What does Jaynes mean by "mollusk-like quality"?

Question

I am trying to read Prior Probabilities (1968), by Edwin T. Jaynes.

In two sections he discusses mollusk-like qualities [of parameter spaces]:

The real problem, therefore, must be stated rather differently [than that of Jeffreys and Bayes]; we suggest that the proper question to ask is: "For which choice of parameters does a given form such as that of Bayes or Jeffreys apply?" Our parameter spaces seems[sic] to have a mollusk-like quality that prevents us from answering this, unless we can find a new principle that gives them a property of "rigidity".

Stated in this way, we recognize that problems of this type have already appeared and have been solved in other branches of mathematics. In Riemannian geometry and general relativity theory, we allow arbitrary continuous coordinate transformations; yet the property of rigidity is maintained by the concept of the invariant line element, which enables us to make statements of the definite geometrical and physical meaning independently of the choice of coordinates.

What does mollusk-like quality actually mean in this context?

Answer 1

The reference is to Einstein's foundational work on relativity theory. He refers to a "mollusc" (non-rigid) space in this work.

For this reason non-rigid reference-bodies are used which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion. Clocks, for which the law of motion is any kind, however irregular, serve for the definition of time. We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the “readings” which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount. This non-rigid reference-body, which might appropriately be termed a “reference-mollusk,” is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusk” a certain comprehensibleness as compared with the Gauss co-ordinate system is the (really unqualified) formal retention of the separate existence of the space co-ordinate. Every point on the mollusk is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusk is considered as reference-body. The general principle of relativity requires that all these mollusks can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusk.

The statement concerns transformation groups of continuous parameter(s) to obtain compactness and invariance. It is a topic of consideration in differential geometry. Jaynes is trying to formulate a hybrid between the considerations of a Bayesian statistician accommodating multiple disparate prior constraints (e.g. we have 90% believe $\theta > 0$ or $\theta$ has a unimodal distribution) and the Bayes' / Jeffreys' approach of having a "naive" / uniform possibly improper prior. His hybrid may be envisioned as being "mollusk" like in shape by having a complex space (impact of prior specifications) but a well-defined measurable compact property (preferences of Bayes and Jeffreys).

Answer 2

I think Einstein's use of the mollusc was simply a counterpoint to the concept of a "rigid body". If you were to rapidly accelerate a titanium anvil (from behind, as in a push) it would physically shorten an infinitesimal amount in the direction of travel. A jellyfish, on the other hand, undergoing the same process would be considerably more distorted... .

Answer 3

The reference mollusc is be understood as a visual picture for a physical implementation of the most general coordinate system that is mathematically conceivable in general relativity. Since the mollusc is non-rigid in space it should have enough degrees of freedom to implement any spatially varying non-linear coordinates. The clock at each point of the mollusc is also non-linearly changing with time.

In actual fact, however, the reference mollusc would only physically implement a 3+1 spacetime coordinate system which can be spatially foliated. The actual most general coordinate system would require 4 independently and completely arbitrarily running non-linear clocks at each point of the mollusc, or a smartphone displaying 4 arbitrarily changing numbers. These 4 ever changing numbers for each point of the mollusc would be able to physically capture virtually every imaginable mathematical coordinate system. Usually, in general relativity you would like to ensure also continuity, so nearby clocks (or smartphones) on the mollusc should show 4 similar numbers.

One may also measure the metric associated with the mollusc coordinate system: according to the equivalence principle it is also possible to construct at each point of the mollusc a local inertial frame in addition to the non-linear coordinates of the mollusc discussed above. The local inertial frame provides rigid (special-relativistic) coordinates and exists for a tiny amount of time and has a tiny spatial extension (freely falling elevator gedankenexperiment). You may then compare the local inertial coordinates with the local non-linear mollusc coordinates. For example, think about the elevator having infinitely thin glass walls so that you can compare the coordinate values of say the elevator walls inside (rigid special relativistic coordinate system) with those outside (non-linear general relativistic mollusc). Once you have both sets of coordinates for enough points on the elevator wall, you may compute all the components of the local metric by transforming from one coordinate system to the other because this is what the metric does: it describes how distances of your locally rigid special relativistic system transform to changes in your locally non-linear coordinate system values. Therefore, this comparison of coordinate values of two different coordinate systems for the same local neighbourhood on the mollusc provides a physical implementation of how to measure the metric associated with the mollusc coordinate system in an actual physical situation.

Incidentally, if you then also measure the local energy tensor you can locally test the validity of the Einstein field equations because you already have measured the metric (and its derivatives by measuring the metric for a little larger local environment).

Therefore, the whole point of the mollusc is to provide a visual picture of how one might physically implement (at least in principle) measurements of the mathematical quantities used in general relativity. Such a picture might help the beginner to understand the theoretical concepts and their possible operational meaning.

NB: as for the generalisation of the mollusc in which you put 4 independent and arbitrarily running clocks at each point of the mollusc, one may even continue this line of thinking and put, say 6 arbitrarily running clocks at each point, or, equivalently, display 6 numbers per smartphone. Then one would find out that the same situation can be also described in a 6 dimensional coordinate system. However, by doing enough experiments you would then find out that this introduces more degrees of freedom than needed to describe every physical phenomenon. This way you could test the dimensionality of our spacetime, and thereby find out that you do not need 6 numbers since 4 numbers per point would be enough to describe everything also in general relativity (or you may find out that they are not enough and win a Nobel prize:)