Accepted for publication in The ICFAI University Journal of Physics, vol II, No. 1, 2009
Towards the reinstatement of absolute space, and some possible cosmological implications Héctor A. Múnera,1,2 1
Centro Internacional de Física (CIF), Bogotá, Colombia E-mail:
[email protected] Retired Professor from Department of Physics, Universidad Nacional de Colombia, Bogotá, Colombia 2
Abstract The paper reviews empirical evidence from several optical experiments on earth showing that it is possible to detect earth’s motion. Four different experiments with different technical means, and from widely separated places in time and space, point towards solar motion in a plane with right ascension 75º, but there is no agreement on the declination, which seems to be large. This evidence contradicts the widely accepted heuristic principle that the motion of earth cannot be detected by terrestrial experiments, and opens the door for a reinstatement of the currently forbidden concept of absolute space Σ. A closed finite Newtonian universe has interesting cosmological implications, as compatibility with the variation of speed with distance (without expansion), and the appearance of new acceleration term of the same order of magnitude as the MOND cut-off, thus offering an alternative explanation to the dark matter and MOND hypotheses. Key words: Michelson-Morley experiment, Miller experiment, dark matter, anomalous cosmic tangential velocity, cosmology models, MOND model
1. Introduction: ontology versus measurement One of the aims of this paper is to draw attention to long-lived problems in modern physics, which must be solved. We want to address here some touchy questions, dealing with some of the most cherished notions of the 20th century physics. By the end of the 19th century, the positivistic school called attention to the importance of measurement in the physical sciences. Following this lead, by the 1930s the Copenhagen interpretation of quantum mechanics went to the extreme view that things only exist when they are measured. It is well known that Einstein was very critical of this philosophical stance. It is worthwhile to recall his famous rhetorical question to one of his collaborators: “Is the moon there when we do not look at it?” Einstein, however, was not equally critical with his own ambiguous position regarding his notions of space and the ether that fills it; see also Kostro (2000). In the context of his general theory, Einstein was for the concepts of ether and absolute space. In his words: “Newton might no less well have called his absolute space ‘Aether’; what is essential is merely that besides observable objects, another thing, which is not perceptible, must be looked upon a s real, to enable acceleration or rotation to be looked upon as something real” (Einstein, 1983), our underlining.
Evidently, Einstein accepted in this context the existence of physical entities, independently of measurement, and even of perception. Note also that acceleration and rotation are set on an equal footing. Since rotation requires the presence of some agent producing the requisite centripetal acceleration, the other word “acceleration” must be understood as linear acceleration. Hence, one of the many possible ways to characterize absolute space Σ is: “Σ is the space where acceleration exists”. Of course, according to Einstein’s principle of equivalence, acceleration of gravitational origin also exists in Σ. We completely concur with this view. But in his special or restricted theory of relativity (RTR in the following), Einstein took, at the beginning of his career, a completely different stance. Up to the beginning of the 20th century, length was a property of bodies, the latter existing in absolute space Σ, or at least in a preferred frame of reference. In that sense, length was an ontological property, independent of the act of measurement. On the contrary, the process of measurement is a technical operation or action, strongly limited by technical means available at a given epoch, and depends of the distance of the observer from the object, and of the state of motion of the observer. For example, let us consider the length L of the longer side of the rectangular top of our desk. It is trivial to note that an observer O, at rest with respect to the desk, may perceive this length by simultaneously touching both ends of the table, or may measure the length L with a ruler such that its zero touches one of his/her hands while the number L at the other end is simultaneously read by O with his/her eyes. After imposing a significant number of restrictions and caveats, this value L provides a first order approximation to the measurement of the ontological property length. Note that the concept of simultaneity used here has its everyday meaning. It is even more trivial to note that another observer O’ at rest with respect to our desk, but located at a remote place will see a smaller apparent size –size that decreases with distance. However, nobody has proposed as yet a physical theory for the length contraction of bodies with distance from the observer. It is evident for everyone that the shrinkage in size is apparent, and that the ontological length L remains the same. Any technically capable observer O’ may readily obtain the ontological value L from the apparent size. It is also evident and trivial that another observer O’’ traveling in a high speed vehicle, and watching our desktop will see a distorted image of the rectangle. The shape and size will depend of the direction of motion of the observer, and its speed of motion. There is no a priori reason to believe, or to postulate, that the ontological length L of the longer side of the table may change when the observer O’’ passes by. Once again, any technically competent O’’ may easily calculate L. Needless to say, the specific method for the calculation depends of the details of the process of measurement. For instance, O’’ may measure the time interval between two light signals reflected from mirrors M1 and M2 placed at each end of our desk, or any other suitable method. However, the young Einstein ran completely askew in the solution of this simple problem. In developing his RTR, Einstein confused measurement with ontology, and even worse, postulated that the ontological length L could not exist. He introduced a meaning for simultaneity that would suite his purposes, completely different from the everyday sense, and developed a kinematical theory of length-contraction.
