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How would the mathematics of electromagnetism and physical interpretation change if the Lorenz gauge used a difference instead of a sum?

In other words, what would this do: $$\frac{1}{c}\frac{\partial \phi}{\partial t} - \nabla \cdot A=0$$ Instead of this: $$\frac{1}{c}\frac{\partial \phi}{\partial t} + \nabla \cdot A=0$$

How would the mathematics of electromagnetism and physical interpretation change if the gauge used a difference instead of a sum?

In other words, what would this gauge do: $$\frac{1}{c}\frac{\partial \phi}{\partial t} - \nabla \cdot A=0$$ Instead of the Lorenz gauge: $$\frac{1}{c}\frac{\partial \phi}{\partial t} + \nabla \cdot A=0$$

How would the mathematics of electromagnetism and physical interpretation change if the Lorentz gauge used a difference instead of a sum?

In other words, what would this do: $$\frac{1}{c}\frac{\partial \phi}{\partial t} - \nabla \cdot A=0$$ Instead of this: $$\frac{1}{c}\frac{\partial \phi}{\partial t} + \nabla \cdot A=0$$

How would the mathematics of electromagnetism and physical interpretation change if the Lorenz gauge used a difference instead of a sum?

In other words, what would this do: $$\frac{1}{c}\frac{\partial \phi}{\partial t} - \nabla \cdot A=0$$ Instead of this: $$\frac{1}{c}\frac{\partial \phi}{\partial t} + \nabla \cdot A=0$$