I'm working my way through Griffith's Introduction to Electrodynamics. In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric potentials $V_0$, we can create a new set of equivalent magnetic and electric potentials given by:
$$ \textbf{A} = \textbf{A}_0 + \nabla\lambda \\ V = V_0 - \frac{\partial \lambda}{\partial t}. $$
These transformations are defined as a "gauge transformation". The author then introduces two of these transformations, the Coulomb and Lorenz gauge, defined respectively as:
$$ \nabla \cdot \textbf{A} = 0 \\ \nabla \cdot \textbf{A}= -\mu_0\epsilon_0\frac{\partial V}{\partial t}. $$
This is where I am confused. I do not understand how picking the divergence of $\textbf{A}$ to be either of these two values actually constitutes a gauge transformation, as in it meets the conditions of the top two equations. How do we know that such a $\lambda$ even exists for setting the divergence of $\textbf{A}$ to either of these values. Can someone convince me that such a function exists for either transformation, or somehow show me that these transformations are indeed "gauge transformations" as they are defined above.
Answer
Comment to the question (v1): It seems OP is conflating, on one hand, a gauge transformation
$$ \tilde{A}_{\mu} ~=~ A_{\mu} +d_{\mu}\Lambda $$
with, on the other hand, a gauge-fixing condition, i.e. choosing a gauge, such e.g., Lorenz gauge, Coulomb gauge, axial gauge, temporal gauge, etc.
A gauge transformation can e.g. go between two gauge-fixing conditions. More generally, gauge transformations run along gauge orbits. Ideally a gauge-fixing condition intersects all gauge orbits exactly once.
Mathematically, depending on the topology of spacetime, it is often a non-trivial issue whether such a gauge-fixing condition is globally well-defined and uniquely specifies the gauge-field, cf. e.g. the Gribov problem. Existence and uniqueness of solutions to gauge-fixing conditions is the topic of several Phys.SE posts, see e.g. this and this Phys.SE posts.
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