The field equation are:
That is the result from last time.
the conservation of charge is:
thus the 4-Laplacian of the F-field is :
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 :
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:
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
that the force is created by the field and the current.
The Electromagnetic stress tensor can also be related with the 4-force by :
Thus, combined with the Field tensor: