|Potential Gradients in Extragalactic Space
The Structure of Space
Astronomical Reversal Zones
The Domain of a Galaxy
The Landscape Around an Astronomical Summit
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Ardue Site Plan
See also:Exercise 3
In order to appreciate the nature of these very distant gravitational fields, let us imagine a space traveller who moves away from the Earth, away from the Solar System, away from our galaxy, far out into lonely extragalactic space. As he rises from the Earth, he first experiences the Earth's gravitational pull. It is so strong that other fields have a negligible effect. But the Earth's field diminishes in accordance with the inverse square law and so a time comes when it is so weak that the field of the more distant Sun, hardly noticed before, preponderates. When the space traveller has journeyed still farther and has left our galaxy behind him, the Sun's pull becomes as weak as that of millions of other galactic stars. Each of these makes its contribution to the field that remains. It is a very feeble field indeed, but a finite one. If the space traveller were to bring his machine to a halt, he would slowly fall back on to our galaxy.
As the ascent proceeds still farther, this feeble field continues to decrease. The climb is like that of a mountain that is steep at its lower slopes and becomes gentle near the top.
During this imaginary ascent into outer space, the space traveller is not only getting farther away from our own galaxy; he is also getting nearer to some other one. He eventually arrives at a region where the faint pull behind him that is exerted by our galaxy equals the forward pull exerted by the galaxy that he is approaching. This region is like that at a mountain ridge; the traveller who reaches it and journeys farther ceases to ascend and begins to descend. The descent is gentle at first and becomes steeper as the second galaxy is approached.
Another name for intensity of the field is potential gradient; for the field intensity is a measure of the rate of change of potential with distance. This is why one can compare the field intensity at any point to the gradient on a mountain side. The analogy is not perfect, but it is helpful. Let us therefore begin by considering gravitational potential gradients in the terrestrial landscape.
One can imagine that the Earth is surrounded by a number of concentric spherical surfaces of differing radii. Their intersection with mountains will be those lines that appear on topographical maps and are called contour lines. If the shells are equally spaced, each represents a given height above sea level. To a very close approximation, each shell is also the locus of all points that have the same potential in respect to the Earth's gravitational field. Only a slight difference is occasioned by variations in the value of g over the Earth's surface. Let the shells be so spaced that the difference in potential represented by adjacent ones is always the same. For small heights, the difference in spacing will then also be nearly constant, just as it is for contour lines. But where the radius of the shells is great compared with that of the Earth, equal potential differences will not correspond to equal differences in radii. The potential gradient decreases with the square of the distance from the centre of the Earth, and so successive shells will be spaced ever more widely as they come to be further out in space. In other words, the vertical journey required to gain a given amount of potential energy increases rather rapidly with increasing distance from the centre of attraction. The diagram in Fig. 1 below illustrates this.
The distance up a sloping mountain side that must be travelled in order to gain a given amount of potential energy is greater the more gentle the slope; and therein lies the analogy. Both in space and up a mountain side, a gradient is defined as the distance that an object must be moved in order for its potential to change by a given amount. But in space, this distance is measured at right angles to an equipotential surface; whereas when the gradient is on a mountain side, it is measured along the sloping ground.
One calls a landscape 'flat' when movement towards any point of the compass does not result in a change of potential; and one calls space 'flat' in the same sense, when movement in any direction, left or right, backward or forward, up or down, does not result in a change of potential. The flatness in space is in three dimensions, and this makes it impossible to represent the topography of space by a map. In cartography, points of equal potential are connected by lines; in spatial topography, they must be represented by curved surfaces. To speak of the spatial landscape is conceptually correct, but it cannot be grasped by the imagination.
If one could do so, one would obtain a picture that differed greatly from any familiar terrestrial landscape.
Fig. 1. represents the Astronomical Landscape near the Earth. The diagram shows the radii of successive shells of equal potential difference (one tenth of the work required to remove a particle from the surface of the Earth to infinity). The lower curve shows the equivalent gradient in two dimensions.
Let the analogy be pursued to the extent of describing the terrestrial landscape that would be equivalent to the actual spatial one. Regions of space near massive stars would look like very deep, circular wells with only a very slight paraboloid taper towards the bottom. Regions of extragalactic space would, on the other hand, look like a vast table-land so flat that one might be inclined to call it absolutely featureless.