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It is worth to stress that we are arguing in this paper from a Newtonian viewpoint, where both time and space are absolute. This philosophical stance is stronger than the position advocated in some recent papers that defend the existence of absolute space in the context of Lorentz or similar transformations of space and time (Selleri, 2005; Guerra and De Abreu, 2006). A striking feature of Einstein’s RTR paper is the lack of references. It may be said that in his early years Einstein did not fully recognize the influence of previous thinkers upon his own ideas. In particular, Einstein insisted that his theoretical postulates were independent of the empirical results of the Michelson-Morley (MM) experiment (Michelson and Morley, 1887). Despite Einstein’s claims, it may be asserted that his RTR ideas did not suddenly come out from his brain, independently of the state of knowledge of his epoch. Quite the opposite, Einstein’s opinions were shaped by the then current views, which was a cumulative result of a long process over the last part of the 19th century. It is well known that Lorentz paid particular attention to the claimed null result of the MM experiment, and proposed a physical theory of length contraction as a function of the motion of the body with respect to the Newtonian absolute space Σ (Lorentz, 1895). The rather recently published Einstein love letters (Renn and Schulmann, 1992) leave in evidence that Einstein read Lorentz during his enrollment at ETH in Zurich, and also studied Drude (1900) in depth. In Drude’s book there is a thorough discussion of the MM and other experiments dealing with the ether. Poincaré was also interested in the empirical evidence, perhaps via Lorentz, and studied the latter in depth (see Poincaré, 1905, pp 171-172). He took one step farther and suggested the hypothesis that the translational motion of earth could not be detected by experiments carried out in our terrestrial laboratory (Poincaré, 1905; Poincaré, 1906; Whittaker, 1987). Is has been known for a long time that Einstein read Poincaré while he was working in Bern at the patent office, as attested by some of his letters, appearing in the collection by Beck and Howard (1995). Critics of Newtonian mechanics have always emphasized that to measure velocity one necessarily requires a reference object. In their view this implies that the notion of absolute motion –that is, relative to absolute space– is meaningless. Hence, according to that view, the only physically relevant motion is relative motion. Once again, there is confusion here between the ontological entity “absolute space”, and the technical question of measuring velocity, its speed in particular. The technical fact that a reference object is needed neither forbids, nor proves the existence of absolute space. As recently stressed by Guerra and De Abreu (2007), in the related context of Lorentz versus Einstein’s interpretations of the RTR, a similar view was defended long ago by Bell (1988): “The facts of physics do not oblige us to accept one philosophy rather than the other.” Lorentz and Fitzgerald were capable of explaining the claimed null result of the MM experiment with coordinate transformations involving speed relative to absolute space. This was a simple theory with a unique speed relative to a proxy for absolute space, the frame where the “fixed stars” are at rest. On the other hand, Einstein proposed an alternative hypothesis, capable of explaining the null result of MM without absolute space, but involving speed relative to the observer. Consistency with the MM experiment alone does not suffice to choose between the two alternatives. From the quotation at the beginning of this section, it appears that Einstein was convinced that the principle of equivalence requires the existence of some absolute space where
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acceleration exists. One may wonder whether Einstein, in his mature years, was a prisoner of his early work on the RTR, and could not develop all the possibilities of his general theory towards his dream of a unified theory. Note that Einstein’s RTR leads to a little discussed paradox in physics: translational motion is relative, but rotational motion is absolute. The latter may be measured by experiments in the laboratory, as Foucault’s pendulum and Foucault’s gyroscope. However, in Lorentz theory both motions are relative to absolute space. A little reflection immediately suggests that Einstein’s dichotomy is not acceptable within a consistent theory of nature. For, consider a particle moving in