Scott A Walter

Poincaré on clocks in motion

Recently-discovered manuscripts throw new light on Poincaré's discovery of the Lorentz group, and his ether-based interpretation of the Lorentz transformation. At first, Poincaré postulated longitudinal contraction of bodies in motion with respect to the ether, and ignored time deformation. In April, 1909, he acknowledged temporal deformation due to translation, obtaining thereby a theory of relativity more compatible with those of Einstein and Minkowski.

Introduction

In the month of May, 1905, Henri Poincaré (1905) was arguably the first scientist to express the principle of relativity in terms of the form-invariance of laws of physics with respect to transformations that form a particular group. This insight had several consequences for Poincaré's understanding of what he called the "new mechanics" of relativity. Notably, following a method outlined by Lie and Scheffers (1893, 669), Poincaré identified invariants of the Lorentz transformation directly from the fact that the transformation is a rotation about the coordinate origin in four-dimensional space (with one imaginary axis). Any transformation of the Lorentz group, he noted further, may be decomposed into a dilation and a linear transformation leaving invariant the quadratic form x 2 + y 2 + z 2 -t 2 , where light velocity is rationalized to unity. Poincaré naturally associated this quadratic form with the propagation of light, and gravitational action (Poincaré, 1906b, § § 4, 8).

While historians agree that Poincaré discovered -and christened -the modern form of the Lorentz transformation, no consensus has formed on how Poincaré interpreted these same transformations in kinematic and dynamic settings. To some extent, the lack of consensus is a consequence of the state of the relevant textual sources, which are inconsistent at crucial junctures. The proximity of Einstein's contributions to relativity and the theory's rapid conceptual and formal evolution in its early years render the project of isolating the approach of either Poincaré or Einstein all the more difficult (Darrigol, 2004). In such a situation, there appears to be little prospect for the emergence of a common reading of Poincaré's theory of relativity.

The situation has changed in the last year, as long-lost manuscripts have come to light which point the way to a new reading of Poincaré's interpretation of the Lorentz group. This reading is presented here chronologically, beginning with a short history of local time, and an account of Poincaré's lectures of 1906, in which he first considered the locus of light wavefronts in the context of Lorentz's electron theory. Next, Poincaré's adoption of a form of time deformation in 1909 is discussed, and is related to a reinterpretation of the Lorentz group employing clocks in motion. The final section takes up Poincaré's mature presentation of the light-locus problem in lectures delivered in June, 1912. The paper closes with a discussion of Poincaré's changing attitude toward the synchronization of clocks in motion, and situates his view in the context of spacetime conventionalism.

From Ortszeit to temps local

The theoretical foundations of classical electron dynamics were worked out by H.-A. Lorentz and Joseph Larmor beginning in 1895, and by 1900, electron theory was understood to represent the most sophisticated approach to a wide range of phenomena at the forefront of physics, from cathode rays and magneto-optics to blackbody radiation and the electrodynamics of moving bodies [START_REF] Buchwald | From Maxwell to Microphysics[END_REF]. After lecturing at the Sorbonne in 1899 on the theories of Lorentz and Larmor, Poincaré made a signal contribution to Lorentz's electron theory, as Darrigol observes (1995). Lorentz's theory explained the absence of measurable effects of uniform motion on electromagnetic phenomena to first order in v/c (velocity over the vacuum speed of light), provided that one transformed the time variable t according to spatial displacement and frame velocity to obtain t :

t = t -vx/c 2 , (1)
or what Lorentz referred to as "Ortszeit" [START_REF] Lorentz | Versuch einen Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern[END_REF], and which Poincaré (1900a, 273), following Alfred Liénard, called "temps local", or local time. In Poincaré's mind, Lorentz's local time took on an operational meaning, as the time read by a clock in uniform motion of velocity v with respect to the ether, synchronized by crossed light signals with other identical comoving clocks, assuming isotropic propagation velocity. The difference between local time and "true" time for terrestrial clocks separated by one kilometer, according to Poincaré's calculation, would be a third of a nanosecond, assuming an ether at rest with respect to the sun (Poincaré, 1901, 530).

In fact, Poincaré's definition implies that measurement of the spatial separation of two clocks at relative rest is performed in the comoving frame, such that the local time transformation would be expressed as follows:

t = t -vx /c 2 , (2)
where x = x + vt. For terrestrial observers, the difference between (1) and

(2) was insensible, which may explain its neglect. First-order agreement with observed optical phenomena was unable to account for Michelson and Morley's precise measurements of fringe shifts, which required the absence of terms up to third order in v/c [START_REF] Michelson | On the relative motion of the earth and the luminiferous ether[END_REF]. G.-F. FitzGerald and Lorentz suggested that a previouslyunnoticed contraction of solids in the direction of their motion with respect to the ether would explain the null result of the Michelson-Morley etherdrag experiment, but it bothered Poincaré that independent hypotheses were required to explain the absence of first-order and second-order effects. He wondered aloud if the principle of relativity would not prove absolutely valid (Poincaré, 1900b(Poincaré, , 1172)).

At the same time, results obtained by a student of Gabriel Lippmann, Victor Crémieu, led Poincaré to doubt the validity of Maxwellian electrodynamics, or at least, the reality of the magnetic effect of electric convection. Lorentz and other theorists were convinced that if Rowland's effect did not exist, this would spell ruin for both Maxwell's theory and its successors, including Lorentz's own electron theory. Crémieu's attempt to reproduce the Rowland effect was unsuccessful, and in Poincaré's considered opinion, the young experimenter had "rendered very doubtful a conclusion that seemed definitively established" (Poincaré, Rapport sur la thèse de V. Crémieu, 30.05.1901, in Walter et al., 2007, §2-62-6).

Poincaré's confidence in Crémieu's results stemmed in part from his conventionalist philosophy of science, which recognized the importance of observation and experiment in determining the direction of scientific research. Unlike his former student Pierre Duhem, Poincaré believed in the "crucial experiment", the result of which determines the viability of theoretical options, or even of entire classes of theories. The investigations of the magnetic effect of charge convection performed by Crémieu, Pender and others provide an example of this type of experiment, inasmuch as a null result might have overturned the Maxwellian tenet of closed current loops, and with it, the electron theories of Larmor, Lorentz, and Max Abraham.

After a dramatic face-off in 1903 with Rowland's student Harold Pender in Edmond Bouty's laboratory at the Sorbonne, where the two experimenters labored in tandem to measure the Rowland effect, Crémieu recognized the effect's reality, bringing the three-year controversy to a close (Walter et al., 2007, §2-17). It was an inglorious moment for Crémieu, and also for Poincaré, who since the early 1890s had been the leading exponent and expert on Maxwell's theory in France [START_REF] Darrigol | The electrodynamic revolution in Germany as documented by early German expositions of 'Maxwell's theory[END_REF].

Following the face-off, Poincaré silently rewrote the passage concerning the Rowland effect in the second edition of his popular collection of essays, La science et l'hypothèse:

Seemingly, the edifice of electrodynamics was definitively elaborated, at least in its principal aspects. This tranquil state of affairs was troubled recently by Mr. Crémieu's experiments, which seemed for a moment to contradict the result obtained some time ago by Rowland. These experiments were disconfirmed by subsequent investigations, and Lorentz's theory emerged victorious from the ordeal. (Poincaré, 1906a, 281) In later editions of Poincaré's book, the cited remark is nowhere to be found.

But then Poincaré had no interest in reminding readers of Crémieu's challenge to Maxwell-Lorentz theory, or of his own role in mounting and overcoming this challenge. While Lorentz's theory survived the charge-convection test, there would be others for it to pass. One of these tests was mentioned by Poincaré in September, 1904, a Poincaré imagined the existence of non-luminous signals with hyperlight velocity, that we would use to verify the synchronization of light-synchronized timekeepers. Any difference in the result of the synchronization process would reveal the common motion of the clocks, in patent contradiction of the principle of relativity. The existence of such signals was not far-fetched, as Poincaré reminded his audience of Laplace's calculation of the propagation velocity of gravitational action, which put the speed of gravitation at a million times that of light (Poincaré, 1904, 312).

Along with the Laplacian estimate of the velocity of gravitational action, the stability of the contractile electron appeared to menace Lorentz's theory in 1904. An analysis of the hypotheses underlying electron theory led the Göttingen Privatdozent Max Abraham to point out that Lorentz's theory was incomplete, insofar as it required a non-electromagnetic force for stability, and none had been proposed (McCormmach, 1970a, 51). With Abraham, a handful of theorists (including Paul Langevin, Alfred Bucherer, and Arnold Sommerfeld) were pursuing what Wilhelm Wien had called an "electromagnetische Weltbild", or electromagnetic worldview, according to which, roughly stated, all physical interactions are of electromagnetic origin (McCormmach, 1970b). Abraham's critique effectively excluded Lorentz's theory from this attractive research program.

The exclusion of Lorentz's electron theory from the electromagnetic worldview had no adverse effect on Poincaré's view of the theory's prospects, which he set out to improve in the spring of 1905. The motivation for Poincaré's renewed interest in Lorentz's theory is unknown, but may well be a consequence of his reading Lorentz's recent papers on the topic, sent to him by Lorentz shortly before his visit to Paris at the end of April, 1905. 1 In addition to Lorentz's papers, in the spring of 1905 Poincaré read Langevin's theoretical study of electromagnetic inertia, where he developed formulas for velocity and acceleration waves issuing from an electron in motion. 2Poincaré did not find the time to meet with Lorentz during the latter's visit to Paris, but he did manage to perfect his electron theory. He announced his findings to Lorentz by letter soon after the latter left Paris. First of all, he corrected Lorentz's current density transformation, such that charge is conserved. In doing so, he tried out three candidate transformations, corresponding to the electron models of Lorentz, Bucherer-Langevin, and Abraham. The latter model, in which an electron is likened to a rigid sphere, implied a property Poincaré was keen on preserving, but felt obliged to abandon: You assume = 1. Langevin assumes k3 = 1. I tried k = 1 in order to preserve the unity of time, but this led me to inadmissible consequences. 3 (Poincaré to Lorentz, ca. May 1905, in Walter et al., eds., 2007, §2-38-3) Abraham's rigid-sphere electron theory preserved the unity of time, and required no non-electromagnetic forces for stability. But it was not compatible with the principle of relativity, and was discarded by Poincaré.

