arXiv:physics/0409010v1 [physics.gen-ph] 1 Sep 2004 ri ihteclsilshr sntfound. not local is the sphere celestial in- in the local time with of ertia local connection and A the time environment. to that zero-rotation sphere from endpoints distant map- geodesic local space a null to of relative ping rotation the earth’s of the actually rotation apparent sphere the celestial to sense related restricted rate the rotation suggested in is holds it that principle Thus Mach’s that stan- space-time. non-rotation throughout local pho- absolute dard or the the geodesics unify null ab- that paths space-time ton concluded of possible then standard all is off local It field a rotation. everywhere zero be solutely space-time to all these has throughout of rel- valid there general one be this to of for formula that use ativity shown the is with It rela- accepted analysed transformations. general The then in is acceleration situation. tivity centrifugal disc simula- for rotating linear formula the a for using trans- by tion rotation introduced relativistic are new formations explained Two is sphere discussed. celestial rotat- and the relativistic and theory about frame ideas ing on of dependence significance its and and meaning The Abstract ∗ [email protected] hr sauieslsadr faslt local absolute of standard universal a is there oee hsaprn oainis rotation apparent this However . odn meilClee etme,2004 September, College, Imperial London, eaiiyTer ofrneIX Conference Theory Relativity hsclItrrttosof Interpretations Physical colo ahmtclSciences Mathematical of School ahsPrinciple Mach’s ahsPicpeII Principle Mach’s nvriyo London of University ...... ue ayCollege Mary Queen coe 9 2018 29, October ae .Gilson G. James by 1 ne nsm omo te u tpopsaphilo- a prompts it but other experi- or has form or sequence some can simple us in very of enced each a which is events This physical of wall. outward cen- the the the from towards level of tre in rise result the concave a causing force a as centrifugal assumes frictional form then of con- and surface wall downwards because The bucket motion the centre. by up drag its spun picks through water is axis water tained vertical containing a usually bucket bucket about is Newton’s A It called is what is. experiment[6]. of defi- problem terms clear the in a techni- what explained give exactly clarified the to of review try be nition us and least, involved let issues Firstly, very physical the philosophically. about and at suggestions cally can, some it give how and of Principle from collection generated Mach’s light the problems of some philosophical basis throw and technical technical to and that attempt nature of the shall analysis on the I in Here us help question. will (2.27), and by (2.11) modified relativistic be additional cen- should an Newtonian formula classical acceleration from the deduction[4] trifugal A that not? relativity it general does theo- or that in structure One operative retical is resolved. theory Principle relativity Mach’s not that general[7] still imply does date, is, to issue up of contentious all is, for and discussed time and that analysed and Mach’s century vigorously a of been almost for has otherwise us or with been validity has the Principle[5] of question The Introduction 1 ∗ γ 2 ( v atr e equations see factor, ) 1 INTRODUCTION 2 sophical question which is difficult to answer. That momentum if it is rotated about an axis. If further such a fundamental physical-philosophical dilemma the body is not subjected to any additional external as exposed by this simple experiment is so immedi- couples, the axis of rotation of the body then main- ately accessible to everyone’s understanding makes it tains a fixed direction with respect to the celestial unique among scientific problems. The question is sphere, with an axis of rotation aligned with respect how does the water in the bucket get the information to a specific galaxy of choice. This line axis locking that it is rotating so as to respond with the concave system is an available gyroscopic device which is also surface? Certainly the wall of the bucket has caused well within existing modern technology and its equiv- the rotation via friction but it has not caused the alent is to be found in all space vehicular guidance centrifugal force response. The situation is further systems. It is important to realise two things about confused when one asks what is the water rotation this coupling of local angular momentum to a distant relative to? Certainly there is the rotation relative object. One is that the locking can be disturbed by to the room in which the experiment takes place but applying couples to the object and the other is that what if that room is itself rotating so slowly as not the strength of the lock does depends on the magni- to have caused a noticeable concavity initially. Is the tude of the angular momentum and only reduces to water responding to just that initially induced room zero that is it becomes unlocked if the magnitude of relative rotation or to some combination of the two the angular momentum is reduced to zero. It is this conceivable contribution to the water angular veloc- gyroscopic type of set up which is usually quoted to ity? It is usually taken to be the case that the water explain one of the aspects of Mach’s Principle. This rotation and the induced centrifugal force are conse- can be expressed as the following question. How is it quential upon a totality of any rotational motions of that a locally rotating object can for some substan- the body about the direction involved. tial finite length of time hook onto a distant galaxy to which it is obviously not connected? This exper- Another much discussed example and the one imental fact is the source of the idea that inertia or which seems most impressive to me is that of the mass controlled influences locally are determined by influence of the rotating state on a physical elastic the mass space fabric of the rest of the universe. An- sphere. It is certainly true that such a sphere set other manifestations of the mystery link between here into rotation will expand in girth about the equa- and the distant universe should be mentioned. The tor of its rotary axis and to a degree dependent on Foucault Pendulum swings in a moving plane with a its angular velocity. Then the question is how does it motion that is locked onto the apparent motion of the acquire the information of the magnitude of its angu- celestial sphere. This gives the impression that the lar velocity unless there is some local standard of zero celestial sphere determines the local state of no rota- rotation? From astronomy technology, we know that tion against which the earth actually rotates. This a telescope can be locked on to a distant galaxy for device can be seen in museums and indeed can be some finite time in order to photograph it. This re- easily constructed by ordinary people. quires mechanical and computer equipment to negate the effect of the earth’s rotation so that the axis of the The two main answers given to these questions are telescope can be kept on a fixed line direction point- what would be Newton’s that there is an absolute ing towards the galaxy so that it remains in view space background everywhere and this defines the lo- at a central fixed position