We measure position, which is the distance from a fixed reference point in different coordinates, by a tape of unit distance from one end point to the other end point or its sub-divisions. We measure mass by comparing it with another unit mass. We measure time, which is the interval between events by a clock, whose ticks are repetitive events of equal duration (interval) which we take as the unit, etc. There is no proof to show that this principle is not applicable to the quantum world. These measurements are possible when both the observer with the measuring instrument and the object to be measured are in the same frame of reference (state of motion); thus without disturbing anything. For this reason results of measurement are always scalar quantities – multiples of the unit. Light is only an accessory for knowing the result of measurement and not a pre-condition for measurement. Simultaneous measurement of both position and momentum is not possible, which is correct, though due to different reasons explained in later pages. Incidentally, both position and momentum are regarded as classical concepts.
In classical mechanics and electromagnetism, properties of a point mass or properties of a field are described by real numbers or functions defined on two or three dimensional sets. These have direct, spatial meaning, and in these theories there seems to be less need to provide a special interpretation for those numbers or functions. The accepted mathematical structure of quantum mechanics, on the other hand, is based on fairly abstract mathematics (?), such as Hilbert spaces, (which is the quantum mechanical counterpart of the classical phase-space) and operators on those Hilbert spaces. Here again, there is no precise definition of space. The proof for the existence and justification of the different classification of “space” and “vacuum” are left unexplained.
When developing new theories, physicists tend to assume that quantities such as the strength of gravity, the speed of light in vacuum or the charge on the electron are all constant. The so-called universal constants are neither self-evident in Nature nor have been derived from fundamental principles (though there are some claims to the contrary, each has some problem). They have been deduced mathematically and their value has been determined by actual measurement. For example, the fine structure constant has been postulated in QED, but its value has been derived only experimentally (We have derived the measured value from fundamental principles). Yet, the regularity with which such constants of Nature have been discovered points to some important principle underlying it. But are these quantities really constant?
The velocity of light varies according to the density of the medium. The acceleration due to gravity “g” varies from place to place. We have measured the value of “G” from earth. But we do not know whether the value is the same beyond the solar system. The current value of the distance between the Sun and the Earth has been pegged at 149,597,870.696 kilometers. A recent (2004) study shows that the Earth is moving away from the Sun @ 15 cm per annum. Since this value is 100 times greater than the measurement error, something must really be pushing Earth outwards. While one possible explanation for this phenomenon is that the Sun is losing enough mass via fusion and the solar wind, alternative explanations include the influence of dark matter and changing value of G. We will explain it later.
Einstein proposed the Cosmological Constant to allow static homogeneous solutions to his equations of General Relativity in the presence of matter. When the expansion of the Universe was discovered, it was thought to be unnecessary forcing Einstein to declare was it was his greatest blunder. There have been a number of subsequent episodes in which a non-zero cosmological constant was put forward as an explanation for a set of observations and later withdrawn when the observational case evaporated. Meanwhile, the particle theorists are postulating that the cosmological constant can be interpreted as a measure of the energy density of the vacuum. This energy density is the sum of a number of apparently unrelated contributions: potential energies from scalar fields and zero-point fluctuations of each field theory degree of freedom as well as a bare cosmological constant λ0, each of magnitude much larger than the upper limits on the cosmological constant as measured now. However, the observed vacuum energy is very very small in comparison to the theoretical prediction: a discrepancy of 120 orders of magnitude between the theoretical and observational values of the cosmological constant. This has led some people to postulate an unknown mechanism which would set it precisely to zero. Others postulate the mechanism to suppress the cosmological constant by just the right amount to yield an observationally accessible quantity. However, all agree that this illusive quantity does play an important dynamical role in the Universe. The confusion can be settled if we accept the changing value of G, which can be related to the energy density of the vacuum. Thus, the so-called constants of Nature could also be thought of as the equilibrium points, where different forces acting on a system in different proportions balance each other.
For example, let us consider the Libration points called L4 and L5, which are said to be places that gravity forgot. They are vast regions of space, sometimes millions of kilometers across, in which celestial forces cancel out gravity and trap anything that falls into them. The Libration points, known as ¨ÉxnùÉäSSÉ and {ÉÉiÉ in earlier times, were rediscovered in 1772 by the mathematician Joseph-Louis Lagrange. He calculated that the Earth’s gravitational field neutralizes the gravitational pull of the sun at five regions in space, making them the only places near our planet where an object is truly weightless. Astronomers call them Libration points; also Lagrangian points, or L1, L2, L3, L4 and L5 for short. Of the five Libration points, L4 and L5 are the most intriguing.
