Chapter 2: The Nature of Time Travel and Choice

An excellent short story by Ted Chiang, author of "Arrival," called "The Merchant and the Alchemist’s Gate" from his collection Exhalation, contemplates time travel within the confines of limited free will. In this narrative, an alchemist in Baghdad possesses a portal that can transport individuals both forward and backward in time, allowing them to revisit their past.

Chiang illustrates how characters can seemingly make free decisions while their outcomes are ultimately predetermined. This idea is supported by a familiar trope in time travel narratives: the universe intervenes to prevent paradoxes from occurring. For instance, if one were to travel back to save someone who had perished, or to eliminate someone who had lived, the universe would subtly thwart those attempts.

Thus, while free will may exist, each choice made is part of a complex dynamic system with repercussions across both past and future. When one attempts to alter history, the universe adjusts events to avoid contradictions. But is this notion genuinely valid?

The second video titled "The Paradoxes of Time Travel" explores the various complexities and theories associated with time travel.

Chapter 3: Analyzing Time Travel with Billiard Balls

To simplify the discussion, let’s consider a scenario involving billiard balls rather than people. In 1991, researchers Echeverria, Klinkhammer, and the later Nobel laureate Kip Thorne published a study examining what might occur on a specially designed billiard table featuring wormholes as pockets, potentially allowing the balls to travel back in time and collide with their earlier selves.

The aim of their research was to determine if the equations governing a billiard ball’s motion through a wormhole to its own past could yield a solution. If no solution exists, the scenario is physically impossible. Conversely, if there is a unique solution, the ball would follow a distinct path based on its initial conditions. However, if multiple solutions exist, this indicates an ill-defined situation, necessitating further constraints or revealing logical inconsistencies.

A wormhole permitting an object to return to its past is termed a Closed Timelike Curve (CTC). Such solutions in Einstein’s general relativity allow for paradoxes. Other wormhole types exist, some of which take so long to traverse that returning to one’s past becomes impossible.

Without a CTC, any initial value problem will yield only one solution, a fundamental principle of high school physics. However, with a CTC, fascinating phenomena occur.

Chapter 4: The Implications of Self-Interaction

Consider a scenario where a billiard ball collides with itself in the past. Initially, one might assume there was no collision, only to find that one has occurred. This leads to a logical paradox that seems to obstruct a solution.

Surprisingly, the researchers found that self-interacting trajectories, which they termed dangerous, do indeed have an infinite number of solutions. This indicates that for every set of initial conditions, if a self-collision occurs, countless trajectories can arise.

A particularly intriguing problem is the self-inconsistent solution, where the ball travels through the wormhole only to strike itself, preventing entry altogether. The existence of infinite solutions suggests that the problem itself is ill-posed and reflects issues with classical mechanics.

The researchers concluded that classical mechanics is merely an approximation of quantum mechanics, which posits that every object adheres to multiple trajectories through time and space, each contributing probabilistically to any measurement. Thus, the billiard ball does not follow a single path but numerous paths at once, each with its own probability.

In this framework, a self-collision creating a paradox merely raises the likelihood of trajectories that avoid entering the wormhole, without negating those that do. The collision itself is also a matter of probability; nothing is certain in quantum mechanics.

Many believe that the Many Worlds Interpretation of quantum mechanics resolves paradoxes by suggesting that paradoxical actions influence alternate realities. However, the resolution actually lies within quantum mechanics itself. The interpretation merely offers a lens through which to view the solution.

Ultimately, quantum mechanics permits the coexistence of classically contradictory events, regardless of interpretation. Paradoxes shape the quantum wavefunction but do not contravene any physical laws.

Chapter 5: Conclusion: The Future of Time Travel

This raises the question: Can we travel back in time? While Closed Timelike Curves exist within the realm of Einstein’s relativity, there is currently no known matter or energy that would allow a person or any macroscopic object to journey into the past or surpass the speed of light. Maintaining such a wormhole would necessitate hypothetical matter or energy that has yet to be discovered, which would violate established principles of quantum physics.

So while the dream of a time machine remains elusive, it's reassuring to know that the grandfather paradox is only paradoxical when viewed through a classical lens rather than a quantum one.

References: - Tobar, Germain, and Fabio Costa. “Reversible dynamics with closed time-like curves and freedom of choice.” arXiv preprint arXiv:2001.02511 (2020). - Echeverria, Fernando, Gunnar Klinkhammer, and Kip S. Thorne. “Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory.” Physical Review D 44.4 (1991): 1077.