Purpose of Tensors, Indices in Tensor Calculus Explained

[AndersF]

TL;DR Summary

I wonder what the purpose of some elements and operations defined in tensor calculus is, such as the index contraction, the covariant and contravariant indexes or the raising and lowering operations.

I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it.

1. Tensors. As I understand it, the main purpose of the Tensor Calculus in Physics is to formulate the laws of nature in a way that they are independent of the reference frame from which they are observed. For this purpose, the concept of tensors is introduced, which are invariant objects under coordinate transformations. Would this be correct?
2. Covariant and contravariant indices. We distinguish two types of indices on the components of a tensor $\mathbb{T}$: covariant ones, placed up, and contravariant ones placed down. What is the purpose of introducing these two types of indices?
3. Raising and lowering indices. I see that the metric tensor allows us to change the covariant or contravariant nature of the indices of a tensor by lowering or raising them, respectively. For example, if we want to raise the index $j$ in $T_{ij}$, we would do so by $g^{a j} T_{ij}=T^{a}_{i}$. What motivates us to want to change the nature of an index on the components of a tensor? (And why do the indices have to be placed in this way? That is, why are the indices $a j$ placed at the top in $g$, in the opposite position to the index $j$ that we want to raise?)
4. Index contraction. I see that this operation allows us to combine two tensors of different orders to obtain a new tensor of lower order. For example, $T_{jk}^i A^k=B_j^i$. What is the purpose of this operation?
5. Spaced notation for indices. Finally, I have seen that some books use a notation that includes spaces between the covariant and contravariant indices, while other books do not follow this rule. For example, this index lifting operation would be written in these two different ways $g^{a j} T_{ijk}=T^{\space a}_{i \space \space \space k}$ and $g^{a j} T_{ijk}=T^{a}_{ik}$. What is the motivation for the first notation? Is one of the two notations better than the other?

Sorry if the question has taken too long. Any help with these questions would be welcome .

 

[Vanadium 50]

You're coming at this from the wrong direction. Tensors weren't invented for GR. GR is just a theory that naturally uses this particular mathematics.

 

[PeterDonis]

I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus.

Tensor calculus was discovered by mathematicians, not physicists. Your questions are really about how tensor calculus is used in physics, and General Relativity in particular. That's a much narrower subject than the uses of tensor calculus in general.

Tensors. As I understand it, the main purpose of the Tensor Calculus in Physics is to formulate the laws of nature in a way that they are independent of the reference frame from which they are observed. For this purpose, the concept of tensors is introduced, which are invariant objects under coordinate transformations. Would this be correct?

Basically, yes. More precisely, scalars, which can be formed by contracting two or more tensors in such a way that there are no free indices, are invariants.

Covariant and contravariant indices. We distinguish two types of indices on the components of a tensor $\mathbb{T}$: covariant ones, placed up, and contravariant ones placed down. What is the purpose of introducing these two types of indices?

Because you have two different types of tensor objects, not one. If we consider objects with just one index, we have vectors (upper index) and covectors or 1-forms (lower index), which are bounded linear maps from vectors to numbers. Contracting a vector with a covector just means applying the bounded linear map that the covector describes to that particular vector to obtain a scalar (i.e., an invariant). The upper and lower indices, as a notation, make it easy to tell when you have correctly formed a contraction. See further comments below.

Raising and lowering indices. What motivates us to want to change the nature of an index on the components of a tensor?

You're not "changing the nature of an index". You're finding a correspondence between two different tensors. For example, if I have a vector, I can lower its index with the metric to get a corresponding covector. Often it's easier to just use the covector in further expressions, rather than having to write out the contraction of the vector with the metric every time.

why are the indices $a j$ placed at the top in $g$, in the opposite position to the index $j$ that we want to raise?)

Because, as above, that makes it easy to tell whether a particular contraction is correct.

Index contraction. I see that this operation allows us to combine two tensors of different orders to obtain a new tensor of lower order. For example, $T_{jk}^i A^k=B_j^i$. What is the purpose of this operation?

Ultimately, to form scalars, since scalars are invariants (see above). Often it is useful to form intermediate contractions with fewer indices in order to simplify expressions, when a particular intermediate contraction occurs very often (an example would be the Ricci tensor, which is a contraction of the Riemann tensor on its first and third indices).

Spaced notation for indices. What is the motivation for the first notation?

