Non-"00....0" components of charge density for a spin-s force field

[ergospherical]

It is given that the charge density of a particle of charge $q_0$, world line $z^{\mu}(\tau)$ (and 4-velocity $u^{\mu}$) in a spin-$s$ force field is a $s$-tensor\begin{align*}
T^{\mu \nu \dots \rho}(x^{\sigma}) = q_0 \int u^{\mu} u^{\nu} \dots u^{\rho} \delta^4[x^{\sigma} - z^{\sigma}(\tau)] d\tau
\end{align*}and that the "Coulomb" part of this charge density is $q \equiv \int T^{00\dots0} d^3 x$, which does make sense because if one works in the local frame of observer of 4-velocity $U^{\mu} = \delta^{\mu}_0$ then the measured charge within some region is\begin{align*}
q \equiv \int T^{00\dots0} d^3 x &= (-1)^s \int T^{\alpha \beta \dots \gamma} U_{\alpha} U_{\beta} \dots U_{\gamma} d^3 x \\
&= (-1)^s q_0 \int \dfrac{d\tau}{dt} (u^{\alpha} U_{\alpha}) (u^{\beta} U_{\beta}) \dots (u^{\gamma} U_{\gamma}) \delta^4[x^{\sigma} - z^{\sigma}(\tau)] d^4 x \\
&= q_0 \int \dfrac{d\tau}{dt} (-\mathbf{u} \cdot \mathbf{U})^s \delta^4[x^{\sigma} - z^{\sigma}(\tau)] d^4 x \\ \\
&= \int \dfrac{q_0}{\gamma^{1-s}} \delta^4[x^{\sigma} - z^{\sigma}(\tau)] d^4 x
\end{align*}I am curious as to what the interpretations of all of the other components of the tensorial charge density $T^{\mu \nu \dots \rho}$ are. Do they have physical meaning?

 

[Ambitwistor]

I would like to provide some insights into the interpretations of the components of the tensorial charge density $T^{\mu \nu \dots \rho}$.

Firstly, it is important to note that the tensorial charge density represents the distribution of charge in space and time, taking into account the 4-velocity of the charged particle. This means that each component of the tensor corresponds to a different aspect of the charge distribution.

The components with indices $\mu = \nu = \dots = \rho = 0$ represent the "Coulomb" part of the charge density, which is the overall charge within a given region. This is the component that is usually measured by an observer in the local frame of the particle.

The components with indices $\mu = \nu = \dots = \rho \neq 0$, on the other hand, represent the anisotropic distribution of charge in space. They describe how the charge is distributed in different directions, taking into account the 4-velocity of the particle. These components are important for understanding the electric and magnetic fields generated by the charged particle.

The components with indices $\mu = \nu = \dots = \rho = i$, where $i$ runs from 1 to 3, represent the electric current density of the charged particle. This is because these components take into account the velocity of the particle in each direction, and the flow of charge in these directions.

Overall, the different components of the tensorial charge density provide a comprehensive description of the charge distribution of the particle, taking into account its 4-velocity. They are all interconnected and play a role in understanding the behavior of the charged particle in a spin-$s$ force field.