- #1
dEdt
- 288
- 2
Are there any resources or texts that treat Minkowski geometry in a purely coordinate-free way?
bcrowell said:I spent some time googling and digging around and didn't find anything that's purely coordinate-free SR. Winitzki, whose book is coordinate-free, lists some recommended books, of which Ludvigsen is the only one that uses abstract index notation. Amazon reviews say Ludvigsen is very difficult (so presumably not for you, if you're looking for an intro to SR) and point to Schutz's Geometrical Methods of Mathematical Physics as an easier alternative. Schutz, judging by the table of contents, is really not specifically about SR, but may be of interest.
dEdt said:That said, here's a simpler request that may be easier to meet: are there any resources or texts that treat Euclidean geometry in a purely coordinate-free way?
ImaLooser said:Isn't the geometry of Euclid coordinate-free? Cartesian coordinates weren't invented for another two thousand years.
Maybe I don't understand the question.
Muphrid said:Perhaps you could post some example of a calculation that uses coordinates in the way you'd prefer to avoid?
It may be that Taylor and Wheeler's book Spacetime Physics is just what you are looking for. On page 10, they make the claim that:dEdt said:Basically, what I'm looking for is a way of defining terms and stating the properties of a geometry without referring to a particular coordinate system. Then, I can say "if you want to create a coordinate system, you have to do blah blah blah" where "blah blah blah" is some procedure for creating the coordinate system. It doesn't have to be very sophisticated either, I can imagine that it would only involve linear algebra.
If I can do this, then I'm sure I can extend the ideas to Minkowski geometry.
To chart all happenings, we need no more than a table of spacetime intervals between every pair of events. That's all we need! From this table and the laws of Lorentz geometry, it turns out, we can construct the space and time locations of events as observed by every laboratory and rocket observer.
In brief, we can completely describe and locate events entirely without a reference frame.
Of course, if we want to use a reference frame, we can do so.
dEdt said:What's the distance between two given points?
Obviously, this can't be answered per se, because the only way to get a numerical answer would be to use a coordinate system. So really, the question would be, is there a coordinate free way of defining distance from which a formula for the distance between two points involving their coordinates in an arbitrary coordinate system can found?
dEdt said:Euclid's geometry, although coordinate free, is also expressed in a language completely unsuitable for SR: geometric constructions, congruence, and so on aren't appropriate for relativity.
Right, in other words, the metric is fundamental, and the output of the metric (the interval) is a scalar, so it's coordinate-independent. Geroch's popular-level book General Relativity from A to B carries this approach through completely, without ever using coordinate systems.ghwellsjr said:It may be that Taylor and Wheeler's book Spacetime Physics is just what you are looking for. On page 10, they make the claim that:
To chart all happenings, we need no more than a table of spacetime intervals between every pair of events. That's all we need! From this table and the laws of Lorentz geometry, it turns out, we can construct the space and time locations of events as observed by every laboratory and rocket observer.
"Coordinate-free special relativity" is a mathematical framework used to describe the laws of physics in a way that is independent of any particular coordinate system. It is based on the principles of special relativity, which states that the laws of physics are the same for all observers in uniform motion.
Traditional special relativity uses coordinates to describe the position and motion of objects in space and time. In contrast, coordinate-free special relativity uses mathematical objects called tensors to describe the laws of physics in a way that is independent of coordinates. This allows for a more general and abstract understanding of the principles of special relativity.
Coordinate-free special relativity is important because it allows for a deeper understanding of the laws of physics, and it can be applied to a wider range of scenarios. It also helps to simplify and unify different theories of physics, making it a valuable tool for scientists studying the fundamental principles of the universe.
Coordinate-free special relativity has many applications in modern physics, including in the study of black holes, cosmology, and quantum field theory. It is also used in engineering and technology, such as in the development of GPS systems and satellite communication.
While coordinate-free special relativity may seem daunting at first, with the proper background in mathematics and physics, it can be understood and applied effectively. It is a widely accepted and used framework in modern physics, and with practice and study, it can become more intuitive and easier to grasp.