What does Einstein mean by “mollusc” in chapter 29 of His book Relativity?

Question

What does Albert Einstein mean by the word “mollusc” and how does it fit in his theory of Relativity? The word can be found in chapter 29 of Relativity: The Special and General Theory.

In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity.

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 of 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-mollusc”, is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusc” a certain comprehensibility as compared with the Gauss co-ordinate system is the (really unjustified) formal retention of the separate existence of the space co-ordinates as opposed to the time co-ordinate. Every point on the mollusc is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusc is considered as reference-body. The general principle of relativity requires that all these molluscs 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 mollusc.

Answer 1

As Dale notes, a mollusc is a kind of animal — specifically, one of the many kinds of animals belonging to the phylum Mollusca, which includes many well known types of animals such as slugs, snails, clams and even squids and octopuses, as well as a large number of less commonly known (mostly marine) lifeforms.

The name "mollusc" comes from the Latin word mollis, meaning "soft", describing their characteristically soft and flexible bodies. Being invertebrates, molluscs have no bones, and they also lack the hard chitinous exoskeleton characteristic of arthropods (insects, spiders, lobsters, crabs, etc.). While some molluscs (such as snails and clams) do possess a hard external shell, many others (such as slugs) do not. The bodies of such shell-less molluscs have essentially no rigid parts at all and can stretch and deform to a considerable extent.

^{A yellow slug (Limacus flavus), a common species of mollusc. Photo by Wikimedia Commons user AnemoneProjectors, used under the CC-By-SA 2.0 license.}

Even the bodies of many shelled molluscs (obviously excluding the shell itself) are notably flexible: for example, snails normally carry their shell on their back, keeping most of their body outside it, but they can pull their entire body inside the shell if threatened. Molluscs, whether shelled or not, also lack rigid limbs and instead commonly move around by undulating or otherwise deforming their flexible bodies.

Given the phrasing and context of your quote, I suspect that it's specifically this characteristic softness and non-rigidity of molluscs — and not their rigid shell — that Einstein was most likely alluding to when coining the term "reference mollusc" to describe a non-rigid reference body.

Answer 2

A mollusk is an animal like a clam, oyster, or snail, with a curved shell, or even an octopus without a shell. Einstein is using the word "mollusk" simply to convey the idea that the coordinates need not be flat Cartesian coordinates but can be curved almost arbitrarily like the shell or skin of a mollusk and even that they can deform over time There is no deeper biological or other similar meaning, simply a familiar visual concept of an arbitrarily curved surface.

Answer 3

I think here Einstein here wants us especially to think of a shell-less mollusc, of which the sea slugs and nudibranchs from the Gastropoda as well as the octopus from the Cephalopoda are the only examples I can think of[1].

He’s being a bit fancy and “polymathematic” in using this word but basically he means something that is “sqidgy” and sqishy, like a ball of Play-Doh. And his main point is that in relativity (special as well as general) there is no such thing as a rigid body that undergoes rigid Euclidean transformations in its motion in response to any force. Simplest example: we try to imagine a “rigid” bar reaching from here to Mars, at least twenty light minutes away. We hit it with a hammer, its end is shoved a few millimeters axially. It’s meant to be rigid and thus undergo a Euclidean translation. But that would mean our stroke would be immediately detectable at the other end on Mars, twenty light minutes away! Our shove is transmitted faster than $c$, which violates the assumption in relativity that there can be no faster that light signaling (we make this postulate to protect causality and avoid such weirdness as the Tachyonic Antitelephone Scenario). So we conclude that the bar can’t be rigid and indeed must have an acoustic wave speed less than $c$.

His statement is also illustrated by the famous Ehrenfest “Paradox”. See my description here. Euclidean Rigidity in relativity can be partially salvaged as a concept known as Born Rigidity, but this is a “squidgy” Rigidity designed to avoid faster than light signaling that is not very intuitive in terms of everyday experiences of “stiffness” and “hardness”.

[1] I’m not a marine biologist but I am a diver of 35 years‘ experience with a particular love of cephalopods. My fave is Sepia Apama of my homeland halfway across the world, which does have a stiff shell (cuttlebone) within, and which I think is just sublimely beautiful, especially with its gorgeous „peplum fin“ that girds its whole lower body and whose wave motion gives this gorgeous creature an astonishingly deft control of its attitude and bearing in its 3D world.

Answer 4

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 (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:)