# What are “local degrees of freedom in gravity”, and why do they lead to fixed energy densities?

I am reading Jan de Boer's review of the AdS/CFT correspondence and I quote from end of page 1, where he is talking about equivalence of $(d+1)$-dimensional gravity to $d$-dimensional field theory

“If true, it implies [...]. If the degrees of freedom in gravity would be local, one would imagine that one can have arbitrarily large volumes with fixed energy density.[...]”

I don't quite understand that. What does it mean for “degrees of freedom to be local”? And how does that lead to fixed energy-density?

To discuss energy, let's take a simple example where we have a field sitting at the minimum of the potential. Then the energy associated with a single lattice point is determined only by the potential energy $V(\phi_{min})$. Here we see that the energy density is fixed over the whole spacetime, in the sense that the energy associated with each lattice point is the same. If there are $N$ points on the lattice, then the total energy of the whole lattice is $NV(\phi_{min})$. The energy scales like the number of spacetime points, which is the volume.