Differential Forms or Tensors for Theoretical Physics Today

[kay bei]

TL;DR Summary

Do theoretical physicists prefer using tensors over differential forms today? What are the advantages of one over the other?

There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor approach, yet Gravitation by Wheeler uses differential forms. Frankel “Geometry of Physics” uses Differential Forms, and Chris Isham “Modern differential geometry for physicists” uses differential forms.
What are the advantages of one over the other? Do theoretical physicists today tend to prefer one over the other and is it field specific? Is there a trend for one approach? Could Kip Thorne book “modern classical physics” be rewritten in differential forms and how much shorter would the book be after that. I know mathematicians have mostly adopted the approach of differential forms. Also if you know anything of these books I mentioned or any others that would be great to hear your comment. I just want to stay current with what tools and approaches theoretical physicists are using today.

 

[martinbn]

Differential forms are tensors.

 

[kay bei]

Differential forms are tensors.

Hi Martin, you are certainly right. Maybe my question gave the impression that I wasn’t aware of this. The main point of the question is that there are differing approaches to theoretical physics such as by differential forms and the tensors with indices. I know it sounds strange to speak of these two as different things even though they are both tensors really. But there is obviously a difference because physicists and mathematicians speak of them both as different approaches. What then do physicists mean when they speak of tensors with indices. If you look at the books above that I mentioned you’ll see what I mean.

 

[martinbn]

Are you asking about coordinate based and coordinate free approaches?

 

[kay bei]

Are you asking about coordinate based and coordinate free approaches?

Maybe? I’m not very experienced so if you can enlighten me that would be great. I thought Tensors and Differential Forms were coordinate free until you choose a coordinate system for them. Would you say that all the books I mentioned above use coordinate free approaches. Is this the trend in Physics? And why would Kip Thorne choose Tensors with indices over differential forms for his book. Is there a difference in the definition of tensor by physicists and differential forms that I am missing?

 

[kay bei]

Are you asking about coordinate based and coordinate free approaches?

Differential forms are tensors.

Martin I want to show that my confusion isn’t unreasonable.
Many authors mention they will introduce modern differential geometry using the approach of differential forms. I mean if you go and read Kip Thorne book “Modern Classical Physics” he explicitly says he will be using Tensors throughout the book and not differential forms. These are his own words. It is hard to argue with Kip Thorne. There is clearly a difference between the two, or at least physicists have an implicit understanding of what is meant by tensor and that differential form is different from it. What are these Tensors they speak of and in what way are they different from differential forms?

 

[romsofia]

People who are pedantic will sit here and tell you that differential forms are tensors, and while that is true, the notation is different and it *does* matter in the field (ease of understanding). So, if you're interested in getting into GR research wise you MUST know the conventional tensor notation as most people still use it. Although I prefer the differential form notation, sometimes other physicists won't understand your notation as easily.

EDIT: And I typed this up before post #6, yes, your concern is valid. You're better off posting these questions in the physics section as mathematicians don't really concern themselves with actual computations, which is where the power of the differential form notation comes from!

So, a quick example on the difference of notation, let's look at the structure equations.

Differential forms: $$\text{First structure equation: } \Theta^i = d\sigma^i + \omega^i_j \wedge \sigma^ i $$
$$\text{Second structure equation (this gives you curvature!): } \Omega^i_j = d\omega^i_j +\omega^i_k \wedge \omega^k_j$$

Where $\sigma^i$ are your basis 1-forms. In GR, we have the Levi-Cievata connection, so you set the 1st one equal to zero, and you're able to compute those $\omega^i_j$ pretty easily. They're called the connection 1-forms, which just talks about how your basis vectors moves from point to point $d\hat{e_i} = \omega^j_i \hat{e_j}$

In the more conventional tensor notation you'll have different names and notation, for what we call the "first structure equation" you'll have the Christoffel symbols and for what we call the "second structure equation", you'll have the curvature tensor which usually looks as follows:
$$\text{Christoffel symbols: } \Gamma^a_{mn} = \frac{1}{2}g^{ab}(g_{bm,n}+g_{bn,m}-g_{mn,b})$$
$$\text{Curvature tensor: }R^b_{mnq} = \Gamma^b_{mq,s} - \Gamma^b_{ms,q} +\Gamma^b_{ns}\Gamma^n_{mq}-\Gamma^b_{nq}\Gamma^n_{ms}$$