orbital motion over a circle of a very large radius, such motion is absolute. In the limit of a very large circle the motion becomes for most practical purposes translational, and hence relative. When, how and why did the intrinsic nature of motion change from absolute to relative? Selleri (1998) advanced similar considerations. There is, however, an alternative way out of this conundrum: The only motion that exists is rotational. In such scenario linear motion would be just an artifact caused by our observations in our local and small scale laboratory. We will return to this matter in section 4. It may be also noted that it is quite possible to develop a really relativistic theory of mechanics, based on the distance between objects and its variation. This was done by Assis (1999) with his relational mechanics, which neither introduces absolute space, nor gives preponderant role to the observer. Since there are several consistent theories of space and time, there is room for choice according to scientific methods. It is our strong opinion that the validity of physical theories cannot be assessed by subjective notions, as beauty and symmetry (defended by Einstein). On the contrary, the validity of theories in the natural sciences is a simple question of empirical testing. This is the subject of next section. To end this opening section, it is worth saying that the pendulum of philosophy in physics has swung back during the last 100 years. It seems that the tenets of the positivistic school are completely forgotten. Theoretical physics is filled now-a-days with notions that cannot be measured, or cannot be observed by definition, like quarks, dark matter and dark energy – reminiscent of the “star wars” film. Here we argue for the closer-to-our-intuition concept of absolute space Σ.
2. Measurement of terrestrial velocity by optical means on the earth
Over the 20th century several researchers reported “anomalous” observations indicating that the velocity of earth relative to absolute space could be detected by experiments on earth, thus violating the widely accepted heuristic principle that the motion of earth cannot be detected by experiments on earth. This principle will be called hereafter Poincaré’s principle. Esclangon (1926a, page 921) conjectured that an absolute motion of the earth, additional to orbital motion, would lead a pendulum to deviate from the local vertical, deviation that –due to earth’s rotation– would have a period of one sidereal day. He analyzed data taken with a Foucault pendulum at Postdam continuously from 1904 to 1909; he hypothesized that the deviation from the vertical would be a quasi-periodic function of the mean solar day, the lunar day, the solar year, the lunar year, and the sidereal day. From the latter effect he calculated that motion is either in the right
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ascension direction α = 4h-35 min, or in direction α = 16h-35 min. A second pendulum oriented at 90° to the former, but with less observations, produced α = 3h-30 min (Esclangon,1926a and 1926b). Previous work prompted Allais (1997, pp.331-372 ) to carry out observations with topographic instruments during July 1958, and in a long series, day and night, from February 23 to March 26, 1959. He detected relative deviations of the order of 1.6 micro radians between measurements taken South-North relative to those taken in the opposite direction North-South; he also found cyclic variations with periods of 25h, 24h, and 12 h, as well as long term cycles associated with the moon’s orbital motion around the earth. In Russia, Baurov (1993) reported a magnetic anisotropy suggesting a preferred direction in space, and recently has discovered a nuclear anisotropy in the beta decay of a couple of nuclei, followed by gamma emission (Baurov, 2001). Let us focus here on observations involving “anomalous” behavior of light in our terrestrial laboratory, chiefly on interferometer experiments attempting to measure the relative motion of the earth with respect to an external entity, which we interpret as absolute space Σ. No further mention is made here of the ether, a possible fluid filling that space. 2.1 Esclangon observations. Let us start with the almost forgotten astronomical observations by Esclangon (1926a), who was the director of the Observatory at Strasbourg (France). He noted significant differences between the angles of incidence and reflection when observing circumpolar stars using direct and reflected modes of observation, and interpreted the differences as due to a motion of earth toward a fixed cosmic direction. As usual, there are two velocities consistent with the model, namely: • Esclangon 1. Motion in the northern celestial hemisphere toward right ascension α = 4h-36 min, declination δ = +44°. He did not estimate the speed V. • Esclangon 2. Motion in the southern celestial hemisphere towards α = 16h-36 min, δ = -44°. 