for the finite time neces- cal angular velocity condition for any physical body. sary. All such viewable galaxies are part of what There is then what would be Mach’s explanation that is called the celestial sphere and a specific known the angular velocity of a body anywhere should be re- galaxy can be used to indicate a definite coordinate ferred to a frame of reference attached to the celestial position on that sphere. The mechanisms and sys- sphere of so called fixed stars. This is a rather con- tems required to construct such a viewing system for densed translation of the two points of view, not ex- studying the celestial sphere is a commonplace avail- actly as expressed by either Newton or Mach. In fact, able technology of the present day. This telescopic what these two researchers actually meant by their viewing system can be regarded as giving an identifi- theoretic constructs is also a matter of much debate. cation and measuring device for the geometry of the One can see this interpretational type of difficulty by celestial sphere. There is another way to lock a local reading the following quotation from Mach’s writings line axis onto an object on the celestial spheres and given by Soshichi Uchii [1] on his website. that is by using a property of angular momentum. The comportment of terrestrial bodies with respect An extended body which is of material construction to the earth is reducible to the comportment of the so that it is composed of some distribution of mass earth with respect to the remote heavenly bodies. If we within its physical boundaries will acquire angular were to assert that we knew more of moving objects 1 INTRODUCTION 3 than this their last-mentioned, experimentally-given led some authors[15] to remark that the change of comportment with respect to the celestial bodies, we geometry that would seem to occur as a result of should render ourselves culpable of a falsity. When, relativity theory when a flat object rotates cannot accordingly, we say, that a body preserves unchanged realistically physically be seen as embedded in the its direction and velocity in space, our assertion is space of three dimensional Euclidean geometry. Here nothing more or less than an abbreviated reference to then, is a working statement of Mach’s Principle, not the entire universe. (ch.2, vi-6, 285-6) as expressed by Mach but, in two sentences covering The considerations just presented show, that it is the two aspects of the principle and in a form as it not necessary to refer the law of inertia to a spa- impinges on the subject matter of this article:- cial absolute space. On the contrary, it is perceived Observers fixed in local non-rotating reference that the masses that in the common phraseology exert frames see the distant celestial sphere as having zero forces on each other as well as those that exert none, rotation relative to themselves. The inertial proper- stand with respect to acceleration in quite similar re- ties of local particles are somehow determined by all lations. We may, indeed, regard all masses as related the rest of the particles of the universe with the very to each other. That accelerations play a prominent distant masses greatly involved. part in the relations of the masses, must be accepted This can also be taken as the definition for such a as a fact of experience; which does not, however, ex- non-rotating frame according to Mach. clude attempts to elucidate this fact by a comparison I have found a relativistic generalization of the of it with other facts, involving the discovery of new usual formula that is used in both classical and rela- points of view. (ch.2, vi-8, 288) tivistic theory to explain the effect of a rotating body However, in spite of these interpretation uncer- on observations from such a platform as opposed to tainties, the description of the two points of view, the view from a non-rotating body. Usually the clas- as I have expressed them above, do encapsulate the sical formula has to be adapted by some sort of ad essence of the problem. The conceptual difficulties hock addition of a second time variables to make it in both points of view become clearer when one tries have some conformity with the two times that are to update them in terms of relativity theory ideas. involved in the Lorentz transformation between two Newton’s point of view seems to fail because abso- inertial frames in linear relative motion. The inade- lute space, as far as velocity is concerned, is ruled quacy in attempts at relativistic analyses of rotating out and Mach’s view seems untenable because it is systems is of course due to a full relativistic rotation difficult to see how objects in deep space can deter- transformation formula having not to date been ob- mine local properties. To prove which view is correct tained. The relativistic rotation transformation for- in modern relativity terms is difficult because up to mula that I am intending to use here will be detailed date there is not a satisfactory relativity theory[15] in the following pages. However, I do not wish to for the rotating rigid body and indeed such concep- claim that this formula is the last word in explain- tual areas are controversial and not finally settled. ing the relativistic rotating state. One reason for not Another difficulty that immediately emerges when making such a claim is that there appear to be a one thinks about a rotating body in relativistic terms number of possibilities with slightly different physi- arises from the length contraction effect[4] that is in- cal meanings which I suspect will have applications to trinsic to special relativity. For a small radius disk slightly different physical situations. So I shall take rotating at high angular velocity or a large radius the rather cautious course of introducing this new disk rotating at small angular velocity there can be formula within a mathematical model of the physi- a radius on the disk that is moving transversally at cal philosophical situation that the Mach Principle the speed of light. Thus it seems that special relativ- dilemma seems to be describing. This model shows ity length contraction implies that the circumference that the Mach-Newton-Einstein[10] conundrum can at this radius will be of zero length on the basis of be greatly clarified. Hopefully it will be possible to measurements made by a stationary observer. Geo- go further and show that our physical universe con- metrically one can try to explain this happening by forms to the pattern of mathematical-physical quan- asserting that the originally flat disc will have to curl tities uncovered in this model. Whether or not Ein- up or down from centre toward edge to form a closed stein’s general[6] relativity conforms to this principle sphere. However, it is difficult to imagine what such still seems to be an open question and indeed as we an occurrence would look like from the point of view have seen it is far from clear just what the Princi- of a non-rotating observer. From his point of view ple is actually about in any precise sense. Einstein is a two dimensional flat object would have become a said to have been guided by Mach’s principal but it three dimension object. This type of difficulty has is unlikely Mach and Einstein would be able to agree 1 INTRODUCTION 4

