Worldlines and Spacetime Topology

Time and space are deeply intertwined, as established by the principles of modern physics. In the context of Circaevum, this relationship is leveraged to reimagine how humans interact with time—not as a series of static events, but as a dynamic, interconnected topology. By incorporating the concept of worldlines and the structure of spacetime, Circaevum provides a framework that aligns with how reality operates at its most fundamental level.

What Are Worldlines?

Definition: A worldline represents the path of an object or entity as it moves through spacetime. It combines spatial location and temporal progression, offering a 4D perspective of existence.

In physics, a worldline visualizes the trajectory of an object, where each point along the line corresponds to the object’s position at a specific moment in time.

Origin: Coined within the framework of Einstein’s theory of relativity, worldlines emphasize the interconnectedness of time and space as a unified continuum.

Examples:

The worldline of a person represents their life, from birth to death, as a continuous thread through spacetime.
The worldline of a planet traces its orbital path around a star across time.

https://youtu.be/hTxWAQGgeQw

What Is Spacetime Topology?

Definition: Spacetime topology refers to the structure and relationships of spacetime, describing how events and objects are organized in the universe.

Topology studies the shape and connectivity of these structures, rather than focusing on precise measurements.

Key Principles:

Connectivity: Events are connected through causal relationships, where one event influences or is influenced by another.
Curvature: The geometry of spacetime is shaped by mass and energy, as described in general relativity. While Circaevum does not simulate gravitational effects, it draws on the concept of interconnected paths to create a meaningful visualization of events.

The Circaevum Coordinate System

The core structure of Circaevum is modeled on the Worldline of the Earth. As we can generally flatten the Orbital Plan without losing much in terms of conceptual understanding, we can simplify the Earth and other planetary motion into a 2D Polar Coordinate System. From here, we construct a vertical axis (perpindicular to the orbital plane), and extrude it based on Time.

Outward from the Sun and Earth’s shared North Poles extends into the Future, and outward from the South Poles extends the Past, with the orbital plane defaulting to the Present. This might be described as a Polar Worldline Coordinate System, or a Polar Spacetime Coordinate System. However, as we construct a recursively mapped version of this for the Earth’s daily rotation, and do the same for other planets, moons, and other space faring objects, we will refer to this unique system as the Circaevum Coordinate System.

Variations of this System

As Circaevum evolves, versions may vary in how this coordinate system is displayed. For example, on the Meta Quest and Windows Desktop versions, users can zoom from the annual view into the daily view, and rotate the entire solar system to help transition between preferred short and long-term views. However, on the Apple Vision Pro, the task of creating a Windowed App added new constraints to build around, resulting in a daily Circadian Rhythm view that is extracted from the Annual Orbital view, and displayed next to it at a zoomed in scale.

Avoiding the Trivial Stuff

At this phase in development we have decided to remove several parameters of Earth’s motion that are trivial for getting the basic experience down. For example, Earth’s orbital eccentricity is 0.0167, meaning it ever-so-slightly deviates from a perfect circle. We just make a perfect circle.

We also don’t include axial tilt, orbital inclination, orbital precession, axial precession, and other Milankovitch cycles. We may bake these back in at some point, when we feel like it’s necessary for a more cohesive user experience, but at this point it just adds trivial complications that will largely cause more headache than actual user experience improvement.

Earth as the Centerpiece

As we will explore in the The Overview Effect page, the Earth and its motion are at the heart of this experience. We love thinking about how this works, and this video does an amazing job of expanding on that:

https://youtu.be/IJhgZBn-LHg?si=mCjc_uu7lUoTSS8D