Because the Minkowski inner product product is not positive definite, the norm induced by an inner product is a norm, and the space is not a metric space strictly speaking.
The name given to this type of space is a pseudometric space.
This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.
By default, we will use the time negative representation unless stated otherwise:but another equivalent one is to use a time positive representation:The matrix is typically denoted by the Greek letter eta.
Why should I care when I can calculate new x and new time with Lorentz transformation?
Answer: it can give some qualitative intuition on what is larger/smaller happens before/after based only on arguably more intuitive geometric considerations, without requiring you to do any calculations, see e.g. Figure "Spacetime diagram illustrating how faster-than-light travel implies time travel".
A subset of Spacetime diagram.
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 coneOtherwise 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.
Animation showing how space-separated events can be observed to happen in different orders by observers in different frames of reference
. Source. 
Mathematically, we can decide if two events are timelike-separated or spacelike-separated by just looking at the sign of the spacetime interval between them.
On the light cone, these are events on the left/right part of the cone.
Different observers might not agree on the order of two spacelike-separated events.
Further discussion at Section "Light cone".
The opposite of those events are timelike-separated events.
In the Galilean transformation, there are two separate invariants that two inertial frame of reference always agree on between two separate events:
- time
- length, given by the Pythagorean theorem
However, in special relativity, neither of those are invariant separately, since space and time are mixed up together.
Instead, there is a new unified invariant: the spacetime-interval, given by:
Note that this distance can be zero for two events separated.
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.
Subtle is the Lord by Abraham Pais (1982) page 22 mentions that when Einstein saw this in 1915, he was so excited he couldn't work for three days.
Maxwell's equations in curved spacetime by 
Ciro Santilli 35 Updated 2025-03-09 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-09 +Created 1970-01-01Combination of electromagnetism and general relativity. Unlike combining quantum mechanics and general relativity, this combination was easier.
TODO any experiments of interest at all?
In 2020 physics, best explained by general relativity.
TODO: does old Newtonian gravity give different force results than general relativity?
Experiments that measure the gravitational constant by Ciro Santilli 35 Updated 2025-03-09 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-09 +Created 1970-01-01As of 2019, the Standard Model and general relativity are incompatible. Once those are unified, we will have one equation to describe the entirety of physics.
There are also however also unsolved problems in electroweak interaction + strong interaction, which if achieved is referred to as a Grand Unified Theory. Reaching a GUT is considered a sensible intermediate step before TOE.
The current state of Physics has been the result of several previous unifications as shown at: en.wikipedia.org/wiki/Theory_of_everything#Conventional_sequence_of_theories so it is expected that this last missing unification is likely to happen one day, potentially conditional on humanity having enough energy to observe new phenomena.
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!
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/derivative - 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 4. Visual Studio Code extension tree navigation.
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.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Internal cross file references done right:
- Infinitely deep tables of contents:
Figure 6. Dynamic article tree with infinitely deep table of contents.Live URL: ourbigbook.com/cirosantilli/chordateDescendant 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