On The Michelson-Morley Experiment

Transcription

1 APPENDIX D On The Michelson-Morley Experiment 1. The Classical Interpretation Of The Michelson-Morley Experiment The negative result of the Michelson-Morley experiment presented early twentieth century physicists with a serious and very perplexing problem. Two explanations seemed possible 1 : -A kinematical one: The velocity of light is constant and the same in every direction, regardless of the motion of the Earth. There is no shortening of the arms of the experimental apparatus, and it is not necessary to invoke some new physical process. -A dynamical one: There is an actual shortening of the arms of the experimental apparatus in the direction of the Earth s motion, so as to compensate for the variable speed of the light ray, and this represents a bona fide physical process. However, as it turns out, this process cannot be described in a traditional Newtonian context. The first explanation is of course consistent with Einstein s Theory Of Special Relativity, and has been adopted by the overwhelming majority of physicists since then. It does however present some serious drawbacks, both theoretical and practical, which we will now discuss. The main problem is that the Michelson-Morley experiment yields a negative result for every observer attached to a so-called Galilean reference frame. For this to be consistent, the traditional Galilean transformation between coordinate systems, leaving the time invariant, must be dropped and replaced by the well-known Lorentz Transformation, where the time plays the role of fourth coordinate. Each observer now has his or her own proper time, for which the velocity of light is constant (the same for every observer) whatever its direction. However, although there is no modification of the experimental apparatus attached to a given observer, he or she can still only explain the negative result of the experiment for all the other Galilean observers by assuming a shortening of the arms of their apparatuses in the direction of their relative motion. The only way to not invoke some new physical process is to assume that this shortening of the arms is some sort of optical illusion (the so-called length contraction ), since it is not in fact observed by any of the observers attached to the moving apparatuses. The above explanation introduces some radical new concepts to traditional physics. First of all, time is elevated to the status of a fourth dimension, alongside the three spatial coordinates. It no longer has to be measured thanks to some physical apparatus that obeys the laws of Newtonian mechanics (as is the case of all traditional clocks): the Michelson- Morley apparatus, where no physical process is necessary to describe the motion of the light ray can itself now be used as a clock. Secondly, the notion of Galilean reference frame (i.e. a frame moving through space with uniform velocity) plays a crucial role. This is all the more remarkable in that such 1 See for example a discussion in [Pau58] 1

2 2 JOHN G. BRYANT frames are difficult to define in practice. For instance an Earthbound observer, in motion through the Solar System (itself in motion through the Milky Way), can hardly be considered attached to a Galilean reference frame! It follows that one can certainly wonder whether the negative result of the Michelson-Morley experiment should not also hold in non-galilean reference frames as well. Finally, even though the above considerations dispense with the notion of physical process, some set of dynamical laws must be reintroduced. It would appear that the equations of motion derived from the electromagnetic field equations (which are invariant under a Lorentz transformation) should guide us, but in fact the laws of particle mechanics used in Special Relativity are far from satisfactory (at least from a theoretical standpoint), since they are not Lorentz invariant. (cf. [Gol81]) It becomes clear, as it became clear to Einstein, that a more general theory is necessary, one where the effects of a gravitational field can be taken into account. The General Theory Of Relativity (1915) was his answer to the problem, and it represents an even more radical departure from classical mechanics. Although it has led to some spectacular successes, and has gained nearly universal acceptance, it has also met with two remarkable failures: the well-known failure resulting from its lack of compatibility with Quantum Mechanics, and the less well-publicized but equally troubling failure to provide an adequate formulation of the N-Body Problem. 2. The Interpretation In The Framework Of The Reformulated N-Body Problem We now propose to reinterpret the Michelson-Morley experiment in light of our reformulation of the N-Body Problem. For this, we first of all assume that the Michelson- Morley experiment has been carried out on a number of free-falling celestial bodies (e.g. the Earth, the Moon, various planets, an inter-planetary space probe, etc.) belonging to a well-defined N-Body Problem (e.g. the Solar System), and we make the following (extremely plausible) hypothesis: In every case, the Michelson-Morley experiment yields a negative result. A purely kinematical explanation is now no longer at our disposal, since none of the various experimental apparatuses can be assumed to be moving with constant velocity with respect to each other, and no generalization of the Lorentz Transformation, valid in the case of variable velocities, can be invoked. On the other hand, the negative result can be explained in a dynamical context only by assuming that there is an actual (and continuous) shortening of the arms of each experimental apparatus in the direction of each body s motion, in order to compensate for the apparently variable velocity of the light ray. To interpret this result, we have as before several possibilities: -The shortening of the arms of each apparatus depends on the velocity of the body with respect to a fixed point in 3-dimensional space, and with respect to which the light ray has constant velocity c. 2 This is unacceptable when we take account of the fact that the properties of the N-Body Problem are independent of its general motion through space (because, as we know, the motion of the center of mass is separate from the rest of the motion), and one would expect this to hold true of all the attached apparatuses as well. In others words, the shortening of the arms cannot depend on the absolute velocities of the bodies. 2 We could call this the Lorentz-Poincaré generalization