on its meaning. I shall not attempt to reference the path from a position L(0) on S at time t = 0. Let ′ ′ vast literary archives on this subject but for those in us choose the X˜1 coordinate axis of S˜ to lie along need of finding out more, I suggest reading the intro- the direction of the normal to the circle at P (t) with ductory article by Soshichi Uchii [1] and follow back its origin coinciding with the origins of S and S′ so ′ references from there. that its second axis X˜2 lies perpendicular to the first ′ In analysing the collection of problems that arise in axis as indicated in the diagram. S˜ is an inertial the quest for an understanding of Mach’s Principle, frame moving in a fixed direction with a constant ve- I shall here make use of a new relativistic rotation locity relative to the inertial frame S. Thus at time t, transformation, (2.9 → 2.15) which is one of four or it simulates the velocity of the rotating disc both in more possibilities that I have found based on a simple magnitude and direction relative to the frame S. It ′ rotation simulation idea. The transformation I place can be used to give a local time variable t (t) to the ′ most emphasis on is the first of the pair displayed in disc at the point P (t) at time t. The inertial frame S˜ the next section This formula can be derived rigor- can acquire it own time keeping by being paired with ously from special relativity under a simple assump- another inertial frame S˜ rigidly attached to S with its ′ tion about rotation. The second relativistic rotation X˜1 axis parallel with that of X˜1 and separated from transformation, (2.16 → 2.24) has a number of very it by the distance P (t)L(0) = v(t)t = r(t)ωt, the ′ attractive mathematical aspects and I suspect that distance that the frame S˜ has separated from the it could have physical significance under some cir- frame S˜ in the time interval 0 → t, assuming that cumstance. It is not directly derivable from special the angle of frame rotation separation over the same relativity and will not be used in this article. time is given by φ(t) = ωt. This same distance is represented on the diagram by the arc P (t)D(0). So far the construction has installed a local time keep- 1.1 Possible Rotation ing from the inertial frame S to the rotating frame Transformations S′ at the simulation point P (t) which is in effect any point along the path P (t). This is the basis of the The two relativistic rotation transformation given by formulae (2.14) and (2.23). The lorentz transforma- equations (2.9 → 2.15) and equations (2.16 → 2.24) tion connecting the two inertial frames of reference below are constructed with reference to a mathemat- frames S˜ and S˜′ has the form ical simulation process that generates the equivalent ′ of a rotating body from the relative linear motions x˜1(t) =x ˜1(t) (1.1) ′ of Lorentz frames of reference S˜ and S˜ depicted in ′ x˜2(t) = γ(v(t))(˜x2(t) − v(t)t) (1.2) The Relativistic Rotating Frame Diagram by coordi- ′ 2 nates with the top ∼ symbol. The construction is t (t) = γ(v(t))(t − v(t)˜x2(t))/c ) (1.3) 2 −1 2 a follows. The basic reference frame S is taken to γ(v) = (1 − (v/c) ) / (1.4) be rectangular, rigorously inertial and to have a syn- v(t) = r(t)ω (1.5) chronised time keeping system throughout its space extent. That is importantly, it is not linearly accel- because the relative motion of these two frames is ′ erating and it is not rotating. To logically continue along the X˜2 direction with magnitude v(t)= r(t)ω. explaining the construction we have to now introduce As the frame S˜′ was chosen so that P (t) always lies ′ a space time path P (t) in S which will be denoted in on the X˜1 axis (1.3) reduces to (2.14). Equation (1.1) polar coordinates by (r(t),θ(t),t) and a second frame is the hidden assumption in special relativity that the of reference S′ which has a common origin with S and space contraction factor γ(v) applies in full and only is rotating with an angular velocity ω with respect to to the dimension of the direction of the velocity vec- ′ S about that common origin. Let us now imagine a tor which here is thex ˜2(t) direction. I shall drop this disc shaped region of the frame S′ centred at the com- assumption in the second possible rotation transfor- mon origin of S and S′. If the path P (t) is at position mation (2.16 → 2.24). P (t) in the diagram at time t this defines a radius of Material related to this general area of ideas can the disc r(t) at time t and also the circle shown in be found in references ([8], [9], [11], [12], [13], [14]) the diagram. It follows that the path P (t) coincides with a point on the disc that is moving tangentially with velocity v(t) = r(t)ω anti-clockwise. Now, we can conjure up a special relativity inertia frame S˜′ moving with this velocity and in the same tangen- tial direction as the motion at P (t) which has come up to coincidence at P (t) at time t along a straight 2 RELATIVISTIC ROTATION SIMULATION 5

