Physics of trans-Planckian gravity

We study the field theoretical description of a generic theory of gravity flowing to Einstein general relativity in IR. We prove that, if ghost-free, in the weakly-coupled regime such a theory can never become weaker than general relativity. Using this fact, as a by-product, we suggest that in a ghost-free theory of gravity trans-Planckian propagating quantum degrees of freedom cannot exist. The only physical meaning of a trans-Planckian pole is the one of a classical state (black hole) which is described by the light IR quantum degrees of freedom and gives exponentially-suppressed contributions to virtual processes. In this picture Einstein gravity is UV self-complete, although not Wilsonian, and sub-Planckian distances are unobservable in any healthy theory of gravity. We then finally show that this UV/IR correspondence puts a severe constraint on any attempt of conventional Wilsonian UV-completion of trans-Planckian gravity. Specifically, there is no well-defined energy domain in which gravity could become asymptotically weak or safe.

Figure 1

Momentum-scale dependence of ${\alpha }_{\mathrm{grav}}$. The dashed line shows a running of the gravitational coupling where gravity becomes weaker in the weakly-coupled regime. In a ghost-free theory this cannot happen. The solid line represents a typical running of ${\alpha }_{\mathrm{grav}}$ usually found within the Asymptotic Safety scenario. Here, gravity first hits the strong-coupling (${\alpha }_{\mathrm{grav}}=1$) at scale $M_{*}$, before turning over to the fixed point scaling. The shaded region indicates the regime in which black hole formation takes place and which hence cannot be probed by experiments.

Figure 2

Trans-Planckian distances are shielded by a black hole barrier. Probing poles at $p^2=L^{-2}\ll M_P^2$ one has to localize energy of order $L^{-1}$ within the distance $L$. The corresponding black hole horizon of this energy, $R_H(L)\ge R_S(L)=2L_P^2/L$, shields the sub-Planckian region ($L<L_P$) from being probed by any physical experiment. The sub-Planckian distance $L$ is mapped to the macroscopic distance $R_H(L)$. On the right-hand side we show a qualitative plot of the energy-distance relation. The grey “blob” around the Planck scale indicates that at the Planck scale itself we do not know how the precise relation between energy and distances is. Also, there is an uncertainty about the far IR black holes, i.e. for energies $E_{\mathrm{IR}}=2L_P^2/L_{\mathrm{IR}}$ as we cannot exclude the possibility that at scales $L\gg L_{\mathrm{IR}}$ gravity is modified.

Figure 3

Gravitational field produced by a source $T$. The wiggled lines represent the emitted gravitons $h_{\mu \nu }$. At the horizon the trilinear and higher order interactions are of the same order as the one-particle exchange.

Figure 4

A heavy scalar (double line) is mediating the interaction between a source $T$ and $k$ gravitons (wiggled line). Integrating-out this scalar at tree-level will induce an effective pointlike interaction between the source and $k$ gravitons.