2.2 Miller experiments. Interferometer experiments to measure the motion of earth relative to some external entity started in the second half of the 19th century, in particular Michelson and Morley (1887). From about 1902 onwards, Miller (1933) claimed that neither the original MM experiment nor his own repetitions were null, but produced a small fringe shift. Invoking kinematical considerations related to the circle described by the apex of the terrestrial motion against the background of stars, Miller introduced a scaling factor to amplify the small observed fringe shift by a factor close to 20 (Miller, 1933, pp. 234-235). Due to the presence of quadratic terms, the observations of any interferometer experiment admit two different velocities of solar motion, namely (α,δ) and (α+π,−δ); for a recent discussion of this issue see Consoli and Constanzo (2007). The velocities consistent with Miller’s observations are: •
Miller 1. Motion towards the northern celestial apex with speed V = 200 km/s, α = 17h, δ = +68º, announced at the Pasadena Conference in 1927 (Miller, 1933, pp. 232).
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•
Miller 2. Motion towards the southern celestial apex with speed V = 208 km/s, α = 4 h-54 m, δ = –70º-33’, obtained by the end of 1932 from a re-analysis of his data (Miller, 1933, pp. 232-234).
Miller finally adopted solar motion towards the southern celestial apex as the velocity most compatible with his data. Whatever the choice, Miller’s work demonstrated that it was possible to measure the velocity of the earth by local optical measurements, thus casting doubts upon Poincaré’s principle. However, when Miller published his final analysis in 1933 Einstein’s RTR was firmly entrenched, and every effort was made to discredit and/or to ignore Miller’s great experimental efforts, see for instance Shankland et al. (1955). By the end of the 1990s several investigators independently revived the interest in Miller’s work (Allais, 1997 and 1998; Vigier, 1997; Múnera, 1998). The latter writer revised all the initial MMtype experiments up to the early 1930’s, and found that the experimenters always observed small variations in the velocity of light, that were consistently interpreted as zero. More detailed analyses uncovered further weaknesses in the design, operation and data reduction of both the MM and Miller experiments (Múnera et al, 2002; Múnera, 2006). 2.3 Múnera experiment. Rather than endlessly discussing previous experiments, Múnera and collaborators chose to repeat the MM experiment at the International Centre for Physics (CIF) in Bogotá, Colombia. Several improvements on the original design were introduced: a stationary table, automatic video capture of the interference patterns, and correction for environmental variations. Assuming that light moves with constant speed c in absolute space, independently of the motion of the emitter, and using Galilean addition of velocity, for a given velocity of the sun (speed and direction), Múnera (2002) obtained the expected fringe shift in a stationary interferometer located at some latitude, as function of the month of the year and the time of day. The model predicts cyclic variation of the fringe shift with a period of 24 sidereal hours,1 superimposed upon annual variations. Of course, Einstein’s RTR predicts no variation in the fringe-patterns. Hence, a consistent and reproducible observation of periodic variations (after subtracting environmental and other periodic effects) falsifies both Poincaré’s principle (because motion of earth is measured in a terrestrial experiment), and Einstein’s second postulate of RTR, because the speed of light adds2 to earth’s motion. The CIF experiment ran from January 2003 to February 2005 in Bogotá, Colombia using a symmetrical interferometer with equal arms (2.044 m each) disposed upon a 13,000 kg concrete table. The arms were oriented W-E and S-N, and the light source was a stable green laser, with wavelength= 532 nm, produced by an Nd:YAG diode pumped laser, model DPY 325/425II, manufactured by Adlas, Germany. Sensors to measure temperature and humidity along each arm were installed. A photograph of the interference pattern was captured with a video camera, and recorded every minute into a computer outside the concrete table, for a total of 1,440 photographs every 24 hours.
