On Space Time And The Fabric Of Nature Part 2

Introduction to the "field" concept and the modelling thereof

To the Mathematician, 2 apples multiplied by 6 oranges is 12.
To the Engineer, the multiplication of apples and oranges makes no sense at all.

The above statement is a good illustration of the difference between Mathematics and Physics. In Physics, Mathematics is a fantastic tool. It allows us to make highly accurate predictions and formal descriptions of the processes we observe to take place in Nature. In a way, Mathematics is a very powerful language, because the symbols it uses and defines, have an enormous expressive power. Perhaps, above all, it is this expressive power, formalized in a language, which makes it so very useful in Physics.

It is this same expressive power, for example, which makes Python such a powerful programming language. One of the reasons Python is such an expressive programming language is because in a way "everything goes anywhere". You can add "strings" and "numbers" in Python, for example, which is a bit like adding apples and oranges in other languages. The disadvantage of such a language, however, is that sometimes errors occur when a program is being run with unexpected inputs.

The same argument can be made about Mathematics. Mathematics doesn't care if you are trying to multiply apples and oranges, even though the obtained result has no meaning in the "real world". So, what makes Mathematics such a powerful tool and language, is that because it defines all kinds of abstract concepts, relationships and calculation methods in a formal symbolic language, it has an enormous expressiveness. Expressiveness, which can be used to accurately describe all kinds of processes and systems and to solve all kinds of problems associated with these. However, it does not give a "real world" meaning to these abstract concepts it studies and describes.

So, it is not up to Mathematicians, but up to Physicists and Engineers to make sure that the Mathematics they use actually produce meaningful results. This is illustrated by Albert Einstein, who wrote in "The Evolution of Physics" (1938) (co-written with Leopold Infeld):

"Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory. The ideas must later take the mathematical form of a quantitative theory, to make possible the comparison with experiment."

As we saw earlier, Freeman Dyson illustrated how Maxwell´s field model, which was founded upon well described and understood Newtonian principles, gradually evolved from a meaningful concept into a pure abstract concept, whereby eventually all connection to the "real world" has been lost:

Maxwell's theory becomes simple and intelligible only when you give up thinking in terms of mechanical models. [...] Fields are an abstract concept, far removed from the familiar world of things and forces.

In his paper "A Foundation for the Unification of Physics"(1996) Paul Stowe described this as follows:

Many of apparent inconsistencies that exist in our current understanding of physics have results from a basic lack of understanding of what are called fields. These fields, electric, magnetic, gravitational...etc, have been the nemesis of physicists since the birth of modern science, and continues unresolved by quantum mechanics. A classical example of this is the problem of an electron interacting with it's own field. This case results in the equations of quantum mechanics diverging to infinity. To overcome this problem, Bethe introduced the process of ignoring the higher order terms that result from taking these equations to their limit of zero distance, in what is now a common practice called renormalization.
These field problems result in a class of entities called virtual, existing only to balance and explain interactions. These entities can (and do) violate accepted physical laws. This is deemed acceptable since they are assumed to exist temporarily at time intervals shorter than the Heisenberg's uncertainty limit. It has been known for some time that such virtual entities necessitate the existence of energy in this virtual realm (Field), giving rise to the concept of quantum zero point energy.
As a result of this presentation I will propose the elimination of both the need for renormalization and any such virtual fields. This will be accomplished by replacing the virtual field with a real physical media within which we define elemental particles (which more precisely should be called structures) and the resultant forces which act between them.

Currently, the field concept has departed so much from Maxwell's down to Earth origins, that there is little room left to distinguish the modern field concept from pseudo-science, if we are to follow Karl Popper's definition:

"In the mid-20th century, Karl Popper put forth the criterion of falsifiability to distinguish science from nonscience. Falsifiability means a result can be disproved. For example, a statement such as "God created the universe" may be true or false, but no tests can be devised that could prove it either way; it simply lies outside the reach of science. Popper used astrology and psychoanalysis as examples of pseudoscience and Einstein's theory of relativity as an example of science."

