User talk:Rschwieb/Physics notes

Tensors in physics

This may be a very blatently obvious and redundant thing to say, but I can't tell so here goes: to understand physical laws mathematically, instead of concentrating on GA why not just tensor calculus?

Of course this branch of mathematics is something that you know inside out (I have only recently been doing tensor algebra, but have had plenty of exposure to the calculus also), but it seems less complicated and confusing than GA. Lots of physical laws are easily and rather simply expressed as tensors (compared to vector feilds, say). So with a knowlage of tensor calculus, you can follow eaily into EM and GR which both make extensive use of tensor calculus, rather than GA as the starting point. Ok - so GA can simplify things more than tensors, using Maxwell's equations for example:

• in tensor calculus they are 2 equations:

$\epsilon ^{\delta \alpha \beta \gamma }{\dfrac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}=0,\quad {\dfrac {\partial F^{\alpha \beta }}{\partial x^{\alpha }}}=\mu J^{\beta }$

where F^αβ is the electromagnetic tensor containing the electric and magnetic fields in a rank-2 tensor (qualitativley "the EM field"), and J^β is the 4-current containing the charge density ρ and electric current density J (qualitativley the amount of charge flowing in space-time), μ is the Vacuum permeability but ε^δαβγ not permittivity, its just the contravariant permutation sumbol.

• in GA they are just 1:

$\left({\dfrac {1}{c}}{\dfrac {\partial }{\partial t}}+\nabla \right)F=\mu cJ$

where F is the EM multi-vector and J the electric current multi-vector, c is speed of light,

• vector calculus has the 4 well-known and simple (sort of) equations for the electric E and magnetic B fields, related to their sources of ρ and J (same as above)

$\nabla \cdot \mathbf {E} =\rho /\varepsilon ,\quad \nabla \cdot \mathbf {B} =0,\quad \nabla \times \mathbf {E} =-{\dfrac {\partial \mathbf {B} }{\partial t}},\quad \nabla \times \mathbf {B} =\mu \mathbf {J} +{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {E} }{\partial t}}$

here ε is the permittivity (or 2 equations for the vector A and scalar ϕ potentials - complicated, but can be simplified becuase of gauge invariance of A and ϕ).

Neither includes the Lorentz force law (EM force on charges), always seperate anyway. Again - I'm not sure if you have already concentrated on tensors in physics, presumably you have to some extent, but I'd say its better than GA. Of course rotations and (dare I geuss spinors!) can be formulated using tensors. F = q(E+v×B) ⇄ ∑_ic_i 19:29, 30 March 2012 (UTC)

I have not mastered either, but here is my impression so far. Tensor calculus is a little easier in the sense that it's "the first method you would think of". The tensor approach also has the benefit of being refined in the mainstream for so long.

On the other hand, it sometimes tend to focus on components to the point of obfuscation. I certainly did not notice anything geometric while I was learning tensors. I think this probably comes in later with bundles and actions and whatever I have not understood yet.

I get the feeling that GA might have the benefit of incorporating the geometric component earlier, without too much fussing with the bundles. Yes, it's learning curve is steeper than tensor algebra, but like the ads say, it might yield some good insights. Rschwieb (talk) 01:56, 31 March 2012 (UTC)

True, tensors are also a flood of indices, despite sources saying there is a geometric interpretation: I guess (weakly) each index corresponds to some associated direction (geometric or physical). Just a suggestion. =) Admittedly I can't be editing WP too much anymore (exams), but can still watch pages time to time. F = q(E+v×B) ⇄ ∑_ic_i 07:06, 31 March 2012 (UTC)

