A law of physics is Galilean invariant if the same formula works both when you are standing still on land, or when you are on a boat moving at constant velocity.
For example, if we were describing the movement of a point particle, the exact same formulas that predict the evolution of must also predict , even though of course both of those will have different values.
It would be extremely unsatisfactory if the formulas of the laws of physics did not obey Galilean invariance. Especially if you remember that Earth is travelling extremelly fast relative to the Sun. If there was no such invariance, that would mean for example that the laws of physics would be different in other planets that are moving at different speeds. That would be a strong sign that our laws of physics are not complete.
Lorentz invariance generalizes Galilean invariance to also account for special relativity, in which a more complicated invariant that also takes into account different times observed in different inertial frames of reference is also taken into account. But the fundamental desire for the Lorentz invariance of the laws of physics remains the same.
Subgroup of the Poincaré group without translations. Therefore, in those, the spacetime origin is always fixed.
Or in other words, it is as if two observers had their space and time origins at the exact same place. However, their space axes may be rotated, and one may be at a relative speed to the other to create a Lorentz boost. Note however that if they are at relative speeds to one another, then their axes will immediately stop being at the same location in the next moment of time, so things are only valid infinitesimally in that case.
This group is made up of matrix multiplication alone, no need to add the offset vector: space rotations and Lorentz boost only spin around and bend things around the origin.
One definition: set of all 4x4 matrices that keep the Minkowski inner product, mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) page 63. This then implies:
All indefinite orthogonal groups of matrices of equal metric signature are isomorphic by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Following the definition of the indefinite orthogonal group, we want to show that only the metric signature matters.
First we can observe that the exact matrices are different. For example, taking the standard matrix of :and:both have the same metric signature. However, we notice that a rotation of 90 degrees, which preserves the first form, does not preserve the second one! E.g. consider the vector , then . But after a rotation of 90 degrees, it becomes , and now ! Therefore, we have to search for an isomorphism between the two sets of matrices.
For example, consider the orthogonal group, which can be defined as shown at the orthogonal group is the group of all matrices that preserve the dot product can be defined as:
Classification of simple Lie groups by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
A bit like the classification of simple finite groups, they also have a few sporadic groups! Not as spectacular since as usual continuous problems are simpler than discrete ones, but still, not bad.
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
This does not seem to go deep into the Standard Model as Physics from Symmetry by Jakob Schwichtenberg (2015), appears to focus more on more basic applications.
But because it is more basic, it does explain some things quite well.
Sometimes, these are more than just mechanics, but also have deeper life analogues. The title of Zen and the Art of Motorcycle Maintenance comes to mind. Sometimes they are just mechanics.
With more philosophical metaphors:
- when your bike breaks, that provides an unique opportunity to learn how to fix it: try to fix your own bike before taking it to the shop
- if the wind is blowing against you on the way out, it will likely blow behind you on the way back. But remember that the other way around also applies.
- always take one extra clothing layer than what you think you will need in your back pocket or sport bag, especially when time is changing fast in Sprint and Autumn. The weather on the road outside of town can change very quickly!
- if you took a turn, and it feels wrong, stop to check the map, and possibly backtrack to safety. When it feels wrong, it usually is a bad idea, e.g. roads where cars are too fast/too many. But if you take a wrong turn and it feels right, then follow it without fear and see what it leads to!
- don't carry a speedometer on your bike. Analysis can be done afterwards on Strava. The only measurement that matters is "how awesome am I feeling right now?". Live in the moment instead of checking your speedometer every 10 seconds.
- cycle when you body calls it out of addiction, not out a goal that you've made up that must be reached
With less philosophical metaphors:
- learn how much water and food you need to take for a trip. Otherwise, you will bonk at some time, when you least expect, it happens very suddenly.And then you better hope to God that you can find a food shop nearby. Luckily this was the case for Ciro's first and only bonk so far.And besides bonking all out, being tired and hungry makes you make stupid decision, especially where traffic is involved!Food is safety. Light is safety. Time is safety. Calm is safety. Chocolate bars and candy cannot count as lunch food, only delay lunch. A sandwich with ham cheese and salad is food. A bag of M&M's with a can of soda can bring you back from the dead.When you are not in familiar grounds, take twice as much as you think you might possibly ever need. Hofstadter's law.You will also learn that, surprise surprise, carbohydrates that you ate one or two days before a ride stay stored in your liver and muscles, and also greatly affect how quickly you will bonk, thus the concept of carbohydrate loading.
- correct saddle hight is fundamental, your legs must be almost fully stretched at the bottom position
- it is impossible to reach the correct tire pressure with (cheap?) hand pumps, their only purpose is to fill up a flat tire so you can get home after a long ride. But a track pump.
- clean and lube your chain. The speed benefit is instantaneous and mind blowing. It also greatly improves gear shifting.This also prevents the chain from rusting, because the lube takes up the place where water would stay, and the muck makes it harder for water to evaporate.This is the most common bike maintenance mistake you see on the streets: people with that high pitched overly dry chain noise.
- when a piece on your bike breaks and has no clear name written on it, you can try to identify it Google images
- the more you watch YouTube maintenance videos without haste, the more you end up learn random new stuff that unexpectedly saves you later
- if you took a turn, and it feels wrong, stop to check the map, and possibly backtrack to safety. When it feels wrong, it usually is a bad idea, e.g. roads where cars are too fast/too many
- public place with lots of people are bicycle parking Hell, because due to anonymity and the large number of distractions, it becomes exponentially more likely that someone will fuck you bike somehow, e.g. by dropping it on the ground. Always search a bit for a reasonable place to park, and avoid overcrowded parking spaces at all costs.
- gear change matters
- when you get on your bike to start riding, start riding slowly and gradually switch up pedal forces and gears. Things may have shifted in a weird position as it gets kicked around in parking. Ciro managed to bend his derailleur like that!
- spin to win, AKA learn to user your gears
- it is not shameful to ride on your lower gears on a hill. You can actually go surprisingly fast with them, and conserve energy for later. Learn when to use each gear ratio.
- learn to identify your suppliers:
- www.wiggle.co.uk/: in Europe, this is best place to buy clothing from, and also good for some bike parts. It is the most organized website, and contains non-generic shit which Amazon is full of.For bike parts Amazon is also worth looking into however. Bike parts a bit different from clothing because you have to make sure that stuff fits, so you hopefully know exactly the part name before before buying it, and therefore website organization is not as crucial.Wiggle is however guilty of shameless: discounts that happen more often than not
- always take your lights off the bike into your bag when you park, anywhere, and for any amount of time, even if a quick stop. Drug addicts are everywhere, always ready love to steal and resell them.
- sometimes you do something stupid like going into a really muddy path, and it is really fun, because you've never been there in your life. But then your bike gets really dirty, and your feet are wet and freezing, and you promise yourself you will never do something that stupid again. But then you do it again in a different location, because it was too much fun. Once more unto the breach just comes to Ciro's mind every such time. Embrace this.
Pinned article: ourbigbook/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!
Intro to OurBigBook
. Source. We have two killer features:
- 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-calculusArticles 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/derivativeVideo 2. OurBigBook Web topics demo. Source. - 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.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.Figure 3. Visual Studio Code extension installation.Figure 5. . 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. - Infinitely deep tables of contents:
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