Emergent time and more from wavefunction collapse in general relativity

This paper proposes a theory where wavefunction collapse in general relativity, driven by stochastic fluctuations of the lapse and shift, generates emergent time and a monotonic scale factor, leading to the suppression of constraint-violating modes while identifying long-lived scalar excitations as a viable dark matter candidate and recovering unitary dynamics for tensor gravitons in the long-time limit.

The Gist

The Big Idea: Time is a "Collapse," Not a River

Imagine the universe isn't a movie playing on a screen where time flows smoothly from the past to the future. Instead, imagine the universe is a giant, foggy cloud of possibilities.

In standard physics, we usually think that for the universe to exist, it must follow strict rules (like a perfect dance routine). But this paper suggests something wild: The universe starts out breaking those rules.

The author proposes that time itself is created by the universe slowly "collapsing" from a chaotic, rule-breaking mess into a neat, orderly state. As the universe fixes its mistakes, it grows bigger. This growth is what we experience as the ticking of a clock.

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The Analogy: The "Fuzzy Photo" Developing

Think of the universe at the beginning of time as a fuzzy, developing photograph.

1. The Messy Start (The Violation):
   When you first put a photo in the developer tray, it's a blur. It doesn't look like a picture yet; it violates the "rules" of being a clear image. In physics terms, the universe starts in a state that breaks the fundamental laws of gravity (called constraints).

2. The Developer Fluid (Stochastic Fluctuations):
   The paper suggests that the "developer fluid" is a random, jittery shaking of the fabric of space and time (called lapse and shift fluctuations). It's like someone gently shaking the photo tray. This shaking isn't random chaos; it's a specific kind of noise that pushes the photo to become clearer.

3. The Emergence of Time (The Collapse):
   As the photo develops, the blur clears up. The "time" we feel is just the process of the photo becoming sharp.

   • The Arrow of Time: Why does time only go forward? Because the photo can only develop from blurry to sharp, not the other way around. The universe is "collapsing" from a state of high violation (blurry) to a state of perfect order (sharp).

4. The Clock (The Scale Factor):
   As the photo develops, it gets bigger. In the universe, as it "collapses" into order, it expands. The size of the universe (the scale factor) acts as the clock. The bigger the universe gets, the more "time" has passed.

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The Three Types of "Gravitons" (The Particles of Gravity)

In this theory, gravity isn't just one thing. When the universe is "developing," it creates three different types of ripples or waves. Think of them as three different instruments in an orchestra that are trying to play a song, but the room is very echoey (non-unitary).

1. The Tensor Graviton (The Star Performer)

• What it is: These are the standard gravitational waves we know about (like ripples in a pond).
• What happens: At first, they are a bit fuzzy, but as time goes on, they become perfectly clear and follow the standard rules of physics (unitary dynamics).
• The Catch: They don't fade away completely immediately. There is a tiny bit of "static" or damping. This means gravitational waves from very far away might be slightly quieter or distorted than we expect, especially at high frequencies.

2. The Vector Mode (The Ghost)

• What it is: A type of vibration that tries to move sideways.
• What happens: This mode is extremely fragile. Because the universe is "collapsing" into order, this specific type of vibration is crushed immediately.
• The Result: It dies out so fast and so completely that it's practically invisible. It's like a ghost that vanishes the moment you try to look at it. It doesn't affect us at all.

3. The Scalar Mode (The Dark Matter Candidate)

• What it is: A vibration that squeezes and stretches space (like a breathing motion).
• What happens: This is the most interesting one.
  • Short waves (Small scale): They die out quickly. If you look at them in a lab on Earth, they vanish before you can measure them.
  • Long waves (Huge scale): They are very tough. They can survive for a very long time if they have a long wavelength.
• The Dark Matter Connection: The paper suggests these long-lived scalar waves might be Dark Matter.
  • Analogy: Imagine a fog that is thick enough to hold up a mountain (gravity) but so thin and fast-moving that you can't see it or touch it in your hand.
  • Because they live so long on cosmic scales, they could be the invisible "stuff" holding galaxies together. But because they die out quickly on small scales, we can't detect them in particle accelerators.