Tensors and GA each seem to have their strong points. The indices of tensors make some algebraic manipulations very starightforward, and allow expression of tensors of high order. GA is more limited in its capabilities, because it makes different tensor coefficients equivalent: it is formally equivalent to the tensor algebra reduced via equivalence in a particular way. This equivalence seems, in many instances (e.g. geometry) to be almost magically the correct way of dealing with it. I tend to think of GA in relation to tensor algebra as a high-level computer language in relation to a lower-level language (in removing the ability to express some things, you're automatically restricted to more compact and useful expressions). So, for example, in tensor algebra one must explicitly indicate a contraction or anti-symmetric part, whereas in GA they "automatically happen" when appropriate. More to the point: GA has an algebraic structure (inverses etc.) that is difficult to find for tensors, and it was only with the advent of an algebra with the structure of GA (Clifford algebra, albeit in the form of matrices) that the Dirac equation was even discovered (I wouldn't even know how to express it in tensor algebra). Consider an example: Rotations of general tensors (including quantities of mixed order, though somewhat artificial) produces a clumsy expression; not so in GA. Many aspects of GA have a direct geometric interpretation, aiding intuition. I would love a notation that combines the best of both. (PS: Rschwieb, how about adding a "Tensors procedure" to your Comparison of methods: rotations page? I'm sure this would be illuminating for all of us.) — Quondum^☏✎ 08:13, 31 March 2012 (UTC)

Forgot to mention: I'm sure everybody appreciates an equation that captures complex physical phenomena, but there is a dichotomy that arises after that. If an easier way to write the same equation arises, one type of person says "wow! that's even better!" and the other type says "boring! I liked it better before". In my experience, the first type is more inclined to be a mathematician, and the second type is more inclined to be in physics or engineering. They are both valid ways of appreciating equations, but they have different values at their core.

@Q: For an entire year now, I have been trying to figure out what the "tensor procedure" for understanding physics is :P While I feel I've made a few steps closer in all these discussions, I'm still mostly in the dark about how to describe how physicists use tensors. I haven't understood the GA approach either, but I think I'm making progress. Rschwieb (talk) 13:30, 31 March 2012 (UTC)

Quick comment: If you have access - an extremley good referance for tensors in GR is Gravitation (book) by C. Misner, K.S. Thorne + J.A. Wheeler. Its at advanced undergrad/PHD level - I presume you will easily follow it.

How do physicists use tensors? Although perhaps an oversimplification; properties which are separate using simpler maths (e.g. vector calculus) but inter-related are amalgamated into tensor components, i.e. physical tensors pack up a collection of physical quantities. Examples are already above - the 4-current J and the EM field tensor F. Instead of dealing with E, B, and J as separate vector fields each with their own components (as in vector calculus in the equations above), the entire EM field tensor has components which are the E and B fields, all into one object. A more extreme example is the stress-energy tensor (cf general relativity) which amalgamates energy and momentum densities and fluxes. So when doing manipulations/calculations, tensors effectively number-crunch the components of a collection of inter-related physical quantities all at once. I'm not sure how to decide on co/contra-variance from the start though, but once a 4-vector or tensor is decided to have co/contra-varaince its easy to change between co/contra-variant components using the metric.

Out of interest - what on Earth does co/contra-variant actually mean!?!? Physically convariance means tensors have the same mathematical form in all coordinate frames (physically obvious), but mathematically: "the components of a tensor transform in the same way as the (given) transformation rule so their components are the same in the new coords" is my (correct? not sure) consensus from books/sources and here on WP. For some reason - its just so painful, daunting and awkward to understand how the physics relates to maths by using partial derivatives in a transformation:

${\bar {u}}_{i}={\dfrac {\partial x^{i}}{\partial {\bar {x}}^{j}}}u_{j}$

where x_i are initial coords, overbarred are the new, and u is the vector, analogous formulae for higher order tensors. [Taken straight from: An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975. p.82, ISBN 0-582-44355-5 - another good reference if you have access, maybe you can get a 2nd-hand copy somewhere like [1], [2], or [3] rather than ebay or Amazon]. But why partial derivatives??? is it like a Jacobian (which is easily and mechanically done mathematically but physically not very comprehensible)???

