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Fra

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**Logic of E-H action, ricci scalar, cosmological constant??**

This crazy thread is mean to stimulate some reflections on the logic of Einsteins Equations. It would be interesting if those who have any ideas can join. Maybe it could be enlightning?

The common way of thinking about GR is that we have the geodesic equations describing what "straight lines" are, and what are the trajectories of test particles - this has a simple geometric interpretation. Ie. the idea that gravity is curved spacetime. This is the simple part.

And we have Einsteins equation describing the feedback of progression into geometry itself. Ie more tricky part to interpret is the dynamics of geometry itself. If anyone have any favourite interpretations of this, let it out.

I am trying to see from my point of view, the logic of reasoning implicit in Einsteins Equations, and it seems the E-H action is one simple place to start.

The EH action in vacuum in compact is

[tex]S_{EH} = k \int R dV - 2k \int \Lambda dV[/tex]

Einsteins expected the cosmological constant to be zero, but now we expect that it's almost zero but not quite. S is an arbitrary measure chosen as -ln(P/Pmax) which measures the probability of our future expectations beeing right (in a special sense).

Now there is a simple comparasion between a choice of measure of probability of probability when applied to a simple case where the retained history can defines a probability distribution of the future, and the question asked here is, the coupling between history and expectations on the future. There is a simple but interesting analogy to EH action here.

[tex]

\textbf{S} = M S_{KL} - ln(w/P_{max})

[/tex]

Here S_KL is the kullback-leibler information divergence, M is the "sample size" of history - corresponding the observers memory capacity. Pmax is a maximum expected "fuzzy probability" of correctly predicting the future. w is a weight factor, that approaces 1 as M -> infinity.

Note here the logic by which it is a correct "expectation" that ln(w/Pmax) -> 0 in the continuum limit, but incompletness suggest that it's not exactly zero.

See https://www.physicsforums.com/showthread.php?t=238501 for some more notes.

Since this is reflections only, the details need refinements another time but the interesting reflection I make here is that a certain type of statistical reasoning suggest that the

1) ricci scalar can be interpreted as a sort of "density of information divergence"

2) The cosmological constant seems to associate to the fact that we have discrete information! In the continuum limit, and a infinitely massive observer one would expect a zero constant, but a finite observer can never conclude a zero value. Rather one would expect it to be "almost zero".

3) In a certain sense (to be clarified) the EH action can be thought of as the "action of the action", and this is immediately realized to be nothing but an induction step.

This is very fuzzy and needs refinement. I'm currently trying to analyse the meaning of this. It does seem that Einsteins equation could be interpreted (alot of details missing though), rightfully as an "expectation" almost in line with that we "expect" the entropy of the universe to always increase, but the fact that we don't know with infinite confidence, suggest first of all the non-zero, but close to zero lambda, but even worse, the whole equation might possibly be just one step in a certain direction (guided by a certain logic of reasoning).

The comparasion here is the statistical interpretation of the second law. Ie. the total entropy (whatever that is - another question) does not have to increase, rather there i a high probability that it will increase (by construction, but as above the details and notions of probabiltiy is unclear - this is the basis of the discretisation of it I argued for in the other thread)

I think that no one that has been thining about this could avoid having some kind of personal associations or "interpretations" of einsteins equations.

What other possible logic do you see behind GR?

I hope this thread isn't too fuzzy for this section. The purpose is not to discuss withing the contact of GR, it's to discussed a possible larger context in which GR is seem to emerge as natural. I feel a need to gain a deeper conceptual understanding of the bits and pieces we are trying to merge in QG, and maybe if others share the same need, we can elaborate the conceptual basis for GR. Of course if someone has all the answers explicitly worked out in a stringent formalisms already, even better. But since I suspect no one has that, the reflections are bound to be somewhat fuzzy :)

/Fredrik