Space Time and Gravitation/Chapter 9

by Arthur Eddington

 

CHAPTER IX

MOMENTUM AND ENERGY

For spirits and men by different standards mete
The less and greater in the flow of time.
By sun and moon, primeval ordinances—
By stars which rise and set harmoniously—
By the recurring seasons, and the swing
This way and that of the suspended rod
Precise and punctual, men divide the hours,
Equal, continuous, for their common use.
Not so with us in the immaterial world;
But intervals in their succession
Are measured by the living thought alone
And grow or wane with its intensity.
And time is not a common property;
But what is long is short, and swift is slow
And near is distant, as received and grasped
By this mind and by that.

Newman, Dream of Gerontius.

One of the most important consequences of the relativity theory is the unification of inertia and gravitation.

The beginner in mechanics does not accept Newton's first law of motion without a feeling of hesitation. He readily agrees that a body at rest will remain at rest unless something causes it to move; but he is not satisfied that a body in motion will remain in uniform motion so long as it is not interfered with. It is quite natural to think that motion is an impulse which will exhaust itself, and that the body will finally come to a stop. The teacher easily disposes of the arguments urged in support of this view, pointing out the friction which has to be overcome when a train or a bicycle is kept moving uniformly. He shows that if the friction is diminished, as when a stone is projected across ice, the motion lasts for a longer time, so that if all interference by friction were removed uniform motion might continue indefinitely. But he glosses over the point that if there were no interference with the motion—if the ice were abolished altogether—the motion would be by no means uniform, but like that of a falling body. The teacher probably insists that the continuance of uniform motion does not require anything that can properly be called a cause. The property is given a name inertia; but it is thought of as an innate tendency in contrast to force which is an active cause. So long as forces are confined to the thrusts and tensions of elementary mechanics, where there is supposed to be direct contact of material, there is good ground for this distinction; we can visualise the active hammering of the molecules on the body, causing it to change its motion. But when force is extended to include the gravitational field the distinction is not so clear.

For our part we deny the distinction in this last case. Gravitational force is not an active agent working against the passive tendency of inertia. Gravitation and inertia are one. The uniform straight track is only relative to some mesh-system, which is assigned by arbitrary convention. We cannot imagine that a body looks round to see who is observing it and then feels an innate tendency to move in that observer's straight line—probably at the same time feeling an active force compelling it to move some other way. If there is anything that can be called an innate tendency it is the tendency to follow what we have called the natural track—the longest track between two points. We might restate the first law of motion in the form "Every body tends to move in the track in which it actually does move, except in so far as it is compelled by material impacts to follow some other track than that in which it would otherwise move." Probably no one will dispute this profound statement!

Whether the natural track is straight or curved, whether the motion is uniform or changing, a cause is in any case required. This cause is in all cases the combined inertia-gravitation. To have given it a name does not excuse us from attempting an explanation of it in due time. Meanwhile this identification of inertia and gravitation as arbitrary components of one property explains why weight is always proportional to inertia. This experimental fact verified to a very high degree of accuracy would otherwise have to be regarded as a remarkable law of nature.

We have learnt that the natural track is the longest track between two points; and since this is the only definable track having an absolute significance in nature, we seem to have a sufficient explanation of why an undisturbed particle must follow it. That is satisfactory, so far as it goes, but still we should naturally wish for a clearer picture of the cause—inertia-gravitation—which propels it in this track.

It has been seen that the gravitational field round a body involves a kind of curvature of space-time, and accordingly round each particle there is a minute pucker. Now at each successive instant a particle is displaced continuously in time if not in space; and so in our four-dimensional representation which gives a bird s-eye-view of all time, the pucker has the form of a long groove along the track of the particle. Now such a groove or pleat in a continuum cannot take an arbitrary course—as every dress-maker knows. Einstein's law of gravitation gives the rule according to which the curvatures at any point of space-time link on to those at surrounding points; so that when a groove is started in any direction the rest of its course can be forecasted. We have hitherto thought of the law of gravitation as showing how the pucker spreads out in space, cf. Newton s statement that the corresponding force weakens as the inverse square of the distance. But the law of Einstein equally shows how the gravitational field spreads out in time, since there is no absolute distinction of time and space. It can be deduced mathematically from Einstein's law that a pucker of the form corresponding to a particle necessarily runs along the track of greatest interval-length between two points.

