Undergrad What is the most correct way to write a vector in GR?

[davidge]

So, most Relativity textbooks (although some famous, like Weinberg's don't) show us that a vector $V$ is properly written as $$V^\mu(x) \frac{\partial}{\partial x^\mu}$$ where $V^\mu(x)$ are its components at the point $x$ and the "base" in which the vector is written in is the operator $\frac{\partial}{\partial x^\mu}$.

But... is it useful to write the basis vectors as in above? Why are vectors written in that way in Special and General Relativity and not in other fields of physics and mathematics? e.g. in elementary vector calculus, vector analysis and linear algebra courses we usually don't write the basis vectors as partial derivative operators. Actually, I have to say the only place I have seen basis vectors represented in that way is in Special Relativity and in General Relativity.

 

[robphy]

A "real" relativity book writes a vector as $V^{a}$, where ${}^a$ is an abstract index [not reference to a component in some coordinate system].

 

[davidge]

A "real" relativity book writes a vector as $V^{a}$, where ${}^a$ is an abstract index [not reference to a component in some coordinate system].

Ok. But what if we are going to represent it in some coordinate system?

 

[pervect]

I believe the most correct and general way to write a vector in terms of its components would be as follows:
$$V^\mu \, e_\mu$$

or possibly

$$V^\mu \, \vec{e}_\mu$$

Here $\mu$ is a dummy index, and summation of it is understood, though one could I suppose explicitly write

$$\sum_{\mu=0..n-1} V^\mu \, e_{\mu}$$

Here n=4 for typical relativity applications, but some other applications might have n not be equal to 4. And sometimes people start with the first index being 1 rather than zero.

The $V^\mu$ are the components of the vector, and the $e_{\mu}$ or the $\vec{e}_{\mu}$ are the basis vectors. So when we write a vector in terms of it's components, we're just saying that a general vector V can be written as the weighted sum of other vectors, called components, which we call "basis vectors".

Note that it is not in general necessary that $e_\mu = \frac{\partial}{\partial x^\mu}$. When $e_\mu = \frac{\partial}{\partial x^\mu}$ we say that we are writing the vector in a coordinate basis, but it's perfectly possible and not uncommon to write vectors in a non-coordinate basis.

It's not necessary to use components to describe a vector either. A vector is an abstract entity that exists regardless of what symbol we choose to denote it. It's a bit less confusing if we use conventions to indicate which symbols are vectors, but it's not strictly necessary. Some texts use boldface to indicate that a variable represents a vector quantity, some texts use arrows $\vec{V}$ or hats $\hat{V}$. And more advanced papers typically leave it up to the reader to discern which quantities are vectors, though it's certainly not bad practice to spend a few words explaining things. A lot depends on the target audience, a textbook will naturally spend more time using consistent notation and explaining it than a paper aimed at people with experience who can be expected to figure out the meaning.

 

[davidge]

Thanks pervect. In addition to what you said, I'm wondering what family of functions we are allowed to operate on with the "operator - basis vectors" $\partial / \partial x^\mu$. I mean in General Relativity, in what functions would we apply a vector $V = V^\mu(x) \partial / \partial x^\mu$?

 

[martinbn]

These are the usual notations in differential geometry.

 

[stevendaryl]

Thanks pervect. In addition to what you said, I'm wondering what family of functions we are allowed to operate on with the "operator - basis vectors" $\partial / \partial x^\mu$. I mean in General Relativity, in what functions would we apply a vector $V = V^\mu(x) \partial / \partial x^\mu$?

The use of \frac{\partial}{\partial x^\mu} as the basis vectors for the coordinate system x^\mu is a little disconcerting when you first see it (I still don't completely like it, although I understand why it's a reasonable choice).

The idea is that given a vector V, and a scalar field \Phi (a scalar field is a function that assigns a scalar--a real or complex number, typically--to each point in space, or spacetime if you're dealing with relativity) there is a corresponding quantity V^\mu \frac{\partial \Phi}{\partial x^\mu} giving the "directional derivative" of \Phi in the direction V. (I assume you know Einstein's convention that the \mu is summed over.) So every vector V corresponds to an operator V^\mu \frac{\partial}{\partial x^\mu}. It's common in differential geometry to just define a vector to be equal to its directional derivative. In that case, the basis vectors in a coordinate basis are just the operators \frac{\partial}{\partial x^\mu}.

As to what they operate on: they operate on scalar fields.

 

[davidge]

Thanks stevendaryl.

they operate on scalar fields

Can you give me some examples of such scalar fields in relativity?

 

[DrGreg]

Thanks stevendaryl.

Can you give me some examples of such scalar fields in relativity?

See Scalar field.

