The gravitational $S$ matrix

We investigate the hypothesized existence of an $S$ matrix for gravity and some of its expected general properties. We first discuss basic questions regarding the existence of such a matrix, including those of infrared divergences and description of asymptotic states. Distinct scattering behavior occurs in the Born, eikonal, and strong gravity regimes, and we describe aspects of both the partial wave and momentum space amplitudes, and their analytic properties, from these regimes. Classically the strong gravity region would be dominated by the formation of black holes, and we assume its unitary quantum dynamics is described by corresponding resonances. Masslessness limits some powerful methods and results that apply to massive theories, though a continuation path implying crossing symmetry plausibly still exists. Physical properties of gravity suggest nonpolynomial amplitudes, although crossing and causality constrain (with modest assumptions) this nonpolynomial behavior, particularly requiring a polynomial bound in complex $s$ at fixed physical momentum transfer. We explore the hypothesis that such behavior corresponds to a nonlocality intrinsic to gravity, but consistent with unitarity, analyticity, crossing, and causality.

Figure 1

Scattering regimes in an impact parameter picture; the question marks denote possible model dependence discussed in Sec. 3c.

Figure 2

The H diagram, which provides a leading correction to the eikonal amplitudes as scattering angles approach $\theta \sim 1$.

Figure 3

Schematic of a black hole as a resonance in $2\to 2$ scattering.

Figure 4

Absorption coefficients at a fixed angular momentum as a function of the c.m. energy.

Figure 5

Absorption coefficients at a fixed c.m. energy as a function of angular momentum, with $L_c\equiv L(E)$.

Figure 6

Phase shift for fixed angular momentum as a function of the c.m. energy.

Figure 7

Phase shift for a fixed c.m. energy as a function of angular momentum, with $L_c\equiv L(E)$.

Figure 8

The complex $s$ plane, indicating some of the relationships entering into the Phragmen-Lindelöf argument for a polynomial bound.

Figure 9

Illustration of scattering by a repulsive interaction of range $R$; the scattered wave at angle $\theta$ has a path that is shorter by $2R\sin \frac{\theta }{2}$ relative to a wave traveling unscattered through the origin, and thus has a relative time advance.

Figure 10

Illustration of scattering of a particle by the gravitational field of an ultrarelativistic source; the scattering angle is negative, corresponding to attraction, and this results in a path for the scattered wave that is longer by $R\sin \theta \sim 2R\sqrt{tu}/s$ as compared to a wave that passes through the scattering center.