3 UNIVERSAL N-BODY PROBLEM 3 -The shortening of the arms of each apparatus depends on the velocity with respect to a designated body, normally the first one playing the role of observer. There would be no shortening of the arms of the observer s apparatus, with respect to which the velocity of the light ray would have constant value c. This would appear to be the natural generalization of one of the postulates of Special Relativity to free-falling reference frames (and not only Galilean ones). 3 As we know however, this viewpoint could only have practical value, since it implies an unrealistic behavior of the light ray, in that it would not then interact gravitationally with any of the other bodies. -The shortening of the arms depends on the relative velocity of each body with respect to the center of mass of the N-Body system, and with respect to which the light ray has constant velocity c. This is of course the interpretation we must adopt to be in agreement with our theoretical model for the N-Body Problem (since as we know all the N bodies will then be able to interact with the massless particles that are presumed to make up light rays), but also implies a radical departure from what is assumed classically. In order understand this we must first of all consider the actual motion of the experimental apparatus taken to be that of an ordinary solid (Σ) moving through space, and whose center of mass has an initial velocity V 0 at some instant t 0. In Newtonian mechanics its total energy would be the sum of: a) its mechanical energy, or energy of motion, comprising its kinetic energy (both translational and rotational) and potential energy; b) its internal energy, which binds together the atoms and molecules that make up (Σ). 4 Now consider the following two examples: -Suppose first of all that at t 0, (Σ) is falling vertically downwards in the Earth s gravitational field, which we assume is uniform and constant (a good approximation near the surface of the Earth). At a subsequent moment t 1, (Σ) s velocity will have increased to a value V 1, and according to a well known result of Newtonian mechanics, its (translational) kinetic energy will also have increased, but not its internal energy, and no modification of the length of the arms will have taken place. On the other hand, an Earthbound observer, with respect to whom light has a constant velocity c (and whose velocity is therefore not constant with respect to (Σ)), seeing the negative result of the Michelson-Morley experiment has to conclude that there is an actual shortening of the arms of (Σ) between the instants t 0 and t 1, and therefore an actual increase in its internal energy as well, in contradiction with the classical theory. -Suppose now that (Σ) is in a circular orbit around the Earth. The length of the (Newtonian) velocity and acceleration vectors of the system s center of mass are then constant (because the vectors are perpendicular), so there is no variation of the system s kinetic energy or internal energy. However the relative direction and velocity of light with respect to the system are variable (we assume that the arms of (Σ) have fixed directions), so the negative result of the Michelson-Morley experiment means there must be a continuous variation in the lengths of both arms of (Σ), as it orbits the Earth: in other words (Σ) undergoes a non-symmetrical continuous deformation, even though its internal energy is constant. The fact that the negative result of the Michelson-Morley experiment cannot be explained by the classical theory leads to the following conclusion: contrary to Newtonian 3 We could call this the Einstein generalization 4 A very telling representation would be to assume that both arms of the apparatus consist of rigid coil springs. Its internal energy is then given by the arms extension or compression.