Lorentz's reply to Poincaré has not been located, but after receiving it, Poincaré announced that his "ideas had changed", and that he had come to a new understanding of the constant referred to in his previous letter. Rationalizing the velocity of light, Poincaré rewrote Lorentz's transformation in a new, compact form:

x = k (x + εt), t = k (t + εx), y = y, z = z,
and affirmed that this transformation forms a group if and only if = 1. Where Lorentz had argued that = 1 on physical grounds, Poincaré expressed the principle of relativity in terms of group invariance: Lorentz's transformation forms a group, and this transformation forms a group if and only if = 1, therefore = 1.

Not long afterwards, Einstein offered a derivation of what Poincaré called the "Lorentz transformations" from kinematic assumptions based on his twin postulates of relativity and light-speed invariance (Einstein, 1905, §3). To convince his readers of the compatibility of his postulates, Einstein showed that the equation of a spherical lightwave in one frame, say x 2 +y 2 +z 2 = c 2 t 2 , has the same form when it is Lorentz-transformed. A spherical wave propagating with the speed of light in one frame, Einstein remarked, is thereby a spherical wave propagating with the speed of light in a second inertial frame. As the British mathematical physicist Ebenezer Cunningham pointed out later, form-invariance of the lightwave equation is all that is required for the derivation of the Lorentz transformation, along with a linearity constraint [START_REF] Cunningham | On the electromagnetic mass of a moving electron[END_REF].

It is remarkable that Poincaré's view of light propagation conflicted with that of Einstein and Cunningham, at least until 1909. Starting in the early 1970s, several commentators have noticed differences between Einstein's and Poincaré's views of light propagation, based on the latter's diagrammatic model of the Lorentz transformation.4 None of these contributions considers Poincaré's view in its full chronological extent, i.e., from 1906 to 1912. To make matters more interesting, archival research turned up a long-lost manuscript, which throws new light on Poincaré's model. The next section explores the latter source in detail.

3 Poincaré's light ellipse and the limits of Newton's laws (1906-07)

A few weeks after the above-mentioned exchange by mail with Lorentz, Poincaré communicated his results to the Paris Academy of Sciences (Poincaré, 1905). Following a trip to Scandinavia, and an audience with King Oscar II of Sweden, Poincaré finished a long memoir on the dynamics of the electron in July, 1905, which appeared in print the following January (Poincaré, 1906b). Curiously, for one who had engaged with the Riemann-Helmholtz-Lie problem of space, Poincaré avoided drawing consequences for the foundations of geometry from the "new mechanics" of the Lorentz group, with one exception. He observed that while previously, measurement of length implied the physical displacement of solids considered to be rigid, this is no longer true in the current theory, if we admit the Lorentzian contraction. In this theory, two equal lengths, by definition, are two lengths spanned by light in the same lapse of time.

Light signals, in other words, constituted the new standard of both temporal and spatial measurement. But how was one to go about measuring lengths in a frame in motion, where measuring rods are Lorentz-contracted? Poincaré's measurement problem called for a solution, and after a while, Poincaré provided one. In lectures at the Sorbonne in 1906-1907, he interpreted the Lorentz transformation with respect to a geometric figure representing a meridional section of the wavefront of a light pulse, as judged by an observer at rest with respect to a luminous source in uniform motion. I will refer to Poincaré's figure as a "light ellipse". The light ellipse is a staple of Poincaré's kinematics of relativity, in that he illustrated his view with this device on three occasions in public during the final six years of his life, from 1906 to July 12, 1912, and four times in print. A major change in Poincaré's model of the Lorentz group took place in 1909, which will be the discussed in the next section. 5 From an historical standpoint, Poincaré's view of the propagation of light from a source in uniform motion merits our attention for two reasons. His light ellipse was, first of all, a graphical illustration of kinematic relations in relativity theory, the first in a long line of techniques designed to display the relations of relativistic kinematics. Secondly, Poincaré's model stands as the first of many attempts to reconcile an assumed Lorentz-covariance of physical laws with Galilean kinematics. These attempts have been largely neglected by historians, belonging as they do to a program that was increasingly marginalized by the success of Einstein-Minkowski relativity theory. 6 While the inspiration for Poincaré's light ellipse remains obscure, it may have been a response to Einstein's derivation of the Lorentz transformation, mentioned above. A more likely source of inspiration is Langevin's paper on electromagnetic inertia, also mentioned above, which features a discussion of the electromagnetic waves produced by an electron in motion with sublight velocity. Inventing such a diagram was not unusual for Poincaré, known for his skillful employment of diagrams to convey new ideas in mathematics and physics. Several of his figures gained notoriety over the years, including the disk and half-plane models of hyperbolic geometry, and the return map in 5 The three public presentations of the light ellipse are the following: (1) lectures at the Paris Faculty of Sciences, 1906Sciences, -1907, student notes of which were edited by Marguerite Chopinet (Poincaré, 1953), (2) a plenary lecture delivered at the annual meeting of the Association française pour l'avancement des sciences, published in two versions (Poincaré, 1909(Poincaré, , 1910a)), and (3) and lectures delivered at the École supérieure des postes et des télégraphes in Paris in June, 1912, published posthumously as (Poincaré, 1913). The light ellipse appeared first in print in a popular article on the dynamics of the electron (Poincaré, 1908a), reedited in the essay collection Science et méthode (Poincaré, 1908b).

6 For a notable exception, see [START_REF] Chang | A misunderstood rebellion: the twin-paradox controversy and Herbert Dingle's vision of science[END_REF].

systems dynamics.

The first known record of the light ellipse is found in notes taken by Henri Vergne, a student of Poincaré's lectures in the 1906-1907 school year at the Paris Faculty of Sciences in the chair of Celestial Mechanics and Theoretical Astronomy. 7 Entitled "The limits of Newton's laws", these lectures were the first to present the principle of relativity in a university setting, but were not immediately edited. As Jean Chazy, a successor of Poincaré in the Chair of celestial mechanics, recounted the story in the preface to the publication, he asked Vergne if he could use the latter's notes for his own lectures on the theory of relativity, and in response, Vergne made him a gift of the notes. Years later, Chazy sought to publish Vergne's notes, and found in Marguerite Chopinet, astronomer at the Bordeaux Observatory, a willing editor. Chopinet's edition of Vergne's notes appeared in the Bulletin astronomique in 1953.

Chopinet's edition of Vergne's notebooks long served as the principal source for study of Poincaré's 1906Poincaré's -1907 lessons. lessons. Until recently, it was the only source available, because the fate of Vergne's notebooks was unknown. This was a particularly frustrating state of affairs, since Chazy had noted in his preface to Chopinet's edition that the "mise au point" of Vergne's notes was "delicate". Comparison with Vergne's recently-rediscovered notebooks reveals Chopinet's numerous editorial additions, suppressions, and alterations, some of which are highly significant for historians of early-twentieth century physics and astronomy. A new, critical edition of Vergne's notebooks is now forthcoming. In what follows, remarks on Poincaré's 1906Poincaré's -1907 lectures refer only to Vergne's notebooks. 8 In his Sorbonne lectures, Poincaré wanted to show that length and time measurements are transitive for inertial observers, transitivity being a sign of objectivity. To do so, he imagined a light source in uniform motion of velocity v that passes through the coordinate origin O at time t 0 = 0. At a later time t 1 > 0, the source reaches a point B = vt 1 , such that the light-wave originating at time t 0 and propagating in all directions with speed c has a spherical wavefront of radius ct 1 . Figure 1, redrawn from Vergne's notes, shows two light circles associated with three successive positions of the source: O, A, and B. The larger light circle has center O, and the smaller has center B, as judged by an observer at rest with respect to the ether, with 7 Henri Vergne (1879-1943) obtained his Ph.D. in mathematical sciences at the Paris Faculty of Sciences in 1909 [START_REF] Vergne | Contribution à la théorie des ondes liquides[END_REF], and edited Poincaré's 1910Poincaré's -1911 lectures on cosmogony for publication (Poincaré, 1911). Vergne later joined the faculty of the École Centrale in Paris, and served as president of the French Mathematical Society in 1939. 8 Vergne's three notebooks were discovered in 2012 in the archives of the Bordeaux Observatory by Jérôme de la Noë, who arranged for their transfer to the Poincaré Archives. Poincaré then considered how a light sphere would appear if it were measured with rigid rods in uniform motion:

Now I take a rigorously spherical surface. I measure it with my meter stick, which will be contracted in the direction of motion by a factor α, [such that] its true length will become 1/α. Therefore, my diameter in the direction of motion will have a measured length of α. The measured length in the perpendicular direction will be 1. So a sphere will appear [as] an ellipsoid elongated in the direction of motion.9 (Vergne, notebook 2, 49-50) Vergne, 1906-1907 (Poincaré Archives). Labels H and A are added for legibility, and notation is modernized.

Lorentz contraction γ = 1/ 1 -v 2 /c 2 Semimajor axis a = OA = γct Semiminor axis b = OH = ct Eccentricity e = 1 -b 2 /a 2 = v/c Focal length f = OF = γvt Light path ρ = F M Apparent displacement x = F P
Vergne illustrated this measurement protocol with a diagram, in which a circle with two orthogonal radii labeled "1" and "α", in the vertical and horizontal directions, respectively (Figure 2).

As it stands, Poincaré's measurement protocol leaves us guessing about just who does the measuring, and this has led to some confusion. 10 It stands to reason, however, that Poincaré meant to measure a light shell with flying meter sticks, at a single instant of true time, as determined by a clock at rest with respect to the ether. 11 In order to demonstrate the transitivity of apparent time, Poincaré required ideal clocks at rest in the ether, rigid rods in motion, and light-point coincidences. The state of motion of the observer was incidental to the demonstration, in that the observer was assumed to be colocated with light-point coincidences. And while Vergne's notes do not make explicit the latter assumption, they do employ the notion of a "point éclairé" or light point, and treat the distance between two such light points as null, or as an absolute invariant (Vergne,notebook 2,51).

With these kinematic elements in place, Poincaré observed how the light path is related to apparent displacement. Poincaré's argument proceeded as follows, based on the ellipse dimensions shown in Figure 3. From the diagram, Poincaré read off the standard relation for an ellipse with focus F :

F M + F P e = a(1 -e 2 ), (3)
sera devenue 1/α. Donc mon diamètre dans le sens du mouvement aura pour longueur mesurée α. Dans le sens perpendiculaire la longueur mesurée sera 1. Donc une sphère paraîtra un ellipsoïde allongé dans le sens du mouvement." The factor α is defined to be α = 1/ 1 -v 2 /c 2 , using modern notation.