2 Relativistic Rotation Simulation

Relativistic Rotating Frame Diagram

′ X2 R3 ∗ X ˜ ′ 2 O v(t) v(t)cos(χ(t)) X1 U++ D + d V-- ++ -- ++ -- + -- + P X˜1 ++ (t) D ++ N2 j D D, + jjjj , + v(t)sinj(jχj(jt)) , + jjjj , + jjj ,, K ++ jju j ,χ(t) + jjjj , + jjjj , + jjj ,, ∗ L(0) ω ′ +j+ jj g ,D(0) ∗ N2 + r(t) , + , ′ + , X1 + c , ll6 + φ(t) , ll L1 + , lll + ^ ,, lll ++ χ(t) , ll R4 ∗ ll ++ lll′ + lllN1 + lll + lll + lll + θ(t) lll X˜2 ++ lll e + lll + ll\ l + lll e lll φ(t) − 1 o l 0 / X1 lll , N1 lll ,, lll , lll , lll , lll , lll ,, lll , 0˜ lll , lll , lll , ll ,, ′ vlll , −e1 , ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,  ′ −e2  −e2

The transformation (1.1 → 1.5) can be used to complete the explanation of the simulation construction ′ as follows. The point L1 has coordinates (r(t), −(r(t) tan(χ(t)) − v(t)t) in S˜ and in S˜ it has coordinates (r′(t), −r′(t) tan(χ′(t)), denoting the view of the angle χ in S′ as χ′. If we substitute the second components of these into (1.2), we obtain, −r′(t) tan(χ′(t)) = γ(v(t))(−(r(t)(tan(χ(t)). However from equation (1.1) ′ ′ ′ and the diagram r (t)=x ˜1(t)=x ˜1(t) = r(t). It follows that tan(χ (t)) = γ(v(t)) tan(χ(t) which re- expressed is equation (2.12) of the rotation equation set. We now only need note from the diagram that ′ ′ ′ ′ x1(t) = r(t)cos(χ (t)) and x2(t) = r(t) sin(χ (t)). Thus all the parts of the first transformation have been derived from the rotation linear simulation idea. 2 RELATIVISTIC ROTATION SIMULATION 6

