Are the definitions of vectors in Liner Algebra and physics contradictory?

[MManuel Abad]

I'm really confused. I study physics, and in Relativity we deal a lot with vectors and tensors.

I know an element x of Rⁿ is a vector, because Rⁿ is a vector space. That's fine.

Now, in physics, we define a (contra-) vector x^a (with 'a' an index running from 1 to n) as a n-tuple of quantities V^a that, under a coordinates transformation from x to x' given by:

$$x^{a}\rightarrow x'^{a}=x'^{a}\left(x^{1},x^{2},...,x^{n}\right)$$

transform as:

$$V^{a}\rightarrow V'^{a}=V^{b}\frac{\partial x'^{a}}{\partial x'^{b}}$$

where Einstein summation convention is used.

Everything is fine... 'till the question: ¿Are coordinates (cartesian, if you like) vectors?

A coordinate is an element of Rⁿ so, as the definition of Linear Algebra says, YES, coordinates are vectors.

But let's see if that is so in physics.

Let's consider the simplest coordinate transformation: that of adding a constant c^ato each coordinate:

$$x^{a}\rightarrow x'^{a}=x^{a}+c^{a}$$

Then, it's easy to see that, using physics definition, the coordinates x^a are not vectors (at least, not contra-vectors), 'cuz the expresionThen, it's easy to see that, using physics definition, the coordinates xa are not vectors (at least, not contra-vectors), 'cuz the expresion

$$\frac{\partial x'^{a}}{\partial x'^{b}}$$gives 1 if a=b and 0 if a=/=b, so the definition would say:

$$x'^{a}=x^{a}$$

which, obviously, is not true.

So, what? Actually, I've never seen, in Relativity books, that the author calls the xa contravectors. But even physicists call them a "Radio vector". What's going on? But I've seen that the differentials dxa ARE contravectors (The physics definition works with the transformation I've used).

Could someone explain it to me? Are the definitions of vectors in Liner Algebra and physics contradictory?

 

[Sam Gralla]

In general coordinates are not vectors. However, in flat space with Minkowski coordinates, there is a natural isomorphism between coordinates and vectors, so you can get away with thinking about coordinates as "position vectors". This is one of the (many) concepts you have to "unlearn" in order to learn general relativity. It sounds like you are well on your way!

 

[MManuel Abad]

Wow, thanks! I hope I'm on the right way :D! When you said "Minkowski coordinates", you meant those of Descartes, plus time (with opposite sign), didn't you?

But I still don't know how to resolv that apparent paradox, between the concept of contravector and that of vector in Linear Algebra. I mean, with that simple transformation I made, didn't I get a contradiction? Cuz, using the "physical" (transformation based) definition of a contravector, I find that the old coordinates are equal to the new coordinates. That does hold, even in Minkowski coordinates. Where am I wrong??

 

[MManuel Abad]

Sorry, I just found an error the computer doesn't allow me (for some reason) to re-edit in my post. In the equation just above the phrase "which, obviously, is not true", the first x has a prime.

 

[Fredrik]

Sorry, I just found an error the computer doesn't allow me (for some reason) to re-edit in my post. In the equation just above the phrase "which, obviously, is not true", the first x has a prime.

There's a bug that makes the wrong images show up in previews. The workaround is to refresh after each preview. You may also have to refresh one more time after saving an edit.

The only definition of "vector" that deserves to exist is the one that says that the members of a vector space are called vectors. The definitions you have encountered are really bad ways to define tangent vectors and cotangent vectors of a manifold. If you want to learn a better way, this post is a good place to start. (Check out the links at the end too).

 

[Sam Gralla]

I wouldn't call this a contradiction, since "member of a vector space" is such a broad definition of vector that it isn't fair to compare with GR definition, which aims to define a much smaller class of objects. For example, the metric tensor is a member of a vector space, but GR would call it a rank 2 tensor, not a vector. These are just two different definitions that are useful in different contexts.

However, there is a very important point that you are bringing up, which is that in GR a vector does not "point from one coordinate location to another" (except possibly infinitesimally). In basic physics we are taught that subtracting two coordinate locations gives a vector. This makes sense if the space is flat, but it's obviously silly in a curved space. For example, consider a 2D example, the surface of a sphere, and make your two points the north and south poles. Which direction on the surface of the sphere does this vector point?? It's clear that the subtraction does not define an object we would like to think of as a direction on the surface of the sphere. Instead, vectors in GR are only defined locally at a point; technically they live in the tangent space at that point. While the coordinate transformation definition is efficient, you might get more understanding by reading about an equiavlent definition in terms of tangent vectors to curves passing through a point, which you can probably find somewhere on the internet.

Once you're comfortable with the GR definition, then try to see why the "subtracting two points" definition only works for flat spaces with Cartesian-like coordinates (which I called Minkowski out of habit in working in 4D spacetimes). Basically the point is there is a natural isomorphism between the tangent space and the coordinates in only this case.

 

[MManuel Abad]

Can anybody tell me how can I quote something?

But thank you both!

"However, there is a very important point that you are bringing up, which is that in GR a vector does not "point from one coordinate location to another" (except possibly infinitesimally). In basic physics we are taught that subtracting two coordinate locations gives a vector. This makes sense if the space is flat, but it's obviously silly in a curved space. For example, consider a 2D example, the surface of a sphere, and make your two points the north and south poles. Which direction on the surface of the sphere does this vector point?? It's clear that the subtraction does not define an object we would like to think of as a direction on the surface of the sphere. Instead, vectors in GR are only defined locally at a point; technically they live in the tangent space at that point. While the coordinate transformation definition is efficient, you might get more understanding by reading about an equiavlent definition in terms of tangent vectors to curves passing through a point, which you can probably find somewhere on the internet."

That really helped me, Sam Gralla!

"The only definition of "vector" that deserves to exist is the one that says that the members of a vector space are called vectors. The definitions you have encountered are really bad ways to define tangent vectors and cotangent vectors of a manifold. If you want to learn a better way, this post is a good place to start. (Check out the links at the end too). "

That's very interesting, Fredrik. I know almost nothing about manifolds an pseudo-Riemannian surfaces, but I've been waiting so long for a course in GR with that mathematical formalism. I'll look at those links, thank you really much!

I must confess I'm only an undergraduate in 4th semester, and I'm studying GR outside my courses, using "The Classical Field Theory, by Landau & Lifgarbagez. They don't use that formalism.

 

[MManuel Abad]

Wow, Fredrik! Your post really cleared me so many stuff. Thanks.

 

[Fredrik]

Wow, Fredrik! Your post really cleared me so many stuff. Thanks.

You're welcome. Glad I could help.

I used to find this stuff extremely frustrating when I first encountered it. I really disliked the way my physics professors tried to teach it. You don't need to understand the exact definition of "manifold" to get a pretty good understanding of tangent spaces, cotangent spaces, partial derivative operators, etc. You just need to know that the definition involves a set M, and a bunch of functions called "coordinate systems" (or "charts") that map open subsets of M onto open subsets of ℝⁿ.

Can anybody tell me how can I quote something?

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