For many purposes it can safely be so regarded; but I propose to show here that the tiny features that occur in these regions should be expected to be significant and, in fact, to determine the origin — even the shape — of galaxies. So instead of thinking of these regions as flat, let us try to think of them as though they were represented by those relief maps in which vertical heights are greatly magnified.
The vector that defines the potential gradient provides a physical means by which to distinguish one part of space from another. Hence the different accelerations with which, and the different directions in which, particles move in different parts of space are said by relativists to define the physical properties of the space in which the particles find themselves. For this reason it will often be found useful to say that the vectors that represent potential gradients define the structure of space.
These vectors determine the movement of all ponderable matter. It is they that enter into the calculation of the movement of the planets, the Earth, the Moon. The Earth's orbit around the Sun is a function of the magnitude and direction of the gravitational potential gradient in which the Earth is from moment to moment. It is a gradient that is obtained by combining vectorially the gradients attributable respectively to the Sun and to those planets that are near enough to the Earth to influence its course.
Similar vectors must combine to determine the movement of molecules of hydrogen in extragalactic space, and these distant vectors become the proper study for anyone who would understand what happens there. But there is a great difference between the order of magnitude of the gravitational vectors within the Solar System and of those in extragalactic space; there is a similar difference in the order of magnitude of the significant time intervals.
The potential gradient near the surface of the Earth, for instance, is such that a body free to move there adds to its velocity nearly ten metres a second during every second. In a potential gradient of such steepness, a short time suffices for a substantial displacement of matter. Distant though the Sun is from the Earth, the potential gradient of its field at the Earth's orbit is still great enough to cause the Earth to complete the orbit within a year, which means that the vector defining its linear velocity is reversed in the short time of half a year.
In the comparatively minute potential gradients that occur in extra-galactic space, it would take a very long time for a molecule of hydrogen to add ten metres per second to its velocity; but then a very long time is available for the processes that occur there. Those who like to adopt hypothesis A2 and believe that the whole Universe began at a finite moment in the past have recently come to place this moment some thousands of millions of years ago. As has been mentioned already, this period of time has been arrived at from the evidence of several observed irreversible processes which are assumed to have begun with the beginning of the Universe.
It will be found here in due course that a time interval of about three and a half million years does have a particular cosmic significance, though not the one attributed to it by supporters of A2. Even a small fraction of this interval would suffice for molecules of hydrogen subjected to very feeble accelerations to acquire very high velocities and to move over very great distances. Any hydrogen that occurs in extragalactic space must follow these gradients and must, in time, be displaced from one region to another very distant one. It must during the process acquire a large quantity of kinetic energy, a quantity that would cause considerable disturbance whenever the moving mass of hydrogen hit anything.
In short, the structure of space everywhere has cosmic significance. In mathematical terms the important quantity is ∫0x E.dx, where E is the potential gradient at a given distance x. The value of this integral can become great in extragalactic space because x can be great there.
An astronomical summit can be defined as a point in space where the potential is a maximum and the potential gradient is zero, in whichever direction it be measured.
Here the analogy to a terrestrial landscape fails. The latter is two-dimensional, while the astronomical landscape is three-dimensional. At a terrestrial mountain top, one could increase one's potential by rising up into the air. But on an astronomical summit, one cannot increase one's potential by moving in any direction. Just as at the North Pole every signpost points South, so at an astronomical summit every displacement is downwards.
The astronomical summits are points on a reversal zone. What may be called 'astronomical passes' are other points on the zone. A pass lies on a straight line between two adjacent galaxies, and the position of the pass is given by the inverse square law if the effect of more distant galaxies is neglected. It would be such that
when D1 and D2 are, respectively, the distances of the pass from the galaxies, and m1 and m2 are the masses enclosed by the reversal zones that surround the two galaxies.
It is clearly not necessary for the masses to be concentrated in the galaxies themselves. Equation (10a) holds however the masses may be distributed within their respective reversal zones. The whole mass may be in the galaxy or a part of it may be diffused as extragalactic hydrogen.
While an astronomical summit is a point on the reversal zone where the potential is a maximum, an astronomical pass is a point on the reversal zone where the potential is a minimum, though the potential is less at any point away from the reversal zone on either side of the pass. In this respect, terrestrial summits and passes are analogous. Between passes and summits there are ridges, and in a terrestrial landscape water will always flow down the side of a ridge and never across it. Similarly, hydrogen will always flow down the side of an astronomical ridge.
