# How did “no prior geometry” father 50 years of confusion?

I've come across this quote attributed to Misner, Thorne & Wheeler from their book, Gravitation:

Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

What was the confusion?

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You ask about the history. Perhaps the easiest answer is to refer you to the long reference in Wikipedia, which will give you more than fifty pages of explanation:

One short answer is that “general covariance” and “no prior geometry” are two separate things, but that was not realized at first. At least general covariance has a very clear mathematical definition. But it was not what Einstein wanted or needed. After it was realized that something else, or something additional was needed, it took a while for that something else to be clearly defined. Of course, the something else turned out to be what MTW call “no prior geometry”. Nowadays it is more often called “background free” or “background independence”. It is still controversial, particularly as to how strong it should be and as to how it applies or does not apply to string theory and loop quantum gravity.

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I didn't know that!:) Thanks for the very clear answer and reference. – John McVirgo Feb 3 '11 at 16:22

I don't know specifically what the confusion they're thinking of was, having not been around during that half-century, but I can easily make some educated guesses.

By "prior geometry," I believe what MTW mean is any aspect of the geometry that is externally fixed or non-dynamical. So any additional constraints on the mathematical structure other than $G = 8 \pi T$ or the underlying structure (metric, connections, etc) really fall under this.

The first obvious case is Mach's principle, which contains speculation about the correct notion of physically meaningful reference frames, and physically meaningful notions of motion. This was thought to be important, and lead to Einstein's ideas behind the equivalence principle. After GR was formulated, people (including Einstein) asked if Mach's principle was consistent with, or derivable from, GR / the equivalence principle, but it turns out that it is not. (There are, however, some remnants of the idea left in GR.)

Before people realized this, it did lead to a lot of confusion, even among some of the big-name people, and some of that confusion seems to have lasted a lot longer than it should have. But trying to impose Mach's principle on top of GR would constitute a "prior geometry" in a sense, and would either render GR inconsistent with observation, or make it mathematically inconsistent, depending on how one does it, so this could be one of the things they had in mind.

The second obvious case that jumps to mind is a little more subtle (but was still certainly understood by MTW's time), and is the issue that, if you want to break up the metric as

$g_{\mu\nu} = \eta_{\mu\nu} + \epsilon h_{\mu\nu}$

or more generally,

$g_{\mu\nu} = \overline{g}_{\mu\nu} + \epsilon h_{\mu\nu}$

for an appropriately defined small $\epsilon$, what are the correct interpretations of $g$, $\overline{g}$, and $h$?

One can surely, without making any additional assumptions, write the metric as the sum of two parts, and write Einstein's equations in terms of these two parts without changing anything. But you want to "decouple" the "background" and "perturbed" parts of the metric in order to allow you to solve some kinds of problems, such as gravitational waves.

It seems, though, once you have done this, you may have done something strange. You now have two independent metrics, which describe two seemingly independent, unrelated ("decoupled") spacetimes, and have two independent sets of symmetries. So it now seems you may have, in addition to the prior geometry of $\overline{g}$, an ambiguity in matching coordinates from $g$ to ones in the components $\overline{g}$ and $h$ due to their independent diffeomorphism symmetries.

This ambiguity lead to some confusion over if this kind of method is applicable (it is), if you have to be careful in using it (mostly, you don't), and if it introduces any fundamental conceptual difficulties (it doesn't). (I don't remember how well MTW explains this, but you can find some detailed descriptions in some of the newer GR books, IIRC, so I won't describe it here.)

As far as I know, this confusion started in considering gravitational waves, and was later picked up by people trying to study quantum gravity as a canonically quantized version of GR in this way. The trouble in the quantization case was, through misunderstandings related to this prior geometry issue, people thought it okay to introduce what amounted to actual "physical" prior geometry, and thought that this problem that had been understood by the gravitational wave people was telling them something important and fundamental about how quantum gravity worked.

This prior geometry, in the quantum gravity case, I believe amounts to problems like selecting a preferred reference frame that would typically be quite obvious macroscopically, or breaking Lorentz invariance to (e.g.) a discrete subset, or causing the theory to become inconsistent. Unfortunately, it seems some of the people working on various "alternative" approaches to quantum gravity still haven't learned this lesson, and continue to try to write down theories like this. But to the rest of the world, this issue has been well-understood for a very long time.

