Lorentz transformation by Ciro Santilli 40 Updated 2025-07-16
The equation that allows us to calculate stuff in special relativity!
Take two observers with identical rules and stopwatch, and aligned axes, but one is on a car moving at towards the direction at speed .
When both observe an event, if we denote:
It is of course arbitrary who is standing and who is moving, we will just use the term "standing" for the one without primes.
Then the coordinates of the event observed by the observer on the car are:
where:
Note that if tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:
This explains why we don't observe special relativity in our daily lives: macroscopic objects move too slowly compared to light, and is almost zero.
Length contraction by Ciro Santilli 40 Updated 2025-07-16
Suppose that a rod has is length measured on a rest frame (or maybe even better: two identical rulers were manufactured, and one is taken on a spaceship, a bit like the twin paradox).
Question: what is the length than an observer in frame moving relative to as speed observe the rod to be?
The key idea is that there are two events to consider in each frame, which we call 1 and 2:
  • the left end of the rod is an observation event at a given position at a given time: and for or and for
  • the right end of the rod is an observation event at a given position at a given time : and for or and for
Note that what you visually observe on a photograph is a different measurement to the more precise/easy to calculate two event measurement. On a photograph, it seems you might not even see the contraction in some cases as mentioned at en.wikipedia.org/wiki/Terrell_rotation
Measuring a length means to measure the difference for a single point in time in your frame ().
So what we want to obtain is for any given time .
In summary, we have:
By plugging those values into the Lorentz transformation, we can eliminate , and conclude that for any , the length contraction relation holds:
The key question that needs intuitive clarification then is: but how can this be symmetric? How can both observers see each other's rulers shrink?
And the key answer is: because to the second observer, the measurements made by the first observer are not simultaneous. Notably, the two measurement events are obviously spacelike-separated events by looking at the light cone, and therefore can be measured even in different orders by different observers.
Light cone by Ciro Santilli 40 Updated 2025-07-16
The key insights that it gives are:
  • future and past are well defined: every reference frame sees your future in your future cone, and your past in your past cone
    Otherwise causality could be violated, and then things would go really bad, you could tell your past self to tell your past self to tell your past self to do something.
    You can only affect the outcome of events in your future cone, and you can only be affected by events in your past cone. You can't travel fast enough to affect.
    Two spacetime events with such fixed causality are called timelike-separated events.
  • every other event (to right and left, known as spacelike-separated events) can be measured to happen before or after your current spacetime event by different observers.
    But that does not violate causality, because you just can't reach those spacetime points anyways to affect them.
Figure 1.
Animation showing how space-separated events can be observed to happen in different orders by observers in different frames of reference
. Source.
General relativity by Ciro Santilli 40 Updated 2025-07-16
Unifies both special relativity and gravity.
Not compatible with the Standard Model, and the 2020 unification attempts are called theory of everything.
One of the main motivations for it was likely having forces not be instantaneous, but rather mediated by field to maintain the principle of locality, just like electromagnetism did earlier.
Fast travel is a gameplay mechanic commonly found in video games, particularly in open-world and role-playing games (RPGs). It allows players to quickly move between locations on the game map without needing to travel the distance in real-time. This feature is often implemented to save time and enhance the gaming experience by allowing players to focus on other aspects of the game, such as quests or exploration.
"Ship load" typically refers to the quantity of goods, cargo, or materials that a ship is designed to carry. It can relate to various measurements, including: 1. **Deadweight Tonnage (DWT)**: This is the maximum weight a ship can safely carry, including cargo, fuel, crew, provisions, and any other items. 2. **Cargo Capacity**: This specifically refers to the volume or weight of goods that can be loaded onto the ship for transport.
Video 1.
Replicating the Fizeau Apparatus by AlphaPhoenix (2018)
Source. Modern reconstruction with a laser and digital camera.
Video 2.
Visualizing video at the speed of light - one trillion frames per second by MIT (2011)
Source. Fast cameras. OK, this takes it to the next level.
Body Mass Index (BMI) is a numerical value derived from a person's weight and height, used as a screening tool to categorize individuals into different weight status categories.

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 2.
    You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
    .
    Figure 3.
    Visual Studio Code extension installation
    .
    Figure 4.
    Visual Studio Code extension tree navigation
    .
    Figure 5.
    Web editor
    . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
    Video 4.
    OurBigBook Visual Studio Code extension editing and navigation demo
    . Source.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact