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manisar
"Whenever you can, share. You never know who all will be able to see far away standing upon your shoulders!"
I write mainly on topics related to science and technology.
Sometimes, I create tools and animation.
Tensors in Physics - Once and For All
Author - manisar
If the idea of tensors ever intrigued you and you wanted to get to the real physical understanding of them, this article might help.
Tensors in physics and mathematics have two different but related interpretations - as physical entities and as transformation mapping. Let's look at the former interpretation below. But before that, if you want to quickly glance at some interesting ways to look at tensors, click on the buttons below.
Behave like a Tensor! Must see.
This description is based on the aphorism - "a tensor is something that behaves like a tensor"!
From a physical entity point of view, a tensor can be interpreted as something that brings together different components of the same entity together without adding them together in a scalar or vector sense of addition. E.g.
If I have 2gm of Calcium and 3gm of Calcium together, I immediately have 5gm of Calcium - this is scalar addition, and we can perceive the resulting substance.
If I am moving at 5i m/s and 6j m/s at the same time, I'm moving at (5i+6j) m/s. This is vector addition, and once again, we can make sense of the resulting entity.
If I have monochromatic pixels embedded in a cube that emit light at different angles, we can define pixels per unit area ($\chi$) in the cube as $\begin{bmatrix} \chi_x&\chi_y&\chi_z \end{bmatrix}$ where $\chi_x$ is the number of pixels emitting light perpendicular to the area in yz plane, and so on.
This entity, $\chi$, has three components, and by writing $\chi$, we are writing the three components together. Apart from that, the three components cannot be added like a scalar or vector, and we cannot visualize $\chi$ as a single entity.
$\chi$ above is an example of a tensor. Though we may not be able to see $\chi$ as a single perceivable thing, it can be used to fetch or understand perfectly comprehensible entities, e.g. for a given area $\vec{s}$, we can get the total number of pixels emitting light perpendicular to it by the product:
$$\begin{bmatrix}\chi_x&\chi_y&\chi_z
\end{bmatrix}\cdot
\begin{bmatrix}s_x\\s_y\\s_z
\end{bmatrix}$$
Change the monochromatic pixels in this example to RGB pixels, and we get something very similar to the stress tensor (a tensor of rank 2), and we can get the traction vector (force per unit area for a given unit area n) by the equation:
$$\begin{align}
\textbf{T}^{(\textbf{n})}
& = \begin{bmatrix} T_x\\T_y\\T_z \end{bmatrix}^{(n)}
= \textbf{n} \cdot \boldsymbol{\sigma}\\
& =
\begin{bmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\
\sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\
\sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\
\end{bmatrix}
\begin{bmatrix}n_x\\n_y\\n_z \end{bmatrix}
\end{align}$$
Though it's difficult to visualize the stress tensor in totality, each of its components tells us something very discrete, e.g. $\sigma_{xx}$ tells us how much force in x-direction is being experienced by a unit surface area that is perpendicular to the x-direction (at a given point in a solid). The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction, we get the traction vector from the stress tensor, or, I do not mean literally though, the stress tensor collapses to the traction vector.
Note that the possibility of interpreting tensors as a single physical entity or something that makes sense visually is not zero. E.g., vectors are tensors and we can visualize most of them (e.g. velocity, electromagnetic field).
Also note that in all the three cases - scalars, vectors and tensors, the resulting entity - the collection or the grouping of the individual components, whether or not it makes any sense to you intuitionally or visually (e.g. the total amount of Calcium, the net velocity, or the pixel density above), remains invariant. If we change the bases, we get different values of the components of these entities in different coordinate systems, but the resulting entities remain the same, and it is possible to find the values of the components of an entity in another system by knowing it in one - by virtue of having transformation rules between the two coordinate systems.
This invariance, forced by the transformation rules, is the meaning of the phrase "behaving like a Tensor".
A Vector with more than one Direction?
We know how a physical entity can have a magnitude, or a magnitude and a direction. But what doesn't seem natural to intuition, but is nevertheless possible, is the fact that entities can have a magnitude and two (or more) directions interwoven together.
The phenomenon being referred here is not of addition or multiplication of directions. Adding is possible only when the directions have the same dimensions, e.g. adding two velocities, and that doesn't give us any new entity. Multiplication of directions is possible and that gives us a vector.
But other operations are possible between dimensions having directions (or vectors), e.g. when we define $\text{stress} = {\text{force} \over \text{area}}$, both the terms on the right have directions in addition to having magnitude, and these directions (each having three components) interweave together to give an entity called the stress tensor having 9 components. The traction, or total force per unit area (for a given unit area) at any point is given by
\[\begin{align}
\end{align}\]
The stress tensor ($\sigma$), in its entirety, may not make intuitional sense to us, but its 9 components can manifest this entity in a way that we can get to know the direction of force on a surface of a given area and direction. These two directions (or vectors) - of force and of the surface are lying hidden in the stress tensor ($\sigma$) that come to surface as and when needed.
Though it's difficult to visualize the stress tensor in totality, each of its components tells us something very discrete, e.g. $\sigma_{xx}$ tell us how much force in x-direction is being experienced by a unit surface area that is perpendicular to the x-direction (at a given point in a solid). The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction of the surface, we get the traction vector, or, I do not mean verbally though, the stress tensor collapses to the traction vector.
Further, if we fix the direction of the force as well, i.e. we are only interested in force in a specific direction, we get a scalar - force in that specific direction on a surface facing a given direction. The traction vector has now collapsed to a scalar!
Dimensions Combined in Interesting Ways!
Every dimension of a physical entity can be comprised of different components, e.g. the three components of the velocity dimension $\vec{v} = v_x\vec{i}+v_j\vec{j}+v_z\vec{k}$. When we combine physical entities that are comprised of different dimensions in order to define new physical entities, the different components of the individual entities can interplay in different ways, e.g.
Each component can be scaled by a constant amount to give the corresponding component of the resulting entity, e.g. combining two physical entities - mass and velocity to get momentum, $\vec{p} = m\vec{v}$.
The individual components can multiply and add (or divide and get subtracted) with one another and result in components that can still look like components of a vector. E.g. force in magnetic field $\vec{F_{magnetic}} = q(\vec{v} \times \vec{B})$.
The individual components can combine with one another in a way that the resulting entity doesn't look like a scalar or vector! This is where tensors come in. They represent such entities. For such entities, their individual components need not adhere to scalar or vector addition among themselves.
What happens mostly is that, we are able to interpret the resulting entity manifested in specific ways, or the individual components themselves. But we may not be able to visualize the resulting physical entity in its entirety.
Nevertheless, we can, and do, write the resulting entity in its entirety, and call them tensors. E.g. electric polarization of a crystal in an electric field is given by:
P & =
\begin{bmatrix} P_x\\P_y\\P_z \end{bmatrix}\\
& = \begin{bmatrix}\alpha_{xx}&\alpha_{xy}&\alpha_{xz}\\
\alpha_{yx}&\alpha_{yy}&\alpha_{yz}\\
\alpha_{zx}&\alpha_{zy}&\alpha_{zz}\\
\begin{bmatrix}E_x\\E_y\\E_z
While the polarization tensor $\alpha$ may not make much sense to us like a scalr or a vector, the way it results in the different components of $\vec{P}$, $\begin{bmatrix} P_x\\P_y\\P_z \end{bmatrix}$ for a given $\vec{E}$ does, and we keep using the tensor representation of $\alpha$.
To say that tensors are physical entities is to say that there are physical entities out there that are represented by tensors, just like there are physical entities that represented by scalars or vectors. So, what is actually a tensor physical entity? In order to understand tensors as physical entities, we must have a fair understanding of the following. These ideas or phenomena are behind any and every physical entity.
Flavor (same same but different things)
Combination of flavor and dimensions
What Makes a Physical Entity Physical
The dimension of any physical entity is the thing that provides it its physics, its measurability. Examples are length, mass, time etc. Anything without a dimension is just a concept, a mathematical one, not physical. E.g. phrases like 'the temperature is 25', 'this is 4 of iron', 'the wire has 2 current', don't have any physical meaning until we add dimensions like 'the temperature is 25o Celsius', 'this is 4 kg of iron', 'the wire has 2 amperes of current'.
There are 7 fundamental dimensions which is basically saying that there are 7 primary dimensions that are not a combination of other dimensions.
Then comes the magnitude. This is the second mandatory requirement for the existence of a physical entity. I don't think I need to explain it.
Flavor. This is my construct! But you'll see how natural and necessary it is in understanding physical entities. Let's say I've 5 apples which are identical in every aspect except one - 3 of them are red and 2 green. Here 5 is the magnitude and apple-ness is the dimension. Now, even though looking at my apples as apples will suffice in many scenarios, there will be situations in which it'll be helpful to differentiate between red and green apples. Here, red and green are the flavors of our entity (apples). Whether we want to look at them as just apples or red and green apples separately depends entirely upon the context.
Also, we may or may not be accustomed to looking at them one way or another. E.g. somebody gravely allergic to green apples will see them as hugely different things whereas somebody who is color blind will find them identical.
You may ask why not just see these as different entities (with different dimensions) altogether. The answer lies in the measurability - if we can measure an entity using the exact same unit, it makes sense to see them having the same dimension.
Once we move from scalar entities to vector ones, things become clearer. In the world of vectors, flavor of an entity is its direction. A velocity of 5m/s in x-direction is not the same as 5m/s in y-direction. But, the dimension for both is m/s. So, these are like, different flavors of the same physical entity - the velocity.
Now comes the last piece of the puzzle - the combination of flavors and dimensions. The flavors of individual dimensions (entities), and different dimensions themselves can combine together to give rise to new physical entities.
And, we may or may not be used to understanding and comprehending the new physical entity in its own right, or conversely, we may or may not able to see the underlying combination. But, from a physical point of view, it always makes sense to able to describe the entities in terms of their components, even if its the combined entity that is more natural for us.
A few Examples of Combination of Flavors and Dimensions
Length and time - these two different dimensions can combine in a way to give rise to the physical entity called speed, and we can very much comprehend it its own right.
Momentum is comprised of two dimensions - mass and velocity, but we can quite fairly see and feel these two dimensions separately from each other, e.g. even with closed eyes, you can somewhat tell if a 5gm penny hit you at 20m/s or a 20gm coin hit you with 5m/s, even though both of these have the same momentum.
Red, green and blue - three flavors of the same dimension - luminosity - can combine together to form a composite color. This time, there are two possibilities - we can see the composite color, or only one or two flavors (colors) in the composite color if there are any constraints like filters or color-blindness.
A velocity comprised of two flavors - 5$\vec{i}$m/s and 4$\vec{j}$m/s, i.e. a velocity in the direction of (5$\vec{i}$ + 4$\vec{j}$)m/s is comprehensible to us as a single entity, and we do not see it as a composition of two individual velocities in real world. But, for a unidirectional being living on x-axis, only the 5$\vec{i}$m/s part of the velocity will be perspicuous.
The gist is that depending upon the context (or the reference frame) and our familiarity, a physical entity can make sense to us either in its own right, or in terms of its components (flavors and dimensions), or in both. If you have understood this correctly, tensors are going to be a piece of cake now.
Pixel Density in a Cube (an Analogy)
Before bringing in tensors, a last piece of analogy. Let's say we have cube that has LED pixels embedded all through its volume. We can create an entity called pixel density, or pixels per unit volume to get a rough idea of how bright or how luminous this cube could be at different points inside it. Let's denote it by $\chi$. We'll be using the term luminosity for brightness (which, in turn, will be equal to the number of pixels) - do not confuse it with the traditional definition of luminosity. In the video below, we are looking at a unit cube inside our pixel cube:
But, now we realize that since these pixels have to be connected with wires and some gluing mechanism, they do not emit light in all directions. In fact, each pixel can only illuminate the volume in front of it. So, our physical entity (pixels/m3) is not doing justice in understanding the brightness of different points of our cube. Because, at any moment, we can look at only one surface inside the cube, that too only from a specific direction. Depending upon how the pixels have been glued at any given location, the brightness in a specific direction can be different from any other.
What we have in reality is something like this:
Pixel Density By Area
So, let's create another entity - pixel density by area, or pixels per unit area. Here, unit area is supposed to have a direction that is perpendicular to the unit area in question, and the pixels counted will be the ones that emit light in that direction. Because the unit areas can be facing different directions, we'll need to have not one but three values for our new entity pixels-per-unit-area.
Let's assume these pixel densities are constant (they do not change at different points of the cube). And, the values to be $\chi_x = 150$, $\chi_y = 23$ and $\chi_x = 85$. So now, we can get the total luminosity ($L^{(s)}$) perpendicular to any surface ($\vec{s}$) within the cube:
L^{(s)}
\begin{bmatrix}150&23&85
Or, for a unit surface ($\vec{n}$):
L^{(n)}
\begin{bmatrix}n_x\\n_y\\n_z
The things to note here are:
The luminosity increases if the size of the area increases (it's obvious)
Luminosity per unit area at any point changes in value if ($\vec{n}$) changes direction
Let's get back to the physical entity we defined - pixels per unit area ($\chi$). Here, we have combined two entities - pixels (p) and area (s) - to generate a new entity. But, it so happens that one of our underlying entities - area - comes in three flavors - $s_x$, $s_y$ and $s_z$. So, in order to have any meaning of our entity pixels per unit area, we need to break it into three flavors - pixels per unit-area-perpendicular-to-the-x-direction, and so on. These individual flavors, in spite of being scalars by themselves, and not even having a direction, cannot be just added together to give the total pixels by unit area at a point!
This entity of ours - pixel density per unit area per direction- is neither a scalar, nor a vector. It has got flavors each of which has a magnitude but not a direction associated to it, and these flavors separately do make sense to us. So, we just write these flavors or components together and understand this thing to represent the total luminosity per unit area at a point (for all directions!). We call this combined thing a tensor. It is something that brings together different components of the same entity together without adding them together in a scalar or vector sense of addition. It exists by definition, but we can (mostly) make sense of only its individual components.
So, what we have seen here - $\begin{bmatrix}150&23&85\end{bmatrix}$, or more generally $\begin{bmatrix}\chi_x&\chi_y&\chi_z\end{bmatrix}$ - is a tensor that represents $\chi$ or pixel density per unit area per direction in our pixel-cube.
Now, let's upscale. Instead of having just one color pixels, let's have RGB pixels. What we are now going to have for $\chi$ is:
\chi
\begin{bmatrix}\chi_{xr}&\chi_{yr}&\chi_{zr}\\
\chi_{xg}&\chi_{yg}&\chi_{zg}\\
\chi_{xb}&\chi_{yb}&\chi_{zb}\\
Now, our total luminosity per unit area (for a given unit area $\vec{n}$) itself is going to have these three flavors - red, green and blue, and we can get it as:
\begin{bmatrix} L_x\\L_y\\L_z \end{bmatrix}^{(n)}
This total luminosity per unit area for a given unit area $\vec{n}$, $L^{(n)}$, can be understood either as the total brightness of all the three colors combined, or as just a convenient way of writing together the different luminosities (of its different colors together). If we were looking at it ourselves in real world, we'll see it more as the total brightness of the combined color (that is, a composite of the three colors or flavors).
And, each component of the tensor $\chi$ can be understood as the number of pixels of a specific color emitting light in a specific direction (x, y or z) at a given point.
The flavors or the components, may not always add or combine together at all, or at least, as seamlessly as colors do. One example where do they add together in a sense that we can interpret their addition in a physical way, is directions.
In the above example, just replace the total luminosity by force, the three colors by the three directions force can have, and the number of pixels by 9 proportionality constants, and we get the total force per unit area (for a given unit area) at a given point in a solid, which is nothing else but the traction vector. And, the 9 constants together, denote the stress tensor (see TL;DR 2 above)! Each of the 9 components of the stress tensor represents the force in a specific direction (x, y or z) for a given unit area facing a specific direction (x, y, or z). By having this information contained in it,the stress tensor can tell us the magnitude and net direction of the force vector (traction vector) given any surface vector in such a way that the direction of the traction vector is not constrained to be parallel or perpendicular to that of the surface area (which would happen in vector scaling or multiplication).
Though it's difficult to visualize the stress tensor in totality, each of its component tells us something very discrete, e.g. $\sigma_{xx}$ tell us how much force in x-direction is being experienced by a unit surface area that is perpendicular to the x-direction (at a given point in a solid). The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction, we get the traction vector, or, I do not mean literally though, the stress tensor collapses to the traction vector.
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