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What is the four-vector related to electric and magnetic dipole moment?

If complex numbers are allowed, then there always exists a transformation matrix between two metric tensors. So I guess complex numbers are not allowed. Could you explain me why?

The components of transformation matrix can be complex number, can't they? Here, I am looking for the transformation matrix between two coordinate systems while the components of metric tensor are given for the two coordinate systems.

Is that true in general and why: $$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$

Complex numbers are not allowed? Why?

So, there are ##\frac{N+N^2}{2}## equations and ##N^2## unknowns (the ##N^2## components of ##\frac{\partial \chi^i}{\partial x^j}##). As ##N^2>\frac{N+N^2}{2}##, there will be infinite number of solutions.

Aren't there ##N^2## unknowns as the transformation matrix ##\frac{\partial \chi^i}{\partial x^j}## has ##N^2## components in ##N##-dimension?

Sorry for the mistake. Is it necessary to know the transformation matrix? The metric tensor defines the geometry. Isn't it sufficient just to know the components of the metric tensor? Moreover, if I know the components of metric tensor in the two different coordinate systems, I can actually...

Is this method correct?

Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...

I am looking for a surface in 'space'.

Does there exist any 2D surface whose metric tensor is, ##\eta_{\mu\nu}= \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}##

Does it matter?

And in these inertial frames, the wire will always be negatively or positively charged (when any extra charge is not added to the system). The net charge will be conserved in a particular reference frame, but may vary from one frame to another. As, there is no absolute reference frame, we...

Thanks for your reply. At first, it seemed very obvious to me. Later on, when I thought whether the distance should be same or not and why, I couldn't reason it out. Would you please give me some clue?

Consider an infinite wire that has no electric current initially. Then current starts to flow in the wire, i.e. the free electron drifts at speed ##v## (and the positive charges are fixed) Applying special relativity, it appears that the distance between the electrons shrinks, i.e, density of...

Is there any straight forward way to prove these equations by using special relativity ?

Are the accelaration and forces in different inertial referance frame equal ?