The track of a particle of matter is thus determined by the interaction of the minute gravitational field, which surrounds and, so far as we know, constitutes it, with the general space-time of the region. The various forms which it can take, find their explanation in the new law of gravitation. The straight tracks of the stars and the curved tracks of the planets are placed on the same level, and receive the same kind of explanation. The one universal law, that the space-time continuum can be curved only in the first degree, is sufficient to prescribe the forms of all possible grooves crossing it.

The application of Einstein's law to trace the gravitational field not only through space but through time leads to a great unification of mechanics. If we have given for a start a narrow slice of space-time representing the state of the universe for a few seconds, with all the little puckers belonging to particles of matter properly described, then step by step all space-time can be linked on and the positions of the puckers shown at all subsequent times (electrical forces being excluded). Nothing is needed for this except the law of gravitation—that the curvature is only of the first degree—and there can thus be nothing in the predictions of mechanics which is not comprised in the law of gravitation. The conservation of mass, of energy, and of momentum must all be contained implicitly in Einstein s law.

It may seem strange that Einstein's law of gravitation should take over responsibility for the whole of mechanics; because in many mechanical problems gravitation in the ordinary sense can be neglected. But inertia and gravitation are unified; the law is also the law of inertia, and inertia or mass appears in all mechanical problems. When, as in many problems, we say that gravitation is negligible, we mean only that the interaction of the minute puckers with one another can be neglected; we do not mean that the interaction of the pucker of a particle with the general character of the space-time in which it lies can be neglected, because this constitutes the inertia of the particle.

The conservation of energy and the conservation of momentum in three independent directions, constitute together four laws or equations which are fundamental in all branches of mechanics. Although they apply when gravitation in the ordinary sense is not acting, they must be deducible like everything else in mechanics from the law of gravitation. It is a great triumph for Einstein's theory that his law gives correctly these experimental principles, which have generally been regarded as unconnected with gravitation. We cannot enter into the mathematical deduction of these equations; but we shall examine generally how they are arrived at.

It has already been explained that although the values of $G_{\mu \nu }$ are strictly zero everywhere in space-time, yet if we take average values through a small region containing a large number of particles of matter their average or "macroscopic" values will not be zero^[1]. Expressions for these macroscopic values can be found in terms of the number, masses and motions of the particles. Since we have averaged the $G_{\mu \ nu}$, we should also average the particles; that is to say, we replace them by a distribution of continuous matter having equivalent properties. We thus obtain macroscopic equations of the form $G_{\mu \nu }=K_{\mu \nu }$, where on the one side we have the somewhat abstruse quantities describing the kind of space-time, and on the other side we have well-known physical quantities describing the density, momentum, energy and internal stresses of the matter present. These macroscopic equations are obtained solely from the law of gravitation by the process of averaging.

By an exactly similar process we pass from Laplace's equation $\nabla ^{2}\phi =0$ to Poisson's equation for continuous matter $\nabla ^{2}\phi =-4\pi \rho$, in the Newtonian theory of gravitation.

When continuous matter is admitted, any kind of space-time becomes possible. The law of gravitation instead of denying the possibility of certain kinds, states what values of $K_{\mu \nu }$, i.e. what distribution and motion of continuous matter in the region, are a necessary accompaniment. This is no contradiction with the original statement of the law, since that referred to the case in which continuous matter is denied or excluded. Any set of values of the potentials is now possible; we have only to calculate by the formulae the corresponding values of $G_{\mu \nu }$, and we at once obtain ten equations giving the $K_{\mu \nu }$ which define the conditions of the matter necessary to produce these potentials. But suppose the necessary distribution of matter through space and time is an impossible one, violating the laws of mechanics! No, there is only one law of mechanics, the law of gravitation; we have specified the distribution of matter so as to satisfy $G_{\mu \nu }=K_{\mu \nu }$, and there can be no other condition for it to fulfil. The distribution must be mechanically possible; it might, however, be unrealisable in practice, involving inordinately high or even negative density of matter.

In connection with the law for empty space, $G_{\mu \nu }=0$, it was noticed that whereas this apparently forms a set of ten equations, only six of them can be independent. This was because ten equations would suffice to determine the ten potentials precisely, and so fix not only the kind of space-time but the mesh-system. It is clear that we must preserve the right to draw the mesh-system as we please; it is fixed by arbitrary choice not by a law of nature. To allow for the four-fold arbitrariness of choice, there must be four relations always satisfied by the $G_{\mu \nu }$, so that when six of the equations are given the remaining four become tautological.