2.1 Relativistic Rotation

Contraction Projections Diagram

v(t) O ′ ′ v(t)sin(χ (t)) l4 l3 v(t)cos(χ (t)) ROo 4 R5 R/O 3 eKK F KK KK KK KK ′ KK χ(t) ′ R4 KK ′ R3 ∗ohO QQ KK R5 ∗/OB QQQ KK ′ Õ QQQ KK ′ d3 ÕÕ ′ QQ KKd4 ′ Õ l1 QQQ KK χ (t) Õ QQ KK ÕÕ QQQ KK Õ QQQ KK ÕÕ QQQ KK Õ QQ KK Õ l1 d4 QQQ KK ÕÕd3 QQQ KK Õ QQ KK π/2 $ Õ QQQ KK ÕÕ QQQKK Õ QQQK Õ Õ R1 o L(0) / R2

l2

The second rotation equation is obtained with the same simulation process so that the transformed time t′(t) comes out the same as with the first equation but the dilatation factor, γ(v(t)), is distributed over the two directions differently. This scheme can be described using the second diagram Contraction Projections Diagram. This scheme is not derived directly from the usual Lorentz transformation but using the first diagram which is non-specific it can be assumed that the transformed geometry can be expected to involve changes of the radial distance r′(t) so that the equality (r′(t)= r(t)) does not hold. Such a situation could reasonable hold in the case of the rotating elastic sphere for example. The second formula can be found by using the second diagram which is an enlarged representation of a rectangular region of the rotating plane. To see this in place, the diagram should be thought of as reduced in size and then the common points R4, R3 and L(0) of the two diagrams should be brought into coincidence. Thus we have a rectangular region of S′ instantaneously in motion in the direction given by the velocity vector v(t) so that the Lorentz contraction of ′ the width rest length l1 is given by equation (2.6) and, as can be read off the diagram, the length contraction in the two perpendicular directions of the rectangular coordinates in the frame S′ are given by equations (2.7) and (2.8) below, Thus if we distribute the contractions over the dimensions according to this prescription, we obtain the second set of rotation transformation equations (2.16 → 2.24).

′ 2 1/2 l1 = l1(1 − (v/c) ) (2.6) ′ ′ 2 1/2 ′ −1 d4 = d4(1 − (v sin(χ )/c) ) = d4γ(vs) (2.7) ′ ′ 2 1/2 ′ −1 d3 = d3(1 − (v cos(χ )/c) ) = d3γ(vc) . (2.8) 2 RELATIVISTIC ROTATION SIMULATION 7

The relativistic transformation I shall use in analysing the centrifugal force problem has the form, 2 2 1/2 r(t) = (b + (vbt) ) (2.25) ′ ′ −1 x1(t) = r(t)cos(χ (t)) (2.9) θ(t) = tan (vbt/b). (2.26) ′ ′ x2(t) = r(t) sin(χ (t)) (2.10) The formula for centrifugal acceleration can be de- 2 −1/2 γ(v) = (1 − (v/c) ) (2.11) rived from general relativity in a number of different ′ − χ (t) = tan 1(tan(χ(t))γ(r(t)ω)) (2.12) ways, see for example Rindler’s book [4], and has the χ(t) = θ(t) − ωt (2.13) form, ′ − t (t) = γ 1(r(t)ω)t (2.14) 2 ′ ′ ′2 2 2 ′ ′ d r (t )/dt = vtrγ (vtr)/r (t ). (2.27) ′ ′ 2 ′ 2 1/2 r (t) = (x1(t) + x2(t) ) = r(t). (2.15) I shall take it that this formula is a very reliable result from general relativity theory and use it as an inter- The second relativistic transformation mentioned pretational guide. The centrifugal acceleration given above but not to be used in this article has the form, by equation (2.27) only differs from the classical ex- 2 ′ pression by the relativity factor γ (vtr) where vtr is x1(t) = r(t)cos(χ(t))γ(vs(t)) (2.16) the polar transverse component of the vector veloc- ′ ′ ′ ′ x2(t) = r(t) sin(χ(t))γ(vc(t)) (2.17) ity dr (t )/dt . The primes on the various quantities − γ(v) = (1 − (v/c)2) 1/2 (2.18) indicate that they are measured by an observer fixed on the rotating frame S′ as also is the un-primed v . v (t) = r(t)ω sin(χ(t)) (2.19) tr s We can make a very significant point about centrifu- vc(t) = r(t)ω cos(χ(t)) (2.20) gal force by using the linear test path (2.25), (2.26) ′ −1 γ(v (t)) alone and without any reference to relativity as fol- χ (t) = tan (tan(χ(t)) c ) (2.21) γ(vs(t)) lows. To that end we recall that the polar radial and χ(t) = θ(t) − ωt (2.22) transverse components, (ara,atr), of acceleration a(t) ′ −1 in two dimensional motion are given by t (t) = γ (r(t)ω)t (2.23) ′ ′ 2 ′ 2 1/2 2 r (t) = (x1(t) + x2(t) ) . (2.24) (¨r − rθ˙ , rθ¨ + 2r ˙θ˙). (2.28)