Likewise contravariant: "tensor components transform "against" the (given) transformation rule so the vector remains the same in the initial coord frame (i.e. apply inverse transformation to the components?)":

${\bar {u}}^{i}={\dfrac {\partial {\bar {x}}^{i}}{\partial x^{j}}}u^{j}$

Perhaps this is a stumbling block for you also? components of a tensor as physical quantites are easy to comprehend, but what do you think? =) F = q(E+v×B) ⇄ ∑_ic_i 13:58, 31 March 2012 (UTC)

Sorry about the edit conflict - I hadn't finished what I had to say.... =(

@R: You're in good company. I have the haunting feeling that physicists have an intuitive feeling for vectors, but beyond that for most it is pretty much just abstract formulae: a black box of tensor methods, as described by F=q(E+v×B) here. Look at how many people think in terms of matrices for the Dirac equation, spin matrices etc., whereas their only property of significance is the Clifford algebra. Physics needs more mathematicians... (I think tensors are too general for efficient use.) — Quondum^☏✎ 14:25, 31 March 2012 (UTC)

@F: On co/contravariance: The best I have been able to figure out is as follows (this is my own deduction, not from any authoritative source): It does not seem that the entities (vectors, tensors) themselves are covariant or contravariant, but it seems the term covariant is abused to mean behaves as an entity, independent of choice of basis (as an abstract entity really invariant) – so it just means is a tensor (your "physically"). The terms covariant and contravariant really seem to mean (your "mathematically") simply this: once one has chosen a basis {e_i} for a vector space V, and has derived the corresponding cobasis {e^j} for its dual (covector) space V^∗, any vector and covector may be written linearly in terms of these. The vector basis is by definition covariant, and the rest follows. Because any vector v=vⁱe_i is a basis-independent entity, any change of basis can be written v=vⁱe_i=v^je_j, and if covariance is expressed by e_i=A_jⁱe_i, then we must use the inverse of A_jⁱ for the coefficients of v and so on. So, the basis itself is covariant (transforms with A_jⁱ), the cobasis is contravariant (transforms with the inverse of A_jⁱ), the vector coefficients vⁱ are contravariant and a covector's coefficients are covariant. You will notice that I have not made any reference to partial derivatives (i.e. it applies in a vector space e.g. to a vector at any chosen point of a manifold). The partial derivatives only arise as a result of a differential structure on a manifold, and a choice of vector basis connected to this differential structure. A particularly natural choice of basis is connected to this differential structure giving rise to the partial differential relationship with the transforms that you mention, but it is (AFAICT) not necessary; the choice of basis is quite arbitrary at every point on a manifold. — Quondum^☏✎ 17:50, 31 March 2012 (UTC)

Thanks Quondum, that really helps. =) Apologies for writing lazily by not including basis vectors. If I could paraphrase + summarize:

• Naturally the basis choice is arbitrary since vectors are coordinate independent objects, that much is easy.
• I understand that a linear combination of basis vectors makes a vector, an element of a vector space V spanned by the basis set. So these basis vectors form a covariant basis by definition, and transform according to the mapping.
• My mental block is that I don't properly understand what dual vectors are ("dual" - no matter how many times I read this up, simply does not enter my comprehension), but whatever they may be, constructing the dual of the basis vectors gives the cobasis set, a linear combination of these cobasis vectors gives a vector in the dual space V*, spanned by the cobasis vectors, and these are contravariant which transform according to the inverse mapping.
• The partial derivatives are not strictly necessary to describe a change of coordinates, they exist for differential geometric reasons.

About the "behaviour of tensors" in terms of covariance, I'm not sure what you mean. I prefer the perspective of tensors interpreted as multi-dimensional arrays of numbers, generalizations of matrices, vectors, scalars. While we can draw vectors and vector fields, we can’t really for tensors. Instead tensors they "mix and multiply" components of other tensors together. The components are numbers (with units) which can be handled easily. For physical tensors the components are scalar physical quantities (like charge density), so as said above the full tensor is a single object unifying a number of inter-related physical quantities. Again, instead of "electric and magnetic vector fields", we just have "EM field". Same for the stress-energy tensor. I might be paraphrasing what you said again, just completing my view. =) F = q(E+v×B) ⇄ ∑_ic_i 19:09, 31 March 2012 (UTC)