The analogy to a terrestrial landscape is, of course, imperfect, as has been noticed already. A mountain ridge ends somewhere in a plain, but an astronomical reversal zone surrounds a galaxy in every direction: left and right, forwards and backwards, up and down. It encloses a three-dimensional volume. In whichever direction one travelled away from a galaxy, one could never get round its astronomical reversal zone. If one travelled far enough, one would always pass through this zone.
It may be worthwhile to point out for the benefit of the non-physicist that all this is not hypothesis but only what can be inferred from the inverse square law. If one knew the masses of the neighbouring concentrations and their distances from each other, one could calculate where the astronomical summit was, what were the shapes of the valleys and ridges that surround it, what the vector of the potential gradient was at every point, how high the pass was over which one could travel from one concentration to its neighbour.
This spatial landscape is not, like terrestrial ones, unchanging. Perhaps one ought to think of it not as a landscape but as the surface of a boiling, viscous liquid. For the topographical features depend on the relative masses and distances of the neighbouring concentrations, and these do not remain constant. In an expanding Universe, distances are always increasing: and a consequence of this must be a smoothing out of the topography. The dome-shaped tops of the astronomical mountains must become flatter as time goes on — at least provided increase of the mass of the neighbouring concentrations does not have a sufficient steepening effect to counteract this tendency. The cosmological significance of this ever-changing extragalactic landscape will become apparent in due course.
In a terrestrial landscape, a hollow is the catchment area for water. Any rain that falls within it finds its way towards the bottom and any rain that falls beyond the enclosing ridge of hills flows away from the hollow and into a different one. Similarly, the volume enclosed by an astronomical reversal zone is the catchment volume for particles of matter that originate within it and for no others.
In the model based on A3, the gross annual income of a domain is directly proportional to its volume and the volume is, in turn, a direct function of the mass within the domain as the following consideration shows.
Consider two adjacent domains, each containing a galaxy at its centre. Let the masses within the domains be, respectively, m1 and m2. However these be distributed, the two masses must act as though they were concentrated at the centres of gravity of their respective domains.
It follows from equation (10a) that (D1/D2) = (m1/m2))1/2. The volumes V1 and V2 of the domains must be roughly proportional to the cubes of the distances D1 and D2, and so one can write the approximate expression
This shows that the domain from which a galaxy can attract hydrogen to itself is greater the greater the mass contained within the domain.
Thus the astronomical summit is by no means analogous to a simple, smooth dome. It has a featured structure. From the top, ravines extend in all directions, and further down develop into deeper and deeper, as well as steeper and steeper, valleys. Between the ravines, there are astronomical shoulders. They descend, like shoulders from a terrestrial mountain, towards the astronomical passes and the gradient rises on the further side of these along the ridges that lead to the next summits.
A moment's thought shows that the number of principal ravines and shoulders that meet at an astronomical summit cannot vary between very wide limits. Every principal ravine points in the direction of one of the nearest galaxies and every principal shoulder points towards an opening between adjacent galaxies. So the number of principal ravines and shoulders is determined by the number of galaxies near enough to have a significant effect on the local structure of space.
Galaxies are scattered about the sky in quite an irregular manner, but if they formed a closely packed regular pattern and were also all equidistant from each other, they could occur on the corners of tetrahedrons. There would then be an astronomical summit inside each tetrahedron and this would be equidistant from four galaxies. A straight line from the summit to each of these four would follow the course of an astronomical ravine. A straight line to the centre of each of the four triangular surfaces that bound the tetrahedron would, on the other hand, follow the ridge of an astronomical shoulder.
With such a regular pattern, there would therefore be four ravines and four shoulders. Gentle ditches along the latter would lie on lines pointing to more distant galaxies, but the nearest of these would be well over twice the distance from one of the corners of the tetrahedron. It will be seen later from equation (12f) in Section 12.4 that the potential gradient varies inversely as the cube of the distance, so the effect of all such more distant galaxies is negligible.
The regular tetrahedron would provide the most compact spacing. It cannot be typical of the actual irregular pattern; and so an astronomical summit must normally be surrounded by more than four neighbouring galaxies. One might expect a more frequent pattern to approximate roughly to a cubic arrangement. From a summit inside a cube, eight galaxies are equidistant and so there would be eight ravines, while six shoulders would point towards the six sides of the cube. A spacing that led to a greater number of ravines and shoulders could occur, but would not be very common.