Two such Libration points sit in the Earth’s orbit also, one marching ahead of our planet, the other trailing along behind. They are the only ones that are stable. While a satellite parked at L1 or L2 will wander off after a few months unless it is nudged back into place (like the American satellite SOHO), any object at L4 or L5 will stay put due to a complex web of forces (like the asteroids). Evidence for such gravitational potholes appears around other planets too. In 1906, Max Wolf discovered an asteroid outside of the main belt between Mars and Jupiter, and recognized that it was sitting at Jupiter’s L4 point. The mathematics for L4 uses the “brute force approach” making it approximate. Lying 150 million kilometers away along the line of Earth’s orbit, L4 circles the sun about 60 degrees (slightly more, according to our calculation) in front of the planet while L5 lies at the same angle behind. Wolf named it Achilles, leading to the tradition of naming these asteroids after characters from the Trojan wars.
The realization that Achilles would be trapped in its place and forced to orbit with Jupiter, never getting much closer or further away, started a flurry of telescopic searches for more examples. There are now more than 1000 asteroids known to reside at each of Jupiter’s L4 and L5 points. Of these, about ⅔ reside at L4 while the rest ⅓ are at L5. Perturbations by the other planets (primarily Saturn) causes these asteroids to oscillate around L4 and L5 by about 15-200 and at inclinations of up to 400 to the orbital plane. These oscillations generally take between 150 years and 200 years to complete. Such planetary perturbations may also be the reason why there have been so few Trojans found around other planets. Searches for “Trojan” asteroids around other planets have met with mixed results. Mars has 5 of them at L5 only. Saturn seemingly has none. Neptune has two.
The asteroid belt surrounds the inner Solar system like a rocky, ring-shaped moat, extending out from the orbit of Mars to that of Jupiter. But there are voids in that moat in distinct locations called Kirkwood gaps that are associated with orbital resonances with the giant planets – where the orbital influence of Jupiter is especially potent. Any asteroid unlucky enough to venture into one of these locations will follow chaotic orbits and will be perturbed and ejected from the cozy confines of the belt, often winding up on a collision course with one of the inner, rocky planets (such as Earth) or the moon. But Jupiter’s pull cannot account for the extent of the belt’s depletion seen at present or for the spotty distribution of asteroids across the belt – unless there was a migration of planets early in the history of the solar system. According to a report (Nature 457, 1109-1111 dated 26 February 2009), the observed distribution of main belt asteroids does not fill uniformly even those regions that are dynamically stable over the age of the Solar System. There is a pattern of excess depletion of asteroids, particularly just outward of the Kirkwood gaps associated with the 5:2, the 7:3 and the 2:1 Jovian resonances. These features are not accounted for by planetary perturbations in the current structure of the Solar System, but are consistent with dynamical ejection of asteroids by the sweeping of gravitational resonances during the migration of Jupiter and Saturn.
Some researchers designed a computer model of the asteroid belt under the influence of the outer “gas giant” planets, allowing them to test the distribution that would result from changes in the planets’ orbits over time. A simulation wherein the orbits remained static, did not agree with observational evidence. There were places where there should have been a lot more asteroids than we saw. On the other hand, a simulation with an early migration of Jupiter inward and Saturn outward – the result of interactions with lingering planetesimals (small bodies) from the creation of the solar system – fit the observed layout of the belt much better. The uneven spacing of asteroids is readily explained by this planet-migration process that other people have also worked on. In particular, if Jupiter had started somewhat farther from the sun and then migrated inward toward its current location, the gaps it carved into the belt would also have inched inward, leaving the belt looking much like it does now. The good agreement between the simulated and observed asteroid distributions is quite remarkable.
One significant question not addressed in this paper is the pattern of migration – whether the asteroid belt can be used to rule out one of the presently competing theories of migratory patterns. The new study deals with the speed at which the planets’ orbits have changed. The simulation presumes a rather rapid migration of a million or two million years, but other models of Neptune’s early orbital evolution tend to show that migration proceeds much more slowly, over millions of years. We hold this period as 4.32 million years for the Solar system. This example shows that the orbits of planets, which are stabilized due to balancing of the centripetal force and gravity, might be changing from time to time. This implies that either the masses of the Sun and the planets or their distance from each other or both are changing over long periods of time (which is true). It can also mean that G is changing. Thus, the so-called constants of Nature may not be so constants after all.