10 For example, in Chopinet's edition of Vergne's diagram, the longitudinal radius is assigned length 1/α. Cuvaj, who referred to Chopinet's edition, considered that measurement of the light shell was performed by comoving observers (Cuvaj, 1970, 74).

11 I agree here with Darrigol (2012).

and then solved for t :

t = γ -1 t -vx /c 2 , (4)
the latter equation revealing the apparent time t to be a linear function of apparent displacement x , as desired. 12 The resemblance between (4) and ( 2) is remarkable, differing only by a factor γ -1 . Poincaré seems not to have paid attention to this factor at first, but he did so in later years, as the next sections will show. By simply rearranging (4), we obtain the transformation

t = γ(t + vx /c 2 ), (5)
and upon substitution for x , we find t in terms of x:

t = γ(t -vx/c 2 ). (6)
The latter two equations appear in the same form (but with different notation) in Vergne's notes, where their meaning is interpreted with respect to the light ellipse as follows:

Likewise, t is the apparent time, since at two points the apparent time differs by a quantity proportional to the abscissa. 13 (Vergne, notebook 2, p. 51)

The latter remark clearly refers to Poincaré's equation (4). Consequently, the light ellipse was a means for Poincaré to convey not only the transitivity of time measurement in moving frames, but also the geometric and physical meaning of the Lorentz group. Insofar as the light ellipse represented a locus of light for a single light pulse, it was ill-suited to represent the exchange of light signals between comoving observers. This was not an obstacle for Poincaré, who simply reversed the sign of the abscissae difference: I imagine that I'm realizing an exchange of signals: for the return [trip] from M to F :

t -= F M k - e k
F P (because the abscissa has a sign).

12 Using the relations specified in Figure 3, we have

a(1 -e 2 ) = a(1 -(1 -b 2 /a 2 )) = a(1 -(1 -c 2 t 2 /a 2 )) = ac 2 t 2 /a 2 = ct/γ.
Rearranging the latter expression in terms of t, we find t = aγ(1 -e 2 )/c, and substituting the value of a(1 -e 2 ) from (3) we obtain Poincaré's expression (4) for apparent time, t .

13 "De même t est le temps apparent puisq'en 2 points le temps apparent diffère d'un quantité proport[ionnel] à l'abscisse."

Here we have

t + -t - 2 = e k F P.
Therefore the difference is rigorously proportional to the abscissae difference, without neglecting the square of aberration. 14 (Vergne, notebook 2, 50, original emphasis, τ and τ 1 replaced by t + and t -)

Updating the notation, denoting the light time-of-flight from F to M by t + , and that from M to F by t -, and using (5), we can express the above expressions as:

t + = γF M/c + γvx /c 2 , t -= γF M/c -γvx /c 2 , t + -t - 2 = γvx /c 2 .
Notice that Poincaré's thought experiment involves observers in motion exchanging light signals, but does not involve clocks in motion, since light time-of-flight is measured in true time, t, not apparent time, t . Comoving observers have no access to true time, of course, which may be one reason why Poincaré never used this particular thought experiment again. In summary, using a thought experiment, Poincaré compared the form of a light pulse for an observer at rest with respect to the ether, at a single instant of time in the ether frame, for two cases: (1) meter sticks at rest in the ether frame, and (2) meter sticks in uniform motion. He judged the form in the first case to be that of a sphere of radius ct, and in the second case, that of an ellipsoid of rotation, elongated in the direction of motion of the meter sticks, with semiminor axis of length ct, and semimajor axis of length γct. He associated the dimensions of the ellipsoid with the length and time variables of the Lorentz group, but did not interpret the light ellipse in terms of clocks in motion.

The foregoing reading of Poincaré's light ellipse is passingly strange, in that it has Poincaré blithely mixing kinematic quantities from different inertial frames. Braving the risk of anachronism for a moment, let's consider Poincaré's thought experiment with a cognitive tool he did not dispose of: the three-dimensional Minkowski diagram. According to the interpretation of the Lorentz group in Vergne's notes, the radius vector of the light ellipse corresponds to light points at an instant of time t as read by clocks at rest 14 "Je suppose que je fasse l'échange des signaux: j'aurai p[ou]r le retour de M en F :

t 1 = F M
k -e k F P (car l'abscisse a un signe). Ici in the ether frame. The representation of this situation on a Minkowski diagram is that of an ellipse contained in a spacelike plane of constant time t (Figure 4). The ellipse center coincides with spacetime point B = (vt, 0, t), and the points E, B, F , and A lie on the major axis, such that BH is a semiminor axis of length ct. The light ellipse intersects the lightcone in two points, corresponding to the endpoints of the minor axis, H and I. There are no moving clocks in this reading, only measuring rods in motion with respect to the ether, and clocks at rest. (The t -axis is suppressed in Figure 4 for clarity).

The abstract nature of Poincaré's early interpretation of the Lorentz group is apparent in the Minkowskian representation, in that there are points on the light ellipse that lie outside the lightcone, these points representing locations in spacetime physically inaccessible to all inertial observers sharing a spacetime origin. In four-dimensional Minkowski spacetime, the intersection of the light sphere with center E and the light ellipsoid with center B, where E and B lie on a spacelike plane, is a circle of radius ct. On a three-dimensional Minkowski spacetime diagram, where one spatial dimension is suppressed, the corresponding circle with center E and ellipse with center B intersect in two points, labeled H and I, such that EH = EI = BH = BI = ct. The diagram shows that in Poincaré's lightellipse model of the Lorentz group, there are physical events bereft of any causal connection to observers that are nonetheless ascribed to these same observers. 15Minkowski diagrams were not at hand when Poincaré delivered his lectures at the Sorbonne, but other cognitive resources were available, which could be applied to Poincaré's thought experiment. For example, for those who had been following the work of Einstein, there was the notion of time dilation. Yet this notion, too, is absent from Vergne's notes. Instead of time dilation, what we find is a recapitulation of Poincaré's operational (firstorder) definition of local time, mentioned above, along with the following comment on Lorentz contraction and clock synchronization: So Lorentz assumes that all bodies undergo a contraction in the direction of motion proportional to the square of velocity. Lengths are then altered, and durations are altered by the impossibility of setting watches truly, such that the apparent velocity of light is constant. Then we perform a Lorentz transformation: the Lorentz transformations must form a group, such that we have identically

x 2 + y 2 + z 2 -t 2 = x 2 + y 2 + z 2 -t 2 . 16 (Vergne, notebook 2, 52)
What Poincaré taught his students, in other words, was that clocks in motion cannot be set properly, and consequently, are unreliable timekeepers. This view of clocks in motion is consistent with his earlier operational definition of local time (mentioned above), according to which light-synchronized clocks in uniform motion are reliable only to first order in v/c. It is also consistent with his light-ellipse interpretation of the Lorentz group, which puts meter sticks in motion to measure a light shell, but which does not put clocks in motion. Perhaps most compellingly, the strange blend of kinematic attributes from different frames of motion manifested in the Poincaré's presentation of the light ellipse faithfully mirrors that of the terms in Poincaré's formula for apparent time (4), where apparent time is the sum of true time and the product of velocity with apparent displacement, divided by the square of the speed of light. In the next section, I will show that Poincaré's confidence in moving clocks increased over time, leading him to embrace time deformation. dimensional (3+1) vector space [START_REF] Walter | Hypothesis and convention in Poincaré's defense of Galilei spacetime[END_REF].

16 "Donc Lorentz suppose que tous les corps subissent dans la direction du mouvement une contraction proport. au carré de la vitesse. Alors les longueurs sont altérées, et les temps sont altérés par l'impossibilité de régler vraiment les montres, ce qui fait que la vitesse apparente de la lumière est constante. Alors nous effectuons une transformation de Lorentz : les transfor. de Lorentz doivent former un groupe, tel que l'on ait identiquement

x 2 + y 2 + z 2 -t 2 = x 2 + y 2 + z 2 -t 2 ."

Poincaré on time deformation

The light ellipse was known only to Poincaré and his students until he published an essay in the March, 1908 issue of Revue générale des sciences pures et appliquées. This essay recapitulated the presentation of the light ellipse found in Vergne's notes, with one important exception. Where Vergne's notes defined "apparent time" according to (4), and identified this time with the temporal variable t appearing in the Lorentz transformation, in the Revue Poincaré called the quantity γt "apparent duration" (Darrigol, 2012, note 69). Despite this change in nomenclature, when discussing the light ellipse, Poincaré kept his clocks firmly planted in the ether, and his meter sticks flying by at constant velocity.

This change in nomenclature is linked to Poincaré's growing appreciation of the fact that all inertial frames are physically equivalent, and that this equivalence extends to timekeepers. His discussion of the light ellipse in the Revue concluded with the following remark:

[I]t is impossible to escape the impression that the principle of relativity is a general law of nature, that we could never make manifest, by any means imaginable, anything other than relative motions, by which I mean not just velocities of bodies with respect to the ether, but velocities of bodies with respect to each other.17 (Poincaré, 1908a, 393) One consequence of the equivalence of inertial frames -including the ether frame -is that if timekeepers can be synchronized in one frame, then they can be synchronized in every frame. Furthermore, if the ether is not to be distinguished by physical means from any other inertial frame, then mixing kinematic attributes from different frames, which is just what Poincaré did in the pages of the Revue, makes little sense. It is curious that Poincaré did not immediately revise his presentation of the light ellipse to reflect this understanding of the principle of relativity, but as I shall show in the next section, such a revision would soon be at hand.

The roots of Poincaré's coming revision of the light ellipse are apparent in Poincaré's change in nomenclature, an echo of which is to be found in his subsequent discussion of relative velocity. Poincaré's approach to the question of relative velocity in the Revue is of particular interest. While he had previously disclosed the relativistic law of velocity addition, which he read off the infinitesimal generators of the Lorentz group (Poincaré, 1906b), he had not shown how to interpret this law operationally. The way to do this, Poincaré explained in a nutshell to the readers of the Revue, was to employ local time, instead of "true" time. The local time in question was not the first-order version Poincaré had advanced in 1900, but that of the modern Lorentz transformation, with a factor α = γ -1 , as in (4). Furthermore, the evaluation of local time was not to be performed with flying rods and a light pulse, but with clocks in motion.