Now, to see how they are the same thing, you make the connection (no pun intended) that the connection 1-forms are related the christoffel symbols by $$\omega^i_j = \Gamma^i_{jk} \sigma^k$$ Thus, we can also make a connection with the second structure equation and what is known as the curvature tensor! $$\Omega^i_j = \frac{1}{2} R^i_{jkl} \sigma^k \wedge \sigma^l$$
If you want a nice introduction to a "purely" differential form approach to GR, here is a paper: https://arxiv.org/pdf/0904.0423.pdf

 

[lavinia]

Modern mathematicians and physicists are fluent in index notation and coordinate free notation as @martinbn suggested. You need to learn both.

BTW: Differential forms are skew symmetric tensors. Other tensors may not be differential forms for instance the metric tensor which is symmetric rather than skew symmetric.

IMO index notation is less geometrically clear.

 

[kay bei]

Modern mathematicians and physicists are fluent in index notation and coordinate free notation as @martinbn suggested. You need to learn both.

BTW: Differential forms are skew symmetric tensors. Other tensors may not be differential forms for instance the metric tensor which is symmetric rather than skew symmetric.

IMO index notation is less geometrically clear.

I definitely understand that differential forms are tensors. Kip Thorne in his book "modern classical physics" uses Tensors with indices throughout. I know these are coordinate-free. He says also that he does not use differential forms in the book and leaves this "richer mathematics" to more advanced books. He writes Gauss's law in 4 dimensions using tensors, while he says he would not attempt to write Stoke's law in 4d with these tools and said it is easiest with differential forms.

Since Tensors are more general, would using differential forms eliminate some of the generality of the laws of physics formulated with them? When is one more appropriate to use over the other?

 

[lavinia]

I definitely understand that differential forms are tensors. Kip Thorne in his book "modern classical physics" uses Tensors with indices throughout. I know these are coordinate-free.

Index notation is coordinate free in the sense that it describes a tensor with respect to an abstract basis for the tangent space - or more generally a vector bundle - rather than with respect to a specific choice of coordinates and the specific basis determined by the differentials of the coordinate functions. Any tensor can be described in this way including differential forms.

While I know little Physics, judging from Leonard Susskind's Lectures on General Relativity tensors are thought of in Physics as arrays of numbers that transform according to certain rules when coordinates are changed. I imagine that this way of looking at things can get cumbersome in situations where there are many indices. Already in General Relativity there are indices all over the place. I wonder how much fun it would be to describe the differential geometry of a 167 dimensional Riemannian manifold using index notation.Maybe this is what Thorne was talking about.

In mathematics dimensions may not be specified and ideas are often expressed for any dimension. In such a case index notation would seem to be a ball and chain.

He says also that he does not use differential forms in the book and leaves this "richer mathematics" to more advanced books. He writes Gauss's law in 4 dimensions using tensors, while he says he would not attempt to write Stoke's law in 4d with these tools and said it is easiest with differential forms.

Since Tensors are more general, would using differential forms eliminate some of the generality of the laws of physics formulated with them? When is one more appropriate to use over the other?

I do not know how one can do Physics only with Differential Forms. So I would guess that you are not completely understanding what Thorne is saying. Perhaps you can provide a quote from his book.

 

[kay bei]

I do not know how one can do Physics only with Differential Forms. So I would guess that you are not completely understanding what Thorn is saying. Perhaps you can provide a quote from his book.

What I wrote was exactly from his book, I just paraphrased a little and didn't put it all in quotation marks. I just thought there are so many books teaching physics from a purely differential form viewpoint. But you are right, I am obviously confused for no reason.

 

[lavinia]

What I wrote was exactly from his book, I just paraphrased a little and didn't put it all in quotation marks. I just thought there are so many books teaching physics from a purely differential form viewpoint. But you are right, I am obviously confused for no reason.

OK. I will see if I can find the book on line.

There is a wonderful book by Flanders that describes how differential forms are used in Physics. It is called Differential Forms with applications to the Physical Sciences. The book is short, well written, and full of examples. A great example which I love is the differential form description of the law of Biot and Savart. It shows the geometric meaning of this law in terms of linking numbers of closed loops.

 

[martinbn]

In the book he is only saying that for Stokes theorem, in its full generality, you need differential forms.

I would say don't worry about this and just study the book.

