From my own point of view I look upon the curvature of space-time as being caused by
gravity - in the rubber sheet analogy for space-time, gravitational force carriers (if they exist)would bend the sheet.Since Einstein opposed quantum mechanics because of its acausal nature, I should think this would be his point of view.Saying "gravity is space-time curvature" sounds to me how a quantum mechanist would interpret reality.
Also a mass placed half way between two other masses would experience no net
space-time curvature but identifying curvature with gravity we would conclude that
there was no gravity present when there is - two gravitational forces acting in opposite directions.

All spacetime curvature is a result of gravitational field. But not all gravitational fields have spacetime curvarture. But the point is that, using your analogy, the rubber sheet demonstrates tidal acceleration whereas the presence of a gravitational field requires only gravitational acceleration.

That is incorrect. At the half way point there is spacetime curvature. There is just no gravitational acceleration/force. In this case it appears that you're confusing gravitational force with gravitational tidal force. They are quite different.

The gravitational potential of such a system, i.e, two point particles, is given by

I hope other people will come here to discuss the paper. I give here another comment raised by reading it :

Doubtlessly geodesics must pertain to any geometrization of spacetime. In modern texts, geodesics are associated to Killing vector fields, which generate groups of isometries of spacetime, thus being the most natural concept in mathematics to study the symmetries. In turn, Killing vector fields become anavoidable when rigourously discussing the causal structure of spacetime. I think nobody would question the relevance of Killing vector fields to study spacetime. But we also know that symmetries studied this way are then classified according to the curvature. Don't you find that this strongly suggest to introduce curvature as the fondamental concept in the theory, and not only make use of related objects such as the connexion or the Christoffel symbol ? Besides, you did not yet comment the fact that, the most elegant formulation of gravitation, in purely geometrical terms, requires the introduction of the curvature too : the lagrangian formulation requires only the integral of the curvature, nothing else. Is there an alternative formulation exhibiting such simplicity ? Please note that I am aware this excludes non-trivial topological objects such as a cosmic string. Non-trivial topological object are what I have earlier been vaguely refering to as pathological situation.

In general, the term "gravitational field" is so vague, I would avoid it when talking about the subject in any but the most general way. If you're talking about the Riemann curvature tensor, say that that's what you're talking about. If you're talking about the Christoffel symbols, then use that less ambiguous name.

I was going to quote MTW on the gravitational field, but I see Pete has already done that in his paper.

"Nowhere has a precise defintion of the term gravitational field been given, nor will one be given. Many different mathematical entities are associated with gravitation...."

So I would basically say that talking about "the" gravitational field is just too vague to be useful. There's another section in MTW that comes to mind, but because the index of this otherwise great book **** ###, I don't think I can find it easily. It points out that charged objects behave differently than uncharged objects, so there is a very clear idea of what the electric field is at any given point by comparing the behavior of a charged particle at that point to an uncharged one. Because *everything* responds to gravity, similar attempts to define the gravitational field acting on a point mass fail, due to the lack of a "neutral" particle that does not respond to gravity.

I'm a bit curious as to the source of your (humanio's) confusion. I have the feeling that perhaps you (humanio) are reading more into Pete's paper than is actually there. Arguing about whether or not one should call a shovel a shovel, or a "field portable entrenching tool", doesn't seem to me to be something that should be likely to cause a significant amount of uncertanity or angst.

Yes pervect, I realize you might be right. I discovered Pete's article yesterday. Amusingly, I bought Wald's book on GR (Chicago Un. Pr.) two days ago, in order to re-read GR introduction course from a more modern point of view. Besides, I am also currently discovering LQG, mainly through Rovelli's recent big book. Now, Rovelli argues in his book that, contrary to the generally accepted idea, the gravitational field ought better be represented by the vierbein (=tetrad=orthonormal basis formalism) than the usual [tex]g_{\mu\nu}[/tex]. When I first read Rovelli's argument a few weeks ago now, it was a shock to me. Indeed, I remember when I discovered GR through Einstein texts, I felt the orthonormal basis choice was simply ommited in his texts. I was of course absolutely anaware at that time that fermions require the vierbein. But yet, when reading Rovelli's argumentention I had the feeling that eventually somebody had strong arguments to motivate this choice of an orthonormal basis formalism. Rovelli had almost convinced me. Wald clearly states that the vierbein formalism is sometime best suited, but that generally speaking, there is no better choice between Einstein's choice, what I refer for convenience here as Rovelli's choice, and Penrose's choice. Penrose formalism relies as you know on twistors, and are better suited everytime there is a prefered null-direction.

So, when I read Pete's article yesterday, I understood that all those elements had to be reconsidered in view of the fact that curvature is not always solely due to gravitation. I realize now that I might have been worrying too much. Pete certainly has an important message here, but yet the essentials of the 3 approach (Einstein vs Rovelli vs Penrose = [tex]g_{\mu\nu}[/tex] vs veirbein vs twistors) remain valid.

If my last post is not total non-sens, I guess I can apologize to Pete : I had a "violent" reaction when reading your paper (which I did not even notice YOU were the author) and maybe misunderstand your message. I thought you were questionning the appareance of curvature itself, as is sometime argued to be non-physical. I think now your argument is a subtle one. If you are saying that physicists have rapidely jumped to the conclusion that they understood all Einstein's theory just because the could perform the technical calculations, and thus they might have forgotten to pay enough attention to the concepts and ideas of GR, then we agree. I think I made the same mistake too, and you helped me here to keep doing physics on safer grounds. That's alright, I'm still young

So Pete, please accept my appologies, and thanks for providing insights into GR. I am here to learn, and I am glad to contemplate that indeed, PF helps me learn physics.

I'm currently just trying to learn the "simple" (and somewhat modern) version of GR, rather than trying to unite it with quantum mechanics, so I've been focused on the geometric approach. BTW, I'm not extremely familiar with Newton-Penrose formalism, though I gather I'm making use of it when I use the schwb metric with GRT-ii (which I do find _extremely_ handy).

There was one paper I skimmed that I think you might find interesting, though - I heard about it in a thread somewhere-or-other here on physics forum, the abstract is below.

This is a totally non-geometric way to approach GR. I'll let the paper speak for itself, especially since I've only skimmed it - you'll have to make your own evaluation. But since you appear to be looking for new ways to approach GR, this seems to have an approach that is not "on your list".

I am not familiar with Newman-Penrose formalism either. I have the two-volumes spinors&spacetime, I read almost only the beginning of volume 1. Thank you very much for the reference, it looks quite interesting. I'll "eat" it right now.

I am supposed to work on QCD for my PhD. Don't tell my adviser

______________________________
EDIT : I'm aware the Newton-Penrose was just a typo. Just in case somebody around googles "Newton-Penrose formalism"

Heh, I'm so unfamiliar, I didn't even notice the typo. But I checked the spelling, that is in fact what GRTensor-II uses for the "basis/object tetrad library" (Newman-Penrose).

I think this is an accurate text on the development of certain concepts in GR. I totally agree with his statement that the metric and the Christoffel symbols should be compared to the gravitational potenial and field respectively. This is in agreement with the weak field approximation, and it makes it easy to compare Einsteins field equations with the Poisson equations of the Newtonian field. This because the Einstein tensor has derivatives to the christoffel symbols, and these contain derivatives to the metric. So just as the classical field equation contain second derivatives to the potential Einsteins equations contain decond derivatives to the metric.

Also the article made clear to me why both the principle of equivalence and statements like 'gravitation is equal to curvature of spacetime' can be true. Keeping in mind the different definitions of a gravitational field in Einsteins and modern view of GR.