Poincaré imagined a pair of observers located at points A and B at relative rest, in motion with respect to a given frame S, with velocity 2×10 8 m/s, and a body with an equal velocity moving in the opposite direction, and passing successively through points A and B, such that the velocity difference of the observer pair and the body is 4 × 10 8 m/s with respect to S, well in excess of the speed of light. To judge the velocity of a body with respect to the pair of observers in the moving frame, Poincaré explained, it was necessary to refer to local time. The velocity of a body in motion relative to these observers was determined by . . . noting the instant it passes through A in the local time of A, and the instant it passes through B in the local time of B, such that the trip duration will be α(ε + h), and the relative velocity: AB/α(ε + h). And this is how the compensation happens.

Since the quantity in parentheses, ε + h, is equal to t -vx /c 2 , it is just what Poincaré defined operationally as local time in 1900. 18 Consequently, what Poincaré referred to here as the "trip duration" is just what he had called "apparent duration" in his discussion of the light ellipse a few pages earlier, or γt . The "compensation" mentioned by Poincaré refers to the difference in rate of the observers' timekeepers, compared to that of timekeepers at rest in the ether, which mark "true" time.

Roughly a year later, Poincaré's confidence in clocks in motion was on the rise. Much had happened in the previous year to motivate a new look at time deformation in relativity theory. Hermann Minkowski had proposed his theory of spacetime, based on the same four-dimensional vector space introduced by Poincaré. According to Minkowski's theory, clocks in motion read "proper time", defined to be the parameter of their worldline in spacetime [START_REF] Minkowski | Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern[END_REF]. Einstein made a number of bold predictions concerning the behavior of atomic vibrations, including the manifestation of time dilation via a transverse Doppler effect in the spectrum of canal rays (Einstein, 1907b), and the redshift of spectral lines in a gravitational field (Einstein, 1907a).

The occasion Poincaré chose for a stronger embrace of time deformation was a significant one for Franco-German relations. In the fall of 1908, the mathematician David Hilbert invited Poincaré to deliver a series of lectures in Göttingen, sponsored by the Wolfskehl Foundation. Poincaré did not plan at first to speak about relativity, a field scientists in Göttingen followed closely, following their colleague Minkowski's signal contributions. Minkowski and Hilbert were great friends, and the former's sudden death in January, 1909, grieved Hilbert deeply, as the latter shared in a letter to Poincaré a few weeks after the sad event. Hilbert then asked Poincaré to consider supplementing the lectures he had scheduled for "Poincaré-week" with a talk on mathematical physics or astronomy. In response to Hilbert's request, Poincaré delivered a talk in French on the "new mechanics", which took up the theory of relativity and its consequences for astronomy. 19On the opening day of Poincaré-week, Hilbert and his wife Käthe entertained Poincaré at their home in Göttingen, in the presence of Felix Klein and other guests. Klein turned sixty that day, and Hilbert jumped on the occasion to honor his colleague, and to pun upon a singular achievement of his recently-lost friend: Your little ship is still moving ahead with youthful vigor at a full clip. Minkowski taught us that the concept of simultaneity (Gleichzeitigkeit) is relative. This applies even more so to the concept of age (Gleichaltrigkeit). (Cited by [START_REF] Rowe | Hilbert on Poincaré, Klein, and the world of mathematics[END_REF] Hilbert's joke about relative simultaneity had an edge to it, in that unlike Minkowski, Poincaré had not yet adopted the notion of frame-dependent time.

Three days later, in the course of his sixth and final Wolfskehl lecture, on 28 April, 1909, Poincaré returned to the topic of relative velocity that he had discussed in the pages of the Revue générale des sciences pures et appliquées, as mentioned above. This time, however, he introduced two observers A and B in relative motion, equipped not just with timekeepers, but with wireless transmitters and receivers. As a member of the French Bureau of Longitudes, and a leading expert on wireless telegraphy, Poincaré naturally equipped his observers with the means of transmitting time-stamped position data on the fly. 20 In spite of all this equipment, comoving observers were still unable to detect their absolute motion:

A can believe he is at rest, and B's apparent speed will be 400000 km/s. If A knows the new mechanics he will say to himself: "B has a speed that he cannot attain, so it must be that I, too, am in motion." It seems that he could determine his absolute situation. But he would have to be able to observe B's motion. To make this observation, A and B begin by setting their watches, then B sends telegrams to A indicating his successive positions; putting these signals together, A can give an account of B's motion, and trace its curve. Well, the signals propagate at the speed of light; the watches marking apparent time vary at every instant and it all will go down as if B's watch were fast. 21 (Poincaré, 1910b, 54-55) The tabulation of telemetric data would, in principle if not yet in practice, suffice to show that the watches of the two observers in relative motion did not run at the same rate. This is the first time that Poincaré explicitly invoked the deformation of time due to translation, although as shown above, a year earlier he employed the notion in his algebraic analysis of relative motion for the Revue générale des sciences pures et appliquées.

The concept of time deformation employed by Poincaré in Göttingen and thereafter was quite distinct from that of Einstein and Minkowski. For the latter theorists, time dilation and length contraction were kinematic effects, 20 For a review of Poincaré's activity at the Bureau of Longitudes, see [START_REF] Galison | Einstein's Clocks and Poincaré's Maps: Empires of Time[END_REF], and for discussion of his engagement with contemporary technological issues, including wireless telegraphy, see Gray (2013, 177).

21 "A peut se croire au repos et la vitesse apparente de B sera, pour lui, 400000 kilomètres. Si A connaît la mécanique nouvelle il se dira: B a une vitesse qu'il ne peut atteindre, c'est donc que moi aussi je suis en mouvement. Il semble qu'il pourrait décider de sa situation absolue. Mais il faudrait qu'il puisse observer le mouvement de B lui-même; pour faire cette observation A et B commencent par régler leurs montres, puis B envoie à A des télégrammes pour lui indiquer ses positions successives; en les réunissant, A peut se rendre compte du mouvement de B et tracer la courbe de ce mouvement. Or les signaux se propagent avec la vitesse de la lumière; les montres qui marquent le temps apparent varient à chaque instant et tout se passera comme si la montre de B avançait."

or consequences of the four-dimensional (3+1) metric of spacetime, respectively. According to Poincaré, the velocity dependence of measured lengths and durations was best understood as a result of compensating deformations of meter sticks and timekeepers. In the wake of Minkowski spacetime, the absolute space and time of Newtonian mechanics took on a conventional nature for Poincaré, for whom the concept of Galilei spacetime had not lived out its utility for science, and would not do so for some time (Poincaré, 1912).

Henceforth for Poincaré, clocks could be put in motion and used to indicate the time of events for comoving observers. The rate of clocks in motion exceeds that of clocks at rest with respect to the ether by just the amount required by the principle of relativity. This was a significant step for Poincaré, and one that led him to adjust his account of the light ellipse. It also brought his theory into an apparent conflict with those of Einstein and Minkowski, where clocks in motion run slower than those at rest. This difference was linked to Poincaré's reading of the light ellipse, as I will show in the next section.

Three months after Poincaré-week in Göttingen, Poincaré had another opportunity to speak to the topic of the new mechanics. He was invited to deliver the opening lecture of the Congress of the French Association for the Advancement of Sciences held in Lille, on the third of August, 1909, and used the occasion to rehearse his analysis of relative motion, and to explain how the light ellipse captures the behavior of clocks in motion.

In Lille, Poincaré once again considered the form of a light pulse for a source in uniform motion, and modeled light paths with the radius vector of an ellipse elongated in the direction of motion. Denoting the head and tail of the radius vector by A and B, respectively, Poincaré explained that clocks in motion could not reveal the motion to comoving observers: Now let's suppose that an observer is comoving with the luminous body. [. . . ] To this observer, wave surfaces would appear to be elongated in the direction of motion; they will appear to him to be ellipsoidal. All these ellipsoids will be mutually homothetic, and the luminous body will occupy a focus.

Under these conditions, a very elementary geometrical theorem shows that the apparent time required for light to travel from A to B, i.e., the difference between the local time at A when the wave leaves A and the local time at B when the wave reaches B, this apparent time, I say, is the same as if the translational motion did not exist, just as required by the principle of relativity.22 (Poincaré, 1909, 173-174, original emphasis) Dans ces conditions, un théorème de géométrie très simple montre que le temps apparent que la lumière mettra à aller de A en B, c'est-à-dire la différence entre le temps local en A au moment du départ de A, et le temps local en B au moment de l'arrivée en B, que ce temps apparent, dis-je, est le même que si la translation n'existait pas, ce qui est bien conforme au principe de relativité."

According to Poincaré's revision in Lille of the light ellipse, the light pulse is measured with comoving clocks and meter sticks, such that the corresponding figure on a three-dimensional spacetime diagram is an ellipse in a plane of constant time, t . The latter x y -plane intersects the lightcone at an oblique angle, as shown in Figure 5, such that their intersection is a Poincaré light ellipse. (The y-axis and the y -axis are suppressed for clarity). By adopting comoving clocks and observers, Poincaré successfully transformed his light ellipse from a physically-untenable interpretation of the Lorentz group to a relativistic interpretation of time dilation and Lorentz contraction. In the next section, I examine Poincaré's most detailed, and final presentation of the subject of time deformation and the light ellipse.

The light ellipse at the École des postes and télégraphes

In the month of June, 1912, Poincaré delivered a series of lectures for the benefit of students of electrotechnology at an engineering school in Paris overseen by the Ministry of the Post and Telegraph, the École supérieure des postes et des télégraphes, or "ESPT" for short. Poincaré, who was engaged by the ESPT as Professor of theoretical electricity in 1902, delivered courses on advanced topics in electrical engineering, which were usually transcribed by student note-takers for publication in the biweekly Éclairage électrique, or its successor, Lumière électrique, the editorial board of which he was a member. Like the topics of earlier lecture series, that of 1912 was suggested by the Ministry of the Post and Telegraph, which financed and managed the ESPT [START_REF] Guillet | École nationale supérieure des PTT: histoire de la naissance et de la formation d'un corps de l'État[END_REF]. Poincaré acknowledged that his topic differed from those of previous years, which saw him cover questions directly relevant to "engineering arts", such as current propagation, telephony, and wireless telegraphy. Apparently, the Ministry had exhausted its stock of engineering topics ready-made for the application of principles of mathematical physics, and was consequently "forced to exit the circle" of such topics, in order to find one suited to Poincaré's theoretical bent. The topic assigned for the 1912 series was thus chosen to be the dynamics of the electron, said Poincaré, adding that he would also examine the principle of relativity. He expressed the hope that telegraphists, in spite of the absence of any immediate practical application of the theory of electron dynamics, would nonetheless find his remarks to be of interest, inasmuch as they represented "modern ideas relative to the theory of electricity" (Poincaré, 1913, 18).

t a = Kt r + C. (7)
By introducing the coefficient K, we admit that mechanical phenomena are accelerated by a motion of translation. The constants K and C will depend on v, and C will continue to depend on the position of the point. This accounts for the compensation, at least in optics.23 (Poincaré, 1913, 44) The acceleration of mechanical phenomena referred to here undoubtedly included those of mechanical timekeepers, which would explain the remark cited earlier about watches in motion marking apparent time, where the watches were given to be running "fast" with respect to a watch at rest. According to (7), timekeepers run fast or slow depending on the value of K.

