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And how can we generalize this to the statement that it has $D-1$ independent components in dimension $D$?

I know that $F_{\mu \nu}$ has six independent components (because of antisymmetry), how do I go from there?

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First, I'll address a language issue: "independent components" can mean two different things:

  • The statement that $F_{\mu\nu}$ has six independent components is a statement about the number of parameters required to specify an antisymmetric matrix.

  • The statement that $\partial^\mu F_{\mu\nu}=0$ has three independent components is a statement about the number of functions required to specify a solution of this differential equation.

The question about the number of independent functions required to specify a solution is answered in https://physics.stackexchange.com/a/20072/206691, so this could potentially be marked as a duplicate question. That other answer uses a different notation, though, so I'll offer a version that uses the current OP's notation.

I'll write $j,k$ for spatial indices and $0$ for the time index. Suppose that we have some solution $F_{\mu\nu}$ to $$ \partial^\mu F_{\mu k}=0 \tag{1} $$ but that we have not made any attempt yet to enforce $$ \partial^\mu F_{\mu 0}=0. \tag{2} $$ Equation (2) is equivalent to $$ \partial^k F_{0k}=0. \tag{3} $$ Equation (3) is a constraint: it is a condition that the initial data must satisfy, rather than a condition on how that data evolves in time. Applying $\partial^k$ to equation (1) gives $$ \partial^\mu \partial^k F_{\mu k}=0. \tag{4} $$ In equation (4), the terms with $\mu\neq 0$ are identically zero because $F$ is antisymmetric, so equation (4) is equivalent to $$ \partial^0 \partial^k F_{0 k}=0. \tag{5} $$ In words, this says that if the functions $F_{\mu\nu}$ that we chose to satisfy equations (1) for all times are also chosen to satisfy the constraint (2) at some initial time, then they automatically satisfy the constraint (2) for all times. In this sense, a solution to $\partial^\mu F_{\mu\nu}=0$ in $D$-dimensional spacetime has only $D-1$ independent components.

Summary: Equation (1), which involves time-derivatives, has $D-1$ components (the number of values of the spatial index $k$). Equation (3) does not involve time-derivatives; it imposes a constraint on the initial data rather than an additional condition on how that data evolves in time.

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There are 3 dofs for the three spatial directions. Then there are two directions - time reversal symmetry. Taking polarisation into account one has 12 independent solutions, per frequency. That is a lot of dofs.

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