These days I am interested in Holographic superconductor, but I do not clearly understand the effect and significance of probe approximation.
The proble limit (or probe approximation) is determined by the relative strengths of the electromagnetic and gravitational couplings. In the simplest case, the scalar field "\Psi" and vector potential "A" are renormalised by a factor of "1/q", of which charge "q" is then taken to be very large. Under such an approximation, it is considered that matter fields embedded within the Lagrangian density have no influence (or back-reaction) upon the metric. It may be helpful to think of this in terms of the Einstein-Maxwell equations - the left hand side (which tells us about the geometry) is affected by the right hand side (which contains all matter fields). Hope this helps!
You're very welcome! To answer your follow up, as soon one works outside the probe limit, we begin to see a new class of black hole solutions which will depend upon additional parameters related to your source terms.
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