10.1: Potential Gradients in Extragalactic Space
It sometimes happens that differences between physical quantities are
insignificantly small from some points of view and of decisive importance
from others. I propose to show here that it is so for the potential gradients
that occur far out in extragalactic space. From the inverse square law of
forces one can infer that there is a finite gravitational field everywhere,
even in regions very remote from concentrations of matter. But for most
purposes the field has a significant intensity only in the vicinity of massive
stars; we can usually ignore its existence elsewhere without reaching any
false conclusions. But it will nevertheless appear during the next few
chapters that small differences in the intensity of the gravitational field
in extragalactic space are of great cosmic significance.
In order to appreciate the nature of the 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 anywhere 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
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. I 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, while
it is measured along the sloping ground when the gradient is on a mountain
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
is in three dimensions here 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. The Astronomical Landscape near the earth. The diagram shows the radii
of successive shells of equal potential difference (one tenth of work required
to remove a particle from surface of 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
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 thin
of them as though they were represented by those relief maps in which
vertical heights are greatly magnified.
10.2: The Structure of Space
Another name for gravitational potential gradient is the acceleration
experienced by a body that is subjected to no other force than that of
gravity. This acceleration is a vector quantity, and so departures from flatness in the astronomical landscape can be described alternatively as differences in the magnitude and direction of the vector that defines the potential
gradient or as differences in the value and direction of the accelerations
that would be experienced by freely falling molecules of hydrogen.
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 round 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
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 in the order of magnitude of the gravitational
vectors within the solar system and of those in extragalactic space, as there
is also in the order of magnitude of the time intervals that are significant.
The potential gradient near the surface of the earth is, for instance,
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:
0∫XE 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.
10.3: Astronomical Summits
Clearly there must be points in extragalactic space at which the
gravitational pulls from all surrounding concentrations of mass exactly
cancel. These are points of true flatness and also points such that one loses
potential when one moves away from them in any direction. They can
aptly be called astronomical summits.
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.
10.4. Astronomical Reversal Zones
I propose to give the name 'astronomical reversal zone' to a surface in
space at which the potential gradient reverses. At such a surface the
acceleration of a particle subjected only to gravitational forces is zero.
A projectile could, of course, be shot through the zone. Its velocity would
then diminish as it approached the zone and would increase after it had
passed through. An astronomical reversal zone is in no sense of the word
impenetrable. But gravitation alone can never cause a particle to penetrate
it, only to move away from it. To gravitational forces it offers a boundary.
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
(D1 / D2)2 = m1 / m2 ...... (10a)
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 was the shape of the valleys and ridges
that surround it, what the vector of the potential gradient was in every
place, 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 extra-
galactic landscape will become apparent in due course.
10.5: The Domain of a Galaxy
The volume enclosed by an astronomical reversal zone limits the region
from which a galaxy draws its supplies and this gives it its cosmological
importance. The domain is analogous to a hollow in a terrestrial landscape.
The galaxy at its lowest point can then be thought of like a town that
nestles at the bottom of the hollow and is surrounded on all sides by a
mountain ridge. Only, of course, 'on all sides' has to be taken literally for
the astronomical landscape. An astronomical domain has no outlet any-
where. Matter within an astronomical reversal zone only disappears by
extinction, not by flowing away to somewhere else.
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
Consider two adjacent domains, each containing a galaxy at its centre.
Let the masses within the domain 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)1/2V1 / V2 = (m1 / m2)3/2 ...... (10b)
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.
10.6: The Landscape Around an Astronomical Summit
A mountain summit in a terrestrial landscape is the meeting place of
several ridges and between the ridges valleys descend from the summit
down to the lower plains. Similarly, an astronomical summit is the
meeting place of several astronomical reversal zones. These separate the
galaxies by which the summit is surrounded. The galaxies themselves lie
at the bottom of astronomical valleys.
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, which develop further down 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
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