1 2
However, the accuracy of the observations do not allow to distinguish between sidereal and solar 24 hr periods. Vector addition.
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Each interference pattern was converted into a brightness intensity profile, where bright fringes correspond to maxima, and dark fringes to minima. The pixel-position of a reference dark fringe was plotted as a function of civil local time, clearly depicting 24-hr periodic variations. The raw amplitude of the variations was as large as 20 fringes, and was correlated to the environmental variations of temperature T, humidity H, and pressure P. It is important to stress that the variation of T, H and P produce a change in the index of refraction along the optical path, which induces a spurious fringe shift. We evaluated these variations for our experimental arrangement and found them to be negligible (Hernández-Deckers, 2005).Therefore, we opted for a stochastic correction of our observations, where the part of the signal that was correlated with the environmental variables was subtracted. The method of stochastic correction may subtract a variation that is not caused by variations of THP (although it is correlated to THP), so that it tends to over-correct, and to decrease the “true” amplitude of the residual. The final residual has amplitude of several fringes, and still shows a 24-hr periodic variation that is no longer significantly correlated with the environmental variables. Our observations are, therefore, consistent with the recent predictions of large frequency shifts, made by Consoli and Constanzo (2004) in the similar case of experiments with resonant cavities in dielectric media. For each month, a several-day session by the middle of the month was selected. A several day session is then averaged over a single 24 sidereal hours period, which is representative of each month. Overall, 24 series were obtained for 24 months between January 2003 and February 2005 (data was not taken in May and June 2004). As in MM and Miller’s experiments, the room for the CIF experiment was not environmentally controlled. It is hoped, however, that the stochastic correction that was applied did eliminate the effect of ambient variations –recall that no correction was applied to their observations by MM and by Miller. Since the solar velocity with respect to absolute space (shown in figure 1) is unknown, the 24 residual curves were used to obtain the velocity of the sun –speed V and direction (α,δ) – that maximizes the correlation between observations and predictions. As usual, in the CIF experiment two possible solutions were obtained: •
•
Múnera 1, with the sun moving within the northern celestial hemisphere. We calculated the average correlation between predicted fringe shift as function of solar velocity and the 12 experimental series, one for each month of the year, where the series for same month over different years were averaged. The maximum correlation is R = 0.71 with standard deviation 0.15, and attains for solar speed V = 365 km/s in the direction α =81º = 5h-24m and δ = +79º (Múnera et al., 2009). Múnera 2, with the sun moving within the southern celestial hemisphere. We calculated the average correlation between predicted fringe shift as function of solar velocity and the average correlation between the 24 experimental series (Múnera et al., 2007). The average correlation over the 24 series was R = 0.55 with standard deviation 0.29, and attains for solar speed V = 500 km/s in the direction α =250º = 16h-40m and δ = –75º.
From the value of the average correlation, it may be concluded that the CIF-experiment is more consistent with solar motion towards the northern pole, contrary to Miller’s choice. Also note that the right ascensions of the velocities found by Múnera are very similar to those of Esclangon, but
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differ of Miller’s values by almost exactly 180º in right ascension. This is an intriguing fact that merits further analysis.3
YΣ
Y
Vh α
Earth
Y
Vh December
March Sense of earth’s orbital motion
X
X
First point of XΣ Aries
Z V δ
Sun
a) Projection of earth’s orbit onto equatorial plane
VZ
X
Y
Vh
b) Decomposition of solar velocity in cylindrical coordinates
Figure 1. Orientation of the system of Cartesian coordinates XΣYΣZΣ in absolute space Σ, with arbitrary origin, and the system XYZ attached to the centre of earth. a) In both systems, the XY plane contains the earth’s equatorial plane, and the Z-axis points out of the page. The orbit shown is the projection of earth’s elliptical orbit on the plane of the ecliptic onto the equatorial XY plane. The direction of the X-axis is fixed by the first point of Aries at noon on the vernal equinox (March 21st). b) Decomposition of solar velocity vector V into Vh and VZ, also described by speed V and angles α, δ.
