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Phrak
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And if so, How?
From the post 15 and 16 of the thread https://www.physicsforums.com/showthread.php?t=474719"
But total charge and total current, Q and I, do form a 4-vector, don't they? There seem to be two ways to solve this, but I can't figure out which one is right.
Properly speaking, charge density in 3 space is a pseudo scalar and current density is a pseudo vector.
[tex]\rho = \rho_{ijk}\ dx^i dx^j dx^k \ [/tex]
[tex]j = j_{ij}\ dx^i dx^j \ [/tex] *
To each of these, there is a corresponding dual.
[tex]\hat{\rho_n}= \epsilon^{ijk}\rho_{ijk}[/tex]
[tex]\hat{j}_i = {\epsilon_i}^{jk} j_{jk}[/tex]
Where j can be though of as current flux density, [itex]\hat{j}[/itex] might be called the "current strength" or "current intensity". It would be nice to call [itex]\rho[/itex] a current flux, but is doesn't sound very good in 3 dimensions of space unlike the case in spacetime.
Raising the index on j and combining to a 4-vector,
[tex](\hat{\rho}, \hat{j}^i )[/tex]
could form a proper Lorentz invariant 4-vector. Integrating over a 4-volume could then yield (Q, I), give or take a negative sign.
The second way would be the inverse sequence of operations above. Integrate rho and j over a 4-volume then raise the index on the spatial components.
* To be precise, these should be a directed volume and a directed area,
[tex]\rho = \rho_{ijk} \ dx^i \wedge dx^j \wedge dx^k[/tex]
[tex]j = j_{ij} \ dx^i \wedge dx^j \ .[/tex]
From the post 15 and 16 of the thread https://www.physicsforums.com/showthread.php?t=474719"
DaleSpam said:yes. Charge is the timelike component of the four-current. So it is relative the same way that the components of any four-vector is.
bcrowell said:Wait, you mean the charge *density*, right? Charge is a Lorentz scalar; this is verified to extremely high precision because the hydrogen atom is electrically neutral.
Properly speaking, charge density in 3 space is a pseudo scalar and current density is a pseudo vector.
[tex]\rho = \rho_{ijk}\ dx^i dx^j dx^k \ [/tex]
[tex]j = j_{ij}\ dx^i dx^j \ [/tex] *
To each of these, there is a corresponding dual.
[tex]\hat{\rho_n}= \epsilon^{ijk}\rho_{ijk}[/tex]
[tex]\hat{j}_i = {\epsilon_i}^{jk} j_{jk}[/tex]
Where j can be though of as current flux density, [itex]\hat{j}[/itex] might be called the "current strength" or "current intensity". It would be nice to call [itex]\rho[/itex] a current flux, but is doesn't sound very good in 3 dimensions of space unlike the case in spacetime.
Raising the index on j and combining to a 4-vector,
[tex](\hat{\rho}, \hat{j}^i )[/tex]
could form a proper Lorentz invariant 4-vector. Integrating over a 4-volume could then yield (Q, I), give or take a negative sign.
The second way would be the inverse sequence of operations above. Integrate rho and j over a 4-volume then raise the index on the spatial components.
* To be precise, these should be a directed volume and a directed area,
[tex]\rho = \rho_{ijk} \ dx^i \wedge dx^j \wedge dx^k[/tex]
[tex]j = j_{ij} \ dx^i \wedge dx^j \ .[/tex]
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