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

• I
• ergospherical
In summary, the components of the tensorial charge density have physical interpretations related to the distribution of charge, electric and magnetic fields, and electric current of the charged particle.
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?

Last edited:

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.

## 1. What are "Non-"00....0" components of charge density for a spin-s force field?

The "Non-"00....0" components of charge density refer to the non-zero values of the charge density in a spin-s force field. These components are important in understanding the distribution of charge within a system and can impact the overall behavior of the system.

## 2. How are "Non-"00....0" components of charge density calculated?

The calculation of "Non-"00....0" components of charge density involves using mathematical equations and models to determine the distribution of charge within a system. This can include factors such as the distance between charges, the strength of the charges, and the orientation of the charges.

## 3. What role do "Non-"00....0" components of charge density play in spin-s force fields?

The "Non-"00....0" components of charge density are important in spin-s force fields as they can influence the strength and direction of the force between particles. They can also impact the stability and behavior of the system as a whole.

## 4. How do "Non-"00....0" components of charge density differ from "00....0" components?

The main difference between "Non-"00....0" components and "00....0" components of charge density is that the former represents non-zero values while the latter represents zero values. This means that "Non-"00....0" components have a significant impact on the overall charge distribution and force in a system, while "00....0" components do not contribute to the charge density.

## 5. Can "Non-"00....0" components of charge density change over time?

Yes, "Non-"00....0" components of charge density can change over time as a system evolves and particles move. The distribution of charge within a system can also be affected by external factors such as temperature or pressure, leading to changes in the "Non-"00....0" components of charge density.

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