 

[robphy]

Since Tensors are more general, would using differential forms eliminate some of the generality of the laws of physics formulated with them? When is one more appropriate to use over the other?

I would agree with your suspicion about differential forms being a more precise tool for physics,
but, at this stage, I can't quite elaborate definitively.

It seems many physical laws can be formulated in terms of differential forms,
suggesting that that formalism incorporates specific symmetries that seem to be realized physically.

For example,
Tevian Dray's https://www.amazon.com/dp/1466510005/?tag=pfamazon01-20
says (in the Preface)

For the expert, the only rank-2 tensor objects that appear in the book are the
metric tensor, the energy-momentum tensor, and the Einstein tensor, all of which are
instead described as vector-valued 1-forms; the Ricci tensor is only mentioned to permit
comparison with more traditional approaches.

(bolding mine)
which, it seems to me, suggests that the two "[abstract] index slots" in those R-valued rank-2 tensors
do not play identical roles physically as its notation might suggest.

https://en.wikipedia.org/wiki/Vector-valued_differential_form

 

[romsofia]

For example,
Tevian Dray's https://www.amazon.com/dp/1466510005/?tag=pfamazon01-20
says (in the Preface)

The differential form part can be found online here: http://sites.science.oregonstate.edu/physics/coursewikis/GDF/book/gdf/start.html
The GR part can be found here:
http://sites.science.oregonstate.edu/physics/coursewikis/GGR/book/ggr/start

I enjoyed this book, and it's a good introduction to this formalism.

 

[lavinia]

People who are pedantic will sit here and tell you that differential forms are tensors, ...

Really?

EDIT: And I typed this up before post #6, yes, your concern is valid. You're better off posting these questions in the physics section as mathematicians don't really concern themselves with actual computations, which is where the power of the differential form notation comes from!

Can you go into more detail? It seems that mathematicians are very good at computation.

Differential forms are used in many ways not just for computation. For example the DeRham cohomology groups are made of closed differential forms modulo exact forms. These groups are key to connecting the Differential Topology of smooth manifolds to their Algebraic Topology.

 

[romsofia]

Really?

Yes, really. It was mentioned twice in this thread already, and I don't see how it helps clarify any statement. It doesn't, it's truly a useless fact in my opinion when someone approaches an expert about differential forms.

In fact, let's look at the OP and see if that answers a single question.

Do theoretical physicists prefer using tensors over differential forms today?

What are the advantages of one over the other?

Do theoretical physicists today tend to prefer one over the other and is it field specific?

Could Kip Thorne book “modern classical physics” be rewritten in differential forms and how much shorter would the book be after that

Which one of the 4 questions does "differential forms are (skew-symmetric) tensors" answer?

 

[martinbn]

Yes, really. It was mentioned twice in this thread already, and I don't see how it helps clarify any statement. It doesn't, it's truly a useless fact in my opinion when someone approaches an expert about differential forms.

I made the comment because in my opinion if someone doesn't understand what tensors and differential forms are, which is suggested by the fact that the question treats them as entirely differenty things, then any answer will be useless. So in my opinion the best thing the OP can do is get a better understanding of what tensors and differential forms are, then he will not need to ask the question that he thinks he is asking. And until then any answer is going to be useless to him.

 

[robphy]

I think the point being made is this:

Suppose summary question had instead read
Do theoretical physicists prefer using tensors over vectors (as in vector calculus) today? What are the advantages of one over the other?

While true, is an answer like "Vectors are tensors"
that helpful?

It seems to me the key question is the second part
What are the advantages [of the formalism or approach] of one over the other?

 

[martinbn]

I think the point being made is this:

Suppose summary question had instead read
Do theoretical physicists prefer using tensors over vectors (as in vector calculus) today? What are the advantages of one over the other?

While true, is an answer like "Vectors are tensors"
that helpful?

It seems to me the key question is the second part
What are the advantages [of the formalism or approach] of one over the other?

My point is that if the person doesn't understand well enough what a tensor is and what a vector is, then any answer will not be helpful.

 

[robphy]

From a skim on Google and on my copy of "Tensor Analysis for Physicists" by J.A. Schouten, I haven't found a definitive statement on the "tensor approach vs differential-form approach"
but maybe this helps (which was inspired by my post referencing Tevian's book above):

Tensors (which Schouten calls affinors) and tensor fields seem like the correct tools for describing geometric objects related by linearity
whereas
differential forms (the totally antisymmetric tensor fields) are the subset that are also more naturally suited for integration... and maybe quantities of physical interest that appear as "integrals of tensors fields" are really "integrals of vector-valued differential forms".

