Differential forms or Tensors for modern theoretical physics?

Question

There many proponents to teaching differential forms and others teach with tensors. This is true for both mathematics and physics education. It seems mathematicians prefer to teach differential geometry using differential forms. I want to know what is the current trend in theoretical physics, do they prefer to develop theory in terms of differential forms, or in terms of tensors (with indices). It seems most authors report that differential forms become more elegant when dimensions of a manifold increase and they also allow to write down equations without the use of indices.

There are the books, "Modern Classical Physics" by Kip Thorne which uses Tensors, "Gravitation" by Wheeler and Thorne which uses differential forms, "Modern Differential Geometry for Physicists" by Chris Isham which use differential forms, and "Geometry of Physics" by Theodore Frankel which uses differential forms. Judging by Isham, Frankel and Wheeler/Thorne(in Gravitation) who are all extremely respected scientists, it would seem differential forms are the standard tool. But I dont understand why Kip Thorne would go with the differential forms approach in Gravitation and yet stick to Tensors in "Modern Classical Physics". Why didnt Thorne use differential forms in his book "Modern Classical Physics". So I thought there was a trend towards differential forms but then Kip Thorne wrote his book "Modern Classical Physics" in terms of tensors, so now that he won a nobel prize, it certainly seems that Tensors are extremely relevant. I just want to know why not differential forms?

From what I read, differential forms seem to be useful for Gauge Theories but then again gravitation is taught in the language of differential forms in Gravitation.

Is it possible to do modern theoretical physics entirely with the use of differential forms and not resort to any tensors? What are the advantages to this? Are there any other more modern alternatives to using differential forms and tensors?

Hoping you theoretical physicists can help lead me on the right path here! Please comment on the textbooks I mentioned, if they are "Modern" in their use and if they are any good. Which is your favourite textbook for differential geometry for physics and do you have any other recommendations?

Answer 1

This is a very good question!y

Let me first try to address the issue of differential forms vs. tensors. First, as Qmechanic already mentioned, differential forms are special type of tensors. However, certainly not all tensors of importance to physics are differential forms. An example is that of vector fields, which are another kind of tensors. These appear everywhere in geometry. Just to mention one, infinitesimal transformations on a physical theory are represented by vector fields on its manifold of states. General tensors however can be constructed by taking tensor products of vectors and 1-forms (which are the simplest kind of differential forms). In coordinates $x^\mu$, vectors are spanned by $\frac{\partial}{\partial x^\mu}$ while 1-forms are spanned by $\text{d}x^\mu$. Examples of these more general tensors are

• General $k$-forms $$\omega=\frac{1}{k!}\omega_{\mu_1\cdots\mu_k}\text{d}x^{\mu_1}\otimes\cdots\otimes\text{d}x^{\mu_k},$$ with $\omega_{\mu_1\cdots\mu_k}$ totally antisymmetric. These are the objects that can be integrated on manifolds of dimension $k$. An example of this is the symplectic form of Hamiltonian mechanics.
• Metrics $$g=g_{\mu\nu}\text{d}x^\mu\otimes\text{d}x^\nu,$$ with $g_{\mu\nu}$ completely symmetric. This is not a differential form. It is however constructed out of 1-forms. They are key in order to define relativistic spacetimes.
• Inverse metric $$g^{\mu\nu}\frac{\partial}{\partial x^\mu}\otimes\frac{\partial}{\partial x^\nu},$$ with $g^{\mu\nu}g_{\nu\rho}=\delta^\mu_\rho$. This is again not a differential form. It is not even built out of 1-forms!

Now, in general relativity it may sometimes seem that everything is built out of differential forms because a large class of tensors (the covariant tensors which are the ones that have all of its indices down) can be built out of $1$-forms. In particular, once we have a metric we can write all tensors as if they where covariant by lowering all of the indices. The same happens in classical mechanics, once we have a symplectic form. However, certainly even in these cases not all tensors are differential forms. Moreover, there are physical situations where one doesn't have metrics or symplectic forms where not all tensors can be built out of 1-forms and one also needs vector fields. This is for example the case of a Newtonian spacetime, where there is no metric and one requires vector fields in order to describe, say, the velocity of a particle.

Having that out of the way, in my experience (which I admit is very reduced), it is more and more common for theoretical physicists to have a very solid understanding of the basics of differential geometry (and much more!). This includes an understanding of tensors in general. I think that due to the immense amount of applications of the subject in physics, it is certainly worthwhile to try to learn the subject.

Recommendations:

1. Take a look at this playlist of Prof. Frederic Schuller on general relativity. This lecture series and the next have become very famous. I've met people all over the world that have learned the subject watching them.
2. This next playlist of the same professor is on general differential geometry. It starts at a more basic level than the previous ones and focuses on other topics of geometry which are of interest in areas other than general relativity. It is certainly more in depth than 1. However, although it is targeted at physicists and certainly all of the topics covered are very important to modern physics, the course doesn't cover many applications. It was therefore more difficult to watch for me. I saw 1. first and then, as the topics appeared in my physics studies, I saw different parts of 2. Some would argue that a modern understanding of particle physics (even at the classical level) already requires all of the material in 2.
3. The book Geometry, Topology and Physics of Nakahara is a classic in this aspect. I however found it too difficult to read at first. However, after watching the lectures above I now enjoy it very much. Moreover, it covers many other topics relevant for physics outside of the realm of differential geometry which are key now a days.
4. I would also mention An introduction to Riemannian Geometry: With Applications to Mechanics and Relativity of Godinho and Natário. Much like reference 1. the objective of this book is Riemannian geometry rather than differential geometry. However, it is still an excellent introduction and I found the application chapters very useful!
5. However, I think that the best thing a student interested in geometry can do is to explore the standard references of the mathematicians. They are the clearest and easiest to use in my opinion. When one needs inspiration on the physical applications, one can always go to the references above. Classic mathematics textbooks are Introduction to manifolds (read this first) of Tu and Introduction to Smooth Manifolds of Lee. This two authors have written other textbooks on geometry that are also very useful.