# Mathematicians’ Tensors vs. Physicists’ Tensors

It seems, at times, that physicists and mathematicians mean different things when they say the word “tensor.” From my perspective, when I say tensor, I mean “an element of a tensor product of vector spaces.”

For instance, here is a segment about tensors from Zee’s book Einstein Gravity in a Nutshell:

We already saw in the preceding chapter
that a vector is defined by how it transforms: $V^{'i} = R^{ij}V^j$ . Consider a collection of “mathematical
entities” $T^{ij}$ with $i , j = 1, 2, . . . , D$ in $D$-dimensional space. If they transform
under rotations according to
$T^{ij} \to T^{'ij} = R^{ik}R^{jl}T^{kl}$ then we say that $T$ transforms like a tensor.

This does not really make any sense to me. Even for “vectors,” and before we get to “tensors,” it seems like we’d have to be given a sense of what it means for an object to “transform.” How do they divine these transformation rules?

I am not completely formalism bound, but I have no idea how they would infer these transformation rules without a notion of what the object is first. For me, if I am given, say, $v \in \mathbb{R}^3$ endowed with whatever basis, I can derive that any linear map is given by matrix multiplication as it seems the physicists mean. But, I am having trouble even interpreting their statement.

How do you derive how something “transforms” without having a notion of what it is? If you want to convince me that the moon is made of green cheese, I need to at least have a notion of what the moon is first. The same is true of tensors.

My questions are:

• What exactly are the physicists saying, and can someone translate what they’re saying into something more intelligible? How can they get these “transformation rules” without having a notion of what the thing is that they are transforming?
• What is the relationship between what physicists are expressing versus mathematicians?
• How can I talk about this with physicists without being accused of being a stickler for formalism and some kind of plague?

What a physicist probably means when they say “tensor” is “a global section of a tensor bundle.” I’ll try and break it down to show the connection to what mathematicians mean by tensor.

Physicists always have a manifold $$MM$$ lying around. In classical mechanics or quantum mechanics, this manifold $$MM$$ is usually flat spacetime, mathematically $$R4\mathbb{R}^4$$. In general relativity, $$MM$$ is the spacetime manifold whose geometry is governed by Einstein’s equations.

Now, with this underlying manifold $$MM$$ we can discuss what it means to have a vector field on $$MM$$. Manifolds are locally euclidean, so we know what tangent vector means locally on $$MM$$. The question is, how do you make sense of a vector field globally? The answer is, you specify an open cover of $$MM$$ by coordinate patches, say $${Uα}\{U_\alpha\}$$, and you specify vector fields $$Vα=(Vα)i∂∂xiV_\alpha=(V_\alpha)^i\frac{\partial}{\partial x^i}$$ defined locally on each $$UαU_\alpha$$. Finally, you need to ensure that on the overlaps $$Uα∩UβU_\alpha \cap U_\beta$$ that $$VαV_\alpha$$ “agrees” with $$VβV_\beta$$. When you take a course in differential geometry, you study vector fields and you show that the proper way to patch them together is via the following relation on their components:
$$(Vα)i=∂xi∂yj(Vβ)j (V_\alpha)^i = \frac{\partial x^i}{\partial y^j} (V_\beta)^j$$
(here, Einstein summation notation is used, and $$yjy^j$$ are coordinates on $$UβU_\beta$$). With this definition, one can define a vector bundle $$TMTM$$ over $$MM$$, which should be thought of as the union of tangent spaces at each point. The compatibility relation above translates to saying that there is a well-defined global section $$VV$$ of $$TMTM$$. So, when a physicist says “this transforms like this” they’re implicitly saying “there is some well-defined global section of this bundle, and I’m making use of its compatibility with respect to different choices of coordinate charts for the manifold.”

So what does this have to do with mathematical tensors? Well, given vector bundles $$EE$$ and $$FF$$ over $$MM$$, one can form their tensor product bundle $$E⊗FE\otimes F$$, which essentially is defined by
$$(E⊗F)p=⋃p∈MEp⊗Fp (E\otimes F)_p = \bigcup_{p\in M} E_p\otimes F_p$$
where the subscript $$pp$$ indicates “take the fiber at $$pp$$.” Physicists in particular are interested in iterated tensor powers of $$TMTM$$ and its dual, $$T∗MT^*M$$. Whenever they write “the tensor $$Tij...kℓ...T^{ij...}_{k\ell...}$$ transforms like so and so” they are talking about a global section $$TT$$ of a tensor bundle $$(TM)⊗n⊗(T∗M)⊗m(TM)^{\otimes n} \otimes (T^*M)^{\otimes m}$$ (where $$nn$$ is the number of upper indices and $$mm$$ is the number of lower indices) and they’re making use of the well-definedness of the global section, just like for the vector field.

Edit: to directly answer your question about how they get their transformation rules, when studying differential geometry one learns how to take compatibility conditions from $$TMTM$$ and $$T∗MT^*M$$ and turn them into compatibility relations for tensor powers of these bundles, thus eliminating any guesswork as to how some tensor should “transform.”

For more on this point of view, Lee’s book on Smooth Manifolds would be a good place to start.