In this core chapter, the one-loop effective action for Matrix theory on 3 + 1 dimensional branes is elaborated, and the Einstein–Hilbert term is obtained in the presence of fuzzy extra dimensions. Some justification for the stability of the background is given.

In this epilogue we place the theories of semiclassical and stochastic gravity in perspective, exploring their linkage with quantum gravity, defined as theories for the microscopic structures of spacetime, not necessarily and most likely not from quantizing general relativity. We distinguish two categorical approaches, ‘top-down’ (Planck energy) and ‘bottom-up’. The tasks of the ‘top-down’ approach, which include string theory and other proposed theories for the microstructures of spacetime, lie in explaining how the micro-constituents give rise to macroscopic structure. They are thus more appropriately called emergent gravity. warnings are issued not to blindly follow the dogma that quantizing general relativity naturally yields a microscopic structure of spacetime, or to accept, without checking the emergent mechanisms, the dictum that some micro-constituent is the theory that gives us everything. Stochastic gravity takes the more conservative ‘bottom-up’ approach. For the linkage with quantum gravity we mention (a) the kinetic theory approach, relying on the structure of a correlation hierarchy and the role played by noise and fluctuations, and (b) the effective theory approach, using large N techniques. The ingredients of both approaches have been developed in earlier chapters systematically. We end with a description of the advantages and limitations of stochastic gravity.