1. On duality (your third bullet): You have a pretty firm grasp of vector spaces, a basis, and the uniqueness of components with respect to a basis. Covectors are nothing other than linear mappings V→R, and the cobasis is the set of such mappings for producing the components for a specific basis when acting on an arbitrary vector. You might also note that there has been no mention of a metric yet. I imagine that confusion starts to creep in when we start treating these linear mappings as belonging to a vector space, and forget what they were defined for. See what you make of this, and I'll see whether I can clarify.
2. On your last bullet, don't confuse coordinates with basis. The coordinates on a manifold (and/or their partial derivatives) are unnecessary to describe a change of basis of a vector space (think of the vector space being all possible values of a vector at a point; it does not matter whether there are neighbouring points).
3. When I say "behaves as an entity independent of choice of basis" I'm just saying that in this sense it is like a vector, exactly as you've described so well in your first bullet. — Quondum^☏✎ 20:34, 31 March 2012 (UTC)

Corresponding responses to each point:

1. When you mention "V→R", that sounds familiar - like row vectors multiplied by column vectors, or bra's multiplied by kets in Dirac notation. So the cobasis vectors are linear mappings which take vectors from the vector space V to the real numbers R... I know functions (aka mappings) can also be treated as elements of vector spaces (aka function spaces), yet still cannot see how it all comes together in this context… I definitely appreciate all your efforts into helping me understand, though for now it may be better if I sit down and thoroughly go through the theory and (concrete) examples pen, paper and book (or draft on whiteboard): at some point the brain-barrier will break to pieces and I'll understand, rather than reading from time to time on this topic (never really concentrated much on dual spaces before)... Also you mention the metric as a mapping: it takes 2 vectors in and churns out their scalar product like a bilinear form, yeh? Presumably so.
2. For point 2, again I wasn't clear, coordinates and basis vectors are indeed not the same, though of course they are hand-in-hand.
3. Ok. =)

F = q(E+v×B) ⇄ ∑_ic_i 23:01, 31 March 2012 (UTC)

Out of all the tensor calculus used in physics, tensor fields are easier for me to grasp. Forces and electric charge, pressure are easily imagined this way. However the stress and momentum stuff begin to lose me :) I like your description, F, of the tensor as "bundling physical quantities".

There are a few comments on User:Rschwieb/Physics notes relevant to this conversation. I think co and contravariance might be only an artifact of our choice of basis (but I could be wrong.)

I can make a comment on partial derivatives. It looks like it is all part of the familiar process of resolving quantities into components. For a fixed set of curvilinear coordinate axes through a point, the partial derivative finds the tangent vectors to the axes at the point. These create a basis for the "flattened" tangent space attached to the manifold at that point. Rschwieb (talk) 13:02, 1 April 2012 (UTC)

co and contravariance might be only an artifact of our choice of basis: I think you are exactly right. There is however the further observation that to perform any calculation, such a choice is unavoidable. So it is an artifact, but a necessary one. On tangent spaces, I think it is possible to define fibre bundles that are not in any sense tangent to the manifold, so there must be additional assumptions/axioms to be examined before considering the differential structure as automatic. — Quondum^☏✎ 16:48, 1 April 2012 (UTC)

Sorry to jump backwards, but I also wanted to make a comment about the Jacobian. It seems pretty comprehensible: loosely speaking, the Jacobian matrix accounts for the distortion of an element of volume caused by a transformation or coordinate change, and the Jacobian determinant records the "divergence" at the point. Maybe this is not what you had in mind for understanding the Jacobian, but I just wanted to throw it out there. Rschwieb (talk) 15:17, 2 April 2012 (UTC)

"Sorry to jump backwards" is simply unreasonable: your comments (combined with Quondum's) are awesome! Always good to have more than one opinion. I have never understood the partial derivatives in a transformation to be thought of in that way (only been taught how to calculate partial derivatives + theory behind them). I did already have the geometric interpretation of partial derivatives as tangential gradients along surfaces, but forgot they form tangent vectors (which is not good...). The image

By Hernlund

at Covariance and contravariance of vectors is starting to make sense. Same for Jacobians - they can be calculated like mad, but your interpretative explanation helps a lot, thanks! F = q(E+v×B) ⇄ ∑_ic_i 18:14, 2 April 2012 (UTC)