4 4 JOHN G. BRYANT theory, a solid s energy cannot be divided into two distinct parts (mechanical and internal), but must be considered as a single entity. This property appears as a logical generalization, on a macroscopic scale, of what we showed when formulating the problem of two bodies in the presence of a distant and extremely massive third one (cf. Chapter 4, Section 1), and in fact an N-Body model for the motion of (Σ) could be built along the same lines as those used for the motion of the Sun-Earth system through the Galaxy: the Earth would now play the role of the Galaxy, and (Σ), considered as a stable (N-1)-Body sub-system, would replace the Sun-Earth sub-system. We would then, as before, have a choice between two Hamiltonian models, i.e. a separate one and a non-separate one. In the first, the Hamiltonian could be split into two terms, each with a constant value during the motion: one describing the (Kepler) motion of the center of mass of (Σ) about the Earth; and the other, the relative (internal) motions of (Σ). Here the distinction between the two types of energy can be clearly made. This is not the case for the second Hamiltonian model, since there would be no separation of the Hamiltonian into two terms allowing us to distinguish clearly between the two forms of energy. The negative result of the Michelson-Morley experiment, which implies either a variation of (Σ) s internal energy, or a continuous redistribution of internal energy from one arm to the other, depending on the type of motion, clearly rules out the first separate model and so leads us to adopt the second non-separate Hamiltonian model, which allows for such variations. 5 The interpretation of the Michelson-Morley experiment in terms of the N-Body Problem also allows us to explain the otherwise very puzzling fact that the variation of the length of the arms of the experimental apparatus compensates exactly the variation of the velocity of the light ray (with respect to the apparatus), thus yielding a negative result for the experiment: for if this were not the case, an observer attached to the apparatus would be able to determine its exact velocity with respect to the center of mass of the N-Body system by simply measuring the variations of the fringe pattern produced by the interferometer attached to (Σ). And as we have seen before (cf. Chapter 3, Section 3: The Twin Paradox), such an exact determination is not possible, owing essentially to the fact that the real motion of massless particles does not separate from the motion of the other bodies, and is therefore not integrable, and cannot be exactly determined. Finally, the notion of a solid s energy being a single indivisible entity can be used to obtain the resolution of the so-called Ehrenfest Paradox, which has to do with the motion of a solid disk rotating about a fixed axis: the paradox stems from the fact that, according to Special Relativity, a ruler used to measure the circumference of the disk undergoes a Lorentz contraction, whereas the same ruler used to measure its radius does not; it follows that the measurement of the ratio of the circumference to the radius is no longer equal to 2π, but in fact depends on the distance from the axis, and this appears incompatible with the solid nature of the disk. 6 On the other hand, by extending the principles brought into play to explain the negative result of the Michelson-Morley experiment, the apparent paradox can be solved in a simple and natural manner: 7 we simply have to assume that any variation in the disk s rotational velocity leads not only to a variation of its (rotational) kinetic energy, but of its internal energy as well, and this results in a uniform variation of the disk s radius which ensures that the shape and integrity of the disk are preserved. 5 The classical Newtonian theory for the motion of solids remains of course an excellent approximation, but can no longer hold in the case of the Michelson-Morley experiment, which, although it involves massless particles, can still be said to belong to everyday physics! 6 See for example [Pau58, p.131] 7 And which is quite incompatible with Special or General Relativity!

5 UNIVERSAL N-BODY PROBLEM 5 This not only guarantees the negative result of any Michelson-Morley experiment carried out with an apparatus attached to the disk, it also means that it is never possible to actually measure the absolute value of the disk s rotational velocity, i.e. its value with respect to 3-dimensional space, but just the disk s relative rotational velocity with respect to, say, another rotating disk about the same axis.

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7 Bibliography [Gol81] H. Goldstein, Classical Mechanics, second ed., Addison-Wesley, Reading, Mass., [Pau58] W. Pauli, Theory Of Relativity, Pergamon Press, London,