These relations must be identities implied in the mathematical definition of $G_{\mu \nu }$; that is to say, when the $G_{\mu \nu }$ have been written out in full according to their definition, and the operations indicated by the identities carried out, all the terms will cancel, leaving only $0=0$. The essential point is that the four relations follow from the mode of formation of the $G_{\mu \nu }$ from their simpler constituents ($g_{\mu \nu }$ and their differential coefficients) and apply universally. These four identical relations have actually been discovered^[2].

When in continuous matter $G_{\mu \nu }=K_{\mu \nu }$ clearly the same four relations must exist between the $K_{\mu \nu }$, not now as identities, but as consequences of the law of gravitation, viz. the equality of $G_{\mu \nu }$ and $K_{\mu \nu }$.

Thus the four dimensions of the world bring about a four-fold arbitrariness of choice of mesh-system; this in turn necessitates four identical relations between the $G_{\mu \nu }$; and finally, in consequence of the law of gravitation, these identities reveal four new facts or laws relating to the density, energy, momentum or stress of matter, summarised in the expressions $K_{\mu \nu }$.

These four laws turn out to be the laws of conservation of momentum and energy.

The argument is so general that we can even assert that corresponding to any absolute property of a volume of a world of four dimensions (in this case, curvature), there must be four relative properties which are conserved. This might be made the starting-point of a general inquiry into the necessary qualities of a permanent perceptual world, i.e. a world whose substance is conserved.

There is another law of physics which was formerly regarded as fundamental—the conservation of mass. Modern progress has somewhat altered our position with regard to it; not that its validity is denied, but it has been reinterpreted, and has finally become merged in the conservation of energy. It will be desirable to consider this in detail.

It was formerly supposed that the mass of a particle was a number attached to the particle, expressing an intrinsic property, which remained unaltered in all its vicissitudes. If $m$ is this number, and $u$ the velocity of the particle, the momentum is $mu$; and it is through this relation, coupled with the law of conservation of momentum that the mass $m$ was defined. Let us take for example two particles of masses $m_{1}=2$ and $m_{2}=3$, moving in the same straight line. In the space-time diagram for an observer $S$ the velocity of the first particle will be represented by a direction $OA$ (Fig. 19). The first particle moves through

a space $MA$ in unit time, so that $MA$ is equal to its velocity referred to the observer $S$. Prolonging the line $OA$ to meet the second time-partition, $NB$ is equal to the velocity multiplied by the mass 2; thus the horizontal distance $NB$ represents the momentum. Similarly, starting from $B$ and drawing $BC$ in the direction of the velocity of $m_{2}$, prolonged through three time-partitions, the horizontal progress from $B$ represents the momentum of the second particle. The length $PC$ then represents the total momentum of the system of two particles.

Suppose that some change of their velocities occurs, not involving any transference of momentum from outside, e.g. a collision. Since the total momentum $PC$ is unaltered, a similar construction made with the new velocities must again bring us to $C$; that is to say, the new velocities are represented by the directions $OB^{\prime }$, $B^{\prime }C$, where $B^{\prime }$ is some other point on the line $NB$.

Now examine how this will appear to some other observer $S_{1}$ in uniform motion relative to $S$. His transformation of space and time has been described in Chapter iii and is represented in Fig. 20, which shows how his time-partitions run as compared with those of $S$. The same actual motion is, of course, represented by parallel directions in the two diagrams; but the

interpretation as a velocity $MA$ is different in the two cases. Carrying the velocity of $m_{1}$ through two time-partitions, and of $m_{2}$ through three time-partitions, as before, we find that the total momentum for the observer $S_{1}$ is represented by $PC$ (Fig. 20); but making a similar construction with the velocities after collision, we arrive at a different point $C^{\prime }$. Thus whilst momentum is conserved for the observer $S$, it has altered from $PC$ to $PC^{\prime }$ for the observer $S_{1}$.

The discrepancy arises because in the construction the lines are prolonged to meet partitions which are different for the two observers. The rule for determining momentum ought to be such that both observers make the same construction, independent of their partitions, so that both arrive by the two routes at the same point $C$. Then it will not matter if, through their different measures of time, one observer measures momentum by horizontal progress and the other by oblique progress; both will agree that the momentum has not been altered by the collision. To describe such a construction, we must use the interval which is alike for both observers; make the interval-length of $OB$ equal to 2 units, and that of $BC$ equal to 3 units, disregarding the mesh-system altogether. Then both observers will make the same diagram and arrive at the same point $C$ (different from $C$ or $C^{\prime }$ in the previous diagrams). Then if momentum is conserved for one observer, it will be conserved for the other.

