Possible link between the electromagnetic field and gravitation- Was Nikola Tesla onto something?

The main philosophical question that is at the root of the fluid model presented here is the following: what is the ontological meaning of the four potential \mathbf{A}? James Clerk Maxwell used the concept of four-potential in his original formulation of Maxwell’s equation (later reformulated by Heaviside in their canonical vector form). However, to this day there is no conclusive agreement on what the entity “really” is. As the magnetic field is assumed to be divergence-free, it is a fact of vector calculus that there exists a vector potential that generates the magnetic field. But this is merely a mathematical theorem, it has no ontological added value in it. To me, the whole idea of describing fluids of space-time was to interpret the four-potential as a velocity field of the fluid. This is an ontological approach which helps me to find analogies and if one wants to interpret is as a potential instead of velocity, by all means do so. But the power of analogy is very useful when one tries to understand nature. So my working hypothesis is still (despite of the fact that the scientific community does not generally accept it) that the four-potential is to be interpreted as a velocity of an aether in 4-dimensional space-time.

As I explained earlier, the most important properties of fluids are the divergence, vorticity and viscosity (which are interrelated). I have already shown that the divergence of the fluid is directly linked to the Lorentz-gauge and that the vorticity of the fluid is essentially the electromagnetic field tensor. What occurred to me recently was the concept of “curvature”. Now curvature is important, as the general relativity describes that mass and energy alter the curvature of the space (i.e. the metric tensor differs from the Kronecker delta). Curvature is a nice heuristic concept more generally as well, because for example we can describe the flow of heat using the heat equation based on the curvature of the temperature field. Similarly, electrostatics and Newtonian gravity are based essentially on Poisson’s equation where charge or mass generate curvature on the potential. So I started thinking about curvature of the velocity field of aether.

So what is curvature? As a first approximation we can understand it through the second derivative or the Laplacian. In differential geometry we have concepts like the Riemann tensor, Ricci tensor and so forth. However, as I want to keep things as simple as possible, I start of with the Laplacian approach, which actually takes me quite far as you shall see. Let us then consider a 3-dimensional space and take the following vector calculus identity

\nabla \times \nabla \times \mathbf{A}=\nabla (\nabla \cdot \mathbf{A})-\nabla\cdot \nabla\mathbf{A}

This is a beautiful identity as it binds together the “curvature” and divergence of the vector field, which together manifest themselves as the curl of the curl of the vector field. What I realized, was that this actually means that the curl of the curl is basically a commutator, i.e. it measures how much the divergence and gradient do not commute locally. Not to mention that this is actually quite close to the definition of the Riemann tensor!

Now I shall the construct a curvature concept for the aether fluid. This is straightforward, but it is rather complicated in 4-dimensions, because the curl operator has to be generalized as I’ve done in my earlier posts. Nevertheless, the end result is a rank 3 tensor, which has 64 components.

I began my interest in this curl of the curl concept also because of some experiments that have to do with rotating magnetic fields. Nikola Tesla was one of the pioneers in the field. Apparently rotating magnetic fields can produce quite interesting effects which have to do with gravity. See for example the following article by the European Space Agency


Obviously it is also quite interesting that the rotation of the magnetic field happens to be linked to the curl of curl of the vector potential! So curvature, rotating magnetic fields and gravitation are all linked together in some mysterious way.

Let us then find what is the “curvature” of the “aether” field. What we already have is the curl of the velocity field, which I obtained using the sort of exterior/Grassmann algebra:

