# What is the Geometry of Space for a Rotating Observer?

 Page 2/7 Date 13.05.2016 Size 116.33 Kb.

## 2.2 What is the Geometry of Space for a Rotating Observer?

At a decisive moment in the course of his discovery of the general theory of relativity, sometime in 1912, Einstein realized that the geometry of space for an accelerated observer may be non-Euclidean. He showed this for the case of a uniformly rotating observer in special relativity by means of a thought experiment concerning a rigidly rotating disk. (See Stachel, 1980.) Einstein imagined that the geometry of the surface of the disk is investigated by the usual method of laying down measuring rods. If the disk's diameter is D, what will we measure for its circumference C? Will it be the Euclidean C = D? The lengths of measuring rods laid radially are not affected by the Lorentz contraction of special relativity since their motion is perpendicular to their length. But rods laid tangentially along the circumference move in the direction of their length and will be contracted. Thus more rods will be needed to cover the circumference than according to Euclidean expectations. That is, we will measure a non-Euclidean C > D. See Figure 2.

Figure 2. What is the geometry of space for a rotating observer?

While Einstein's thought experiment gives a non-Euclidean geometry with C > D, an anti-thought experiment gives the opposite result of a geometry with C < D. The alternative was proposed, for example, by Joseph Petzold in a letter to Einstein of July 26, 1919. (See Stachel, 1980, p. 52.) It is, in effect, that the rotating disk be conceived as concentric, nestled, rotating rings. The rings are uncontracted radially, so the diameter of the disk is unaffected. But the rings are contracted in the circumferential direction, the direction of motion due to the rotation; so their length is less that the corresponding Euclidean length. That is, the lengths on the disk conform to C < D.

Another anti - thought experiment, investigated by Ehrenfest in 1910 and Varicak in 1911, gives the Euclidean result C = D. The positions of distance markers on the rotating disk are transferred at some instant to superimposed but non-rotating tracing paper and the geometric figures on the disk reconstructed. The result, Varicak urged, would be recovery of Euclidean figures because the surface of non-rotating tracing paper conforms to Euclidean geometry.5

## 2.3 What is the Lift of an Infinite Rotor at Rest?

Imagine a helicopter rotor. When it rotates, it generates lift as a reaction force resulting from the momentum imparted to the current of air it directs downward. If the rotor moves a mass m of air in one second at speed v, then the lift L generated is just mv. What would happen if we double the radius of the rotor? To answer, let us assume that it is part of the design of rotors of varying size that the speed of the air currents they generate is proportional to the rotational speed of the rotor. (This can be achieved by flattening the rotor blades more, further from the center.) Since the area swept by the rotor has increased by a factor of 22 = 4, if we leave the rotational speed of the rotor fixed, in one second the rotor will move a mass 4m of air at speed v. So the lift will have increased by a factor of 4 to 4mv. To keep the lift constant at L = mv we should now also reduce the rotational speed of the rotor by a factor of 2. That halves the speed of the air to v/2 and also halves the mass moved from 4m to 2m. The lift is now (2m).(v/2) = mv = L, which is the original lift.6

In short, the lift stays constant at L as we double the rotor size and halve its speed. Repeat this process endlessly in thought, indefinitely doubling the rotor size and halving the rotational speed. In the limit of infinitely many doublings, we have a rotor of infinite size that is not rotating but still generates the original lift L.

Figure 3. Effect of increased size and reduced speed on a helicopter rotor.

The obvious anti-thought experiment yields no lift for an infinitely large rotor at rest. A finitely sized rotor that does not turn generates no lift. This is true if we double its size. In the limit of infinitely many doublings we have an infinitely large rotor that does not rotate and generates no lift.

### The Challenge

It is to hard resist the puzzle of determining which (if either) of the members of a pair gives the correct result and what is wrong with the other one. That sort of exercise is part of the fun of thought experiments. But it is not my principal concern here. My concern is to ask how different epistemologies diagnose the existence of the competing pairs; how they explain why one succeeds and the other fails; and how the epistemologies can do this while still preserving the reliability of thought experiments as instruments of inquiry.

## 3. Thought Experiments are Arguments

### Why Arguments?

My account of thought experiments is based on the presumption that pure thought cannot conjure up knowledge, aside, perhaps, from logical truths. All pure thought can do is transform what we already know. This is the case with thought experiments: they can only transform existing knowledge. If thought experiments are to produce knowledge, then we must require that the transformations they effect preserve whatever truth is in our existing knowledge; or that there is at least a strong likelihood of its preservation. The only way I know of effecting this transformation is through argumentation; the first case is deductive and the second inductive.

Thus I arrive at the core thesis of my account:

(1) Thought experiments are arguments.

which forms the basis of my earlier account of thought experiments. (Norton, 1991, 1996)7 To put it another way, if thought experiments are capable of producing knowledge, it is only because they are disguised, picturesque arguments. That does not assure us that all thought experiments do produce knowledge. They can fail to in just the same way that arguments can fail; that is, either may proceed from false premises or employ fallacious reasoning.