It follows from the form (2.28) that, if a particle is The two transformations above (2.9 → 2.15) and in motion at a constant speed and in a straight line (2.16 → 2.24) are constructed so as to be from an perpendicular to the radius vector at any instant so inertial frame S to a frame S′ rotating with an an- that its radial acceleration, a (t), is zero, the quan- gular velocity ω relative to the inertial frame S. The ra tityr ¨(t) takes the place of radial acceleration in the transformations are not just expressed in terms of 2 rotating frame and, further, it is equal to r(t)θ˙(t) . the coordinate change from one frame of reference Thus forgetting relativity for the moment, we can cal- to another because their expression requires the in- culate the value ofr ¨(t) directly using the straight line put of some definite path (r(t),θ(t)) which is usually test path (2.25), (2.26) and evaluate its value at time to be taken to be a straight line in space traced out t = 0. We find that by giving the velocity parameter at a constant speed against time. In other words, a v the value ωb and direct calculation that path that any free particle should follow according p to Newton’s first law in the inertial frame S. If any (¨r(0) = r(0)θ˙(0)2 = bω2. (2.29) other path not conforming to this condition in S is encountered it would imply that the particle is either This is the usual formula for the centrifugal accel- not free or the inertial character of the frame S is eration experience by a particle fixed to a disc at a somehow compromised by being in a state of rota- distance b from its centre and which is rotating at an- tion itself, for example. The representation of the gular velocity ω. However, the particle on this path two transformations may strike the reader as number is moving in a straight line with no connection to redundant but there are so many equations because any rotating reference frame. All that has been done both collections of formulae include the transformed here is to give the constant velocity test path a dis- version in rectangular Cartesian and polar forms and, tance b from the origin at time t = 0 the same value ′ of course, the extra time variable t relation to t for that the rotating disc has at the radial distance b the transformed system. from its centre of rotation. Thus a straight line non- I shall make use of a specific straight space-line test accelerating path passing perpendicularly to a radius path in S defined by, of the disc and instantaneously having the velocity 3 RELATIVISTIC CENTRIFUGAL ACCELERATION 8