\nabla^*\times \mathbf{A}=\begin{bmatrix} 0&-\frac{i}{c}\frac{\partial \phi}{\partial x}-\frac{i}{c}\frac{\partial A_1}{\partial t}&-\frac{i}{c}\frac{\partial \phi}{\partial y}-\frac{i}{c}\frac{\partial A_2}{\partial t}&-\frac{i}{c}\frac{\partial \phi}{\partial z}-\frac{i}{c}\frac{\partial A_3}{\partial t}\\ \frac{i}{c}\frac{\partial A_1}{\partial t}+\frac{i}{c}\frac{\partial \phi}{\partial x}&0&\frac{\partial A_1}{\partial y}-\frac{\partial A_2}{\partial x}&\frac{\partial A_1}{\partial z}-\frac{\partial A_3}{\partial x}\\ \frac{i}{c}\frac{\partial A_2}{\partial t}+\frac{i}{c}\frac{\partial \phi}{\partial y}&\frac{\partial A_2}{\partial x}-\frac{\partial A_1}{\partial y}&0&\frac{\partial A_2}{\partial z}-\frac{\partial A_3}{\partial y}\\ \frac{i}{c}\frac{\partial A_3}{\partial t}+\frac{i}{c}\frac{\partial \phi}{\partial z}&\frac{\partial A_3}{\partial x}-\frac{\partial A_1}{\partial z}&\frac{\partial A_3}{\partial y}-\frac{\partial A_2}{\partial z}&0\\ \end{bmatrix}

We can state this in a more simple notation using the symbols for electric and magnetic field

\mathbf{F}=\nabla^*\times \mathbf{A}=\begin{bmatrix}0 & iE_x & iE_y & iE_z\\ -iE_x & 0 & -B_z & B_y \\ -iE_y & B_z & 0 &-B_x \\ -iE_z & -B_y & B_x & 0\\ \end{bmatrix}

How to then apply the curl to a second rank tensor? We can do it by proceeding to each column vector separately. First we take the column vector and form a gradient tensor out of it. The first column vector is

\mathbf{F}^1=\begin{bmatrix} 0 \\ -iE_x \\ -iE_y \\ -iE_z \\ \end{bmatrix}

So that the gradient tensor is

\nabla ^* \mathbf{F}^1=\begin{bmatrix} 0 & 0 &0 &0 \\ \frac{1}{c}\frac{\partial E_x}{\partial t} & -i\frac{\partial E_x}{\partial x} & -i\frac{\partial E_x}{\partial y} & -i\frac{\partial E_x}{\partial z} \\ \frac{1}{c}\frac{\partial E_y}{\partial t} & -i\frac{\partial E_y}{\partial x} & -i\frac{\partial E_y}{\partial y} & -i\frac{\partial E_y}{\partial z} \\ \frac{1}{c}\frac{\partial E_z}{\partial t} & -i\frac{\partial E_z}{\partial x} & -i\frac{\partial E_z}{\partial y} & -i\frac{\partial E_z}{\partial z} \\ \end{bmatrix}

One should note btw that the diagonal contains the divergence of the electric field.

In similar manner, one has the second column vector,

\mathbf{F}^2=\begin{bmatrix} iE_x \\ 0 \\ B_z \\ -B_y \\ \end{bmatrix}

What one needs to do then is to take the transpose of the each gradient tensors and take the difference of the gradient tensor and its transpose. The result is a rank 3 tensor which contains just so much information that it needs another posting…

Gauge invariance, Lorentz factors and Elasticity of Spacetime

In this piece of writing, I will try to analyze further the main features of my main equation. I will talk about gauge invariance and the importance of imaginary time as well, and the link to the Lorentz factor. Finally, I present a specific form for the general equation, which interprets potential energy as elasticity of space-time.

The main equation is

\mathbf{H}=\frac{1}{2}\rho \left( \mathbf{A}\otimes \mathbf{A}\right) +\nu \left( \nabla^*\mathbf{A}+\nabla^* \mathbf{A}^T\right) +\mu \left(\nabla^*\mathbf{A}+\nabla^* \mathbf{A}^T\right)^2 + p\mathbf{I}

What sort of Gauge invariance do we have?

As the total energy of the system is conserved, the divergence of the tensor \mathbf{H} should disappear, that is

\nabla^* \cdot \mathbf{H}=0

This in turn means that when one considers the actual dynamic equation, we can have any such total energy tensor, whose divergence vanishes. This is the gauge freedom in the model. By taking the divergence of the main equation, one ends up with fairly complicated nonlinear system of partial differential equations. But the main point is to understand that the total energy tensor must be divergence free. This will allow us to choose conveniently that


which is probably the most trivial choice, but I tend to believe that nature works in the most simple ways as is possible. So then our main equation reads