### How Experience Enters a Thought Experiment

Thought experiments need not produce knowledge of the natural world. There are, for example, thought experiments in pure mathematics (for examples, see Brown, 1992, pp.275-76) and these, I have argued, are merely picturesque arguments (see Norton, 1996, pp. 351-53). However the thought experiments that interest me here are those of the natural sciences that do yield contingent knowledge of the natural world. According to empiricism, they can only do so if knowledge of the natural world is supplied to the thought experiment; that is, if this knowledge comprises a portion of the premises upon which the argument proceeds. It may enter as explicitly held knowledge of the world. We assert on the authority of an empirical theory, special relativity, that a moving rod shrinks in the direction of its motion. Or it may enter as tacit knowledge. We just know that the space of our experience never runs out; we have never seen a boundary in space beyond which we could not pass, unless there is already something past the boundary to obstruct us.

I do not seek here to argue for empiricism; the debate between empiricism and other epistemologies is as ancient as philosophy itself and not likely to be advanced fundamentally here. However, empiricism is overwhelmingly the predominant epistemology in philosophy of science, so that an account that accommodates thought experiments to empiricism in a simple and straightforward manner ought to be accepted as the default, as opposed so some more extravagant account. I claim this default status for the view advocated here.

### Two Forms of the Thesis

The thesis that thought experiments are arguments requires some elucidation. Is the claim merely that thought experiments can do no more than argumentation when it comes to justifying claims? Or is it in addition that the actual execution of a thought experiment is just the execution of an argument? Following Norton (1996, p. 354) I intend the stronger version and urge both:

(1a) (Context of justification)8 All thought experiments can be reconstructed as arguments based on tacit or explicit assumptions. Belief in the outcome-conclusion of the thought experiment is justified only in so far as the reconstructed argument can justify the conclusion.

(1b) (Context of discovery) The actual conduct of a thought experiment consists of the execution of an argument, although this may not be obvious since the argument may appear only in abbreviated form and with suppressed premises.

### Justifying (1a)

As indicated above, the first thesis (1a) derives from the assumption that pure thought cannot conjure up new knowledge. There is a second and more practical justification. As far as I know, all thought experiments can in fact be reconstructed as arguments and I have little hope of finding one that cannot. Indeed this expectation supplies a quite stringent test of thesis (1a). It can be defeated merely by finding a thought experiment that cannot be reconstructed as an argument. Norton (1991, 1996) contain many examples of reconstruction of typical thought experiments from various different areas of the physical sciences, including those have been offered as opaque to such reconstruction. The ease of their reconstruction suggests that a counterexample will not be found. The reconstructions are generally rather straightforward and often differ little from the original narrative of the thought experiment. Einstein's rotating disk thought experiment is a typical example. It can be reconstructed in summary as:

(D1) In Euclidean geometry, the measured circumference of a disk is  times its diameter. (Premise)

(D2) The geometry of a non-rotating disk is Euclidean. (Premise)

(D3) The motion of a radial element on a rotating disk is perpendicular to its length, so that (according to special relativity) the length is unaltered. (Premise)

(D4) The motion of a circumferential element on a rotating disk is along its length, so that (according to special relativity) the length is contracted. (Premise)

(D5) Therefore the measured circumference of a rotating disk is more than  times the measured diameter. (From D2, D3, D4)

(D5) Therefore the geometry of a rotating disk is not Euclidean. (From D1, D5)

### Justifying (1b)

The situation with the second thesis (1b) is not so straightforward. It is both a thesis in the philosophy of thought experiments and also a thesis in empirical psychology. Perhaps prudence should instruct us to assert only (1a) and remain agnostic on (1b), awaiting the verdict of empirical work in psychology. Indeed (1a) with agnosticism on (1b) already amounts to a strongly empiricist restriction on what thought experiments can teach us. However it seems to me that this contracted account is unnecessarily timid. There are several indications that favor (1b).

In spite of their exotic reputation, thought experiments convince us by quite prosaic means. They come to us as words on paper. We read them and, as we do, we trace through the steps to complete the thought experiment. They convince us without exotic experiences of biblical moment or rapturous states of mind. At this level of description thought experimenting does not differ from the reading of the broader literature in persuasive writing. A long tradition in informal logic maintains that this activity is merely argumentation and that most of us have some natural facility in it. The text prompts us to carry out arguments tacitly and it is urged that reconstructing the arguments explicitly is a powerful diagnostic tool. I merely propose in (1b) that matters are no different in thought experimenting. Parsimony suggests that we make this simplest of accounts our default assumption.

Thesis (1a) supplies a stronger reason for accepting (1b). Whatever the activity of thought experimenting may be, if we accept (1a), we believe that the reach of thought experimenting coincides exactly with the reach of argumentation. If thought experimentation opens up some other channel to knowledge, how curious that it should impersonate argumentation so perfectly! How are we to explain this coincidence if not by the simple assumption that thought experimenting merely is disguised argumentation? Analogously, we would accord no special powers to a clairvoyant whose prognostications coincide precisely with what could be read from one's high school graduation year book. We would strongly suspect a quite prosaic source for the clairvoyant's knowledge.

### Thought Experiment-Anti Thought Experiment Pairs

This account of the nature of thought experiments can readily accommodate the existence of these pairs. We can have two arguments whose conclusions contradict. It then follows that at least one of the arguments is not sound; it has a false premise or a fallacious inference. The diagnosis is the same for a pair of thought experiments that produce contradictory outcomes. The argument of at least one of them has a false premise or fallacious step and we resolve the problem by finding it. Thus the existence of these pairs presents no special obstacle to the reliability of thought experiments. If they fail, they do so for an identifiable reason, although finding the false premise or fallacy may not be easy. Thought experiments have the same transparency and reliability as ordinary argumentation.