To help keep track of the order of the "slots" on the tensor, i.e., the order in which vectors and covectors are applied when forming contractions. In many cases the contractions commute, so the order doesn't really matter; but in some cases the order does matter, so it's useful to have a way to keep track of it.

Two examples to illustrate this are:

The stress-energy tensor, $T_{ab}$, has two slots, both for vectors. Typically we contract it with vectors that are physically meaningful for the problem we are dealing with; for example, if we contract it twice with the 4-velocity of some observer, $T_{ab} u^a u^b$, we obtain the energy density measured by that observer. Since the tensor is symmetric, if we have two different vectors to contract it with (for example, the 4-velocity $u^a$ of some observer and a spacelike vector $e^b$ describing a particular direction--this contraction gives the momentum density in that direction measured by that observer), we get the same scalar regardless of in which order we contract the vectors (for example, $T_{ab} u^a e^b = T_{ab} e^a u^b$).

The Riemann tensor, by contrast, has a different physical meaning for each of its slots; in its most fundamental form it is written $R^a{}_{bcd}$, so it has one slot for a covector and three slots for vectors. If you contract it with the tangent vector $u^a$ to a field of geodesics and the normal vector $n^b$ that connects neighboring geodesics, you get a 1-form describing the geodesic deviation (tidal gravity) between the neighboring geodesics: $w^a = R^a{}_{bcd} u^b n^c u^d$. But you can't switch around the $n$ and the $u$ vectors here; the Riemann tensor is not symmetric on those indices and the physical meaning of the contraction is changed (to something that doesn't make sense).

 

[FactChecker]

Tensors. As I understand it, the main purpose of the Tensor Calculus in Physics is to formulate the laws of nature in a way that they are independent of the reference frame from which they are observed. For this purpose, the concept of tensors is introduced, which are invariant objects under coordinate transformations. Would this be correct?

Yes.

Covariant and contravariant indices. We distinguish two types of indices on the components of a tensor $\mathbb{T}$: covariant ones, placed up, and contravariant ones placed down. What is the purpose of introducing these two types of indices?

I think of it as the difference between "x seconds" and "x per second". When you change from seconds to minutes, the two will be affected differently. We must keep track of which is which.

Raising and lowering indices. I see that the metric tensor allows us to change the covariant or contravariant nature of the indices of a tensor by lowering or raising them, respectively. For example, if we want to raise the index $j$ in $T_{ij}$, we would do so by $g^{a j} T_{ij}=T^{a}_{i}$. What motivates us to want to change the nature of an index on the components of a tensor? (And why do the indices have to be placed in this way? That is, why are the indices $a j$ placed at the top in $g$, in the opposite position to the index $j$ that we want to raise?)

Like any mathematical manipulation, it is to simplify or convert to match something else and make it more understandable.

1. Index contraction. I see that this operation allows us to combine two tensors of different orders to obtain a new tensor of lower order. For example, $T_{jk}^i A^k=B_j^i$. What is the purpose of this operation?

Same as above. To simplify or make it match something else and be more understandable.

Spaced notation for indices. Finally, I have seen that some books use a notation that includes spaces between the covariant and contravariant indices, while other books do not follow this rule. For example, this index lifting operation would be written in these two different ways $g^{a j} T_{ijk}=T^{\space a}_{i \space \space \space k}$ and $g^{a j} T_{ijk}=T^{a}_{ik}$. What is the motivation for the first notation? Is one of the two notations better than the other?

Just keeping track of the indices. They are equivalent. The whole system is to help in "bookkeeping" with shorthand notation.

 

[pervect]

Here's an incomplete and non-rigorous version of some aspects of tensor calculus that may answer some (though not all) of your questions.

1) Scalar quantities exist which are independent of coordinates. The transformation laws of invariant scalar quantites are particularly simple - they don't change with coordinates.

2) Vectors exist. Vectors must satisfy the axioms of a vector space. They have a commutative addition, (X+Y) =(Y+X) are associative (X+Y)+Z = X+(Y+Z), can be multiplied by scalars, and the scalar multiplication is distribuitive a(X+Y) = aX + aY, (a+b)X = aX + bX. There are a few other requirements as well, see for instance https://mathworld.wolfram.com/VectorSpace.html. The definition is purely mathematical, the application to physics is that they are useful for representing physical quantites with a direction and magnitude, such as velocities. In physics, vectors are usually graphically represented as little arrows.

