Still no second part of the spectral action post. But instead a litte puzzle that came up over coffee yesterday: What is the Kaluza-Klein description of a black hole?
To be more explicit: Take pure gravity on R^4xK for compact K and imagine that it is large (some parsec in diameter say). Then you could imagine you have something that looks like a blackhole in this total space-time. What is its four dimenional description in KK theory?
With KK theory I mean the 4d theory with an infinite number of fields. I want to include the whole KK tower. This theory should be equivalent to the higher dimensional one since both are related by a (generalised) Fourier transform on K. One might be worried that a black hole is so singular that this Fourier transform has problems, does not converge or something. But if that is your worry, take a black hole that is not eternal but one that is formed by the collision of graviational waves say. In the past, those waves came in from infinity and if you siufficiently go back in time all fields are weak. This weak field configuration should have no problem being described in KK language and then the evolution is done in the 4d perspective. What would the 4d observer see when the black hole forms in higher dimensions?
The question I would be most interested in is if there is always a black hole in terms of the 4d gravity or if the 4d gravity can remain weak and the action can be entirely in the other fields.
One scenario I could imagine is as follows: The 4d theory has besides the metric some gauge fields and some dilatons. If the black hole is well localised in K then many higher Fourier modes of K will participate. From the 4d perspective, the KK momentum is the charge under the gauge fields and the unit is dependent on the dilaton. So could it be that there is a gauge theory black hole, i.e. a charged configuration that is confined to small region of space time where the coupling is strong with all the causality implications of black holes in gravity?