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Why Does This Matter?

1. Solving the "Time" Problem: In standard quantum gravity, it's hard to explain what "time" is because the equations say the universe is static. This theory says time is the process of the universe fixing its own errors.
2. New Physics: It predicts that gravity isn't perfectly "unitary" (perfectly reversible) at the very beginning. It leaves a tiny fingerprint on gravitational waves that we might be able to detect with future telescopes.
3. Dark Matter: It offers a new, elegant explanation for Dark Matter. It's not a mysterious new particle; it's just a leftover vibration from the universe's "development" process that happens to be very long-lived.

Summary in One Sentence

The universe is like a developing photograph that starts blurry and breaks the rules, but as it "collapses" into a sharp, orderly image through random shaking, it expands (creating time), and the leftover vibrations from this process might be the invisible Dark Matter holding our cosmos together.

Technical Summary

1. Problem Statement

The paper addresses the "Problem of Time" in quantum gravity. In standard canonical quantum gravity, physical states must be invariant under spacetime diffeomorphisms (satisfying the Hamiltonian and momentum constraints). Consequently, a single physical state represents an entire spacetime history, making the extraction of a temporal evolution sequence difficult.

• Relational Time Issues: Existing relational approaches (defining time via correlations between dynamical variables) suffer from conceptual ambiguities, such as the "many-fingered time" problem, where different clock choices yield different spatial geometries and entanglement structures.
• Constraint Violation: The author argues that to define a notion of time evolution, one must consider states that break diffeomorphism invariance.
• Extension to Full GR: Previous work by the author applied a wavefunction collapse theory to a minisuperspace model (cosmology). This paper aims to extend that framework to full-fledged General Relativity (GR) with propagating degrees of freedom (tensor, vector, and scalar modes) to determine if emergent unitarity and a time arrow persist.

2. Methodology

The paper employs a non-unitary, stochastic approach to time evolution based on two postulates:

1. Extended Hilbert Space: The physical Hilbert space $\mathcal{H}_{phys}$ is enlarged to include states that violate the Hamiltonian ($\hat{H}$) and momentum ($\hat{P}_\mu$) constraints.
2. Stochastic Collapse: Time evolution is defined as a process where an initial non-invariant state gradually collapses toward a diffeomorphism-invariant state. This collapse is driven by stochastic fluctuations of the lapse ($N$) and shift ($N^\mu$) functions.

Technical Framework:

• Path Integral Formulation: The evolution is modeled as a path integral over phase space variables ($g_{\mu\nu}, \Pi^{\mu\nu}$) averaged over a distribution of lapse and shift functions.
• Action Construction: The authors derive an effective action $S'$ where the constraints are squared ($\hat{H}^2, \hat{P}^2$) and multiplied by imaginary coefficients. This results in a non-Hermitian evolution operator.
  $$S' = \int dl \int dx \left( \Pi_{\mu\nu}\partial_l g^{\mu\nu} + i \frac{1}{\sqrt{g}} [\xi \hat{H}^2 + \xi' \hat{P}^2] \right)$$
• Saddle-Point Approximation: The analysis is performed in the limit of large spatial dimension ($d \to \infty$) to control calculations. The path integral is evaluated using a saddle-point approximation around a homogeneous background.
• Perturbation Theory: Fluctuations around the saddle point are decomposed into:
  • Tensor modes ($h_{\mu\nu}$): Gravitational waves.
  • Scalar mode ($\chi$): Associated with the scale factor and Hamiltonian constraint violation.
  • Vector modes ($\gamma_\mu, \Phi$): Associated with momentum constraint violation.