Furthermore, maybe this quote by Geroch (Mathematical Physics, pg 1 in the Introduction) is useful:

What one often tries to do in mathematics is to isolate some given structure for concentrated, individual study: what constructions, what results, what definitions, what relationships are available in the presence of a certain mathematical structure—and only that structure? But this is exactly the sort of thing that can be useful in physics, for, in a given physical application, some particular mathematical structure becomes available naturally, namely, that which arises from the physics of the problem. Thus mathematics can serve to provide a framework within which one deals only with quantities of physical significance, ignoring other, irrelevant things. One becomes able to focus on the physics. The idea is to isolate mathematical structures, one at a time, to learn what they are and what they can do. Such a body of knowledge, once established, can then be called upon whenever it makes contact with the physics.

A similar question would apply to why use "geometric algebra and geometric calculus" vs
differential forms vs tensor-calculus vs vector-calculus vs component-based calculus for physics applications.

 

[romsofia]

"tensor approach vs differential-form approach"

The "differentinal form approach" would be more commonly known in the field as the Cartan Formalism or usually just something with Cartan (although, this usually leads to Cartan gravity). When you start to consider the one forms as the fundamental ingredient, in my opinion, easier computations arise leading to new insights but people have more robust reasons to use this formalism.

EDIT: Also, Palatini's name also pops up a lot due to his variation being with respect to the connection.

 

[robphy]

Apart from gravitation,
differential forms (exterior forms) also appear in electromagnetism, mechanics, fluids, thermodynamics, and outside of physics: economics, computer graphics, and differential equations.

 

[wrobel]

I just do not understand how theoretical physicists can prefer using one mathematical tool over other one. Problem dictates mathematical tool.

 

[robphy]

I just do not understand how theoretical physicists can prefer using one mathematical tool over other one. Problem dictates mathematical tool.

I would think it depends on the specific goal of the research
and the levels of preparation of the particular researcher and of the target audience.

 

[lavinia]

I just do not understand how theoretical physicists can prefer using one mathematical tool over other one. Problem dictates mathematical tool.

I think now a days you are completely correct. While I am not a student of the history of Physics or Mathematics I have the impression that earlier in the 20'th century Mathematics and Physics used different formalisms in some areas although they were talking about the same mathematical structures. A famous but perhaps apocryphal story is that CN Yang was talking to James Simons about his research and Simons said 'Oh. Your talking about a connection'. I would guess that he was talking about a connection on a principal Lie group bundle. Here is a quote from CN Yang

"The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years."
— CN Yang, 1979

Simons was a student of Chern's. Chern was a Differential Geometer and his early work on the geometry of fiber bundles I think was largely done in the 1930's and 40's. Chern has a paper on the mutual recognition by mathematicians and physicists that they were both talking about connections on principal Lie group bundles.

Personal opinion: I think one of the morals is that there is a unity of mathematics and physics. IMO the idea that mathematics is just a tool of physics is passe at best. It reminds me of my sister's ballet teacher who viewed music solely as accompaniment to dance.

 

[giulio_hep]

I would also quote

the gauge degree of freedom in EM are related with a local phase degree of freedom in QM ... and the mathematical concept of a connection 1-form and curvature 2-form are related with the physical counter parts of vector potential and field strength tensor, respectively.

From "Fiber bundles, Yang and the geometry of spacetime." (by Federico Pasinato)

or more simply Wikipedia "Applications in physics" paragraph under "differential form":

The EM form is a special case of the curvature form on the U(1) principal bundle on which both EM and general gauge theories may be described... equations can be written very compactly in differential form notation... Also Yang–Mills theory, in which the Lie group is not abelian, is represented in a gauge by a Lie algebra-valued one-form A.

As William O. Straub noticed in "Differential Forms for Physics Students"

Differential forms point to a profound connection between general relativity, electromagnetism and quantum physics. This connection, which is difficult to see without the formalism, is provided by the Cartan structure equations, which all physics students should at least be aware of.

