F_{ij} =\begin {pmatrix} 0 & D_1 & D_2 & D_ 3\\ -D_1 & 0& H_3 & -H_2 \\ -D_2 & - H_3 & 0 & H_1 \\ -D_3 & H_2 & -H_1 & 0 \end {pmatrix}

G_{ij} =\begin {pmatrix} 0 & B_1 & B_2 & B_ 3\\ -B_1 & 0& -E_3 & E_2 \\ -B_2 & E_3 & 0 & -E_1 \\ -B_3 & -E_2 & E_1 & 0 \end {pmatrix}

The field equation are:

\partial /\partial x_i F_{i j } = - J_i

\partial /\partial x_i G_{i j } =0

That is the result from last time.

the conservation of charge is:

\partial/\partial x_i J_i = 0

thus the 4-Laplacian of the F-field is :

\partial^2/\partial x_i^2 F_{ij} = 0

The physical meaning of the simplification of the field equation by the field tensor is, a gradient in the tensor field is equation to the minus of 4-current, or zero. recall that the gradient in 3-D vector space, the conservation of charge density is :

\nabla \cdot \vec{J} = - \partial \rho/\partial t

we have the same form in the 4-D tensor space. the creation of field is conservation of the 4-charge displacement, if we integrate the 4-current. i dun know what physical meaning of the G-field. personally, i believe that the F-field and G-field can be related by some transform.

we have another interesting things. we can write the Lorentz force into the field tensor:

\vec{f}_j = \frac{d\vec{P}_j}{d\tau}=q \frac{d\vec{X}_j}{d\tau}F_{ij}

the reason why we can write this, i don’t know. any physical meaning? i don’t know. may be we can think in this way, the force depends on the motion of the 4-vector and the field and the charge. thus, it is natural to multiple them together to get the force. But why not the G field? never the less, the field tensor reduce the number of Field qualities into 2.

the Lorents Force can be more simple

\vec{f}_j = J_i F_{ij}

that the force is created by the field and the current.

The Electromagnetic stress tensor can also be related with the 4-force by :

\vec{f}_j = - \partial/\partial x_i T_{ij}

Thus, combined with the Field tensor:

\partial/\partial x_iT_{ij}+ J_iF_{ij} = 0

 

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