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Complexity Growth in Flat Spacetimes

Reza Fareghbal, Pedram Karimi

TL;DR

This work extends the Complexity=Action (CA) proposal to BMSFTs dual to asymptotically flat spacetimes by evaluating the on-shell action on a carefully chosen Wheeler-DeWitt patch. The authors derive the growth rate of complexity $\dot{\mathcal{C}}$ in both $d\ge3$ and $d=2$ cases, showing zero growth at initial times and a late-time behavior that, for $d\ge3$, approaches Lloyd's bound from above, while in $d=2$ it remains constant with a logarithmic deviation. The flat-space results reproduce the corresponding AdS/CFT limits, with the null joint scale $\mathfrak{a}$ playing a key role in the boundary terms. This work clarifies how holographic complexity behaves in ultrarelativistic, flat spacetimes and provides a framework for exploring complexity formation and further flat/BMSFT holography.

Abstract

We use the complexity equals action proposal to calculate the rate of complexity growth for field theories that are the holographic duals of asymptotically flat spacetimes. To this aim, we evaluate the on-shell action of asymptotically flat spacetime on the Wheeler-DeWitt patch. This results in the same expression as can be found by taking the flat-space limit from the corresponding formula related to the asymptotically AdS spacetimes. For the bulk dimensions that are greater than three, the rate of complexity growth at late times approaches from above to Lloyd's bound. However, for the three-dimensional bulks, this rate is a constant and differs from Lloyd's bound by a logarithmic term.

Complexity Growth in Flat Spacetimes

TL;DR

This work extends the Complexity=Action (CA) proposal to BMSFTs dual to asymptotically flat spacetimes by evaluating the on-shell action on a carefully chosen Wheeler-DeWitt patch. The authors derive the growth rate of complexity in both and cases, showing zero growth at initial times and a late-time behavior that, for , approaches Lloyd's bound from above, while in it remains constant with a logarithmic deviation. The flat-space results reproduce the corresponding AdS/CFT limits, with the null joint scale playing a key role in the boundary terms. This work clarifies how holographic complexity behaves in ultrarelativistic, flat spacetimes and provides a framework for exploring complexity formation and further flat/BMSFT holography.

Abstract

We use the complexity equals action proposal to calculate the rate of complexity growth for field theories that are the holographic duals of asymptotically flat spacetimes. To this aim, we evaluate the on-shell action of asymptotically flat spacetime on the Wheeler-DeWitt patch. This results in the same expression as can be found by taking the flat-space limit from the corresponding formula related to the asymptotically AdS spacetimes. For the bulk dimensions that are greater than three, the rate of complexity growth at late times approaches from above to Lloyd's bound. However, for the three-dimensional bulks, this rate is a constant and differs from Lloyd's bound by a logarithmic term.

Paper Structure

This paper contains 11 sections, 39 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. BMSFT Complexity Gowth in $d\geq3$
  4. Initial time
  5. Late time
  6. Numerical results for the Schwarzschild black hole
  7. BMSFT Complexity Growth in $d=2$
  8. Initial time
  9. Late time
  10. Discussion
  11. Non-symmetric evolutions in past and future null infinities

Figures (5)

  • Figure 1: Initial time for $d>3$. Gravitational action is evaluated on the gray region.
  • Figure 2: Late time for $d>3$. Gravitational action is evaluated on the gray region.
  • Figure 3: The rate of complexity growth for different $\mathfrak{a}$.
  • Figure 4: Initial time for FSC. Gravitational action is evaluated on the gray region.
  • Figure 5: Late time for FSC. Gravitational action is evaluated on the gray region.