The Horizon

The high frequency quantum fluctuation of the location of the D3-branes are invisible to a low frequency probe. Roughly speaking this is insured by the renormalization group as applied to the SYM description of the branes. The renormalization group is what insures that our bodies are not severely damaged by constant exposure to high frequency vacuum fluctuations. We are not protected in the same way from classical fluctuations. An example is the thermal fluctuations of fields at high temperature. All probes sense thermal fluctuations of the brane locations. We use the thermal field fluctuations of φ. For each of the N² components the thermal fluctuations have the form

 (1.1)

and

 (1.2)

It is clear that the thermal fluctuations will be strongly felt out to a coordinate distance z = T_sym. In terms of r the corresponding position is

1 − r ∼ 1/T_sym    (1.3)

In fact this coincides with the location of the horizon of the AdS black hole.