Return to Question

I don't understand, how the $\vec{A}$ potential is defined in the covariant formulation of the Maxwell-Equations.

I am using the metric $(+,-,-,-)$.

The electric field (SI-Units) is defined as: $$ \vec{E}=-\vec{\nabla}\phi-\partial_t\vec{A}. $$ In index notation, I would write this as: $$ E^i/c=-(\vec{\nabla}A^0)^i-\partial_0A^i $$ And with the definition of the 4-Gradient: $$ E^i/c=-E_i/c=-\partial_iA_0-\partial_0A^i=\partial^iA^0-\partial^0A^i=F^{i0} $$ But this is exactly the opposite of the field tensor in Wikipedia: $$ E_i/c=F_{0i}=-F^{0i}=F^{i0} $$

What is going wrong here?

I don't understand, how the $\vec{A}$ potential is defined in the covariant formulation of the Maxwell-Equations.

I am using the metric $(+,-,-,-)$.

The electric field (SI-Units) is defined as: $$ \vec{E}=-\vec{\nabla}\phi-\partial_t\vec{A}.\tag{1} $$ In index notation, I would write this as: $$ E^i/c=-(\vec{\nabla}A^0)^i-\partial_0A^i.\tag{2} $$ And with the definition of the 4-Gradient: $$ E^i/c=-E_i/c=-\partial_iA_0-\partial_0A^i=\partial^iA^0-\partial^0A^i=F^{i0}.\tag{3} $$ But this is exactly the opposite of the field tensor in Wikipedia: $$ E_i/c=F_{0i}=-F^{0i}=F^{i0}.\tag{4} $$

What is going wrong here?

I don't understand, how the $\vec{A}$ potential is defined in the covariant formulation of the Maxwell-Equations.

I am using the metric $(+,-,-,-)$.

The electric field (SI-Units) is defined as: $$ \vec{E}=-\vec{\nabla}\phi-\partial_t\vec{A}. $$ In index notation, I would write this as: $$ E^i/c=-(\vec{\nabla}A^0)^i-\partial_0A^i $$ And with the definition of the 4-Gradient: $$ E^i/c=-E_i/c=-\partial_iA_0-\partial_0A^i=\partial^iA^0-\partial^0A^i=F^{i0} $$ But this is exactly the opposite of the field tensor in Wikipedia: $$ F^{\mu\nu}=\begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \\ \end{pmatrix} $$

What is going wrong here?

I don't understand, how the $\vec{A}$ potential is defined in the covariant formulation of the Maxwell-Equations.

I am using the metric $(+,-,-,-)$.

The electric field (SI-Units) is defined as: $$ \vec{E}=-\vec{\nabla}\phi-\partial_t\vec{A}. $$ In index notation, I would write this as: $$ E^i/c=-(\vec{\nabla}A^0)^i-\partial_0A^i $$ And with the definition of the 4-Gradient: $$ E^i/c=-E_i/c=-\partial_iA_0-\partial_0A^i=\partial^iA^0-\partial^0A^i=F^{i0} $$ But this is exactly the opposite of the field tensor in Wikipedia: $$ E_i/c=F_{0i}=-F^{0i}=F^{i0} $$

What is going wrong here?

I don't understand, how the $\vec{A}$ potential is defined in the covariant formulation of the Maxwell-Equations.

I am using the metric (+,-,-,-).

The electric field (SI-Units) is defined as: $$ \vec{E}=-\vec{\nabla}\phi-\partial_t\vec{A} $$ In index notation, I would write this as: $$ E^i/c=-(\vec{\nabla}A^0)^i-\partial_0A^i $$ And with the definition of the 4-Gradient: $$ E^i/c=-E_i/c=-\partial_iA_0-\partial_0A^i=\partial^iA^0-\partial^0A^i=F^{i0} $$ But this is exactly the opposite of the field tensor in Wikipedia: $$ F^{\mu\nu}=\begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \\ \end{pmatrix} $$

What is going wrong here? Thanks

I don't understand, how the $\vec{A}$ potential is defined in the covariant formulation of the Maxwell-Equations.

I am using the metric $(+,-,-,-)$.

The electric field (SI-Units) is defined as: $$ \vec{E}=-\vec{\nabla}\phi-\partial_t\vec{A}. $$ In index notation, I would write this as: $$ E^i/c=-(\vec{\nabla}A^0)^i-\partial_0A^i $$ And with the definition of the 4-Gradient: $$ E^i/c=-E_i/c=-\partial_iA_0-\partial_0A^i=\partial^iA^0-\partial^0A^i=F^{i0} $$ But this is exactly the opposite of the field tensor in Wikipedia: $$ F^{\mu\nu}=\begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \\ \end{pmatrix} $$

What is going wrong here?