Asymptotic Charges Cannot Be Measured in Finite Time

TL;DR

The paper shows that conserved charges defined at future null infinity, such as the Bondi mass and Bondi electric charge, cannot be measured in finite retarded time over a finite region at arbitrarily large radius due to quantum fluctuations constrained by asymptotic entropy bounds. By analyzing massless QED and a non-minimally coupled scalar in the gravitational setting with time-smearing, the authors compute the fluctuations of these charges and demonstrate that, even with optimally chosen measurement protocols, the fluctuations diverge as the observation region is pushed to infinity. Consequently, these charges cannot be associated with finite neighborhood observables on cuts of ${\mathscr{I}}^+$; access to them requires measurements that scale with the radius, effectively tying them to semi-infinite regions of null infinity. The findings have implications for flat-space holography and the interpretation of BMS charges, suggesting a fundamental nonlocality of asymptotic observables in quantum gravity.

Abstract

To study quantum gravity in asymptotically flat spacetimes, one would like to understand the algebra of observables at null infinity. Here we show that the Bondi mass cannot be observed in finite retarded time, and so is not contained in the algebra on any finite portion of ${\mathscr{I}}^+$. This follows immediately from recently discovered asymptotic entropy bounds. We verify this explicitly, and we find that attempts to measure a conserved charge at arbitrarily large radius in fixed retarded time are thwarted by quantum fluctuations. We comment on the implications of our results to flat space holography and the BMS charges at ${\mathscr{I}}^+$.

Figures (3)

• Figure 1: If distant observer Bob could measure the Bondi mass of Alice's planet, then Bob could receive information from Alice, without receiving energy. This would contradict recently proven bounds on distant communication channel capacities. In our example, Alice has radiated away some portion of her planet, but Bob does not intercept this radiation (yellow arrows). Instead, Bob later tries to measure how much mass is still left, in some fixed amount of time $\delta u$, at arbitrarily large radius $r_B$. We resolve the contradiction by showing that quantum fluctuations ruin Bob's measurement. The Bondi mass cannot be observed in finite time.
• Figure 2: Penrose diagram of the process we consider. The red line represents Alice's worldline. The yellow arrows are the radiation emitted by Alice and reaching $\mathscr{I}^+$ without interacting with Bob (blue worldline) whose detectors are only on for a retarded time interval $\delta u.$
• Figure 3: In the $\Delta^0$ integral (left diagram), the contour avoids the branch points on the real axis and picks up a residue at the simple pole at $\Delta^0=x^0+i \delta t.$ In the $x^0$ integral (right diagram), a similar contour is used. It now picks up a residue at the simple pole at $x^0=i \delta t.$