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Maxwell's equations in vacuum are symmetric bar the problem with units that you have identified. In SI units $$\nabla \cdot {\bf E} = 0\ \ \ \ \ \ \nabla \cdot {\bf B} =0$$ $$\nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}\ \ \ \ \ \ \nabla \times {\bf B} = \mu_0 \epsilon_0 \frac{\partial {\bf E}}{\partial t}$$ If we let $\mu_0=1$, ...

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In Gaussian units, we set $\epsilon_0 = \frac1{4\pi}$ (and so $\mu_0 = \frac{4\pi}{c^2}$) and change the units of $B$ so both electric and magnetic fields have the same dimension. In these units, Maxwell's equations are as follows: \begin{align} \nabla \cdot \mathbf{E} &= 4\pi \rho \\ \nabla \times \mathbf{E} &= - \frac1{c} \frac{\partial ...

2

AdS black holes exist in various dimensions, $p=3$ is not the only choice. The parameter can take on values above or below $3$. One famous example is the three-dimensional BTZ black hole, and higher dimensional ones are also frequently used in the correspondence. Furthermore, I think there is a misunderstanding on the concept of a "gravity dual". The metric ...

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