- #1

ergospherical

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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?

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?

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