This involves a modified definition of momentum. Momentum must now be the mass multiplied by the change of position $\partial {x}$ per lapse of interval $\partial {s}$, instead of per lapse of time $\partial {t}$. Thus ${\begin{aligned}{\text{momentum}}&=m{\frac {\partial {x}}{\partial {s}}}\\{\text{instead of momentum}}&=m{\frac {\partial {x}}{\partial {t}}}{\text{,}}\end{aligned}}$ and the mass $m$ still preserves its character as an invariant number associated with the particle.

Whether the momentum as now defined is actually conserved or not, is a matter for experiment, or for theoretical deduction from the law of gravitation. The point is that with the original definition general conservation is impossible, because if it held good for one observer it could not hold for another. The new definition makes general conservation possible. Actually this form of the momentum is the one deduced from the law of gravitation through the identities already described. With regard to experimental confirmation it is sufficient at present to state that in all ordinary cases the interval and the time are so nearly equal that such experimental foundation as existed for the law of conservation of the old momentum is just as applicable to the new momentum.

Thus in the theory of relativity momentum appears as an invariant mass multiplied by a modified velocity $\partial x/\partial s$. The physicist, however, prefers for practical purposes to keep to the old definition of momentum as mass multiplied by the velocity $\partial x/\partial t$. We have $m{\frac {\partial x}{\partial s}}=m{\frac {\partial t}{\partial s}}\cdot {\frac {\partial x}{\partial t}}$, accordingly the momentum is separated into two factors, the velocity $\partial x/\partial t$, and a mass $M=m\partial t/\partial s$, which is no longer an invariant for the particle but depends on its motion relative to the observer's space and time. In accordance with the usual practice of physicists the mass (unless otherwise qualified) is taken to mean the quantity $M$.

Using unaccelerated rectangular axes, we have by definition of $s$ $\partial s^{2}=\partial t^{2}-\partial x^{2}-\partial y^{2}-\partial x^{2}$, so that ${\begin{aligned}\left({\frac {\partial s}{\partial t}}\right)^{2}&=1-\left({\frac {\partial x}{\partial t}}\right)^{2}-\left({\frac {\partial y}{\partial t}}\right)^{2}-\left({\frac {\partial z}{\partial t}}\right)^{2}{\text{,}}\\&=1-u^{2}{\text{,}}\end{aligned}}$ where $u$ is the resultant velocity of the particle (the velocity of light being unity). Hence $M={\frac {m}{\sqrt {1-u^{2}}}}$. Thus the mass increases as the velocity increases, the factor being the same as that which determines the FitzGerald contraction.

The increase of mass with velocity is a property which challenges experimental test. For success it is necessary to be able to experiment with high velocities and to apply a known force large enough to produce appreciable deflection in the fast-moving particle. These conditions are conveniently fulfilled by the small negatively charged particles emitted by radioactive substances, known as β particles, or the similar particles which constitute cathode rays. They attain speeds up to 0.8 of the velocity of light, for which the increase of mass is in the ratio 1.66; and the negative charge enables a large electric or magnetic force to be applied. Modern experiments fully confirm the theoretical increase of mass, and show that the factor $1/{\sqrt {(1-u^{2})}}$ is at least approximately correct. The experiment was originally performed by Kaufmann; but much greater accuracy has been obtained by recent modified methods.

Unless the velocity is very great the mass $M$ may be written $m/{\sqrt {(1-2^{2})}}=m+{\tfrac {1}{2}}mu^{2}$. Thus it consists of two parts, the mass when at rest, together with the second term which is simply the energy of the motion. If we can say that the term $m$ represents a kind of potential energy concealed in the matter, mass can be identified with energy. The increase of mass with velocity simply means that the energy of motion has been added on. We are emboldened to do this because in the case of an electrical charge the electrical mass is simply the energy of the static field. Similarly the mass of light is simply the electromagnetic energy of the light.

In our ordinary units the velocity of light is not unity, and a rather artificial distinction between mass and energy is introduced. They are measured by different units, and energy $E$ has a mass $E/C^{2}$ where $C$ is the velocity of light in the units used. But it seems very probable that mass and energy are two ways of measuring what is essentially the same thing, in the same sense that the parallax and distance of a star are two ways of expressing the same property of location. If it is objected that they ought not to be confused inasmuch as they are distinct properties, it must be pointed out that they are not sense-properties, but mathematical terms expressing the dividend and product of more immediately apprehensible properties, viz. momentum and velocity. They are essentially mathematical compositions, and are at the disposal of the mathematician.

