Newtonian force as a covariant or contravariant quantity

Ben Niehoff

I wanted to expand on this, because it might seem mysterious, given that the equation usually called "Killing's equation" has nablas in it.

Remember what Killing's equation actually means. The solution K is a vector field that generates an isometry of the manifold. So for example, on a sphere, the vector K points along lines of constant lattitude, and describes the velocity at each point on the sphere if the sphere as a whole were to be rigidly rotated about its axis. Notice that the flow of K along itself does NOT follow parallel transport (especially near the poles).

Of course, in order for K to solve a meaningful differential equation, there must be some way to relate nearby tangent spaces to each other. It is the rigid motion itself that links tangent spaces via the pullback. The connection is never needed.

The point is that ANY vector field generates some 1-parameter flow, called Lie dragging. The flow lines are the integral curves of that vector field. So, using pullbacks along this 1-parameter flow, you can relate different tangent spaces and compare vectors at different points. If that 1-parameter flow is a rigid motion, then you have an isometry (which essentially means, the manifold can be rigidly moved into itself). The conditions for a flow being a rigid motion are precisely Killing's equation*.

* Note: "Rigidity" is really the wrong concept here, as rigidity is an extrinsic property. Consider the rigidity of a triangular lattice in R^2, and now put the same triangular lattice into R^3; it becomes quite floppy. The correct concept is "metric-preserving", which is the intrinsic property that most closely corresponds to rigidity.

Ben Niehoff

It's important to remember that the indices on [itex]\partial_a, \partial_b,[/itex] etc. are indices that indicate "which vector", not "which component". So to indicate contravariant vectors, the indices must be down in order for [itex]X = X^a \partial_a[/itex] to make sense.

DrGreg

Edit: in the time it took me to write this, I see several others have already made the same point, but I might as well post this anyway.

What you say is true of the second meaning I listed. I agree that
[tex]
\begin{bmatrix}
\frac{\partial}{\partial t} &&
\frac{\partial}{\partial x} &&
\frac{\partial}{\partial y} &&
\frac{\partial}{\partial z}
\end{bmatrix}
[/tex]transforms as a co-vector, where the symbol [itex]\partial / \partial t[/itex] denote one component of a covector.

But the symbol is not being used in that sense. It's being used (in an abuse of notation denoting an isomorphism to another space) to denote a vector, not a component.

It's like writing[tex]
\textbf{V} = V^\alpha \textbf{e}_\alpha = V^0 \textbf{e}_0 + V^1 \textbf{e}_1 + V^2 \textbf{e}_2 + V^3 \textbf{e}_3
[/tex]where [itex]\textbf{e}_0[/itex], [itex]\textbf{e}_1[/itex], ... are four different vectors, not covectors, nor 4 components of one covector.

bcrowell

Actually, I think yours was the only post that actually cleared up the issue, so I'm glad you posted it anyway!

IMO this hits the nail on the head and clears up something that I'd found confusing and that IMO #135 and #136 failed to deal with by resorting to appeals to authority. I think the key words here are "abuse of notation."

Let me see if I can lay out my present understanding and see if others agree with me.

We want a bunch of different things:

(1) We want to be able to use upper and lower indices to notate the real physical differences between two different types of vectors, and as physicists we define these types by their transformation properties.
(2) We want the transformation properties of an object to be apparent from the way we write its indices.
(3) We want to have grammatical rules that make it apparent when we're writing nonsense, e.g., we want to be able to recognize that there's something wrong in an equation like [itex]u_a=v_a w_{abc}[/itex].
(4) We want a notation that's compact and expressive, so we'd like to use Einstein summation notation and avoid writing sigmas.
(5) We want a notation that is widely accepted and understood by other physicists.

Appeals to authority are useful only with respect to #5. Since we don't all have the same printed books on our bookshelves, it's useful to have an online reference that's accessible to all of us, so let's use the online version of Carroll: http://ned.ipac.caltech.edu/level5/M.../Carroll2.html

This violates #2, because the l.h.s. is written without indices, so we identify it as a scalar, and yet it's not a scalar, it's a covector. It satisfies #4 by being written in compact Einstein summation notation. It obeys #2 by writing the covector as [itex] \omega_{\mu}[/itex], but violates it by writing the basis covector as [itex]dx^\mu[/itex].

