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Physics/0409010V1 [Physics.Gen-Ph] 1 Sep 2004 Ri Ihteclsilshr Sntfound Physical Interpretations of Relativity Theory Conference IX London, Imperial College, September, 2004 ............... Mach’s Principle II by James G. Gilson∗ School of Mathematical Sciences Queen Mary College University of London October 29, 2018 Abstract 1 Introduction The question of the validity or otherwise of Mach’s The meaning and significance of Mach’s Principle Principle[5] has been with us for almost a century and and its dependence on ideas about relativistic rotat- has been vigorously analysed and discussed for all of ing frame theory and the celestial sphere is explained that time and is, up to date, still not resolved. One and discussed. Two new relativistic rotation trans- contentious issue is, does general[7] relativity theory formations are introduced by using a linear simula- imply that Mach’s Principle is operative in that theo- tion for the rotating disc situation. The accepted retical structure or does it not? A deduction[4] from formula for centrifugal acceleration in general rela- general relativity that the classical Newtonian cen- tivity is then analysed with the use of one of these trifugal acceleration formula should be modified by transformations. It is shown that for this general rel- an additional relativistic γ2(v) factor, see equations ativity formula to be valid throughout all space-time (2.11) and (2.27), will help us in the analysis of that there has to be everywhere a local standard of ab- question. Here I shall attempt to throw some light solutely zero rotation. It is then concluded that the on the nature and technical basis of the collection of field off all possible space-time null geodesics or pho- arXiv:physics/0409010v1 [physics.gen-ph] 1 Sep 2004 technical and philosophical problems generated from ton paths unify the absolute local non-rotation stan- Mach’s Principle and give some suggestions about dard throughout space-time. Thus it is suggested how it can, at the very least, be clarified techni- that Mach’s principle holds in the restricted sense cally and philosophically. Firstly, let us review the that there is a universal standard of absolute local physical issues involved and try to give a clear defi- rotation rate related to the apparent rotation of the nition of exactly what the problem is. It is usually celestial sphere. However this apparent rotation is explained in terms of what is called Newton’s bucket actually the earth’s rotation relative to a local map- experiment[6]. A bucket containing water is spun ping of null geodesic endpoints from that time and about a vertical axis through its centre. The con- space distant sphere to the local time in the local tained water picks up motion because of frictional zero-rotation environment. A connection of local in- drag by the bucket wall and then assumes a concave ertia with the celestial sphere is not found. downwards surface form as a result of the outward centrifugal force causing the rise in level from the cen- tre towards the wall. This is a very simple sequence of physical events which each of us can or has experi- ∗[email protected] enced in some form or other but it prompts a philo- 1 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.
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