\frac{1}{2}\rho \left( \mathbf{A}\otimes \mathbf{A}\right) +\nu \left( \nabla^*\mathbf{A}+\nabla^* \mathbf{A}^T\right) +\mu \left(\nabla^*\mathbf{A}+\nabla^* \mathbf{A}^T\right)^2 + p\mathbf{I}=0

This actually means that the generalized Lagrangian is zero as well. As the Lagrangian of a system is the difference of the kinetic and potential energies, we actually have in this gauge that kinetic energy equals the negative of potential energy (the sign is not important). This is in turn interesting as it is somehow related to systems where time averages and ergodicity are considered. Think of a simple pendulum without friction. The pendulum divides its  total energy between kinetic and potential energies, but the time average converges to the gauge where kinetic and potential energies are equal (halfway in terms of y-cooordinate and halfway in terms of x-coordinate).

The role of imaginary time

It is worth mentioning that the imaginary time is crucial in the model. I already showed how the inner product of the space-time event  vector  corresponds to the correct relativistic measure. Now we should also consider proper velocities of the space-time event vector. Remember that the space-time event vector is defined as

\vec{r}=\begin{bmatrix} -ict \\ x \\ y \\ z \\ \end{bmatrix}

Now let us define the proper velocity as

\vec{V}=\frac{\partial \vec{r}}{\partial (-ict)}

this is then

\vec{V}=\begin{bmatrix} 1 \\ \frac{i}{c}\frac{\partial x}{\partial t} \\ \frac{i}{c}\frac{\partial y}{\partial t} \\ \frac{i}{c}\frac{\partial z}{\partial t} \\ \end{bmatrix}

Then we can consider the length of this proper velocity, which is a velocity in space-time:

\sqrt{\vec{V}\cdot \vec{V}}=\sqrt{1-\frac{\vec{v}\cdot \vec{v}}{c^2}}

Now this shows that with the use of imaginary time, we can infer naturally the Lorentz factor from special relativity; so that

\sqrt{\vec{V}\cdot \vec{V}}=\frac{1}{\gamma}

It is important to realize that this space-time velocity is a “proper” one, as the partial differentiation is taken with respect to -ict instead of t. Notice that I have used capital letters for space-time and ordinary letter for space-velocity.

What about the Green-Lagrange Strain tensor?

The last thing is this posting I wanna talk about is the role of the stress tensor. Now we have used for the time being the very general form, where the stress tensor is quadratic in the symmetric tensor. That might be indeed the most general case, but for intuitive reasons I want to discuss the more specific stress tensor of the form

\mathbf{T}=\nu \left( \nabla^*\mathbf{A}+\nabla^*\mathbf{A}^T +\nabla^*\mathbf{A}\nabla^*\mathbf{A}^T \right)

In the field of continuum mechanics, in ordinary space without the time-component this tensor is called the Green-Lagrange strain tensor, which provides the strain of elastic bodies. Think of a tennis ball which is squeezed, the amount of squeeze or strain is potential energy stored in the tennis ball. So my analogy is that maybe there is a similar elasticity of space-time (it can be indeed the same as in General relativity, but I don’t know that, so I just proceed with my intuition ).

Now if we indeed proceed with this assumption, the tensor equation reads

\frac{1}{2}\rho \left ( \mathbf{A}\otimes \mathbf{A}\right ) +\nu \left( \nabla^*\mathbf{A}+\nabla^*\mathbf{A}^T +\nabla^*\mathbf{A}\nabla^*\mathbf{A}^T \right) +p\mathbf{I}=0

The equation can be read as :

“The total kinetic energy of the system plus the potential energy stored in the elasticity of the space-time must add up to zero”

On vorticity and divergence of the space-time fluid

In my first posting I proposed the following general equation for the space-time fluid which just states that the total energy of the fluid at a point in space-time is the sum of “kinetic energy” and “potential energy”.