With the current definitions of, for example, the electric-, magnetic- and gravitational fields, there are both units of measurement as well as instruments with which one can measure the strength of these fields. In hindsight, we might argue that these units of measurements are defined somewhat arbitrarily, but they are measurable nonetheless and thus "make possible the comparison with experiment".

But what about "virtual particles", "dark matter", "weak nuclear forces", "strong nuclear forces" and even "10 to 26-dimensional string theories"? Isn't there a strong sceptic argument to be made here that, at the very least, these kinds of unmeasurable concepts are bordering on the edge of pseudo-science?

Either way, even though we are well aware of the wave-particle duality principle and should have concluded that therefore there can be only One fundamental force and therefore only one field, a plethora of fields have been defined, none of which has brought us any closer to the secret of the "old one." So, if there can be only one fundamental physical field of force, what is it's Nature? How do we use this wonderful Mathematical and abstract concept of a "field" and use it in a physically meaningful way?

Let us simply go back to something that stood the test of time: the original foundation Maxwell's equations were based upon, which is to postulate the existence of a real, physical fluid-like medium wherein the same kind of flows, waves, and vortex phenomena occur as which we observe to occur in, for example, the air and waters all around us. We shall do just that and then work out the math, using nothing but "classic" Newtonian physics, meanwhile making sure that the mathematical concepts we use have a precisely defined physical meaning and produces results with well defined units of measurement.

And since it is the field concept which has made a life of it's own, let us first consider what we mean by a physical field of force and define it's units of measurement, so that we can clearly distinguish a physical field of force from the more general mathematical abstract field concept we use to describe our physical field. That way, we can do all kinds of meaningful calculations, predictions and experimental verifications. And just like 2*6=12 has no meaning in and of itself in Physics, the abstract mathematical field concept has no meaning in and of itself in Physics. To sum this up:

In a way, Physics is the art of using abstract Mathematical concepts in a way that is meaningful for describing and predicting the Physical phenomena we observe in Nature.

In practice, that comes down to a book-keeping exercise. All we really need to to is to keep track of which mathematical concept we use to describe what. For example, if we use the abstract field concept to describe something we call a physical field of force, we should unambiguously associate a unit of measurement to the abstract mathematical concept used. This way, we have clearly defined what the abstract concept means within a certain context. In Software Engineering, this is what's called type checking:

In programming languages, a type system is a collection of rules that assign a property called type to various constructs a computer program consists of, such as variables, expressions, functions or modules. The main purpose of a type system is to reduce possibilities for bugs in computer programs by defining interfaces between different parts of a computer program, and then checking that the parts have been connected in a consistent way.

Just read "unit of measurement" for "type" and we are talking about the exact same concept.

Introduction to vector calculus and fields of force

"The partial differential equation came into theoretical physics as a servant, but little by little it took on the role of master." - Albert Einstein (1931)

While this statement by Albert Einstein may or may not ring true to you, many people will ask the question: "What does it mean?" Since it is precisely that question that concerns us when we want to define what we mean by a "physical field of force", let us consider this statement a little bit further, because we will be using partial differential equations to describe our model, although we will make use of the expressiveness vector calculus offers us in order to keep things understandable and to express the concepts we are considering in a meaningful way.

As an illustration of the expressive power of vector calculus, let us consider the definition of "divergence":

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field F = Ui + Vj + Wk is defined as the scalar-valued function:

{$$ \operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (U,V,W) = \frac{\partial U}{\partial x} +\frac{\partial V}{\partial y} +\frac{\partial W}{\partial z}. $$}

At the left, we have the notation in words, followed by a notation using the $\nabla$ operator ($\nabla$ is the Greek letter "nabla"), while at the right, we have the same concept expressed in partial differential notation. So, when we use this $\nabla$ operator in equations, we are actually using partial differential equations. However, with the notation we will use, we can concentrate on the physical meaning of the equations rather than to distract and confuse ourselves with the trivial details.