Carroll (and by proxy physicists in general) is trying to simultaneously satisfy incompatible desires 1-5. A clear way of seeing this is that he wants to write [itex] \omega_{\mu}[/itex] as a synonym for [itex]\omega[/itex]. Now suppose that (in old-fashioned non-abstract index notation, indicated by the Greek indices), we have [itex]\omega_0=1[/itex] for [itex]\mu=1[/itex] and all the other components zero. Then we have [itex]\omega=dx^\mu[/itex]. But [itex] \omega_{\mu}[/itex] is a synonym for [itex]\omega[/itex], so can substitute in and make it [itex]\omega_{\mu}=dx^\mu[/itex]. Oops, this violates #3.

I think what's really happening here is that in a misguided attempt to satisfy 3 and 4, we write expressions like [itex]\omega_\mu dx^\mu[/itex]. This reads like the scalar product of a vector and a covector, which isn't what it is. Maybe a better notation would be something like [itex]\omega_{(.)}=\sum_\mu\omega(\mu) \partial_\mu[/itex]. It violates 4 and 5, but it obeys 1-3.

atyy

I don't agree. One has to distinguish between a representation of an object in particular coordinates, and the object itself. The typical physics index gymnastics notation refer to representations (components) of an object.

It is true that you could say that a tuple of of basis covariant vectors transforms as the components of a single contravariant vector. But this is not standard in physics, where we refer to transformation of the components. (See the Warning on p42 of http://books.google.com/books?id=DUn...gbs_navlinks_s.)

Ben Niehoff

No, you can't write that. But you can write [itex]\omega = dx^1[/itex], which is perfectly sensible.

That is also nonsense. At best you can write [itex]\omega_\mu = \delta_\mu{}^1[/itex].

There is nothing wrong or abusive about writing [itex]\omega = \omega_\mu \, dx^\mu[/itex]. The summation rule is being followed ("one upper and one lower index are summed"), and the indices are in places that indicate their purpose:

1. Lower indices on real-number-type quantities indicate which component of a covariant object.

2. Upper indices on 1-form-type quantities indicate which 1-form. Such indices are written upper in order to be consistent with the summation rule.

If you were to consistently stick with index notation (either abstract or not), you would never write something like [itex]\omega = \omega_\mu \, dx^\mu[/itex]. You would just talk about "the 1-form [itex]\omega_\mu[/itex]".

At any rate, the constant arguing about what exactly indices "mean" is one of the reasons I hate this notation. Such discussions don't lead anywhere useful. It's even worse when you try to use orthonormal frames and you get into "flat space indices" vs. "curved space indices".

WannabeNewton

Just like we think of vectors at a point as a linear combination of derivations, we can write the dual vectors at a point as a linear combination of the dual basis to the derivations. We write [itex]V = V^{i}\partial _{i}[/itex], [itex]\omega = \omega _{i}dx^{i}[/itex] so that [itex]\omega(V) = \omega _{i}dx^{i}(V^{j}\partial _{j}) = \omega_{i}V^{j}dx^{i}(\partial _{j}) = \omega_{i}V^{j}\delta ^{i}_{j} = \omega _{i}V^{i}[/itex]. The initial two expressions are not meant to be inner products of some form. The indices on the derivations and their dual represent the elements of the basis and dual basis respectively, not components.

micromass

Well, your version of Carroll says the same thing as we have been saying:

and

dx

The constraint surface can be represented by an equation of the form f(Xⁱ) = 0. The constraint surface in your second example is

f(φ, θ) = θ - φ = 0

A constraint force which is parallel to the constraint surface can be written as a multiple of df, since df is parallel to the constraint surface.

F = λdf = λ(dθ - dφ)

stevendaryl

Yeah, these conventions are confusing, but they make some sense if you think about it.

