# Coordinate-free special relativity

## Main Question or Discussion Point

Are there any resources or texts that treat Minkowski geometry in a purely coordinate-free way?

Related Special and General Relativity News on Phys.org
bcrowell
Staff Emeritus
Gold Member
Interesting question.

Winitzki's book is on GR, not SR, but it's aggressively coordinate-free and may be of some use:

I assume from your question that you're looking for something fairly sophisticated that introduces a lot of the notational machinery of coordinate-free methods. However, the following books of the "relativity for poets" type do include some substantial discussion of SR in which coordinates don't play a special role:
-Penrose, The Road to Reality (SR and GR)
-Geroch, General Relativity from A to B (completely coordinate-free)

I don't know how much of the apparatus of coordinate-free methods retains any of its interest when spacetime is flat...

Here is a thread where I explored some attributes for arbitrary coordinatization in SR.

bcrowell
Staff Emeritus
Gold Member
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.

You need some basic coordinate free results to build off of, as well as a way of talking about tensors without resorting to index notation. Geometric algebra through the "Spacetime Algebra" or STA offers something that may do this, in that it makes statements like $\nabla \cdot s = 4$ (where $s$ is a position vector). You can also see that objects like the stress-energy tensor are linear operators on vectors and can be written like $\underline T(a)$ for some vector $a$ without reference to a particular basis, just in terms of dot products or other linear functions that involve coordinate free expressions.

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.
Thanks for taking the time to do some research. I've looked through Winitzki, and unfortunately it's too advanced for me to really understand. Maybe I'll have better luck with Schutz, but I'll have to check out my library for that one...

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? 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.

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?
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.

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.
That's absolutely true. I should clarify my question.

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. So a better way to state my question would be, are there any resources or texts that treat Euclidean geometry in a purely coordinate-free way that use the sort of math that would be appropriate for actually doing relativity? For example, by using vector spaces and inner products.

Perhaps you could post some example of a calculation that uses coordinates in the way you'd prefer to avoid?

Perhaps you could post some example of a calculation that uses coordinates in the way you'd prefer to avoid?
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?

ghwellsjr
Gold Member
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.
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.
Then they say:
In brief, we can completely describe and locate events entirely without a reference frame.
And finally:
Of course, if we want to use a reference frame, we can do so.

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?
The distance between two points $a,b$ is $\sqrt{(a-b)\cdot(a-b)}$.

Coordinate free resuls involve dot products or other products which we trust not to depend on the coordinate system.

bcrowell
Staff Emeritus
Gold Member
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.
I'm not so sure that's true. Take a look at Mermin's book It's About Time if you get a chance. It reads very much like Euclid turned loose on Minkowski space.

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.
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.

Based on the OP, I originally thought dEdt wanted to learn a whole bunch of the mathematical machinery of coordinate-free methods, at the level presented in Winitzki. Stuff like differential forms, Lie derivatives, pullback and pushforward, and fibre bundles. I don't know this stuff myself, and would like to learn it at some point.

But now that we've discussed things a little more, I think Mermin and Geroch would be just about right for what dEdt wants.

At an intermediate level, it's really not a big deal simply to learn abstract index notation. Just read the wikipedia article: http://en.wikipedia.org/wiki/Abstract_index_notation Even if you're reading a book that predates or doesn't intentionally use abstract index notation, with a little practice you can recognize that certain equations can be interpreted as valid abstract index notation, and are therefore coordinate-free, even though it *looks* like they're stated in some coordinates because of the presence of the indices.

Last edited:
robphy
Homework Helper
Gold Member
Here's a more advanced treatment by Geroch (connecting his GR from A-to-B presentation to vectors in Minkowski space)
"Example: Minkowski Vector Space" in Mathematical Physics (Robert Geroch)

Some aspects of that viewpoint are further treated in
Classical General Relativity (David Malament) (Malament was also on the faculty at U. Chicago before moving to Irvine)
http://arxiv.org/abs/gr-qc/0506065
... in particular, how an observer uses his four-velocity to decompose vectors and tensors into components (i.e. express a tensor in his coordinate system).

Part Two of Ludvigsen's General Relativity
https://www.amazon.com/dp/052163976X/?tag=pfamazon01-20
also develops observer-dependent vector and tensor decompositions

For intro treatments, try

Ch 2 of Kip Thorne's forthcoming text
http://www.pma.caltech.edu/Courses/ph136/yr2011/

Introduction to Spacetime: A First Course on Relativity (Bertel Laurent)
https://www.amazon.com/dp/9810219296/?tag=pfamazon01-20
http://www.worldcat.org/title/introduction-to-spacetime-a-first-course-on-relativity/oclc/247365199

You also might find this old thread on Minkowski space useful:

Last edited by a moderator:
Sorry for the late reply, I've just had a chance to go through the suggested texts. They were great, helped me out a lot. Thanks.

bcrowell
Staff Emeritus