\mathbf{H}=\frac{1}{2}\rho \left( \mathbf{A}\otimes \mathbf{A}\right) +\nu \left( \nabla^*\mathbf{A}+\nabla^* \mathbf{A}^T\right) +\mu \left(\nabla^*\mathbf{A}+\nabla^* \mathbf{A}^T\right)^2 + p\mathbf{I}

Now, there are two main features of this fluid which are of main interest:

1) The vorticity of the space-time fluid

2) The compressibility or divergence of the space-time fluid

The vorticity of the space-time fluid

Let us analyze the vorticity. In 3 dimensions the vorticity of the fluid is just the curl of velocity field

\vec{\omega}=\nabla \times \vec{v}

In 4 dimensions the cross product does not work in the same sense. We can, however, define an entity which is analogous to curl in 4 dimensions:

\nabla^* \times \mathbf{A}=\nabla^*\mathbf{A}-\nabla^*\mathbf{A}^T

This entity is a skew-symmetric matrix. Note that if the Jacobian of the fluid is a symmetric matrix, the curl will vanish. The direct analogy to three dimensions is that the matrix entries are just the components of the curl. We can actually write the matrix explicitly. First consider the Jacobian of the field:

\nabla^*\mathbf{A}=\begin{bmatrix} \frac{1}{c^2}\frac{\partial \phi}{\partial t}&-\frac{i}{c}\frac{\partial \phi}{\partial x}&-\frac{i}{c}\frac{\partial \phi}{\partial y}&-\frac{i}{c}\frac{\partial \phi}{\partial z}\\ \frac{i}{c}\frac{\partial A_1}{\partial t}&\frac{\partial A_1}{\partial x}&\frac{\partial A_1}{\partial y}&\frac{\partial A_1}{\partial z}\\ \frac{i}{c}\frac{\partial A_2}{\partial t}&\frac{\partial A_2}{\partial x}&\frac{\partial A_2}{\partial y}&\frac{\partial A_2}{\partial z}\\ \frac{i}{c}\frac{\partial A_3}{\partial t}&\frac{\partial A_3}{\partial x}&\frac{\partial A_3}{\partial y}&\frac{\partial A_3}{\partial z}\\ \end{bmatrix}

This is a completely normal Jacobian, one must just bear in mind that we are in 4 dimensions and that the nabla operator involves the speed of light and the imaginary unit. The time-like component of the four-velocity is also imaginary. Now, the transpose of the Jacobian is straightforward. If we take the difference, we thus obtain the curl for our space-time fluid:

\nabla^*\times \mathbf{A}=\begin{bmatrix} 0&-\frac{i}{c}\frac{\partial \phi}{\partial x}-\frac{i}{c}\frac{\partial A_1}{\partial t}&-\frac{i}{c}\frac{\partial \phi}{\partial y}-\frac{i}{c}\frac{\partial A_2}{\partial t}&-\frac{i}{c}\frac{\partial \phi}{\partial z}-\frac{i}{c}\frac{\partial A_3}{\partial t}\\ \frac{i}{c}\frac{\partial A_1}{\partial t}+\frac{i}{c}\frac{\partial \phi}{\partial x}&0&\frac{\partial A_1}{\partial y}-\frac{\partial A_2}{\partial x}&\frac{\partial A_1}{\partial z}-\frac{\partial A_3}{\partial x}\\ \frac{i}{c}\frac{\partial A_2}{\partial t}+\frac{i}{c}\frac{\partial \phi}{\partial y}&\frac{\partial A_2}{\partial x}-\frac{\partial A_1}{\partial y}&0&\frac{\partial A_2}{\partial z}-\frac{\partial A_3}{\partial y}\\ \frac{i}{c}\frac{\partial A_3}{\partial t}+\frac{i}{c}\frac{\partial \phi}{\partial z}&\frac{\partial A_3}{\partial x}-\frac{\partial A_1}{\partial z}&\frac{\partial A_3}{\partial y}-\frac{\partial A_2}{\partial z}&0\\ \end{bmatrix}

Now this curl of the fluid is a matrix, which has 6 independent components. It is actually the transpose of the electromagnetic field tensor, with the electric field rotated in the complex plane:

\mathbf{F}=\begin{bmatrix} 0&-\frac{E_x}{c}&-\frac{E_y}{c}&-\frac{E_z}{c}\\ \frac{E_x}{c}&0&-B_z&B_y\\ \frac{E_y}{c}&B_z&0&-B_x\\ \frac{E_z}{c}&-B_y&B_x&0\\ \end{bmatrix}

With the canonical definitions for Electric field and Magnetic field respectively:

\vec{E}=-\frac{\partial \vec{A}}{\partial t}-\nabla \phi

\vec{B}=\nabla \times \vec{A}

The divergence of the space-time fluid

The divergence of the fluid is somewhat simpler compared to the vorticity, we have:

\nabla^*\cdot \mathbf{A}=\frac{1}{c^2}\frac{\partial \phi}{\partial t}+ \nabla \cdot \vec{A}

which is quite nice, because then an incompressible space-time fluid is actually equivalent to fixing a Lorentz gauge. 