 

[stevendaryl]

Thanks stevendaryl.

Can you give me some examples of such scalar fields in relativity?

For the purposes of differential geometry, a scalar field is just a function of location---any way of assigning a real number to each point in space (or spacetime) counts as a scalar field. It doesn't have to be physically measurable. Any time you set up a coordinate system (x,y,z,t), you are actually picking 4 different scalar fields. On the surface of the Earth, latitude, longitude and altitude are three scalar fields---given any point on the Earth, each of those gives an associated real number.

As for physically measurable scalar fields: there's the Higgs field, there's the Ricci scalar, there's temperature, etc.

 

[Dale]

A "real" relativity book writes a vector as $V^{a}$, where ${}^a$ is an abstract index [not reference to a component in some coordinate system].

Do you know of a good (free) online one? I tend to point to Carroll's notes, but they use the component approach

 

[robphy]

Do you know of a good (free) online one? I tend to point to Carroll's notes, but they use the component approach

These notes
http://home.uchicago.edu/~geroch/Course Notes
got published by the Minkowski Institute Press
http://www.minkowskiinstitute.org/mip/books/ln.html (via Amazon).

 

[pervect]

It might be helpful to give a simple example of when we might NOT use $\partial_{x^i} = \frac{\partial}{\partial x^i}$ as a basis vector. Note that I'm using a simpler format to indicate partial derivatives, it's easier to write and it shouldn't be terribly confusing.

Suppose we have a two-dimensional euclidean plane with polar coordinates $r, \theta$. Then the line element is $ds^2 = dr^2 + r^2\, d\theta^2$

The vector $\partial_\theta$ is NOT a unit vector, as can be seen from the metric. If we let $u = \partial_\theta$, the square of the length is given by $g_{\mu\nu} u^\mu u^\nu = r^2$, so the square of the length of $\partial_\theta$ is $r^2$ and it's length is r.

If we want to interpret physical quantities in the neighborhood of some point, we'd typically use an orthonormal basis rather than a coordinate basis. In the notation we've been using, these orthonormal vectors would be $e_1 = e_r = \partial_r$ and $e_2 = e_\theta = (1/r) \partial_\theta$.

 

[Orodruin]

If we want to interpret physical quantities in the neighborhood of some point, we'd typically use an orthonormal basis rather than a coordinate basis. In the notation we've been using, these orthonormal vectors would be $e_1 = e_r = \partial_r$ and $e_2 = e_\theta = (1/r) \partial_\theta$.

Well, this is a matter of the description. If you want your components to be the actual physical quantities, sure. But whenever you want to compute a measured value, it should (and will) not depend on the basis you are using. Essentially, the metric will save you if you use a coordinate basis, it is just that the metric takes a particularly simple form in an orthonormal basis.

 

[pervect]

With experience, one chooses the basis that is the most suited for the problem at hand. The point I wanted to make to the OP is that one has the freedom to choose a basis other than the coordinate basis, and try and give a simple example of when one might want to use such a non-coordinate basis, to emphasis the point that one's definition of a vector should be flexible enough to allow choosing a non-coordinate basis.

Hopefully the OP shares some common experience with having seen unit vectors $\hat{r}$ and $\hat{\theta}$ being used in polar coordinates in non-tensor notation. The point that I wanted to make is that the vector $\frac{\partial}{\partial \theta}$ isn't the same thing as the unit vector $\hat{\theta}$, because $\frac{\partial}{\partial \theta}$ doesn't have a unit length.

The point robphy mentioned that one can write vectors using abstract index notation rather than component notation is a good one.

Some wiki links if the OP is ambitious enough: <abstract index notation>, <vector notation>. I didn't see anything specific in wiki about index-free notation, though there's some implied discussion in the second link. Math pages had a short write-up on the difference between abstract index notation and component notation (which they called ordinary index notation) <here>.

I believe that component, abstract, and index-free notation are the "big three" of vector notation - I can't think of any others offhand.

 

[Orodruin]

I agree of course. I just wanted to stress that you can compute the physical observables using whatever notation you please as long as you do it consistently and that the components in an orthonormal basis just might have more direct physical interpretations.

 

[davidge]

Thanks pervect and Orodruin.

I see your point on the "unitary-length problem" pervect

Well, this is a matter of the description. If you want your components to be the actual physical quantities, sure. But whenever you want to compute a measured value, it should (and will) not depend on the basis you are using. Essentially, the metric will save you if you use a coordinate basis, it is just that the metric takes a particularly simple form in an orthonormal basis.

It's interesting to see this observation Orodruin, because when I started studying the math of the General Relativity, this was one thing that I noticed before I go through the other topics that textbooks usually follow.