2.4 Cahill experiment. Another estimate of earth’s cosmic motion was recently obtained by Cahill (2006) in Adelaide, Australia, using the one-way speed of electromagnetic waves in a coaxial cable. This experiment is reminiscent of Allais observations in 1958 (see above) and is consistent with De Witte observations in Brussels in the 1980s. Cahill only reports one value: • Cahill. Earth motion towards the southern pole with V = 400 ±20 km/s in the direction α = 5.5 ±2 h, δ = –70º±10º.
3
The difference could be related to the amplifying factor and the direction of the fringe shift, that were introduced by hand by Miller.
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This value is very similar to Miller 2 velocity. It may be noted that Cahill analysis is based on a mixed theory that accepts Lorentz contraction.
3. Earth’s motion in absolute space
Other discoveries over the second half of the 20th century point towards the existence of some special frame of reference. For instance the discovery of the background radiation by Penzias and Wilson (1965), or the dipole anisotropy by the COBE project (Kogut et al, 1993). Table 1 summarizes some of the findings (there is no claim about completeness). It is remarkable that there is consistency in the speed V of earth’s motion obtained by different methods, in the range 200-500 km/s, using vastly different methods and technology. For instance the recent COBE measurement yielded V = 370 km/s, which is similar to the value of 390 km/s reported by Smoot et al. (1977) using astronomical methods. However, the direction of motion of the optical experiments seems to be quite different of the COBE and related measurements. Figure 2 shows the RA of the motion obtained by the various optical measurements. It is remarkable that four different experiments from widely separated places in time and space yield solar motion in a restricted region of space compatible with a plane that crosses the origin from RA = 75º to RA= 255º (i.e. the plane includes both RA = 75º and RA = 255º). This direction is also consistent with α = 4h-35 min or α = 16h-35 min, obtained by Esclangon (1926a; 1926b) from the motion of the pendulum at Postdam. The RA of motion found by Smoot et al (1977), and Kogut et al. (1993) are approximately at a right angle with respect to the optical measurements. Table 1. Velocity of solar system respect an external frame Experiment Esclangon 1 (1926) Esclangon 2 (1926) Miller 1 (1927) Miller 2 (1932) Múnera 1 (2009) Múnera 2 (2007) Cahill (2006) Kogut et al (1993) Smoot et al (1977) Galactic center
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V, km/s --200 208 365 500 400 369.5 390 --
RA, α, º 69 249 255 73.5 81 250 82.5 168.235 165 266
Dec, δ, º +44 –44 +68 –70.55 +79 –75 –70 +6.964 +6 –29
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Motion of sun in absolute space Right ascension
Motion of sun in absolute space Declination
1
sin (Dec)
.
