Gravitational entropy is observer-dependent

TL;DR

The paper argues that gravitational entropy assigned to a spacetime subregion is inherently observer-dependent when accounting for quantum reference frames (QRFs). By generalizing the Page-Wootters reduction and CLPW approach to multiple clocks, it shows that different observers assign different entropies to the same subregion through dressed observables and QRF transformations, even in a fixed global state. The authors derive exact density operators for subsystems relative to arbitrary clock sets, classify the relevant algebras as Type II under suitable conditions, and obtain a semiclassical entropy formula that separates contributions from the quantum field and the clocks. They illustrate entropy-relativity with a gravitational interferometer example, highlighting how clock spectra and entanglement structures drive observer-dependent entropy. The work bridges QRFs and quantum gravity, clarifying that horizon entropy remains observer-independent while interior entropies are frame- and clock-dependent, and points to rich future directions including generalized second laws and dynamical edge modes.

Abstract

In quantum gravity, it has been argued that a proper accounting of the role played by an observer promotes the von Neumann algebra of observables in a given spacetime subregion from Type III to Type II. While this allows for a mathematically precise definition of its entropy, we show that this procedure depends on which observer is employed. We make this precise by considering a setup in which many possible observers are present; by generalising previous approaches, we derive density operators for the subregion relative to different observers (and relative to arbitrary collections of observers and for arbitrary global physical states), and we compute the associated entropies in a semiclassical regime, as well as in some specific examples that go beyond this regime. We find that the entropies seen by distinct observers can drastically differ. Our work makes extensive use of the formalism of quantum reference frames (QRF); indeed, as we point out, the `observers' considered here and in the previous works are nothing but QRFs. In the process, we demonstrate that the description of physical states and observables invoked by Chandrasekaran et al. [arXiv:2206.10780] is equivalent to the Page-Wootters formalism, leading to the informal slogan "PW=CLPW". It is our hope that this paper will help motivate a long overdue union between the QRF and quantum gravity communities. Further details will appear in a companion paper.

Figures (2)

• Figure 2.1: We consider a low energy gravitational system made up of field degrees of freedom described by an effective QFT, and an arbitrary number of clock degrees of freedom, each of which is carried by a quantum reference frame $C_i$. The clocks are taken to measure time along the modular flow of some fixed QFT state $\ket{\psi_S}$ with respect to the algebra $\mathcal{A}_{\mathcal{U}}$ of QFT operators in a fixed spacetime region $\mathcal{U}$. Heuristically (but not always literally), one may imagine that the clocks evolve along some worldlines resembling the integral curves of a boost in $\mathcal{U}$ or its complement $\mathcal{U}'$. By the timelike tube theorem Borchers1961Araki1963AGOWitten:2023qsvStrohmaier:2023opz, each QRF in $\mathcal{U}$ has access to the full regional QFT algebra $\mathcal{A}_\mathcal{U}$.
• Figure 4.1: We consider a state where, in the perspective of a particular clock $C_1$, the state of a second clock $C_2$ is in a superposition of clock states. One can imagine starting with $C_2$'s time reading $0$, and then exposing it to a 'gravitational interferometer', which splits the state of the clock into two branches; in one branch we bring the clock close to a massive object, while in the other we keep it far away (or we accelerate them by different amounts). Gravitational redshift causes the two branches to experience different amounts of time, i.e. Shapiro time delay, so after recombining the branches the final state is in the desired superposition.