The usual gravitational wave memory effect can be understood as a change in
the separation of two initially comoving observers due to a burst of
gravitational waves. Over the past few decades, a wide variety of other,
“persistent” observables which measure permanent effects on idealized detectors
have been introduced, each probing distinct physical effects. These observables
can be defined in (regions of) any spacetime where there exists a notion of
radiation, such as perturbation theory off of a fixed background, nonlinear
plane wave spacetimes, or asymptotically flat spacetimes. Many of the
persistent observables defined in the literature have been considered only in
asymptotically flat spacetimes, and the perturbative nature of such
calculations has occasionally obscured deeper relationships between these
observables that hold more generally. The goal of this paper is to show how
these more general results arise, and to do so we focus on two observables
related to the separation between two, potentially accelerated observers. The
first is the curve deviation, which is a natural generalization of the
displacement memory, and also contains what this paper proposes to call drift
memory (previously called “subleading displacement memory”) and ballistic
memory. The second is a relative proper time shift that arises between the two
observers, either at second order in their initial separation and relative
velocity, or in the presence of relative acceleration. The results of this
paper are, where appropriate, entirely non-perturbative in the curvature of
spacetime, and so could be used beyond leading order in asymptotically flat

Persistent Observables and General Results: A Roadmap for the Future

As the study of gravitational waves continues to advance, researchers have introduced a variety of “persistent” observables that measure permanent effects on detectors. These observables probe distinct physical effects and can be defined in any spacetime where radiation is present. However, many of these persisten observables have only been considered in asymptotically flat spacetimes, limiting their applicability.

The goal of this paper is to reveal the deeper relationships between these observables and provide a more general understanding of their origins. In particular, the focus is on two observables related to the separation between two potentially accelerated observers.

1. Curve Deviation: Drift Memory and Ballistic Memory

The first observable, known as curve deviation, is a natural extension of the displacement memory. The curve deviation contains what this paper refers to as drift memory, which was previously known as “subleading displacement memory,” as well as ballistic memory. By studying these components, this paper aims to shed light on the connections between them and their general applicability beyond asymptotically flat spacetimes.

2. Relative Proper Time Shift

The second observable focuses on the relative proper time shift that occurs between two observers. This shift can arise either at second order in their initial separation and relative velocity or in the presence of relative acceleration. Understanding this proper time shift is crucial for accurately measuring and interpreting gravitational wave data.

What sets this paper apart is its non-perturbative approach to studying these observables. The results presented here do not rely on perturbation theory and provide insights that extend beyond leading order in asymptotically flat spacetimes. This opens up new possibilities for future research and applications of these observables.

Roadmap for Readers: Challenges and Opportunities

For readers interested in exploring the field of persistent observables and their generalizations, this paper offers a roadmap for further study. However, there are some challenges and opportunities on the horizon that readers should be aware of.


  1. The non-perturbative nature of the results presented in this paper may require advanced mathematical techniques and a solid understanding of general relativity.
  2. The applicability of these observables beyond asymptotically flat spacetimes needs further investigation and validation.
  3. Experimental verification of these observables and their effects on real detectors is an ongoing challenge.


  1. The general results and insights provided here pave the way for more comprehensive studies of persistent observables in various spacetime scenarios.
  2. Further research can explore the connections between different observables and uncover new relationships that enhance our understanding of gravitational waves.
  3. The non-perturbative approach opens up possibilities for developing more accurate models for detecting and interpreting gravitational wave signals.

In conclusion, this paper provides a valuable contribution to the field of persistent observables in gravitational wave research. By uncovering deeper relationships and presenting non-perturbative results, it sets the stage for future advancements and applications in this exciting field.

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Introduction to Quantum Cosmology

Quantum Cosmology stands as the forefront of unraveling the profound secrets of our universe. Merging the principles of Quantum Mechanics and General Relativity, this advanced field seeks to explain the cosmos’s very early stages, focusing on the Planck era where classical theories of gravity no longer suffice. We delve deep into the realms of spacetime, singularity, and the initial conditions of the universe, exploring how Quantum Cosmology reshapes our understanding of the cosmos’s birth and evolution.

The Birth of the Universe: The Big Bang and Beyond

At the heart of Quantum Cosmology is the intriguing narrative of the universe’s inception, commonly referred to as the Big Bang. Traditional models depict a singular point of infinite density and temperature. However, Quantum Cosmology introduces a more nuanced picture, suggesting a quantum bounce or other quantum phenomena that avoid the singularity, offering a revolutionary perspective on the universe’s earliest moments.

Unraveling the Planck Era

The Planck era represents the universe’s first






seconds, a time when the classical laws of physics cease to operate. Quantum Cosmology strides into this enigmatic epoch, employing quantum gravity theories like Loop Quantum Gravity or String Theory. These theories aim to provide a coherent description of spacetime’s fabric at this fundamentally small scale, potentially uncovering new insights about the universe’s structure and behavior.

The Role of Quantum Fluctuations

In the primordial universe, quantum fluctuations are believed to play a pivotal role. These minute variations in energy density, amplified by cosmic inflation, are thought to lead to the large-scale structures we observe today, such as galaxies and clusters. Quantum Cosmology seeks to quantitatively understand these fluctuations, deciphering their implications for the universe’s overall architecture and destiny.

Navigating through Cosmic Singularities

One of the most tantalizing challenges in contemporary physics is understanding cosmic singularities—points where the laws of physics as we know them break down. Quantum Cosmology proposes various scenarios to address these enigmas, suggesting that quantum effects may smooth out singularities or even connect our universe to others through cosmic gateways known as wormholes.

The Quantum Landscape of the Universe

The concept of a quantum landscape has emerged, depicting a vast, complex space of possible universes each with their own laws of physics. This landscape offers a staggering vision of a multiverse, where our universe is but one bubble in a frothy sea of countless others. Quantum Cosmology explores these ideas, examining their implications for fundamental physics and our place in the cosmos.

Advanced Theories and Models

To tackle these profound questions, Quantum Cosmology utilizes several advanced theories and models. Loop Quantum Cosmology offers insights into the very early universe, suggesting a bounce instead of a big bang. String Theory proposes a universe composed of tiny, vibrating strings, potentially in higher dimensions. These and other models are at the cutting edge, each contributing valuable perspectives to our understanding of the cosmos.

Empirical Evidence and Observational Challenges

While Quantum Cosmology is a field rich with theoretical insights, it faces the significant challenge of empirical verification. As researchers devise ingenious methods to test these theories, from observations of the cosmic microwave background to the detection of gravitational waves, the field stands at a thrilling juncture where theory may soon meet observation.

Future Directions and Implications

As we advance, Quantum Cosmology continues to push the boundaries of knowledge, hinting at a universe far stranger and more wonderful than we could have imagined. Its implications stretch beyond cosmology, potentially offering new insights into quantum computing, energy, and technology. As we stand on this precipice, the future of Quantum Cosmology promises not just deeper understanding of the cosmos, but also revolutionary advancements in technology and philosophy.

Conclusion: A Journey through Quantum Cosmology

Quantum Cosmology is more than a field of study; it’s a journey through the deepest mysteries of existence. From the universe’s fiery birth to the intricate dance of quantum particles, it offers a compelling narrative of the cosmos’s grandeur and complexity. As we continue to explore this fascinating frontier, we not only uncover the universe’s secrets but also reflect on the profound questions of our own origins and destiny.

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