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I am reading the origin of gravity written by Erik Verlinde

I have the same problem with a paper written by Jacobson on the same subject.

They want to deduce gravity from the holographic principle and thermodynamics.

I suppose that they cannot talk about geodesic at the beginning. the word metrics cannot be found in Verlinde's paper. But Verlinde and Jacobson use Killing vectors and Killing horizon.

In wikipedia Killing vectors are defined as tangent to a Riemannian manifold. so there is a metric , norms, scalar product and so on. A Killing horizon contains null Killing vectors.

Does it make sense to introduce a metric to deduce emergent gravity?

thanks