Similarly nLab about Connections in physics describes EM field as connection on U(1), Yang-Mills field more generally on U(n) and

The field of gravity is encoded in a connection on the orthogonal group-principal bundle to which the tangent bundle is associated.

We can conclude with M. Gasperini:

Thanks to the language of differential forms, we can rewrite all equations in a more compact form, where the tensor indices of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations).

 

[kay bei]

Modern mathematicians and physicists are fluent in index notation and coordinate free notation as @martinbn suggested. You need to learn both.

BTW: Differential forms are skew symmetric tensors. Other tensors may not be differential forms for instance the metric tensor which is symmetric rather than skew symmetric.

IMO index notation is less geometrically clear.

Yes differential forms are defined as antisymmetric tensors. I write here the definition 5.4.1 (page 52) in Michio Nakahara's book. " A differential form of order r or an r-form is a totally anti-symmetric tensor of type (0, r ).". Now if we quote also Gravitation by Wheeler, on page 83 they say "Any tensor can be symmetrized or antisymmetrized by constructing an appropriate linear combination of itself and it's transposes" (they give this as exercise 3.12). Now this is where I get confused, if we put these two facts together, since any tensor can be antisymmetrized, does this also mean we can rewrite any tensor as a differential form? And if this is certainly always possible to do, why would we not want to turn a tensor into differential form notation?

 

[giulio_hep]

Now if we quote also Gravitation by Wheeler, on page 83 they say "Any tensor can be symmetrized or antisymmetrized by constructing an appropriate linear combination of itself and it's transposes" (they give this as exercise 3.12).

Constructing an appropriate linear combination of itself and its transposes is not the same as the original tensor.

 

[kay bei]

Constructing an appropriate linear combination of itself and its transposes is not the same as the original tensor.

i don’t understand. Could you maybe provide an example.Maybe what the author was trying to say was a linear combination of a symmetric and antisymmetric form of itself?

 

[giulio_hep]

i don’t understand. Could you maybe provide an example.Maybe what the author was trying to say was a linear combination of a symmetric and antisymmetric form of itself?

An example from quantum physics: for the canonical symplectic 2-form in the cotangent bundle of the configuration space Q, the minus in the permutation sign for the anti-symmetry comes from the minus sign in the Hamilton equation.

 

[Ibix]

i don’t understand. Could you maybe provide an example.Maybe what the author was trying to say was a linear combination of a symmetric and antisymmetric form of itself?

Consider the Minkowski metric tensor, $\mathrm{diag}(1,-1,-1,-1)$. Antisymmetrize it (you can always do this for any tensor, as MTW say). Can the result possibly contain the same information as the original tensor? Antisymmetrize any other symmetric tensor. You should get the same result - so antisymmetrization is not invertible in general so, again, the result cannot contain the same information as the original.

 

[giulio_hep]

The differentiable structure on a manifold is not enough to define the so called covariant derivative on an arbitrary tensor. But on forms, this totally antisymmetric tensors, there is a notion of exterior derivative that can be defined just in terms of the differentiable structure of the manifold.

 

[weirdoguy]

since any tensor can be antisymmetrized

Why do you think that antisymmetrization of a tensor will give us exactly the same tensor? Why do you expect to do all those transformations and get exactly the same thing? What would be the point in doing them in the first place?

 

[robphy]

Gravitation by Wheeler, on page 83 they say "Any tensor can be symmetrized or antisymmetrized by constructing an appropriate linear combination of itself and it's transposes"

means that one can form the "totally-symmetric part" of a tensor and the "totally-antisymmetic part" of a tensor (not unlike finding the real-part of a complex number and the imaginary-part of a complex number).

For example,
$M_{ab} =\displaystyle \frac{(M_{ab} +M_{ba})}{2} + \frac{(M_{ab} -M_{ba})}{2} = M_{(ab)} + M_{[ab]}$,
where $M_{(ab)}$ is the symmetric-part and $M_{[ab]}$ is the antisymmetric-part.

If $M_{ab}=M_{(ab)}$ (or equivalently, $M_{[ab]}=0$), then $M_{ab}$ is said to be symmetric.

If $M_{ab}=M_{[ab]}$ (or equivalently, $M_{(ab)}=0$), then $M_{ab}$ is said to be antisymmetric.

But in general, since $M_{ab}=M_{(ab)} + M_{[ab]}$,
the general tensor $M_{ab}$ is neither symmetric nor antisymmetric.