# A question about emergent gravity

• A
• Heidi

#### Heidi

Hi Pf
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

Does it make sense to introduce a metric to deduce emergent gravity?
I think so. I think the idea is that given a spacetime manifold and Lorentzian metric ##\left(M,g\right)##, Einstein's equation (i.e., "the law of gravity") follows from coarse graining and something like thermodynamics arguments.

If there is a $g_{\mu \nu}$ field at the beginning this gives the geometry and the left side of the Einstein equation. this field gives the curvature, the geodesics. Verlinde says that gravity and space emerge but it seems that the g metrics is defined on space containing the screens. What am i missing in what he says?

Maybe the metrics is not on space time is not directly given. Velinde says that there is a screen at infinity with no gravitation (Minkowski's metrics). this screen is a 2 + 1 dimensional space time containing bits. Coarse graining will define a new screen "inside" the infinity screen. It is in this sense that he says that space emerges. Maybe between the infinity screen and the first coarse grined sceen the metrics minkowskian too.
and corse graining goes on.
Does Verlinde see things like that?

the problem is to understand the point of Verlinde.
One way would be to consider that there is One 2+1 dimensional screen containing bits of information in which there is no gravity. and nothing else not a boundary of something else. As it is minkowskian there is a Killing field on it representing how things stay on place. One can transform the information on it in a coarse graining procedure. we have less "cells" on it containg the bits. We can imagine that we have two different closed screens. the second being inside the first one with a smaller area. they would be the borders of a slice of space time containing a Killing field and a metric coming from the border.
Do you think that it is his point of view?

There is in GR no way to define a density so that integrated on space would give the total energy. But in his General Relativity book 1984 Wald write this:
However despite the absence of energy density, there is a notion of total energy of an isolated system in an asymptotically flat spacetime.
In special relativity with a particle we have $$E = -P_a \xi^a$$ where xi is a killing vector fill defining the frame work of orbits where things stay at place...

Here Verlinde uses the same strategy. The force applied to a test mass is the force that should be exerted on it to keep it "in place" along the Killing orbit

If there is a $g_{\mu \nu}$ field at the beginning this gives the geometry and the left side of the Einstein equation. this field gives the curvature, the geodesics. Verlinde says that gravity and space emerge but it seems that the g metrics is defined on space containing the screens. What am i missing in what he says?
Maybe it's similar to string theory. In string theory textbooks one often considers strings in curved backgrounds, and only later shows that these backgrounds can be "build as coherent states" by inserting vertex operators on the worldsheet.

As we have F dx = T dS and a metric on the screen , can we mix this dx to the 2+1 metric to get a metric in the bulk?

There is something that i am not sure to understand in Verlinde's paper. It is the aim of this paper.
Is it to derive the physics law of gravitation by using geometry + thermodynamics identifications. In this case there is no problem to introduce the metrics on space time and to consider screens embedding bulks on space time with given temperature and so on.
Or else is it to say that these laws can be obtained from a 2+1 screen at infinity on which a coarse graining is done?
In this case the metrics would have to emerge "between" the first screen and its coarse grained ones seen as inside the non grained ones.
It seems that he navigates unclearly between both

• bdrobin519
Well would there be thermal energy in a isolated singularity? E.g. black holes.

the main objects are the screens not what is in the bulks.