1
0
0
-1 -1
0 -1
0
1
Positive X axis towards first point of Aries
1 cos (Dec) Horizontal axis on equatorial plane
Esclangon 1 (1926)
Esclangon 2 (1926)
Esclangon 1 (1926)
Esclangon 2 (1926)
Miller 1 (1927)
Miller 2 (1932)
Miller 1 (1927)
Miller 2 (1932)
Múnera 1 (2009)
Múnera 2 (2007)
Múnera 1 (2009)
Múnera 2 (2007)
Cahill (2006)
Kogut et al (1993)
Cahill (2006)
Kogut et al (1993)
Smoot et al (1977)
Galactic center
Smoot et al (1977)
Galactic center
Figure 2. Right ascension and declination of the direction of solar motion in Table 1. Note that all optical experiments are close to a plane across the origin with RA = 75º = 5 h. The direction of the galactic center is also included for comparison; its RA is also compatible with the same plane. Also note that the declinations obtained from the interferometer experiments tend to be close to the celestial poles, while the declination inferred from the dipole anisotropy is very low, close to the equatorial plane. An explanation of the differences between the directions found by optical means and by observation of the sky is an open question that merits further analysis. But it may be immediately noted that neither the optical, nor the pendulum, methods require observation of the sky, measurement of cosmic distances, or cosmic time of travel, whereas the astronomical methods, and the inferences from the dipole anisotropy, strongly rely upon direct observation of the sky.4 There are also many recent experiments with resonant cavities that are interpreted as giving support to the standard model extension, and/or to Lorentz invariance, for instance Antonini et al. (2005) and Herrman et al. (2008). However, Consoli and Constanzo (2005) have suggested that such experiments may be also interpreted without imposing consistency with the COBE velocity, thus obtaining different values of velocity, consistent with the notion of absolute space. These authors obtained a declination of 43º (similar to Esclangon value in table 1), but did not report a value for right ascension.
4
Observation of the sky is severely limited by the presence of our Milky Way galaxy close to the equatorial plane.
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4. A novel Newtonian finite and closed universe From the foregoing, it appears that the postulate of the existence of an absolute space Σ, containing all bodies of the universe may be consistent with the observations discussed here. But there are some other intriguing gravitational anomalies, beyond the scope of this paper, as the unknown acceleration perturbing the orbit of some satellites as Pioneer and Lageos, or the apparent failure of the Gravity Probe B experiment, which did not receive further financial support from NASA.5 And there is, of course, the long-standing problem of the anomalously high speeds in some cosmic bodies, first observed by Zwicky (1959), that has led to the introduction of two daring ad hoc hypotheses: the invisible dark matter hypothesis, and the Modification of Newtonian Dynamics, MOND, hypothesis (Milgrom, 2001; Milgrom, 2002), which manifestly violates the principle of conservation of linear momentum. Let us consider instead a classical Newtonian alternative. Let the whole universe be a sphere of total mass M and radius R with a homogeneous average large-scale matter density ρU = 3M/4πR3. The universe is, so-to-speak, a giant gaseous planet contained within absolute space Σ, for which it is possible to calculate the center of mass CMU. Let us locate the origin of the absolute Cartesian system of coordinates XΣYΣZΣ at the CMU. From elementary Newtonian theory of gravitation it is known that a test particle located at distance r from the center of a homogeneous sphere is subject to a gravitational field of magnitude gU, pointing towards the center of the sphere given by 4πGρ U r 4πGρ U 4πGρ U r gU = − r =− rrˆ = −Ω U2 rrˆ where Ω U2 = 3 3 3
(1)
In such universe all test particles will follow cyclic trajectories with period τ = 2π/ΩU. The orbit is elliptic, except in rare occasions when the initial angular momentum is zero. For instance, for ρU = 8 x 10-28 kg/m3 the period is τ = 4 x 1011 years, which is of the order of magnitude of the age of the universe. It is remarkable that in a closed universe all objects will be in continuous rotation with the same period. In such universe, linear motion simply does not exist, except as a local approximation. Therefore, in a closed universe all motion is absolute, and the basic law of physics is conservation of angular momentum. A test particle in a closed universe conserves kinetic plus potential energy along its trajectory according to V 2 Ω U2 r 2 V02 Ω U2 r02 + = + 2 2 2 2
(2)
Another important feature of a closed universe is the variation of speed with distance, which is compatible with the usual Hubble model, and with current observations, usually interpreted as expansion of the universe. As an illustration, table 2 shows V = ΩUr, the speed of test particles at the turning points where radial speed is zero (i.e. the apoastron and the periastron) when it happens at distance r from the CMU, where the calculation was made using elementary 5
According to the New Scientist webpage accessed on 05 December 2008 this experiment was ranked last by Nasa for the 2008 fiscal year.
