{-2}\ln|u|$3 terms. Its Fourier transform gives the universal $u^{-2}\ln|u|$4 tail (Sahoo, 2020).

A complementary development concerns the no-deflection or infinite-impact-parameter limit,

$u^{-2}\ln|u|$5

In that regime the memory term vanishes, but an infinite tower of logarithmic modes survives: $u^{-2}\ln|u|$6 The main result is that the $u^{-2}\ln|u|$7-leading tree-level soft graviton operators generate the corresponding classical $u^{-2}\ln|u|$8 contributions, while the non-factorizing remainders present for $u^{-2}\ln|u|$9 do not contribute to these leading logarithms (Akhtar, 2024).

For general $1/r$0 gravitational scattering, the low-frequency waveform has also been written in a kinematically exact form,

$1/r$1

with explicit PM-expanded results through sub-subleading PM order and a conjecture for the higher tower $1/r$2 (Alessio et al., 2024).

A distinct structural refinement is log translation invariance. The residual gauge transformation

$1/r$3

acts on the logarithmic deviation vectors by $1/r$4. The first two universal logarithmic soft coefficients are invariant under this transformation, and the expected higher leading logs must satisfy the recurrence

$1/r$5

This explains, in particular, why outgoing massless particles cancel from the known universal log soft factors (Boschetti et al., 25 Aug 2025).

6. Regimes, vanishing cases, and phenomenological scope

The theorem is universal only within clearly stated regimes. Some derivations assume a heavy remnant with small escaped energy, so that outgoing particles move in the remnant’s long-range field and mutual forces among the ejecta may be neglected at leading logarithmic order (Laddha et al., 2018). Others treat fully general scattering or explosion processes, but always in four-dimensional asymptotically flat gravity and always for the universal non-analytic part of the radiation field (Saha et al., 2019).

A prominent vanishing result is that the $1/r$6 tail disappears when all final-state particles are massless or ultra-relativistic. In the explicit tail coefficient of the decay/scattering analysis, the factor

$1/r$7

vanishes for $1/r$8, so $1/r$9. In that case there is no $D=4$0 correction to the memory step at the order considered (Laddha et al., 2018). This agrees with the statement that binary black hole mergers, where the energy is carried primarily by gravitational radiation and the final state contains only one massive remnant, do not exhibit the corresponding $D=4$1 and $D=4$2 tail memories at that order, whereas processes with massive ejecta can (Sen, 2024).

By contrast, neutron-star mergers, supernova explosions, and hyper-velocity star production were identified as settings in which nonzero finite gravitational tail memory can occur, precisely because massive asymptotic matter remains in the problem (Sahoo, 2020). A related but different limit is the no-deflection regime, where the memory vanishes but the logarithmic tower $D=4$3 survives, showing that vanishing memory and nontrivial logarithmic soft structure are not contradictory (Akhtar, 2024).

In this sense, the classical log soft graviton theorem is best understood not as a single isolated formula but as the four-dimensional infrared completion of the classical soft graviton expansion: the leading $D=4$4 term gives memory, the $D=4$5 term gives a $D=4$6 tail, the $D=4$7 term gives a $D=4$8 tail, and the same logarithmic hierarchy governs the asymptotic geometry of null infinity, including nontrivial $D=4$9 structures and violation of smooth peeling (Laddha et al., 2018, Das, 2023).