Some speculation on the origin of mass and possible consequences

We should, however stop to think for a while what the divergence would mean in our space-time fluid. Analogically, if a normal fluid is compressible, we say that the density is not equal everywhere. We then talk about the mass density of the fluid. 

In the case of space-time fluid, the fluid transfers energy and therefore we could guess that the divergence of the fluid field is related to mass somehow, as after all, mass is compressed energy. So we could postulate that the divergence is related to the mass density by

\frac{1}{c^2}\frac{\partial \phi}{\partial t}+ \nabla \cdot \vec{A}=\kappa\rho(t,x,y,z)

Introduction to the fluids of space-time

Pierre-Simon Laplace was one of the earliest thinkers who had considered celestial forces as a result of a fluid flow of some sort. He has written the following :

“If gravitation be produced by the impulse of a fluid directed towards the centre of the attracting body, the preceding analysis… will give the secular equation depending on the successive transmission of the attractive force.”.

This blog is about trying to understand various classical and perhaps quantum phenomena found in nature by utilizing a fluid flow analogy. To be more exact , in modern terms, I’m thinking of an analogy stemming from the field of continuum mechanics, and various phenomena therein.

I will show how electricity can be seen as an internal friction in the space-time fluid flow. This is work in progress, so I do not know yet what I will end up with.

Most of the physical laws are related somehow to the idea of conservation of things. For example, one has the conservation of linear momentum, conservation of angular momentum and so forth. Of course we also have the conservation of energy principle, which is probably the most intuitive in a prosaic sense. Usually these mathematical statements then produce in one way or another some mathematical equations that represent e.g. Newtons’s laws of motion.

The other very popular approach is the dynamic optimization paradigm, where one sees the mother nature as an optimal controller of some sort, where typically some functional is to be minimized over a path. The optimality conditions (Euler-Lagrange equations) then usually resemble Newton’s laws of motion again. Analogically, one has for example in the field of economics some behavioral laws in the same spirit.

This all is physics from the 19th century, before Einstein and before the quantum revolution. Personally I find two ontological issues with 20th century physics: the first one is my lack of understanding what quantum physics really is. In other words, I fail to find a reasonable explanation why energy and momentum can be represented through differential operators in Hilbert space. The second problem I have is that I basically do not think that General Relativity is ontologically correct. I do not like the idea of curved space.

From these rather unpleasant feelings, I began to think of some new ways to describe some rather basic concepts in physics. I became fascinated with fluids, as the mathematics of fluids is extremely complicated and therefore allow for rich behavior. First and foremost: fluid mechanics is non-linear. This makes things hard.

In my opinion, fluids have three main features that are especially important

1) The vorticity of the fluid (curls and eddies, spirals)

2) The compressibility of fluids

3) The viscosity properties of fluids (friction, like water vs. oil)

So I started to think about fluids of space-time. Moreover, even though I do not like the ideas of General Relativity, I think that the incorporation of time in the coordinate space is essential. To have a nice theory or a model, we must have solid assumptions. My assumption of the form of space-time is the following (Poincaré, Minkowski): the coordinate vector is

\vec{r}=\begin{bmatrix}  -ict\\  x \\  y \\  z\\  \end{bmatrix}

So that we have the three usual spatial coordinates and one imaginary time-component. This inclusion of the imaginary unit is nice, because we can see directly that inner product has the correct form

\vec{r}\cdot \vec{r}=-c^2t^2+ x^2 +y^2 +z^2

This inclusion of the imaginary unit gives also rise directly to the following differential operator (just think of partials with respect to space-time coordinates):

\nabla^* =\left(\frac{i}{c}\frac{\partial}{\partial t}, \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right )

So. Great. We have a 4D nabla operator and a space. Let’s play around. We need some more ingredients, though. Drawing ideas from the potential representation of Maxwell’s equations, let us define a 4-potential:

\mathbf{A}=\begin{bmatrix}  -\frac{i}{c}\phi\\  A_1 \\  A_2 \\  A_3\\  \end{bmatrix}

The potentials can take values in the complex field, but for now it is okay to think of them as functions from space and time to the real numbers.

