spacetime

A real(istic) time simulator.

spacetime attempts to model time propagation of clocks across the Earth-Moon system. It focuses on how network topology and time synchronization evolve in a growing lunar ecosystem, evaluating both centralized and decentralized Positioning, Navigation, and Timing (PNT) service network configurations.

Relativity Intuitions

This section builds intuitions about relativity and spacetime.

Here are some excellent videos to get started:

• Special Relativity Intuitions
  • Speed of light is constant in all reference frames
  • Everything moves at the speed of light
• General Relativity Intuitions
  • How gravity bends light even though it has no mass
  • Gravity comes from time curving space
  • Curved spacetime changes trajectories of light and matter

Intuitions for Special Relativity

Special Relativity is a theory of space and time that describes how space and time are affected by motion. It follows the natural consequences of the constancy of the speed of light in all reference frames. The special in special relativity refers to the fact that it is a special case of general relativity: situations when objects are moving at constant velocities in a vacuum.

Special Relativity Intuitions

The speed of light is constant in all inertial reference frames, regardless of the motion of the observer or the source of the light. This is a fundamental postulate of special relativity.

Inertial reference frames

The first key insight is to approach the problem with the understanding that motion is relative to the inertial reference frame of the observer. In any situation, we can always choose an inertial reference frame in which one object is at rest and everything else is moving relative to it.

From Brock’s perspective:

• The train is moving towards the station at speed $v$.
• The station is at rest.

Ash and Brock experience the world from their respective inertial reference frames. Each of their observations are true, and they are all valid.

The second key insight is to understand that the speed of light is constant for every observer regardless of their motion or the source of the light.

The speed of light is constant

The first postulate of special relativity is that the speed of light is constant in all inertial reference frames.

Another way to think about this is to consider a perfect “light clock” that each observer watches. The light clock consists of a photon that bounces between two mirrors once per second. Since the speed of light is constant in all inertial reference frames, an observer watching their clock will always observe the photon moving at the speed of light, $c$.

Let’s give a light clock to Ash and Brock and see what happens.

From Ash’s perspective:

• The station is moving towards the train at speed $v$.
• Light clock A is at rest.
• The photon in the light clock A is moving at speed $c$.
• Light clock B is moving towards the train at speed $v$.
• The photon in the light clock B is moving at speed $c$.

From Brock’s perspective:

• The train is moving towards the station at speed $v$.
• Light clock A is moving towards the station at speed $v$.
• The photon in the light clock A is moving at speed $c$.
• Light clock B is at rest.
• The photon in the light clock B is moving at speed $c$.

Light clock A is moving at speed $v$ relative to Brock, but he observes the photon inside to be moving at speed $c$, not $c+v$. How can this be?

The emergence of time dilation

Let’s follow the math and see what happens when we assume all of the observations are true.

$$\text{speed}=\frac{\text{distance}}{\text{elapsed time}}$$

                distance traveled in one second
Brock  * 0 m
Ash    *--* 100 m
Light  *~~~~~~~~~~~~~~~~~~~~~~~~~~~~~* 300_000_000 m

$$\begin{array}{rl}c& =\frac{300,000,000\text{ m}}{t_B\text{ s}}\\ c& =\frac{299,999,900\text{ m}}{t_A\text{ s}}\end{array}$$

In order for the speed of light to be constant in both cases, the elapsed time must be different for each observer!

$$\begin{array}{rl}\text{if}3.00\times 10^8\text{ m/s}& =\frac{300,000,000\text{ m}}{t_B}⟹t_B=1.00\text{ s}\\ \text{then}3.00\times 10^8\text{ m/s}& =\frac{299,999,900\text{ m}}{t_A}⟹t_A=0.999999667\text{ s}\end{array}$$

The time experienced by an observer at rest is called its proper time. Remember that it an observer only ever sees themself at rest. The proper time is always “normal” for the observer. (This is where the twin paradox comes from.)

Intiutions for General Relativity

General Relativity is a theory of gravity that describes how gravity affects the curvature of spacetime. It builds on the principles of Special Relativity, following the natural consequences of the constancy of the speed of light further and leading to the emergence of gravity and curved spacetime.

Space-time coordinates

Instead of thinking only in spatial coordinates, we can think in space-time coordinates. One axis is movement through time, and the other is movement through space. This coordinate system is called Minkowski space. Before continuing, it is critical to remember that in this coordinate system we think of an object moving at any speed as having a larger component along the space axis. So forget the direction of motion for now.

Let’s first consider the spatial component for our example. From Brock’s perspective, he is at rest—his space-time vector has zero space component. Ash is moving at speed $v$ relative to Brock, so his space-time vector has a component along the space axis proportional to $v$.

   * Brock
   *------* Ash
+---------------→ Space

Now we consider the time component—this is where the magic happens. Let’s “tick” time forward by one instant. From the last section, we know that Ash will experience slightly less time than Brock.

Time
▲     Brock
│  *  (1,11)
│  |
│  |        Ash
│  |      * (8,8)
│  |     /
│  |    /
│  |   /
│  |  /
│  | /
│  |/
│  * (1,1)
+---------------→ Space

Working out the math, we find that the length of both space-time vectors is exactly $c$.

Timekeeping Intuitions

Positioning, Navigation, and Timing Networks

Devlog

2024-12-08

I switched the docs to use mdbook today and I’m feeling good about diving back in to this project. Previous devlogs are available on the repository’s archive.

Since the last devlog I’ve gotten a lot more experience with Rust and Bevy. I have a strong sense of how to structure a Bevy project and I’m comfortable with its features now, so I’m ready to drop krabmaga and start building the project from scratch.

To start, I’m going to build the simulation without any rendering or graphics. Just enough to see what’s happening through logs or maybe egui. Unlike previous attempts, I’m going to focus on implementing the math from the research papers I found, rather than trying to jump straight to the end goal.

2024-12-15

Here are two videos that really helped me find intuitions for the math of general relativity:

• https://youtu.be/OpOER8Eec2A?si=22VBvAKu8VV6ye5E
• https://youtu.be/S78h8zQwQe0?si=1ptIvQqqiFlSOqdG

I think it would be helpful to make this app demonstrate these intuitions. Let’s follow the videos’ trains of thought and see if we can implement them along the way. One thing these videos do very well is to build up ideas bit by bit. Similarly, I think we can build up the app’s features bit by bit.

While the math could be implemented with real physical constants, it would be much easier to see what’s happening if we use simple scale factors, especially if we can change them on the fly.

I’ve said this before and I’ll say it again: I am a huge fan of the “retro radar” aesthetic. It would be fun to lean into that. Here is a Pinterest board for inspiration: https://pin.it/CgMRWzHFl

2024-12-22

I watched a very insightful video series that explains special relativity in a way that I think is very intuitive. I think it would be helpful to implement these intuitions here.

• Special Relativity Intuitions
  • Speed of light is constant in all reference frames
  • Everything moves at the speed of light
• General Relativity Intuitions
  • How gravity bends light even though it has no mass
  • Gravity comes from time curving space
  • Curved spacetime changes trajectories of light and matter