From what follows in the notes of Poincaré's ESPT lecture, it appears that Poincaré associated K not with γ, as in ( 6), but with γ -1 , as in (4). Consequently for Poincaré, timekeepers in motion run faster, not slower, than timekeepers at rest. Had he associated ( 7) with (6), he would certainly have come to the conclusion that timekeepers in motion run slower than timekeepers at rest, in agreement with Einstein and Minkowski. While the mathematical form of time deformation is the same here as that employed in Poincaré's discussion of relative motion in the Revue générale des sciences pures et appliquées, four years earlier (letting K = α, see note 18), a more transparent expression of the basis for time deformation is not to be found in Poincaré's writings. The proof that the light ellipse is an isochronous figure proceeded much in the same way as in his previous discussions of the light ellipse, by characterizing the difference between two observer stations located on the ellipse, one of which coincided with a focus F . This time around, however, Poincaré introduced a directrix P Q and a point M 0 such that M 0 F is orthogonal to the major axis (Figure 6).

The light ellipsoid (for which the ellipse is a meridional section) corresponded now as ever to a real light sphere of radius b = ct. In the "ideal" space of comoving observers, Poincaré explained, the apparent time t a depends on true time, t r = b/c, such that:

t a = αb/c + β, (8)
where b denotes the semiminor axis length, and the term β is a function of the difference of abscissae of the ellipse point M and the focal point F . Consequently, we can write, using slightly-modified notation:

β = φ(M P -F Q), (9)
where φ is an unknown constant. Since M 0 F is equal to the semilatus rectum p, M 0 F is related to F Q by the eccentricity, e:

M 0 F = p = eF Q. ( 10
)
With these elements in place, Poincaré announced his objective (letting c = 1):

We want the apparent time to be the same as if [the observer] were at rest; in other words, we want to have αb + β = M F . 24 (Poincaré, 1913, 46) What Poincaré wanted to show his audience, was that the apparent time t a is measured by the radius vector ρ of the ellipse, t a = ρ/c = M F/c, just as the true time t r is measured by the radius b of the corresponding light circle, t r = b/c. True time was measured against the ordinate of the light ellipse, just as apparent displacement was measured against the abscissa. A quick calculation shows that a judicious choice of the parameter φ preserves the principle of relativity, as Poincaré wished. 25 In harmony with Poincaré's presentation of the light ellipse in Lille, his lesson on the light ellipse at the École supérieure des Postes et des Télégraphes in Paris, three years later, featured comoving observers who employ comoving clocks and Lorentz-contracted meter sticks to discover the form of the light pulse emitted by a comoving source. Poincaré showed that the comoving observers are unable in principle to detect their motion, filling a gap left in his presentation in Lille, and in full agreement with the principle of relativity.

24 "Nous voulons que le temps apparent soit le même que si l'on était au repos, c'est-à-dire que l'on ait : αb + β = M F ."

25 From (9) and αb + β = M F , we can write:

M F = αb = β = φ(M P -F Q),
and by inspection of Figure 6: M F = eM P.

Letting φ = e, we obtain:

αb = φF Q = eF Q. ( 11
)
Recalling the identity p = b √ 1 -e 2 , along with (11), we find:

α = 1 -e 2 . ( 12
)
The published notes of Poincaré's lectures at the ESPT do not develop the argument beyond ( 12), but if we recall the values of the parameters (reintroducing c): e = v/c, b = ct, and α = 1 -v 2 /c 2 , we find the transformation: t a = (αb + β)/c = (αct r + vx /c)/c = αt r + vx /c 2 .

Recalling that t a = t , and t r = t, the latter equation is seen to be equivalent to (4), which can be rearranged to give the modern Lorentz transformation (6), as shown above.

Discussion

Historical and philosophical analyses of the discovery and early development of the theory of relativity have shown that the scientific community did not welcome the relativity revolution, but required a great deal of convincing before it abandoned the competing worldviews. 26 The energetic and vocal promotion of Einstein's theory on the part of Max Planck and others brought it significant attention, but many questions remained concerning the theory's logical and mathematical cogency, empirical validity, and potential utility in theoretical and experimental realms. In 1908, Minkowski's theory of spacetime resolved several of the quandaries of the first sort, while the results of Bucherer's electron-deflection measurements did much to establish the empirical credentials of relativity theory. Poincaré's review in March, 1908, of the outlook for the principle of relativity participated in this movement, by affirming the general validity of the principle of relativity for all inertial frames.

Poincaré's message, however, was quite different from that of Einstein or Minkowski. While Poincaré presented the first geometric interpretation of the Lorentz group, the image he presented -that of a light ellipse -was mathematically impeccable, but flawed from a physical standpoint. The next year, in 1909, Poincaré invoked the notion of translation-induced time deformation, and introduced comoving observers and clocks to his light-ellipse interpretation of the Lorentz group. His reasons for doing so are open to speculation, but my guess is that, like so many other theorists in physics and mathematics, Poincaré was impressed by Minkowski's spacetime theory, where the time read by clocks is a path-dependent quantity. In any case, once Poincaré had revised his presentation of the light ellipse so that it featured comoving clocks, meter sticks and observers, the ellipse could be represented in a spacetime diagram as an oblique section of a three-dimensional light cone, thereby bringing his own theory of relativity closer to that of Minkowski. An important distinction remained between the theoretical views of Poincaré, Einstein and Minkowski, despite Poincaré's embrace of clocks in motion. As far as Einstein and Minkowski were concerned, if a light wave was spherical in one inertial frame, it was spherical in all such frames. Not so for Poincaré, who held that, because of Lorentz-contraction of solids in the direction of motion, comoving observers would necessarily conclude that a light shell measured as a sphere was in fact an ellipsoid of rotation, elongated in the direction of motion. This fact was of no use to comoving observers in objectively determining their state of motion, and it appears to have misled Poincaré into believing that clocks in motion run faster than clocks at rest.

None of Poincaré's contemporaries appears to have employed, or even commented upon the light ellipse, and in this sense, it is safe to say that the diagram was a faithful expression of Poincaré's understanding of the Lorentz group. The light ellipse was a valuable pedagogical device for Poincaré, since he rarely missed an occasion to employ it. The radical revision of his presentation of the light ellipse in 1909, replacing observers and clocks at rest in the ether by comoving clocks, meter sticks and observers, brought his theory into compliance with the Einstein-Minkowski interpretation of the Lorentz group. At the same time, it suggested that one could adopt the principle of relativity without sacrificing Galilei spacetime. Poincaré convinced himself that such was the case, while recognizing that one could just as well adopt Minkowski spacetime as a basis for reasoning about physical processes. The choice facing scientists, Poincaré argued in 1912, was not between two theories, or two classes of theories, but a conventional one between spacetimes [START_REF] Walter | Hypothesis and convention in Poincaré's defense of Galilei spacetime[END_REF].

  little more than a year after the resolution of the Crémieu-Pender controversy. Along with Gaston Darboux, Émile Picard, Paul Langevin and others, Poincaré was a member of the French delegation to the Congress of Arts and Science held during the World's Fair in St. Louis. What threatened Lorentz's theory in St. Louis was the notion of local time.

Figure 1 :

 1 Figure 1: A lightsource in uniform motion from O to B, after Vergne's notes (Poincaré Archives).

Figure 2 :

 2 Figure 2: Poincaré's light sphere measurement protocol, after Vergne's notes (Poincaré Archives).

Figure 3 :

 3 Figure 3: Poincaré's light ellipse, after manuscript notes by HenriVergne, 1906-1907 (Poincaré Archives). Labels H and A are added for legibility, and notation is modernized.

Figure 4 :

 4 Figure 4: Spacetime model of Poincaré's light ellipse (1906) in a spacelike plane (t =const.).

Figure 5 :

 5 Figure 5: Spacetime model of Poincaré's light ellipse (1909) in a spacelike plane (t =const.).

Figure 6 :

 6 Figure 6: The light ellipse presented at the ESPT, after Poincaré (1913).

  The lectures delivered by Poincaré at the ESPT are among the last that he would have the chance to deliver, and he passed away before he had the opportunity to review the notes transcribed by Jean-Baptiste Pomey, one of Poincaré's fellow professors. In what follows, I focus on just two of the many topics treated in detail by Poincaré: time deformation and the light ellipse.Poincaré began his discussion of time deformation by recalling his operational definition of local time, which guaranteed the relativity of optics to the first order of approximation. Denoting apparent time by t a and "real" time by t r , Poincaré rewrote the expression for apparent time: t a = t r + C, where C is a constant depending on position (in Poincaré's original formulation, C = -vx/c 2 ). He continued: But this is insufficient, and we have to suppose further that

Poincaré to Lorentz, ca. April

27, 190527, (Walter et al., 2007, §2-38-2), §2-38-2).

See[START_REF] Langevin | Sur l'origine des radiations et l'inertie électromagnétique[END_REF], referred to in(Poincaré, 1906b, 151).

"Vous supposez = 1. Langevin suppose k 3 = 1. J'ai essayé k = 1 pour conserver l'unité de temps, mais cela m'a conduit à des conséquences inadmissibles." The symbols k and were defined to be numerical constants:k 2 = c 2 /(c 2 -v 2 ), where c is the vacuum speed of light, and v is frame velocity, while was defined by Lorentz to be a function of v, = 1 for v = 0, differing from unity for small velocities by a second-order quantity at most(Lorentz, 1904, 813).