This proof of the variation of mass with velocity is much more general than that based on the electrical theory of inertia. It applies immediately to matter in bulk. The masses $m_{1}$ and $m_{2}$ need not be particles; they can be bodies of any size or composition. On the electrical theory alone, there is no means of deducing the variation of mass of a planet from that of an electron.

It has to be remarked that, although the inertial mass of a particle only comes under physical measurement in connection with a change of its motion, it is just when the motion is changing that the conception of its mass is least definite; because it is at that time that the kinetic energy, which forms part of the mass, is being passed on to another particle or radiated into the surrounding field; and it is scarcely possible to define the moment at which this energy ceases to be associated with the particle and must be reckoned as broken loose. The amount of energy or mass in a given region is always a definite quantity; but the amount attributable to a particle is only definite when the motion is uniform. In rigorous work it is generally necessary to consider the mass not of a particle but of a region.

The motion of matter from one place to another causes an alteration of the gravitational field in the surrounding space. If the motion is uniform, the field is simply convected; but if the motion is accelerated, something of the nature of a gravitational wave is propagated outwards. The velocity of propagation is the velocity of light. The exact laws are not very simple because we have seen that the gravitational field modifies the velocity of light; and so the disturbance itself modifies the velocity with which it is propagated. In the same way the exact laws of propagation of sound are highly complicated, because the disturbance of the air by sound modifies the speed with which it is propagated. But the approximate laws of propagation of gravitation are quite simple and are the same as those of electromagnetic disturbances.

After mass and energy there is one physical quantity which plays a very fundamental part in modern physics, known as Action. Action here is a very technical term, and is not to be confused with Newton's "Action and Reaction." In the relativity theory in particular this seems in many respects to be the most fundamental thing of all. The reason is not difficult to see. If we wish to speak of the continuous matter present at any particular point of space and time, we must use the term density. Density multiplied by volume in space gives us mass or, what appears to be the same thing, energy. But from our space-time point of view, a far more important thing is density multiplied by a four-dimensional volume of space and time; this is action. The multiplication by three dimensions gives mass or energy; and the fourth multiplication gives mass or energy multiplied by time. Action is thus mass multiplied by time, or energy multiplied by time, and is more fundamental than either.

Action is the curvature of the world. It is scarcely possible to visualise this statement, because our notion of curvature is derived from surfaces of two dimensions in a three-dimensional space, and this gives too limited an idea of the possibilities of a four-dimensional surface in space of five or more dimensions. In two dimensions there is just one total curvature, and if that vanishes the surface is flat or at least can be unrolled into a plane. In four dimensions there are many coefficients of curvature; but there is one curvature par excellence, which is, of course, an invariant independent of our mesh-system. It is the quantity we have denoted by $G$. It does not follow that if the curvature vanishes space-time is flat; we have seen in fact that in a natural gravitational field space-time is not flat although there may be no mass or energy and therefore no action or curvature.

Wherever there is matter^[3] there is action and therefore curvature; and it is interesting to notice that in ordinary matter the curvature of the space-time world is by no means insignificant. For example, in water of ordinary density the curvature is the same as that of space in the form of a sphere of radius 570,000,000 kilometres. The result is even more surprising if expressed in time units; the radius is about half-an-hour.

It is difficult to picture quite what this means; but at least we can predict that a globe of water of 570,000,000 km. radius would have extraordinary properties. Presumably there must be an upper limit to the possible size of a globe of water. So far as I can make out a homogeneous mass of water of about this size (and no larger) could exist. It would have no centre, and no boundary, every point of it being in the same position with respect to the whole mass as every other point of it—like points on the surface of a sphere with respect to the surface. Any ray of light after travelling for an hour or two would come back to the starting point. Nothing could enter or leave the mass, because there is no boundary to enter or leave by; in fact, it is coextensive with space. There could not be any other world anywhere else, because there isn't an "anywhere else."

The mass of this volume of water is not so great as the most moderate estimates of the mass of the stellar system. Some physicists have predicted a distant future when all energy will be degraded, and the stellar universe will gradually fall together into one mass. Perhaps then these strange conditions will be realised!

The law of gravitation, the laws of mechanics, and the laws of the electromagnetic field have all been summed up in a single Principle of Least Action. For the most part this unification was accomplished before the advent of the relativity theory, and it is only the addition of gravitation to the scheme which is novel. We can see now that if action is something absolute, a configuration giving minimum action is capable of absolute definition; and accordingly we should expect that the laws of the world would be expressible in some such form. The argument is similar to that by which we first identified the natural tracks of particles with the tracks of greatest interval-length. The fact that some such form of law is inevitable, rather discourages us from seeking in it any clue to the structural details of our world.

