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.


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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  • $\begingroup$ +1 for the excellent referencing to related questions as well as related topics - as always $\endgroup$ – Danu Aug 7 '14 at 0:33

The Coulomb and Lorenz gauges are gauge fixing conditions, not gauge transformations. Your question still makes sense, but it should be phrased more like this: how can you prove that for arbitrary $\mathbf A_0$ and $V_0$ there always exists a gauge transformation to fields $\mathbf A$ and $V$ that satisfy these conditions? In fact many such transformations always exist (these conditions don't completely fix the gauge).

For the Coulomb gauge condition, we need $0 = \nabla\cdot\mathbf A = \nabla\cdot(\mathbf A_0 + \nabla\lambda) = \nabla\cdot\mathbf A_0 + \nabla^2 \lambda$. This is Poisson's equation, which can be solved for $\lambda$ with Green's functions. The Lorenz case is easier if you use four-vectors, in which case you get a 3+1 dimensional version of Poisson's equation.

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  • $\begingroup$ For the sake of completeness, could you expand your answer to cover the Lorenz gauge in a more comprehensive way? $\endgroup$ – Danu Aug 7 '14 at 0:28

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