We now make a brief incursion to the ideas of conservation of energy. The total energy of a system (called the Hamiltonian) is composed of the potential energy of the system and of the kinetic energy of the system. Energy is usually thought as a real number like “3 kilo-watt-hours”. I’m extending the concept of energy into matrix values. So potential energy is a matrix and kinetic energy is a matrix. The total energy is therefore simply the sum of these matrices. In symbols:


The matrices are 4 times 4 square matrices. Moreover, I require them to be symmetric. Symmetry is beautiful and symmetric matrices have lots of nice properties (like real eigenvalues). Moreover, symmetry of the matrices is related to the conservation of angular momentum.

So we have the most basic equation at hand. Statement of total energy. We need to add some structure in order to build a meaningful model.

Let us start by thinking of kinetic energy. Usually it is something involving velocity squared. So we need velocities. Here is the crucial assumption: let us think of the 4-potential as a 4-velocity field. The natural way to induce a square matrix from the 4-velocity is to take the outer product of the “velocities” as follows:

\mathbf{K}=\frac{1}{2}\rho \begin{bmatrix}  -\frac{\phi \phi}{c^2} & -\frac{i}{c}\phi A_1 &-\frac{i}{c}\phi A_2 &-\frac{i}{c}\phi A_3\\  -i\phi A_1 & A_1 A_1 &A_1 A_2&A_1 A_3\\  -i\phi A_2 & A_2 A_1 &A_2 A_2&A_2 A_3\\  -i\phi A_3 & A_3 A_1 &A_3 A_2&A_3 A_3\\  \end{bmatrix}

So this is the kinetic energy matrix of our fluid in space-time. In a similar manner, we need to define the potential energy matrix. This is trickier,  as we need a symmetric matrix which meaningfully represents potential energy. Well, we can steal some ideas from continuum mechanics. In continuum mechanics, the potential is related to pressure and frictional forces, which can be summarized in the so-called Cauchy Stress Tensor :


where the so-called deviatoric (first term on the right side) part is defined as

\begin{bmatrix}  \sigma_{00} & \sigma_{01}& \sigma_{02}&\sigma_{03}\\  \sigma_{10} & \sigma_{11}& \sigma_{12}&\sigma_{13}\\  \sigma_{20} & \sigma_{21}& \sigma_{22}&\sigma_{23}\\  \sigma_{30} & \sigma_{31}& \sigma_{32}&\sigma_{33}\\  \end{bmatrix}

The second term is just pressure function times an identity matrix.

Now, obviously at this time, the stress tensor is just a collection of symbols.

The most general form which retains symmetry and has dependence on the space-time velocity gradient is the following:

\mathbb{T}=\nu \left(\nabla^* \mathbf{A}+\nabla^* \mathbf{A}^T\right) +\mu(\nabla ^* \mathbf{A}+\nabla^*\mathbf{A}^T)^2

We can however add more structure by assuming that the frictional forces are linear in the gradient of the velocity field. This constitutes a Newtonian fluid. One can think of this as two layers of fluid moving next to each other, the friction is produced by the velocity differences, think of rubbing your hands together.

This Newtonian fluid assumption can be stated compactly by saying

\mathbb{T}=\nu \left(\nabla^* \mathbf{A}+\nabla^* \mathbf{A}^T\right)

The matrix is obviously symmetric, as if you add the transpose, this does the trick.

So in the end we have the most general equation

\mathbf{H}=\frac{1}{2}\rho \mathbf{A}\otimes \mathbf{A}+\nu \left(\nabla^* \mathbf{A}+\nabla^* \mathbf{A}^T\right)+\mu(\nabla ^* \mathbf{A}+\nabla^*\mathbf{A}^T)^2+p\mathbf{I}

This equation will conclude my first posting on this topic. In the next post I will show how electricity is the friction in this space-time fluid.