Action is one of the two terms in pre-relativity physics which survive unmodified in a description of the absolute world. The only other survival is entropy. The coming theory of relativity had cast its shadow before; and physics was already converging to two great generalisations, the principle of least action and the second law of thermodynamics or principle of maximum entropy.

We are about to pass on to recent and more shadowy developments of this subject; and this is an appropriate place to glance back on the chief results that have emerged. The following summary will recall some of the salient points.

1. The order of events in the external world is a four-dimensional order.

2. The observer either intuitively or deliberately constructs a system of meshes (space and time partitions) and locates the events with respect to these.

3. Although it seems to be theoretically possible to describe phenomena without reference to any mesh-system (by a catalogue of coincidences), such a description would be cumbersome. In practice, physics describes the relations of the events to our mesh-system; and all the terms of elementary physics and of daily life refer to this relative aspect of the world.

4. Quantities like length, duration, mass, force, etc. have no absolute significance; their values will depend on the mesh-system to which they are referred. When this fact is realised, the results of modern experiments relating to changes of length of rigid bodies are no longer paradoxical.

5. There is no fundamental mesh-system. In particular problems, and more particularly in restricted regions, it may be possible to choose a mesh-system which follows more or less closely the lines of absolute structure in the world, and so simplify the phenomena which are related to it. But the world-structure is not of a kind which can be traced in an exact way by mesh-systems, and in any large region the mesh-system drawn must be considered arbitrary. In any case the systems used in current physics are arbitrary.

6. The study of the absolute structure of the world is based on the "interval" between two events close together, which is an absolute attribute of the events independent of any mesh-system. A world-geometry is constructed by adopting the interval as the analogue of distance in ordinary geometry.

7. This world-geometry has a property unlike that of Euclidean geometry in that the interval between two real events may be real or imaginary. The necessity for a physical distinction, corresponding to the mathematical distinction between real and imaginary intervals, introduces us to the separation of the four-dimensional order into time and space. But this separation is not unique, and the separation commonly adopted depends on the observer's track through the four-dimensional world.

8. The geodesic, or track of maximum or minimum interval-length between two distant events, has an absolute significance. And since no other kind of track can be defined absolutely, it is concluded that the tracks of freely moving particles are geodesics.

9. In Euclidean geometry the geodesics are straight lines. It is evidently impossible to choose space and time-reckoning so that all free particles in the solar system move in straight lines. Hence the geometry must be non-Euclidean in a field of gravitation.

10. Since the tracks of particles in a gravitational field are evidently governed by some law, the possible geometries must be limited to certain types.

11. The limitation concerns the absolute structure of the world, and must be independent of the choice of mesh-system. This narrows down the possible discriminating characters. Practically the only reasonable suggestion is that the world must (in empty space) be "curved no higher than the first degree"; and this is taken as the law of gravitation.

12. The simplest type of hummock with this limited curvature has been investigated. It has a kind of infinite chimney at the summit, which we must suppose cut out and filled up with a region where this law is not obeyed, i.e. with a particle of matter.

13. The tracks of the geodesics on the hummock are such as to give a very close accordance with the tracks computed by Newton's law of gravitation. The slight differences from the Newtonian law have been experimentally verified by the motion of Mercury and the deflection of light.

14. The hummock might more properly be described as a ridge extending linearly. Since the interval-length along it is real or time-like, the ridge can be taken as a time-direction. Matter has thus a continued existence in time. Further, in order to conform with the law, a small ridge must always follow a geodesic in the general field of space-time, confirming the conclusion arrived at under (8).

15. The laws of conservation of energy and momentum in mechanics can be deduced from this law of world-curvature.

16. Certain phenomena such as the FitzGerald contraction and the variation of mass with velocity, which were formerly thought to depend on the behaviour of electrical forces concerned, are now seen to be general consequences of the relativity of knowledge. That is to say, length and mass being the relations of some absolute thing to the observer's mesh-system, we can foretell how these relations will be altered when referred to another mesh-system.

1. It is the $g$'s which are first averaged, then the $G_{\mu \nu }$ are calculated by the formulae in Note 5.
2. Appendix, Note 13.
3. It is rather curious that there is no action in space containing only light. Light has mass ($M$) of the ordinary kind; but the invariant mass ($m$) vanishes.