So, what Einstein actually said was something like that the concepts we will use, too, gradually morphed from being useful and meaningful tools into essentially taking all branches of physics hostage. As early as 1931, Einstein already recognised that it was no longer reasoning and fundamental ideas that guided scientific progress, but rather a number of abstract concepts which drifted ever further away from having any physical meaning at all, a destructive process which still continues this very day and age.

Now let us briefly introduce the main mathematical concepts we will use: divergence, curl, gradient, some "identities" and the Laplace operator:

divergence:

In vector calculus, divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.

As an intuitive explanation, one can say that the divergence describes something like the rate at which a gas, fluid or solid "thing" is expanding or contracting. When we have expansion, we have an out-going flow, while with contraction we have an inward flow. As an analogy, consider blowing up a balloon. When you blow it up, it expands and thus we have a positive divergence. When you leave air out, it contracts and thus we have a negative divergence.

It can be both denoted as "{$ div $}" and by using the "nabla operator" as "{$ \nabla \cdot $}".

In fluid dynamics, divergence is a measurement of compression. And therefore, by definition, for an incompressible medium or vector field, the divergence is zero, like for example with the magnetic field, which is called Gauss's law for magnetism:

{$$ \nabla \cdot \mathbf{B} = 0 $$}

curl:

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point.
The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid.

[...]

The alternative terminology rotor or rotational and alternative notations {$ rot \, \mathbf{F} $} and {$ \nabla \times \mathbf{F} $} are often used (the former especially in many European countries).

[...]

Intuitive interpretation
Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the centre of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.

Let us note that, unlike the divergence, the curl as both a length and a direction, which means that it gives you a vector, while the gradient gives you a single number, which is called a scalar.

gradient:

In mathematics, the gradient is a generalization of the usual concept of derivative to functions of several variables. [...] Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction.

The gradient concept is very similar to that of Grade or slope:

The grade (also called slope, incline, gradient, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the gradient in calculus where zero indicates gravitational level. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance and rise is the vertical distance.

Intuitively, the gradient gives you the direction and size of the biggest change of a function. In the mountain analogy, the gradient points in the direction a ball put on a mountain surface would start rolling. The steeper the surface, the bigger the gradient.

Let us note, that unlike the divergence which takes a vector and gives you a scalar value, the gradient takes a scalar and gives you a vector. So, the gradient and the divergence are complementary to one another.

physical field of force:

The three concepts just introduced (divergence, gradient and curl), are what is mathematically called derivatives:

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced.

When working in 1 dimension, the derivative is intuitively very similar to the gradient concept, as can be seen from this WikiPedia picture, which also illustrates the mountain analogy used to describe the gradient:

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

So, the mathematical concept of derivative says something about the "thing" it's the derivative of. In 1 dimension, 1D, this is always the rate of change of the function the derivative is calculated of.

Now this concept can be applied multiple times. For example, like the derivative of the position of a vehicle gives it's speed, the derivative thereof on it's turn gives you the rate of change of speed, which is called "acceleration". And since it's the second derivative of position, it's called the second derivative, or the second order derivative:

In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing with respect to time. In Leibniz notation:

{$$ \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\boldsymbol{x}}{dt^2}, $$}

where the last term is the second derivative expression.

Now let us consider Newton's second law of motion:

The second law states that the rate of change of momentum of a body, is directly proportional to the force applied and this change in momentum takes place in the direction of the applied force.

{$$ \mathbf{F_N} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \frac{\mathrm{d}(m\mathbf v)}{\mathrm{d}t} $$}

The second law can also be stated in terms of an object's acceleration. Since Newton's second law is only valid for constant-mass systems, it can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,

{$$ \mathbf{F_N} = m\,\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = m\mathbf{a}, $$}

where FN is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it.

From this, we can make an intuitive, first explanation for what a physical field of force actually is:

A physical field of force is the 3D version of the 1D concept of acceleration.