of the disc at the point of least distance from the planted on the assumed inertial frame S. This cal- origin appears from the disc view to have a centrifu- culated centrifugal acceleration can then be used to gal acceleration of the exactly correct classical form. see how it has to be changed so as to be consistent This certainly means that the form of the centrifugal with or exactly equal to the known correct formula acceleration formula is determined only by kinemat- for centrifugal acceleration from general relativity. ics, there is no dynamics involved and so classically The question that arises is how dependent is the we must conclude that there is no obvious link be- form of the relativistic centrifugal acceleration on the tween centrifugal acceleration and inertia. This re- linearity of our test path formula P ′. On carrying out inforces the generally held idea that centrifugal force the calculation which involves a full use of the rela- is a fictitious force. It seems to me that force only tivistic rotation transformation and consequent re- comes into this story when a definite valued mass m expressing the transformed variables into their form is to be held attached at a fixed radial distance from of dependence on the t′ variable we need the results the origin of a rotating object. The true force then from equations (3.1), (3.2) and (3.3). The calculation needed arises at least classically as the centripetal is much simplified by evaluating all the quantities at force fcp = −mara(t) = −fcf , the negative of the time t = 0 which is implied by using the |0 symbol. centrifugal force discussed above. Clearly, for a par- We find that the formula involves the ω0 associated ticle to be held in a circular orbit, the free particle with the questionable path and is the last of, tendency to travel in a straight line at constant speed must be overcome. Of course this is just the ortho- 2 ′ ′2 ′ ′ 2 doxy and it appears from the point of view taken here d r (t)/dt |0 = r(0)(dθ (t)/dt |0) | (3.1) ′ ′ ′ ′ to be correct. fcp = −ara(t) is measured in the ro- dθ (t)/dt |0 = (dθ (t)/dt|0)/(dt (t)/dt|0) (3.2) tating frame but here the force is determined by the ′ 2 1/2 dt (t)/dt|0 = (1 − (bω/c) ) (3.3) mass rather than the inertial mass being determined 2 by some mysterious influence of other masses in the 2 ′ ′2 b(ω0 − vb/b) d r (t)/dt |0 = 2 2 vicinity or far away. We can use the new transforma- (1 − (bω/c) )(1 − (bω0/c) ) tion ((2.9 → 2.15)) to check whether relativity gives (3.4) the same conclusion. However, the last few sentences above do seem to leave some small uncertainty about and if this is to be consistent with the form that the relation between distant and local mass values. comes from general relativity considerations then tak- ing the moving particle as fixed instantaneously to Perhaps someone can produce a more definitive case ′ one way or the other. the x1 at time t = 0 at a distance b from the ori- gin of rotation so that vb = bω and also denoting ′ vtr = bω it is necessary that 3 Relativistic Centrifugal ′2 2 Acceleration ω (ω0 − ω) ′ 2 = 2 2 . 1 − (bω /c) (1 − (bω/c) )(1 − (bω0/c) ) It is thus imperative that we should check out (3.5) whether or not the same conclusion follows from a relativistic version of the above argument. In order Suppose now that the angular velocity ω′ is the to carry forward this analysis we need to look at what apparent angular velocity of of the celestial sphere happens if the base frame S, originally assumed to as observed from a flat disk with its centre rigidly be inertial, is some way compromised and actually attached to the earth’s surface in a tangent plane contains particle tracks that would indicate that it configuration centred on the earth’s axis of rotation. is actually rotating. We can achieve this by reading Earth here, can be replaced by any solid rotating from the transformation how the rotating frame S′ platform anywhere. There are two identifications ′ assumed rotation at some angular velocity value ω0, for ω that preserves the simple factor form on both say, would see our linear test path. This is achieved sides of equation (3.5) and simultaneously identify a by substituting the original test path (2.25), (2.26) local frame of reference that is in a state of abso- into the transformation and obtaining its image as lute zero rotation itself. These two local frames are seen in S′. This gives us a new test path called the S and S′ of the transformation formulae (2.9) → ′ ′ ′ ′ ′ P ≡ (r (t),χ (t),t (t)), say, which can now be sub- (2.15). The identifications are ω = ω ⇒ ω0 = 0 or ′ stituted back as an input into the transformation. ω = ω0 ⇒ ω = 0. It is easy to check that any other Then the centrifugal acceleration can be calculated identifications for ω′ destroy this simple factor struc- as before but using the non-linear questionable path ture and further are complicated and aesthetically 4 CONCLUSIONS 9

