Electromagnetic field tensor components

Keep $\eta=\operatorname{diag}(1,-1,-1,-1)$ and $x^0=ct$. Define the contravariant potential $A^\mu=(\phi/c,\mathbf A)$, so $A_\mu=(\phi/c,-\mathbf A)$. Here phi is the electric potential in volts and the spatial vector potential has units of tesla metres. Its time component has the same units. The tensor is $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$. Remember that the spatial components of the raised derivative are negative; omitting this sign reverses the identification of the electric field.

For example, its time-space component is $F^{0i}=c^{-1}\partial_t A^i+c^{-1}\partial_i\phi=-E_i/c$, where $A^i$ is the ith spatial component of the contravariant potential. To avoid ambiguity we use bold symbols for spatial vectors and reserve spacetime subscripts for metric lowering. The purely spatial components involve the curl of the vector potential. Antisymmetry is automatic because exchanging the two derivative indices changes the sign; every diagonal entry therefore vanishes.

Lower both indices by multiplying the tensor by the metric on both slots. A time-space entry picks up one spatial minus sign, whereas a space-space entry picks up two. Consequently $F_{0i}=+E_i/c$ and $F_{ij}=F^{ij}$. Raising just one index gives yet another matrix. The component array is therefore meaningless without its index positions, metric signature, coordinate ordering, and factors of c. Two books can print different arrays and describe identical physics if their conventions differ consistently.

The field tensor is unchanged by a smooth gauge transformation $A^\mu\mapsto A^\mu+\partial^\mu\chi$, since the extra mixed second derivatives commute. With our signs this means $\phi\mapsto\phi+\partial_t\chi$ and $\mathbf A\mapsto\mathbf A-\nabla\chi$. The frequently used opposite sign for the gauge function simply renames chi. Gauge changes relate different potentials within one frame; Lorentz transformations relate different observers' components. They are distinct operations even though both can alter a displayed potential.

A reliable sign audit uses a positive test charge at rest. Its lowered four-velocity is $(c,0,0,0)$, so $qF^{i\nu}u_\nu=qE_i$. The force points along the electric field, as it must. For motion along positive x in a magnetic field along positive z, the ordinary magnetic force points along negative y. Our array gives $F^{21}=B_z$ and $u_1=-\gamma v_x$, reproducing that negative sign. This physical audit catches errors that antisymmetry alone cannot detect.

The tensor represents the field at one event; it does not include the test charge's velocity. Magnetic force appears only when the tensor contracts with a moving particle's four-velocity. This separation explains why a field tensor and a force four-vector are different kinds of object. Maxwell's equations constrain how these local field values vary across spacetime, but are additional dynamical statements. Constructing a valid antisymmetric array at an event does not by itself prove that an arbitrary field assignment solves those equations.