I like the diagram – nice and intuitive. On the topic of Jacobians (which, I understand, can be useful in a change of variables for an integral), in differential forms and GA this distortion is automatically taken care of by the wedge product e.g. dx∧dy∧dz. (Not that I know anything about it; just half-remembered from Penrose.) — Quondum^☏✎ 18:57, 2 April 2012 (UTC)

I've always found the Hernlund diagram incomprehensible :( Maybe I would understand it with colors or a two or three stage animation. Rschwieb (talk) 19:03, 2 April 2012 (UTC)

Really? Perhaps there are too many arrows which look the same? You mentioned that "the partial derivative finds the tangent vectors to the axes at the point", which is in the diagram. (Rschwieb - if you don't already have Penrose's Road to Reality, you haven’t a clue what you're missing! This is another fantastic book, and should be easily available! Furthermore you might summarize the sources somewhere, they are recommended. I'll leave you to it of course though). =) F = q(E+v×B) ⇄ ∑_ic_i 19:30, 2 April 2012 (UTC)

That's exactly it, there are just too many arrows. I can pick out the tangent arrows instantly, but it's still difficult to see what I'm supposed to understand about the contravariant arrows. Another thing is that I'm not really sure it's sensical to represent the dual basis on top of the tangent basis. I really wish there were at least three colors, one for the original arrow, and one for each set of bases. What do you two take away from the picture? Rschwieb (talk) 17:11, 3 April 2012 (UTC)

Admittedly this diagram only makes sense in the context of a metric space, it is not accurate enough with respect to the summation of vectors (nor does it explicitly show the summation, perhaps as parallelograms), and it shows the tangent vectors which are more relevant to the differential structure than to the co/contravariance. A GIF (with different colours) animating how contravariant and covariant bases and multipliers change (albeit still in a metric space), without the coordinates, would have been better. — Quondum^☏✎ 18:18, 3 April 2012 (UTC)

Perhaps I could try and re-colour it? F = q(E+v×B) ⇄ ∑_ic_i 19:27, 3 April 2012 (UTC)

That would be a start, though as you can see there are probably other tweaks worth doing. Rschwieb please comment on what would be an improvement. — Quondum^☏✎ 20:00, 3 April 2012 (UTC)

It has been re-coloured, unfortunately it seems it can only be uploaded as a GIF (with a speckled colour effect), PNG would be better, better yet SVG. I know someone (Maschen) who can produce SVG images really well, so maybe I could ask if he could re-draw it and upload. It would be easier to maintain from then on. Sorry for jumping the gun... =( F = q(E+v×B) ⇄ ∑_ic_i 20:08, 3 April 2012 (UTC)

This page was brought to my attention just now. It seems F=q(E+v^B) has requested me to re-draw the above GIF as SVG (ignore the un-intensional ryhme). What features would you like to add/subtract to the current version? Maschen (talk) 20:41, 3 April 2012 (UTC)

Its done, and has been uploaded for now in SVG format, but for some reason beyond my control (for now) the spacing of the symbols is out of pplace (using SerifDrawPlus X4, also checked with Inkscape, the image was perfect). When any suggested modifications arise I'll update it to a new version and try to fix the problems.

Maschen (talk) 22:12, 3 April 2012 (UTC)

Looks fine for now, thanks for trying hard and altruistically. =) F = q(E+v×B) ⇄ ∑_ic_i 22:13, 3 April 2012 (UTC)

Co-/contra-variance, orientation and other duality

You say the fact that you can reexpress tensors so that they are "all covariant" suggests that had we chosen our basis carefully at the beginning, we would never have noticed "contravariance". I suspect that there may be some misconceptions lurking here, as thus does not sound right. In particular, the concept of co/contravariance applies even in the absence of a metric, and your statement simply does not apply in this case. Even when a metric is present, it is still impossible to globally "choose our basis carefully" when the metric is either intrinsically curved or is indefinite. I would rather have said co- and contravariance are an unavoidable effect of the arbitrary choice of a basis. This freedom of choice induces certain powerful consequences (it is an exact symmetry – albeit reduced by the presence of a metric), and Emmy Noether had something to say about this). — Quondum^☏✎ 16:34, 1 April 2012 (UTC)