unattractive. Thus can we can make the following 4 Conclusions interpretation of this result. If the general relativ- ity formula for centrifugal acceleration is accepted as correct throughout the universe then it must be used against and relative to a local background refer- ence frame that is in a local definite absolute state of It seems that Mach’s principle is true in the limited non-rotation coincident with the state of non-rotation sense of the existence of a universal standard of abso- that can be determined by observation of the celes- lute local rotation relating the celestial sphere to local tial sphere. Expressed otherwise, there is in general conditions. This does not, of course, preclude the ex- relativity everywhere to be found a local reference istence of relatively rotating systems with a relative frame in a state with absolutely no rotation relative angular velocity. Thus this limited Mach’s Princi- to which any local rotating state will have a defi- ple partially restores Newton’s absolute space back- nite absolute value of angular velocity. However, this ground in terms of absolute angular velocity while seems to have nothing very directly to do with iner- leaving the relative relativistic status of linear veloc- tia. The further very important consequence follows ity unchanged. Thus it is the universal network of ac- from this section. If a rigid physical object with a di- tual or potential null geodesics that unite all the local rectional arrow marked on it invariably placed with states of absolute rotation. When we observe the ap- respect to its geometrical form moves through space parent rotation of the celestial sphere, we are actually time either by propulsion or freefall its directional ar- looking at some local mapping of the celestial sphere row can be kept un-rotating with respect to a local on some local present time two dimensional represen- absolute non-rotating local reference frame which is tation of the terminal points of null geodesics that the same thing as keeping it pointing to a fixed iden- originated in the greatly distant past from greatly tified object on the celestial sphere. Of course this distant objects. The two dimensional representation navigation process can be helped by having an ac- at one level could be a telescopic viewing screen at tive gyroscope with rotation axis coincident with the another level, the retina of the eye or at a deeper level telescope so as to hold the telescope on its object and it is an image in the human brain. Always though, here again the gyroscope keeps its constant orienta- we are actually seeing local movement against the lo- tion by reference to the local non-rotating frame. In cal standard of no rotation, actually just the earth’s fact, this is just stating the physical basis for space rotation against this local standard. navigation in practice. We note also in this context that keeping a distant object in the line of sight of such a marked arrow involves locking a local tele- Finally, I would like to mention my motivation for scope in line with the local space part of the null this study which was work([2], [3]) in quantum me- geodesic which is the channel along which the infor- chanics on the fine structure constant problem. I had mation from the distant object is coming. In gen- long been puzzled by the fact that calculations on hy- eral relativity, the passage through space-time of a drogen like atom electronic orbits seemed to assume free photon is considered to take place along a null that the nucleus or a centre of gravity was fixed on geodesic and in quantum theory such a photon is a true inertial frame and one could get from theory also considered to carry a unique quantum of vec- the values of two perfectly definite quantized quan- tor angular momentum of magnitude ~ lying exactly tities, the Bohr radii,λ ¯C /(Zα), and the velocities, in its direction of propagation. Thus the photon in Zαc, of the electrons on such orbits. This implies its journey through space-time will always be locked that in such orbits the electron moves with a definite 2 ~ onto the local state of absolute non-rotation. Alter- angular velocity, me(Zαc) / . It seemed to me that natively expressed, this can be regarded as a process this only made sense if these angular velocities were in which the free photon field connects all the local absolute as we have been discussing above. Further non-rotation states throughout the whole electromag- more, I had managed to derive a very accurate value netic connected universe. A photon exists on the null for the fine structure constant using the equation of geodesic between its creation and destruction events centrifugal force. This implies that centrifugal force is because there is zero proper time between these two true even under condition when the coupling is large, events. 137α approximately unity, as in the case of hydrogen- 137, way outside the quantum electrodynamics realm of the small expansion parameter, α ≈ 1/137. The work described in this article seems to reinforce my view that there is a local standard of zero rotation everywhere. REFERENCES 10

5 Acknowledgements

I am greatly indebted to Professors Clive Kilmister and Wolfgang Rindler for help, encouragement and inspiration over many years.

References

[1] Soshichi Uchii at http://www.bun.kyoto-u.ac.jp/%7Esuchii/mach.pr.html [2] Gilson, J.G. 1996, Calculating the fine structure constant , Physics Essays, 9 , 2 June, 342-353 [3] Gilson, J.G. 1999, The fine structure constant, http://www.fine-structure-constant.org/

[4] Rindler, W. 2001, Relativity: Special, General and Cosmological, Oxford University Press [5] Mach, H. 1893, The Science of Mechanics, Chicago, IL: Open Court [6] Barbour, J. and Pfister, H. 1995, Mach’s Princi- ple: From Newton’s Bucket to Quantum Grav- ity. Boston, MA: Birkhuser [7] Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. 1973, Gravitation. Boston, San Francisco, CA: W. H. Freeman [8] Wolfgang Pauli 1958, Theory of Relativity, 130– 134, Pergamon Press [9] Abraham Pais, 1990, Subtle is the Lord: the Science and Life of Albert Einstein, p 214 [10] J. S. Bell 1987, Speakable and Unspeakable in Quantum Mechanics, p 67, Cambridge Univer- sity Press [11] Dover 1980, The Principle of Relativity, Dover [12] G. Cavalleri 1987, Nuovo Cimento 53B, p 415 [13] O. Gron 1975, AJP, Vol. 43, No. 10, p 869 [14] C. Berenda 1942, Phys. Rev. 62, p 280 [15] M. Weiss 1988, http://www.physics.adelaide.edu.au/∼ dkoks/Faq/Relativity/SR/rigid disk.html