See Cuvaj (1970, 74),Wright (1975, 453),Darrigol (1995;[START_REF] Damour | [END_REF][START_REF] Duplantier | [END_REF],[START_REF] Damour | Si Einstein m'était conté[END_REF],[START_REF] Pierseaux | Special relativity: Einstein's spherical waves versus Poincare's ellipsoidal waves[END_REF], and[START_REF] Rougé | Relativité restreinte : la contribution d'Henri Poincaré[END_REF].

"Alors je prends une surface rigoureusement sphérique. Je la mesure avec mon mètre: dans la direction du mouvement, mon mètre sera contracté de α : sa longueur vraie

In 1912, Poincaré singled out the set of events which can be neither the cause nor the effect of other given events, and attributed its discovery to the employment of a four-

"[I]l est impossible d'échapper à cette impression que le principe de relativité est une loi générale de la Nature, qu'on ne pourra jamais par aucun moyen imaginable mettre en évidence que des vitesses relatives, et j'entends par là non pas seulement les vitesses des corps par rapport à l'éther, mais les vitesses des corps les uns par rapport aux autres."

Poincaré denoted the true time at which the body reached point A by t, and the true time at which the body attained point B by h, such that the elapsed true time for the body's displacement between points A and B is (t + h) -t = h. By comparison with (4), α = γ -1 , and since h is the true elapsed time, t, we have ε = -vx /c 2 . Likewise, Poincaré denoted the times of passage of the body in terms of true time, t. The time at which the body reaches point A is αt for observer A and α(t + ε) for observer B. The time at which the body reaches point B is α(t + h) for observer A, and α(t + h + ε) for observer B. Subtracting the time of the body's passage through A as marked by observer A from the time of passage of the body through B as marked by observer B results in Poincaré's value for trip duration from A to B, α(ε + h).

Hilbert toPoincaré, 25 February, 1909, henripoincarepapers.univ-lorraine.fr/ chp/text/hilbert-1909-02-25.html. On research traditions in mathematics and mathematical physics in Göttingen at the turn of the twentieth century, seeRowe (1992) and[START_REF] Corry | From Grundlagen der Geometrie to Grundlagen der Physik[END_REF].

"Supposons maintenant un observateur entraîné dans la même translation que le corps

"Mais ceci n'est pas suffisant, il faut encore supposer que l'on a: t a = Kt r + C. L'introduction du coefficient K revient à admettre que les phénomènes mécaniques sont accélérés par un mouvement de translation. Les constantes K et C dépendront de v, et C continue à dépendre de la position du point. Ceci rend compte de la compensation tout au moins en optique."

For an overview, see[START_REF] Staley | Einstein's Generation: The Origins of the Relativity Revolution[END_REF].

Acknowledgments

The impetus for this paper was provided by the discovery of Vergne's longlost notebooks by Jérôme de la Noë. The third section of the paper benefited from discussions with Olivier Darrigol.


Peixiang Ma

Paul Schanda

We review recent advances in methodologies to study microseconds-to-milliseconds exchange processes in biological molecules using magic-angle spinning solid-state NMR (MAS ssNMR) spectroscopy. The particularities of MAS ssNMR, as compared to solutionstate NMR, are elucidated using numerical simulations and experimental data. These simulations reveal the potential of MAS NMR to provide detailed insight into short-lived conformations of biological molecules. Recent studies of conformational exchange dynamics in microcrystalline ubiquitin are discussed.

INTRODUCTION

The three-dimensional structure of a biomolecule is dictated by a subtle balance of numerous interactions within the molecule and with its surrounding. These interactions are all individually weak, and at physiologically relevant temperatures they are continuously rearranged. As a consequence, biomolecules populate a multitude of different conformations, differing in their structural and thermodynamic properties, constantly exchanging in a dynamic equilibrium. Molecular functions, such as enzymatic activity, allosteric regulation, or transport across membranes, often critically depend on the relative populations of these states, as well as the rates of the interconversion. 1,2 In order to understand biomolecular function, it is therefore crucial to determine not only a single, three-dimensional structure, that may be considered as the "lowest-energy" conformation, but also to understand what alternative conformations are sampled, and at which rates they interconvert. Many biomolecular functional processes take place within microseconds to seconds, and this time scale is therefore of particular interest when relating protein motion to activity. An increasing number of studies reveal a direct link between the functional turnover of the activity of a protein and the dynamic interconversion between conformational substates; this link has been exemplified for cases of enzyme catalysis, membrane transport and allostery. [3][4][5][6][7] Solution-state NMR spectroscopy is among the most powerful techniques for elucidating microsecond-to-millisecond (µs-ms) dynamics in proteins and nucleic acids. In solutionstate NMR, the interconversion between different conformers on µs-ms time scales can be probed through the difference of the isotropic chemical shifts of different conformers. The fluctuation of the chemical shift results in line broadening or, equivalently, increased apparent transverse relaxation rate constants, which can be quantitatively probed. In particular, so-called relaxation dispersion (RD) techniques and chemical-exchange saturation transfer (CEST) have proven extremely powerful for visualizing conformational exchange dynamics and thus provide insight into short-lived conformations, even if they are populated to only a few percent. [8][9][10][11][12] The methodology in solution-state NMR has been continuously refined over the last two decades. Nowadays, the available techniques provide a wealth of information about the kinetics of dynamic processes, the relative populations of states as well as dynamics 13 , hydrogen-exchange rates 14 and even atomicresolution structures of the short-lived states. [15][16][17] Magic-angle spinning solid-state NMR (MAS ssNMR) rapidly emerges as a tool in structural biology. Its particular power lies in the fact that it does not rely on the molecular tumbling -that is the basis of high-resolution solution-state NMR -for achieving highlyresolved spectra. This opens possibilities to study proteins that are insoluble (aggregates, fibers, membrane proteins in lipid bilayers etc), or which are very large and slowlytumbling molecular systems (large molecular machines, entire cells or cell walls etc) and thus have unfavorable relaxation properties in solution-state NMR. Currently the methodologies for studying µs-ms dynamics by solid-state NMR are less developed and less widespread than their solution-state NMR counterparts. Yet, for theoretical reasons outlined below, ssNMR offers in principle a greater variety of observables that can report on such motion. In this article, we want to review some of the theory related to measurements of µs-ms dynamics by magic-angle spinning ssNMR. For reasons of space we focus here exclusively on spin-1/2 nuclei, thus leaving out the large body of work performed with spin >1/2 nuclei (in particular deuterium). [18][19][20][21][22][23] We also refer the reader to reviews 10,24 for a more general treatment of biomolecular dynamics measurements by NMR.

NMR OBSERVABLES OF MICROSECOND-TO-MILLISECOND DYNAMICS

When a molecule undergoes exchange between different conformations, this dynamic process leads to changes of several NMR-relevant parameters in the molecule, namely the dihedral angles and electronic environment around the involved nuclei, and the relative orientations and distances between nuclei. These structural changes lead to (i) fluctuations of the chemical shift tensors of the involved nuclei, both in terms of the isotropic component and the orientation and magnitude of the anisotropic chemical shift, (ii) fluctuations of the magnitudes and orientations of the dipolar coupling tensors, and (iii) the scalar couplings (although the latter are generally less important). It is through these fluctuations of NMR observables that one can obtain information about molecular dynamics. Figure 1 schematically represents such exchange dynamics between two differently populated protein conformations.

In solution-state NMR, molecules undergo rapid overall reorientation ("tumbling", typically on a time scale of < 10-100 ns). This overall motion averages the anisotropic (i.e. orientation-dependent) interactions, chemical-shift anisotropy (CSA) and dipolar-coupling tensors, to zero. Consequently, if we consider an exchange process taking place on a time scale of microseconds or longer, then all the chemical-shift anisotropies and dipolar couplings are zero in all the involved states. Therefore, as these anisotropic interactions are equal (zero) in the exchanging states, these quantities cannot be exploited for probing motion. Consequently, only the orientation-independent, i.e. isotropic quantities can provide information about the presence of exchange dynamics in solution-state NMR.

Thus, all available techniques in solution-state NMR for detecting µs-s exchange processes require that the exchanging states differ in their isotropic chemical shifts. ‡ The situation is different in MAS ssNMR, where molecules do not undergo overall rotational motion. In order to achieve high resolution in MAS NMR, the sample is spun at a constant frequency along the magic angle, which averages anisotropic interactions over an entire sample rotation period. The resulting averaging is different to the situation in solution-state NMR: while in solution the tumbling is fast (nanoseconds), spatially isotropic and stochastic, magic-angle spinning occurs on a time scale of tens microseconds and it is a periodic and deterministic process, rather than a stochastic one. This has important consequences in the present context. First, MAS does not perfectly average out the anisotropic interactions, notably in the case of multiple (non-commuting) interactions, in contrast to solution-state NMR. This latter feature means that the effect of these interactions is only scaled down, proportional to the inverse spinning speed, and spin coherences evolve still under the remaining action of anisotropic interactions. In practice, this evolution, in particular the evolution due to dipolar couplings (termed dipolar dephasing), appears as a decay of spin coherence, similar to relaxation (i.e. stochastic fluctuations due to dynamics). This situation creates problems when attempting to access exclusively the dynamics-related part, as shown below. Second, the periodic nature of the sample rotation can possibly lead to interference effects, namely when the sample rotation (tens of microseconds) is on a similar time scale as the dynamic process and/or a radiofrequency (RF) irradiation applied to the sample. These features of MAS ssNMR bring a ‡ This requirement for isotropic-chemical shift differences between exchanging states is released when partially aligned (liquid-crystalline) solutions are used, where anisotropic interactions are re-introduced as residual dipolar couplings and residual CSAs. [START_REF] Cavanagh | Protein NMR Spectroscopy[END_REF][START_REF] Reif | [END_REF] This type of approaches is not within the scope of this review. number of experimental challenges, but also new opportunities with respect to the measurement of µs-ms dynamics, as compared to the more established solution-state NMR techniques.

In this section, we want to investigate the properties of some selected ssNMR experiments that were recently proposed to measure µs-ms exchange dynamics in proteins. We first outline the challenges that one faces when attempting to extract dynamic information from relaxation-type observables. Then we use numerical simulations of a spin system undergoing stochastic conformational exchange and magic-angle spinning to highlight how ssNMR can probe µs-ms motions, and how the experimental observables differ from more widespread solution-state counterparts.

Transverse relaxation parameters in solid-state NMR

The fluctuation of isotropic chemical shifts arising from a dynamic process leads to line broadening, whenever the rate of the dynamic process (in units of s -1 ) is comparable to the fluctuation of the chemical shift (in units of Hz), i.e. in the µs-ms regime. 10,[START_REF] Cavanagh | Protein NMR Spectroscopy[END_REF] Quantifying line broadening induced by conformational exchange, or equivalently transverse relaxation parameters (R 2 , R 1ρ ), is the primary technique to study µs-ms motion in solution-state NMR. Relaxation-dispersion experiments measure line broadening as a function of a variable RF irradiation. Before investigating how dynamic information may be extracted from such measurements in MAS ssNMR, we need to consider whether one is able to extract such parameters in ssNMR without systematic errors. Indeed, in solid-state NMR it is challenging to interpret line widths in terms of dynamics, due to the fact that line widths contain two contributions, (i) a contribution arising from the stochastic ("incoherent") molecular motion that alters the local fields -this is the interesting part in the present context -, and (ii) a contribution from "coherent" dephasing processes, that arises primarily from the evolution of spin coherence under the action of dipolar couplings. This "dipolar dephasing" is a consequence of the fact that MAS cannot simultaneously average out multiple (non-commuting) interactions, and it is thus particularly pronounced in protonrich environments (i.e. nuclei with large gyromagnetic ratio γ), such as a protein. Line widths (or coherence decay rate constants) are therefore generally unsuitable to study dynamics by solid-state NMR. This being said, an obvious solution to alleviate this problem of dipolar dephasing is (i) to observe heteronuclei ( 15 N, 13 C) rather than 1 H spins, (ii) to use deuterated proteins in order to dilute the proton coupling network, [START_REF] Reif | [END_REF] and (iii) to employ high magic-angle spinning frequencies (as anisotropic interactions are scaled down with the inverse of the MAS frequency).

Figure 2 investigates the coherence decay rate constants of 15 N nuclei along the backbone of the protein ubiquitin under MAS. The red curve shows the expected R 2 , which was calculated based on the backbone dynamics that have been determined through an extensive set of NMR observables (but not R 2 ). 27,28 Average rates are approximately 3-5 s - 1 , and site-to-site variations reflect differences in local motion. Black data points show experimentally measured R 2 ' decay rate constants § , i.e. the decay under a Hahn echo sequence, obtained in a highly deuterated protein at high MAS frequency (ν MAS =57 kHz). 29 These data clearly show that even under these favorable conditions (fast MAS and deuteration), 15 N R 2 ' rates are significantly above their expected values. This finding immediately shows that by measuring 15 N R 2 ' rate constants one cannot unambiguously extract information about dynamics, as dipolar dephasing contributes to these rates. The pattern of the black data in Figure 2 shows that some information about dynamics is § The term R 2 ' is used for the observed decay rate in a simple coherence echo experimentcontaining thus coherent and incoherent contributions, to distinguish it from R 2 , which is reserved for coherence decay arising from incoherent stochastic processes (dynamics). nonetheless encoded in R 2 '. For example residues 23 and 55 have much larger-thanaverage R 2 ', and as we will show below they undergo µs motions. It is noteworthy that deuteration still helps reducing the apparent line widths, as R 2 ' rate constants in protonated proteins are generally significantly higher, and require high-power 1 H decoupling. 30 The situation is greatly improved when 15 N R 1ρ is measured, i.e. the decay under a 15 N spin-lock RF field. 15 N R 1ρ data shown in blue are obtained on the same deuterated protein at a MAS frequency of 39.5 kHz. In this case the observed rate constants closely match the expected ones. Two groups have independently proposed to use 15 N R 1ρ rate constants, 31,32 and have advocated deuteration or fast-MAS (above ~40 kHz) to suppress dipolar dephasing in such experiments. Of course, these approaches can be combined, 28 as shown here on a deuterated protein at 39.5 kHz MAS.

To summarize, dipolar dephasing represents an experimental challenge that hampers the extraction of dynamic information from coherence life times/line widths. The solutions are either to find a parameters which is not sensitive to dipolar -the 15 N R 1ρ is such a parameter -, or to measure several observables, which individually are not "clean", but of which the difference reflects only the dynamics. As we will show below, differential decay of zero-and double quantum coherence is such a case; furthermore, measurements of the differences of R 2 ' rates, obtained under different CPMG frequencies also can be interpreted in terms of dynamics to some reasonable approximation. We next consider how these parameters, R 1ρ relaxation dispersion, CPMG relaxation dispersion and differential ZQ/DQ decay, can provide insight into dynamics.

R 1ρ relaxation-dispersion MAS NMR

We first consider dynamics measurements through 15 N R 1ρ relaxation dispersion experiments. In this approach, widely used in solution-state NMR, the decay of 15 N coherence is measured under a 15 N spin-lock RF field; the dependence of the R 1ρ rate constant on the RF field strength reflects µs dynamics. To analyze the properties of this experiment, we perform numerical simulations of a spin system exchanging between two discrete states, as depicted in Figure 1. The two-state exchange model was chosen due to its simplicity and tractability in numerical simulations, but it also seems to reflect µs-ms dynamics in proteins rather realistically: numerous studies indeed showed that few (two or three) conformational substates (and fast local motion within these basins), suffice to model NMR-derived µs-ms dynamics data. 5,6,16,17,33,34 For simplicity we consider only a two-spin system, neglecting remote proton spins; similar simulations with a remote spin are shown elsewhere, 35 and the difference to the present calculations are inconsequential for the present discussion.

Figure 3 shows the simulated R 1ρ rate constant for a 15 N spin undergoing a two-site exchange process between two states populated to 10% and 90%, respectively. We assume here different cases, in which this exchange process alters the isotropic chemical shift of the 15 N spin, the orientation of the 15 N CSA and HN bond orientation, or both. We investigate the dependence of R 1ρ on the strength of the spin-lock RF field, i.e. the relaxation dispersion of R 1ρ. In the case shown by black curves, only the 15 N isotropic chemical shift is altered, while the two conformations have identical CSA and bond orientations. The resulting relaxation dispersion curve in this case shows a decrease of R 1ρ when the RF field increases up to about 10 kHz (a zoom of this part is shown in the bottom of the figure), and is flat for larger RF fields. This dispersion is exactly identical to the case in solution-state NMR, i.e. it can be exactly described by the Bloch-McConnell formalism. 10,36 Of course, when the isotropic chemical-shift difference between the two states becomes zero (dashed line), then no dispersion is observed, as expected because the two states are then exactly identical.

In the more general case that the exchange process also involves a change of the anisotropic interactions, i.e. the HN bond orientation (and possibly length) and CSA tensor orientation (or magnitude), the situation gets more complex. Red, green and blue curves in Figure 3 exemplify such cases, assuming different jump angles. From these curves it becomes evident that the isotropic chemical-shift difference still gives rise to a dispersion of R 1ρ , according to the Bloch-McConnell formalism (see zoom region in Figure 3). However, there is overall an increase of the R 1ρ rate constants, which comes from the fact that the fluctuating anisotropic interactions induce relaxation. This increase of R 1ρ is, however, not uniform, but it is particularly pronounced when the spin-lock RF field strength approaches the sample rotation frequency or twice this frequency, i.e. close to the so-called rotary resonance conditions (ν 1 =ν MAS or ν 1 =2ν MAS ). At these conditions the anisotropic interactions, which are averaged by MAS, become re-introduced, as the spinlock RF frequency and the MAS frequency match. This so-called recoupling occurs even without dynamics. However, the range of RF fields over which this rotary resonance match occurs is significantly broadened by the exchange process, and this broadening depends on the dynamics 35,37,38 . That the exchange process impacts the R 1ρ dispersions around these rotary-resonance conditions is readily seen by comparing the red, green and blue curves in Figure 3 (different jump angles). In addition to the jump angle, these curves furthermore depend on the population of the involved states and the exchange rate constant (shown elsewhere 35,37 ). Taken together, in the general case of a R 1ρ relaxation dispersion experiment under MAS, the dispersion profile is the sum of a dispersion arising from the isotropic chemical-shift difference between the states, and a dispersion arising from interference between MAS, RF and the reorientation of the anisotropic interactions (orientations and magnitudes of dipolar couplings and CSAs). The former part (isotropic chemical-shift difference) can generally be found at RF field strengths of a few kHz, while the latter part (broadened rotary resonance recoupling) is found in the vicinity of the rotary resonance conditions ν 1 =ν MAS or ν 1 =2ν MAS ; when the MAS frequency is sufficiently fast, these regimes are well separated.

It is also important to remember that the isotropic chemical-shift difference is irrelevant for the relaxation dispersion profile close to these conditions. Even in case the states have identical chemical shifts, there would still be a large R 1ρ dispersion observable, which would be informative of the geometric rearrangement of the dynamic process (dashed lines in Figure 3). 37 This possibility to obtain geometrical information about the exchange process -which would be impossible in isotropic solution-state NMR -can be of great advantage to structurally characterize the exchanging states.

The possibility to extract information from R 1ρ relaxation dispersion has been discussed by Krushelnitsky et al 32 , Lewandowski et al 31 , and more recently in a more systematic manner, including the behavior close to the recouping conditions, by Ma et al. 35

Differential decay of ZQ/DQ coherences

As we have shown above, R 1ρ relaxation dispersion experiments provide rich information about µs-ms dynamics. The particular advantage is that dipolar dephasing seems to be sufficiently suppressed by deuteration and MAS (≥ 20 kHz seem to suffice 35,39 ), such that experimentally observed data can be considered as "clean", reporting only on dynamics.

Other parameters may also provide insight into dynamics, but in MAS ssNMR they are "polluted" by contributions other than dynamics. An example of such an observable is the R 2 ' rate, which, as we have shown above, contains a significant contribution from dipolar dephasing. A possible solution to this problem is to measure two observables, both of which are equally impacted by dipolar dephasing, such that the in their difference the unwanted part cancels.