3) Maps for vectors to scalars exist. These are called dual vectors. The map from a vector space to a scalar is another vector space of the same dimension as the first vector space, but while it has the same dimensions as the original vector space, it is not the same space. For instance, once we introduce coordinates (which we haven't gotten around to, yet), vectors and dual vectors transform differently under a change of coordinates. Dual vectors can be represented in physics by "stacks of plates". Misner, Thorne, and Wheeler discuss this in their textbook "Gravitation", not all textbooks discuss this particular though very useful graphically representation of dual vectors as stacks of plates.

4) The duality principle. A map from a dual vector to a scalar can be identified as a vector, rather than a map from a (map from a vector to a scalar) to a scalar. You'll typically find a proof of this duality principle in linear algebra textbooks, I personally don't find it very intuitive, but at some point one needs to accept and internalize this principle. Almost all of what I'm talking about here can be found in a good linear algebra textbook, so if you're struggling with tensors, it might be helpful to review your linear algebra.

These are rather general statements, so far vector spaces and their dual spaces don't necessarily have a unique relationship. We can specialized general vector spaces into metric spaces, where we associate invariant lengths with vectors. This might either be a metric vector space or a normed metric space, I'm a bit unclear of the exact details of these defimotopms. However, the sort of vector space we use most in physics is actually the inner product space, where we not only have the length of a vector, but we have an inner product of any two vectors, a map from two vectors to a scalar that gives the squared length of a vector when we take the inner product of a vector with itself.

I won't attempt to justify it in this post, but the existence of an inner product logically implies that there is a natural mapping from vectors to dual vectors, a map which a vector space that does not have an inner product may lack.

Indices, raising and lowering indices, and other such questions in your post arise after we've incorporated coordinates and basis vectors into the more general concepts above. Thus, the existence of a natural mapping from vectors to dual vectors is a pre-requisite for the idea of raising or lowering indices to covert covariant vectors to contravariant vectors, and vica-versa.

Matrix notation also has structures that are equivalent to covariant and contravariant vectors, but they're typically introduced as column vectors and row vectors in matrix notation. The inner product in tensor notation is the dot product in matrix notation.

Matrix notation can deal with rank 1 and rank 2 tensors , but it struggles (and as far as I know , fails) to deal with higher rank tensors, such as the rank 4 curvature tensor, the Riemann tensor. Since the rank 4 Riemann curvature tensor is very important to special relativity, matrix notation is not sufficient and one uses tensor notation instead.

 

[AndersF]

Okay, thank you very much, I have found your answers very helpful :)

 

[cianfa72]

The Riemann tensor, by contrast, has a different physical meaning for each of its slots; in its most fundamental form it is written $R^a{}_{bcd}$, so it has one slot for a covector and three slots for vectors. If you contract it with the tangent vector $u^a$ to a field of geodesics and the normal vector $n^b$ that connects neighboring geodesics, you get a 1-form describing the geodesic deviation (tidal gravity) between the neighboring geodesics: $w^a = R^a{}_{bcd} u^b n^c u^d$. But you can't switch around the $n$ and the $u$ vectors here; the Riemann tensor is not symmetric on those indices and the physical meaning of the contraction is changed (to something that doesn't make sense).

Sorry, even in case the Riemmann tensor is written as $R^a_{bcd}$ it shouldn't be any problem in the contraction $w^a = R^a_{bcd} u^b n^c u^d$ providing the order of $n$ and $u$ vectors is not switched (since as you said Riemann tensor is not symmetric on those indices).

Maybe I missed your point

 

[PeterDonis]

Sorry, even in case the Riemmann tensor is written as $R^a_{bcd}$ it shouldn't be any problem in the contraction $w^a = R^a_{bcd} u^b n^c u^d$

Not in that particular contraction, perhaps. But what happens if you lower the $a$ index? The way you've written it, the ordering of the $a$ index would then not be well-defined with respect to the other indexes.

 

[cianfa72]

Not in that particular contraction, perhaps. But what happens if you lower the $a$ index? The way you've written it, the ordering of the $a$ index would then not be well-defined with respect to the other indexes.

Do you mean lowering the $a$ index of the Riemmann tensor using the tensor metric $g_{ae}$ ?

Note the use of the letter $e$ for the second index of the tensor metric just to avoid confusion.

 

[PeterDonis]

Do you mean lowering the $a$ index of the Riemmann tensor using the tensor metric $g_{ae}$ ?

Yes.

 

[vanhees71]

I think @PeterDonis is referring to the horizontal placement of the indices, which is as important as the vertical one. So it's better to write {R^a}_{bcd}, resulting in ${R^a}_{bcd}$.