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Newtonian theory (no apologies for ignoring RTR). The last column shows the velocity relative to an arbitrary observer located at distance r = 3 x 1026 m. Note that most test particles would be receding from that observer, i.e. positive relative speed. Table 2. Velocity of test particle at turning points Distance from V at turning CMU, m points, m/s 1E+26 2E+26 3E+26 4E+26 5E+26 6E+26 7E+26 8E+26 9E+26
4,7E+07 9,5E+07 1,4E+08 1,9E+08 2,4E+08 2,8E+08 3,3E+08 3,8E+08 4,3E+08
V, km/s
Relative velocity, km/s
47000 95000 142000 189000 236000 284000 331000 378000 425000
-95000 -47000 0 47000 95000 142000 189000 236000 284000
Consider now two test particles M1 = 2 x 1030M2 and M2 = 2 x 1030 kg at vector positions r1 and r2 relative to the CMU. The Newtonian equations of motion respectively are
r d 2 r1 r GM 1 M 2 ρ U r M 1 2 = − M 1Ω U2 r1 + R dt R3 r GM 1 M 2 ρ U r d 2 r2 2 r M2 = − M Ω r − R 2 U 2 dt 2 R3
r r r where R = r2 − r1
(3)
where the relative distance between the test particles is R. Adding and subtracting eqs. (3) we get
r d 2 rCM r (M 1 + M 2 ) = −( M 1 + M 2 )Ω U2 rCM 2 dt
r r M 1 r1 + M 2 r2 r where rCM = M1 + M 2
(4)
r r GM T r d 2R 2 = − Ω R − R = − g U Rˆ − g L Rˆ U dt 2 R3 (5) GM T where g U = Ω U2 R, g L = , MT = M1 + M 2 R2 where rCM is the center of mass of the two test particles. As usual, eq. (4) says that the two-body system behaves as a single particle located at it center of mass. Eq. (5) describes the dynamics of the relative distance R controlled by the usual local gravitational field gL, plus a novel universal gravitational field gU. For short distances gL dominates, but for very large separations R (or for high density ρ) the linear term gU may become important. This term may represent an alternative approach to the MOND and dark matter hypotheses. As a numerical example suppose that our galaxy is at rCM = 3 x 1026 m, so that gU = 7 x 10-11 m/s2. This may be compared to the field of the Milky Way at our sun estimated as gL = V2S/RSG = 2 x 10-10 m/s2, where the solar tangential speed is VS = 220 km/s (Kerr and Lynden, 1986), and our sun is located at RS = 25,000 light-years from the center of the galaxy. Héctor A. Múnera
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In our example the values of gU and gL are of the same order of magnitude, and are similar to 1.2 x 10-10 m/s2, which is the value of the limiting acceleration proposed by the MOND hypothesis (Milgrom, 2001; Milgrom, 2002) as a cut-off for the validity of Newtonian theory. This hints that the MOND hypothesis works because the term gU is currently ignored. Detailed calculations and analyses will be published elsewhere.
5. Concluding remarks Summarizing, it is remarkable that four different optical experiments from widely separated places in time and space yield solar motion in a restricted region of space compatible with a plane that crosses the origin at RA = 75º = 5 h (i.e. the plane includes both RA = 75º and RA = 255º). This direction is also consistent with α = 4h-35 min or α = 16h-35 min, obtained by Esclangon (1926a; 1926b) from the motion of the pendulum at Postdam, and it is rather close to the plane containing the galactic center, that pulls the solar system in that direction (see figure 1). It was noted that the optical evidence discussed herein contradicts the heuristic principle that it is impossible to detect earth’s motion by local terrestrial experiments. On the contrary, the empirical evidence presented here is consistent with the existence of an absolute space Σ. Classical Newtonian theory requires that all accelerations relative to Σ be taken into account. Failure to do so may be at the root of many “anomalous” observations relative to current theories. Finally, it was suggested that a finite closed universe is consistent with current observations indicating variation of velocity of cosmic objects with distance from the earth, and may be a viable alternative to explain by classical Newtonian theory the “anomalous” high tangential velocities of cosmic objects without recurring to ad hoc notions (dark matter and MOND).
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