An example of such an approach is the difference between relaxation rates of the two doublet components of a 15 N coupled to 1 H. As shown by Reif, Chevelkov and Skrynnikov, even though the line widths of each of the doublet components are not by themselves reporting on dynamics alone, the difference of the relaxation rates probes the cross-correlated relaxation, which is sensitive to the fluctuation of dipolar coupling and CSA tensor on time scales of nanoseconds to microseconds. [40][41][42] We focus here on another parameter, namely the differential relaxation of double-and zero-quantum coherences. The presence of dynamics that involve fluctuations of the chemical shifts of both involved nuclei alters the rates at which double-quantum (such as H + N + , H -N -) and zero-quantum (H + N -, H -N + ) coherences relax. [43][44][45] Figure 4 shows the differential relaxation rate of DQ and ZQ coherences (ΔR MQ =R DQ -R ZQ ) in a 1 H- 15 N spin pair in MAS ssNMR. In these simulations, which are done very similarly to the above simulations of R 1ρ , it is assumed that the 1 H and 15 N chemical shift undergo a simultaneous change due to the dynamic exchange process. Fluctuation of both isotropic chemical shifts (blue line) leads to differential relaxation whenever the dynamic process occurs on a time scale of microseconds. Fluctuation of the CSA tensors, either in terms of their magnitude or (shown here in green) their orientation, gives rise to differential relaxation whenever the process occurs within tens of nanoseconds to hundreds of microseconds. In the general case both the isotropic and anisotropic component are likely subject to fluctuation upon a dynamic exchange event (red curve). As long as this process occurs on a time scale between tens of nanoseconds to milliseconds, the differential MQ decay will sense such motions through either the isotropic or the anisotropic chemical-shift fluctuations.

It is interesting to note here again, that in solution-state NMR the CSA fluctuation does not sense this time window. Instead, the differential relaxation occurring due to simultaneous CSA/CSA modulation reflects primarily the tumbling and fast internal motion on picoseconds to nanoseconds time scales, but not slower motional processes. This is because any motion beyond the tumbling correlation time cannot be detected through anisotropic interactions.

CPMG relaxation dispersion in MAS ssNMR

CPMG relaxation dispersion NMR is a widespread approach to quantify µs-ms motion by solution-state NMR. It quantifies the R 2 rate constant (most often of 15 N or 13 C) as a function of the repetition rate at which 180° refocusing pulses are applied during the relaxation delay. Figure 5a shows the dependence of R 2 on the CPMG frequency, simulated for the case where the two exchanging states differ only in their 15 N isotropic chemical shift, but not their anisotropic interactions (which means that the behavior is exactly like in solution state). Several factors complicate the situation in MAS ssNMR, as compared to solution.

First, as shown by the data in Figure 2, R 2 ' rate constants in solids are generally not directly exploitable, i.e. the absolute value of R 2 ' is to a large portion reflecting factors not related to motion (dipolar dephasing). This by itself is not necessarily a problem for CPMG relaxation dispersion, where the absolute level of R 2 rate constants is not relevant, but only the differences of R 2 at different CPMG frequencies matter. Second, in the general case where the exchange process also involves fluctuation of anisotropic interactions (bond orientations, CSA tensors), this fluctuation may interfere with the CPMG pulse train and MAS, adding an additional dependence of R 2 ' on the CPMG frequency, on top of the behavior shown in Figure 5a. Numerical simulations in Figure 5b and shown in more detail elsewhere, 46 reveal that these factors alter the Bloch-McConnell type of CPMG RD profile. Under realistic assumptions, the change in the CPMG RD profiles are R 2 rate constants of about <3 s -1 . This means that rather small relaxation-dispersion profiles arising from isotropic chemical-shift fluctuations may be masked by this effect. Third, in a solid sample the volume is not entirely uniformly filled, due to inhomogeneous packing of protein (crystals, liposomes, etc) and solvent. This leads to an anisotropic bulk magnetic susceptibility, which may also give rise to CPMG relaxation dispersion. 47 Taken together, the situation of CPMG MAS ssNMR is more complex than its solutionstate counterpart, and the analysis may be complicated by the fluctuation of anisotropic interactions. As long as the dispersions due to the isotropic chemical-shift fluctuations are large, it is nonetheless possible to extract information about conformational exchange, as shown below.

DETECTION OF CONFORMATIONAL EXCHANGE IN UBIQUITIN

In this last section we show how the above methods can be used to gain insight into conformational dynamics in proteins, using microcrystalline ubiquitin (8 kDa) as an example. Figure 6 shows experimental data from different approaches that provide evidence for conformational exchange; panels (a) and (b) show R 1ρ relaxation dispersion, and panel (c) shows CPMG relaxation-dispersion data. As a first important finding, the majority of residues, here exemplified in the uppermost panels, show flat dispersion profiles. This is expected for residues that do not exhibit conformational exchange, and is a good indication that artefactual factors, such as the dipolar dephasing discussed above, do not lead to erroneous dispersion profiles under the chosen conditions (highly deuterated samples, fast MAS). A subset of residues, clustering in a well-defined part of the protein including the N-terminal part of the α-helix and the adjacent loop, show non-flat relaxation dispersion profiles. Panels (a) and (c) focus on the part of the relaxation dispersion profiles that are sensitive to isotropic chemical-shift fluctuations, i.e. rather low spin-lock RF fields at fast MAS (cf the box highlighted in Figure 3). The residues that show exchange in this regime also display a strong R 1ρ relaxation dispersion under conditions where the RF field strength comes close to the MAS frequency (cf. Figure 3). This latter observation is indicative of a fluctuation of the orientations of the involved bond (and 15 N CSA tensors).

Lines in panels (a)-(c) indicate numerical fits of the cluster of exchanging residues to a global two-site exchange model. 35,46 Such fits reveal that this part of the protein exchanges with a minor conformer, populated to about 10%, at a rate of about 3000 s -1 , and also provides site-specific chemical-shift changes between the major and minor conformers (plotted onto the structure in panel e). Panel (b) shows that it is possible to estimate the angle by which the NH bonds fluctuate through the exchange process (plotted onto the structure in panel f). Finally, Figure 6d shows differential ZQ/DQ decay data, obtained on the same crystals. 46 This experiment shows large differential decay for residues located in the same part of the protein, and the three residues that have increased differential ZQ/DQ decay data set also reveal relaxation dispersion (in both CPMG and R 1ρ experiments).

Taken together, three different methods all point to an exchange process in a well-defined part of ubiquitin's structure (Figure 6 e,f). What is this exchange process mechanistically, i.e. what does the minor conformational state look like? Answers come from both comparison of different ubiquitin structures and from the structure-related parameters obtained from the above experiments, as well as from solution-state NMR experiments detecting exchange dynamics. The available structure of ubiquitin, determined by NMR in solution or X-ray crystallography, fall in two classes, which differ in the conformation of the peptide plane D52/G53 (Figure 7). In the microcrystals used in the presented solid-state NMR experiments, the carbonyl of D52 points outward, while the NH of G53 points towards the helix and forms a hydrogen bond with the side chain of E24, i.e. the loop forms a so-called type-II β-turn (Figure 7a). In solution, by contrast, as well as in many other crystal structures, the D52/G53 peptide plane is rotated by roughly half a turn, forming a type-I β-turn with a different hydrogen-bonding arrangement (Figure 7b). There is evidence from mutation studies, 48 and solution-state relaxation-dispersion experiments [49][50][51] that the exchange observed corresponds to the transition between these two structures, i.e. between type-I and type-II β-turn conformations. This is also our interpretation of the ssNMR dynamics data. If the flip of D52/G53 is occurring along the exchange, then the bond orientations of this region should change most when the molecules undergoes its exchange process. Figure 6f shows that this is indeed the case, i.e. the largest angle changes occur for the peptide planes neighboring D52/G53. The exchange process in solution therefore seems to be present also in microcrystals.

Interestingly, however, the exchange process in solution-state occurs more than one order of magnitude faster than in microcrystals. This can be understood by contacts with neighboring molecules in the crystal lattice (Figure 7a), which constitute additional freeenergy barriers to be overcome in the exchange process. It is also interesting to note that the minor state in solution is the major conformer in microcrystals and vice versa, as seen by inspection of the respective structures determined on microcrystals by X-ray diffraction and solution-state NMR, respectively. Thus, the relative order of the states is inverted in microcrystals as compared to solution, and the energy barriers are altered by the crystal contacts (Figure 7c).

CONCLUSIONS

We have briefly shown in this review some of the approaches developed recently to study µs-ms exchange processes using MAS ssNMR. Using experimental data on ubiquitin and numerical simulations of an exchanging spin system under MAS, we have highlighted the particularities of MAS NMR approaches. The primary challenge resides in the difficulty of accessing "clean" transverse relaxation parameters that solely reflect dynamics. Deuteration, fast MAS and the use of appropriate observables (R 1ρ or differences of coherence decay parameters rather than these parameters themselves) are the key to determining motion. Besides these challenges, the possibility to create interferences between the rates of MAS, RF fields and the dynamic process hold promise for obtaining a wealth of data of exchanging systems, including structural insights into the short-lived transient conformations that may be important for function. Solid-state NMR's potential to provide structural information about such short-lived states even in very large or insoluble proteins will be very valuable for understanding the function of membrane proteins or large functional assemblies. The available studies have so far focused on amide 15 N sites, but it can be foreseen that very similar approaches will become available for other sites, such as additional probes along the backbone (C α , carbonyl), or side chains (e.g. methyl groups). Advances in isotope labeling approaches will facilitate such developments. We believe that methods such as those reviewed herein will help shed light on short-lived states of protein which are currently out of reach for atomic-resolution studies of dynamics. 15 N transverse relaxation parameters obtained in microcrystalline ubiquitin. Red: Backcalculated 15 N R 2 rate constants, based on a detailed analysis of backbone dynamics using HN dipolar order parameters, 15 N R 1 rates at multiple fields and 1 H- 15 N dipolar/ 15 N CSA cross-correlated relaxation rates. 27,28 Black: measured 15 N R 2 ' rate constants with a perdeuterated sample back-protonated at amide sites to 20%, at a MAS frequency of 57 kHz. 29 Blue: 15 N R 1ρ relaxation rates obtained with a perdeuterated, 50% amide-protonated sample at 39.5 kHz MAS. 28 Note the strongly increase R 2 ' rates of residues 23 and 55, which in part point to exchange dynamics (see main text).

Figure 1 .

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Figure 3 .

 3 Figure 3.Numerical simulations of the15 N decay under a continuous-wave spin-lock of variable amplitude, in a spin system undergoing conformational exchange. The 15 N isotropic

Figure 4 .

 4 Figure 4. Simulated differential decay of 1 H-15 N ZQ and DQ coherences in a system undergoing

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 5 Figure 5.

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 6 Figure 6.

Figure 7 .FIGURESFigure 1 .

 71 Figure 7.

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  rf field / kHz spin-lock rf field / kHz CPMG frequency / kHz

Figure 7

 7 Figure 7.

Conformational Exchange Processes in Biological Systems: Detection by Solid-State NMR