In differential geometry, a vector [itex]V[/itex] is defined via its effects on scalar fields. Specifically, if we choose a particular coordinate basis, we can write:

[itex]V(\Phi) = V^\mu \partial_\mu \Phi[/itex]

with the implicit summation convention. Now, if [itex]V[/itex] happens to be a basis vector [itex]e_\sigma[/itex], then that means that [itex]V_\sigma = 1[/itex], and all the other components are equal to 0. So we have, in this case:

[itex]e_\sigma(\Phi) = \partial_\sigma \Phi[/itex]

Since this is true for any [itex]\Phi[/itex], it's an operator equation:

[itex]e_\sigma = \partial_\sigma[/itex]

stevendaryl

Yeah, this stuff is pretty confusing. Even though [itex]\frac{\partial}{\partial x^\mu}[/itex] is a vector, the prototypical covector is [itex]\nabla \Phi[/itex] for some scalar field [itex]\Phi[/itex], which has components [itex]\frac{\partial}{\partial x^\mu} \Phi[/itex]

To complete the confusion, even though [itex]dx^\mu[/itex] is a 1-form, the prototypical vector is a tangent vector to a parametrized path [itex]P(\lambda)[/itex], which has components [itex]\dfrac{dx^\mu}{d\lambda}[/itex].

TrickyDicky

It is confusing, but even more if one doesn't distinguish the geometrical object from its component representation. [itex]dx^\mu[/itex] is a 1-form, but transforms contravariantly, however if it had components other than unity they would transform covariantly.

So I'm still waiting for someone to explain why the rate of change of momentum is not naturally a tangent vector.

stevendaryl

If momentum is defined to be [itex]\vec{P} = m \frac{\vec{dU}}{dt}[/itex], then it is naturally a vector. If it is defined via a Lagrangian [itex]L[/itex] by [itex]P_\mu = \frac{\partial}{\partial U^\mu}L[/itex], then it is naturally a covector.

Another way to see that momentum should be considered contravariant might be:

In curvilinear coordinates, the equations of motion for a free particle is

[itex]\dfrac{d}{dt} U_\mu = 0[/itex]

not

[itex]\dfrac{d}{dt} U^\mu = 0[/itex]

bcrowell

I think the basic cause of the ugly inconsistency in Einstein notation is not that there's anything wrong with Einstein notation but that it's being mixed inappropriately with Sylvester's notation. Sylvester was the one who originally introduced terms like "contravariant" and "covariant." The pure Sylvester notation is described in this WP article: http://en.wikipedia.org/wiki/Contravariant_vector Unfortunately Sylvester's notation and terminology are much more cumbersome than Einstein notation, and it doesn't correspond to the way physicists customarily describe the objects themselves as having transformation properties, rather than the objects staying invariant while their representations transform.

In Einstein notation, a sum over up-down indices represents a tensor contraction that reduces the rank of an expression by 2. In Sylvester notation, a vector is expressed in terms of a basis as a sum over up-down indices, which is not a tensor product. Mixing Einstein and Sylvester notation produces these silly-looking things where it looks like we're expressing a covector as a sum of vectors or a vector as a sum of covectors.

The abstract index form of Einstein notation is carefully designed so that you *can't* inadvertently refer to a coordinate system; anything written in abstract index notation is automatically coordinate-independent. This means that in abstract index notation, you can't express a vector as a sum over basis vectors. You don't want to and you don't need to.

There is also an issue because in Einstein notation we refer to vectors as covariant and contravariant, whereas in Sylvester notation a given vector or 1-form is written in terms of components and basis vectors that are covariant and contravariant. Neither is right or wrong, but mixing them leads to stuff that looks like nonsense.

atyy

Ok, I actually can't understand the Lewis notes I linked to in #127.

These guys http://www.google.com/url?sa=t&rct=j...41867550,d.b2I do it differently. In proposition 7.8.1 they do define a force via contraction with a vector field. However they contract with a symplectic 2 form, not a metric. I think the relevant definition of the 2 form is on p161.

Incidentally the Lewis notes say ad hoc "motivation" is that the Euler-Lagrange equations transform as the components of a one form. He say this is hand-waving because the EL equations are not a one-form.

atyy

I guess the Lewis notes http://www.mast.queensu.ca/~andrew/t...f/439notes.pdf are fine after all. Force is a one-form. What I was confused by is their notation F: R X TQ -> T*Q. All they mean is that a force is an assignment of a one form to every velocity at every position. Just as they say a vector field is V: Q -> TQ by which they mean a vector field is an assignment of a vector at every position.