Good point, if raising and lowering indices is not possible in some cases, then that reason might break down. Maybe we should talk about the "handedness" idea a bit more too. I don't know if "handedness" is the right word. What's going through my mind is "In noncommutative algebra, we have "left" and "right" and this is all there is, because for binary operations, you cannot have a "third side". So, it may be that you can be this way and that way with respect to a basis transformation, and there are no other ways to be. I get a similar feeling for orientation. (Are there objects where more than two orientations are talked about?)

The reason we only have binary operations, left/right and clockwise/counterclockwise may also be an artifact of the way humans think. I really hope to not go that far afield, though :P Rschwieb (talk) 13:08, 2 April 2012 (UTC)

I think a more appropriate term is sidedness. In physics and geometry, handedness has an unrelated meaning and both will be needed, and thus would lead to confusion. More than two sides makes sense, but requires other notation: the general concept of arity covers this. Einstein notation or the equivalent Penrose graphical notation is an example; n-ary operations in general are used. In geometric contexts, which relates to handedness (or orientation (vector space), as you correctly call it), there are exactly two handednesses, irrespective of the dimension (except for 0-D ) and signature. When one considers co- and contravariance; this too is a (yet further unrelated) duality. There are many (usually unrelated or "orthogonal") types of duality, but in general they come in exact pairs (as implied by the term). Given these distinctions, can you be more specific about anything you want to explore here? — Quondum^☏✎ 13:47, 2 April 2012 (UTC)

I think after a few more statements we can drop it and go back to the harder stuff. I'd like to put out the idea that sidedeness and orientation might be phenomena with a common origin. I'm specificially thinking of the fact that the order of multiplication in GA of vectors (say v, w) vw has opposite orientation to wv in the geometric sense and algebraic sense. However since this does not hold in the entire algebra, maybe it's just a geometric ghost haunting the algebraic subspace V!

Here's a question I hope we can decide: which of the concepts "co/contravariant" "left/right handed" "left/right sided" are concretely related?

From what's been said so far, I feel like the first two are the same thing but maybe the third is independent. The first two both say "I deem this way to be baseline, and everything goes the same way or the opposite way." Rschwieb (talk) 15:07, 2 April 2012 (UTC)

My feeling would be "none of them", other than being vaguely isomorphic to the group ℤ₂. Sidedness is a bit like S_n for some n. Though I'll add a wrinkle: some dualities are symmetric (as with the imaginary unit and its additive inverse, or left and right-handedness), and some are not (as with +1 and −1, even and odd parts of a Clifford algebra, and proper and and improper (reflected) rotations). But for me the algebraic case, being n-ary, is the one that fits worst of all. I remember being introduced to several dualities in tensors years back – and it took me a while to realize that not only were the dualities referred to different, but that they were unrelated. — Quondum^☏✎ 15:41, 2 April 2012 (UTC)

I think you're connecting them without realizing it. Do you have a formal definition of duality you are using? Rschwieb (talk) 17:18, 2 April 2012 (UTC)

Uh-oh. I've been caught out. Technically, no, I don't. I am simply familiar with several exact dualities: vector/covecter (i.e. co-/contravariance), Hodge duality, E–M duality (if you ignore the slight problem of quantization of charge and missing monopoles). Intuitively, I would say any mathematical structure that is exactly isomorphic under an exchange of a pair of variables. I have a haunting feeling that position and momentum are similarly dual in quantum physics, in the sense that time and frequency are dual through the Fourier transform (is the Fourier transform of the universe an identical universe?). In the case of a specific binary operation, your left- and right-sided multiplication would qualify. I should exclude my "asymmetric dualities". — Quondum^☏✎ 17:50, 2 April 2012 (UTC)

All good input. Judging from the many ununified meanings of duality at Duality (mathematics), we could get pretty far sidetracked :) Let's table it for awhile until one of us has an epiphany about it. Rschwieb (talk) 18:58, 2 April 2012 (UTC)