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This is an interesting question, in part about the history of Einstein's formulations of General Relativity and about its foundations.

To deal with "prior geometry" first here is an example metric that was current in Einstein's time (and mentioned in MTW) as a possible form of a geometric gravity theory: $g=\phi^2\eta$ where $\eta$ is the Minkowski metric. So $g$ is the spacetime metric, but there is really a prior metric around too $\eta$. All the variation in $g$ just comes from the $\phi$ function defined on the prior geometry. So Einstein wanted a "prior geometry free geometry" in effect.

The "confusion" referred to by MTW and by this Question is the physical status of the principle known as "General Covariance". Einstein claimed much for this principle but almost immediately Kretschmann (1917) described it as "physically meaningless". A very closely related question was whether "General Relativity" was correctly named - had there actually been a physical discovery here which generalised the relativity principle of Special Relativity?

Kretschmann made the prescient claim in his argument that any (worthwhile) physical theory could be made "generally covariant" - even Newtonian Gravitation. MTW chapter 12 describes the modern differential formulation of the Newton-Cartan theory which seems to vindicate Kretschmann.

The confusion lasted sixty odd years because physicists somewhat lost interest in GR until the 1960s: the textbooks prior to that time did not include the modern language of differential geometry, which somewhat explains what Einstein was getting at.

However even after MTW debate continued into aspects of the foundations of GR. Einstein had initially said it was based on three principles:

1. General Covariance
2. Principle of Equivalence
3. Mach's Principle

The three form a somewhat inter-related package, and all have been challenged or rewritten in various ways post Einstein. What often happens in modern mathematical texts on GR is that (some or all) of these are ignored and the theory is presented as a mathematical theory with Tensors on Manifolds. In an undergraduate GR text sitting on my desk right now, the role of "fundamental principles" above has actually been replaced by the need for equivalence to the Newtonian theory in the weak field limit: they are not mentioned.

A modern student of GR probably should learn it that way and then learn what later authors have done to these principles. It turns out to be an interesting story for those interested in fundamentals: "weak" versus "strong" covariance, mathematical restrictions on Tensor forms, etc. The contemporary significance of this work would be whether it has significance in "Quantum Gravity" - there is a "Quantum Equivalence Principle" for example in the literature.

For the history of this problem in GR I have found this reference.

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"Prior geometry" in the sense of MTW is any kind of geometry that is not influenced by the distribution of gravitational sources.

The unique selling point of general relativity is that the geometry of spacetime is determined dynamically by the "self consistency" Einstein field equation. Especially there is no a priori meaning of a point in spacetime, space and time have no meaning until the gravitational field is specified.

1) a coordinate-free description of geometric objects was not well developed (vector spaces, manifolds), although I suspect that e.g. Gauss already thought this way. The theory of differential forms for example was not developed (Edit: that's not quite right, see the comments) and not well known for several decades after the publication of GR. (In fact, Feynman did not understand differential forms, which motivated his approach to GR as a (quantum) field theory in his "gravitation lectures" where he re-discovered the graviton.) For a long time the distinction of coordinate singularities and true singularities was not clear to many physicists, which you can see for yourself when you study papers about gravitational waves from the 1950ties.

2) Einstein himself had severe problems with the concept of general covariance that manifested in his struggle with the "hole problem". For a detailed discussion from a modern viewpoint please have a look at spacetime on the nLab. This is closely related with the Mach principle(s), of course, you'll find a very interesting discussion of this in the book

• Carlo Rovelli: "Quantum Gravity"
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Actually, Cartan invented differential forms sometime from 1894 to 1904. (Source) The article also states that his work did not gain influence until the 1930s, though. – Greg Graviton Feb 3 '11 at 8:24
Thanks for the correction - I should have looked it up myself :-) I think it is safe to say that most physicists did not learn about differential forms even in the 1950ties... – Tim van Beek Feb 3 '11 at 9:23
when talking about most physicists you can safely change that to 2050 :) – Marek Feb 3 '11 at 10:12
Funny :-) I should specialize to theoretical physicists which I was implicitly thinking about: Every theoretical physicist has to learn differential forms! – Tim van Beek Feb 3 '11 at 10:33
Nah, you can get along fine in GR with good old Riemannian tensor notation. – Gordon Feb 3 '11 at 16:31

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