3. Key Contributions

• Emergent Time and Arrow: The paper demonstrates that the scale factor ($a$ or $\alpha$) acts as a clock. Due to the asymmetry of the Hilbert space (specifically the behavior of the effective mass in the large scale factor region), the stochastic random walk of the wavefunction causes the scale factor to monotonically increase, thereby generating an arrow of time.
• Emergent Unitarity: While the fundamental evolution is non-unitary (due to the collapse mechanism), the paper shows that unitarity emerges for tensor gravitons in the late-time limit.
• Differential Damping of Modes: A crucial distinction is made between the modes:
  • Tensor modes: Satisfy constraints approximately; their decay rate vanishes exponentially as time progresses, restoring unitarity.
  • Vector modes: Strongly damped at all scales; they decay rapidly and are effectively unobservable.
  • Scalar modes: Exhibit "marginal" damping. Their decay rate is proportional to their wavevector ($k$). Long-wavelength modes survive for long periods, while short-wavelength modes decay quickly.

4. Key Results

A. Dynamics of the Scale Factor

The scale factor grows monotonically with the evolution parameter $L$ (interpreted as time). In the limit of large $L$, the scale factor behaves as:
$$e^{\bar{\alpha}(L,L)} \sim (L)^{1/d}$$
This expansion is driven by a boundary term in the action arising from the non-self-adjoint nature of the Hamiltonian operator.

B. Excitation Spectrum and Complex Energies

The effective Hamiltonian governing the fluctuations is non-Hermitian, leading to complex energies $\Omega = \text{Re}(\Omega) + i\text{Im}(\Omega)$.

• Tensor Gravitons:
  • $\text{Re}(\Omega) = |k|$ (Standard dispersion).
  • $\text{Im}(\Omega) \propto -e^{-\text{const} \cdot t} |k|$.
  • Result: The imaginary part (damping) decays exponentially with time. In the late-time limit, tensor gravitons behave unitarily. However, at finite times, high-frequency gravitational waves are suppressed more than low-frequency ones.
• Vector Modes:
  • $\text{Re}(\Omega) = 0$.
  • $\text{Im}(\Omega) \propto -e^{\text{const} \cdot t}$.
  • Result: These modes decay exponentially fast at all scales, effectively suppressing momentum constraint violations.
• Scalar Mode (Dark Matter Candidate):
  • $\text{Re}(\Omega) = -v|k|$ (Negative energy).
  • $\text{Im}(\Omega) \propto -v|k|$.
  • Result: The decay rate is proportional to the wavevector ($k$).
    • Stability: Despite having negative energy (which usually implies instability), the finite lifetime (imaginary part) smears the resonance condition, preventing the runaway production of particles.
    • Dark Matter: Long-wavelength scalar excitations have extremely long lifetimes (scaling with wavelength), allowing them to survive over cosmological distances. Short-wavelength modes decay too quickly to be detected in local experiments. This makes the scalar mode a viable candidate for Dark Matter, contributing to long-range interactions while evading local detection.

C. Gravitational Wave Suppression

The theory predicts a frequency-dependent suppression of gravitational waves from distant sources. The amplitude is suppressed by a factor $F_k(t)$ which depends on the wavevector $k$. High-frequency waves are suppressed more strongly than low-frequency waves, a signature that could potentially be tested against standard GR predictions.

5. Significance

• Resolution of the Time Problem: The paper provides a concrete mechanism where time is not a fundamental parameter but an emergent property of wavefunction collapse, with a clear arrow of time derived from the dynamics of the scale factor.
• New Phenomenology: It predicts specific deviations from General Relativity:
  1. Frequency-dependent gravitational wave damping.
  2. A new form of Dark Matter composed of long-lived scalar gravitons arising from constraint violations.
  3. Suppression of vector modes, explaining why no vector gravitational degrees of freedom are observed.
• Theoretical Consistency: It demonstrates that a theory with non-unitary fundamental dynamics can yield an effectively unitary theory for observable degrees of freedom (tensor gravitons) at late times, reconciling the measurement problem with the emergence of classical spacetime.
• Dark Matter Mechanism: The proposal offers a novel, purely gravitational origin for dark matter without introducing new matter fields, relying instead on the long-lived tail of constraint-violating scalar modes.

In summary, Lee proposes that the universe's evolution is a stochastic collapse toward diffeomorphism invariance. This process naturally generates an expanding universe, an arrow of time, and distinct physical behaviors for different gravitational modes, with the scalar mode offering a compelling explanation for dark matter.