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)

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