From the book "Causality, Electromagnetic Induction, and Gravitation" by Oleg D. Jefimenko (pp. 189-202)

A GRAVITATIONAL AND ELECTROMAGNETIC ANALOGY

**BY OLIVER HEAVISIDE.**
**[Part I, The Electrician, 31, 281-282 (1893)]**

**To form any notion at all of the flux of gravitational energy, we
must first localise the energy. In this respect it resembles the legendary
hare in the cookery book. Whether the notion will turn out to be a useful
one is a matter for subsequent discovery. For this, also, there is a well-known
gastronomical analogy.**

**Now, bearing in mind the successful manner in which Maxwell's localisation
of electric and magnetic energy in his ether lends itself to theoretical
reasoning, the suggestion is very natural that we should attempt to localise
gravitational energy in a similar manner, its density to depend upon the
square of the intensity of the force, especially because the law of the
inverse squares is involved throughout.**

**Certain portions of space are supposed to be occupied by matter,
and its amount is supposed to be invariable. Furthermore, it is assumed
to have personal identity, so that the position and motion of a definite
particle of matter are definite, at any rate relative to an assumed fixed
space. Matter is recognised by the property of inertia, whereby it tends
to persist in the state of motion it possesses; and any change in the motion
is ascribed to the action of force, of which the proper measure is, therefore,
the rate of change of quantity of motion, or momentum.**

**Let p be the density of matter, and e the intensity of force,
or the force per unit matter, then**

(1)

**expresses the moving force on ,
which has its equivalent in increase of the momentum. There are so many
forces nowadays of a generalised nature, that perhaps the expression "moving
force" may be permitted for distinctness, although it may have been formerly
abused and afterwards tabooed.**

**Now the force , or
the intensity , may have
many origins, but the only one we are concerned with here is the gravitational
force. This appears to depend solely upon the distribution of the matter,
independently of other circumstances, and its operation is concisely expressed
by Newton's law, that there is a mutual attraction between any two particles
of matter, which varies as the product of their masses and inversely as
the square of their distance. Let now
be the intensity of gravitational force, and the
resultant moving force, due to all the matter. Then e is the space-variation
of a potential, say,**

**
(2)**

**and the potential is found from the distribition of matter by**

**
(3)**

**where is a constant.
This implies that the speed of propagation of the gravitative influence
is infinitely great.**

**Now when matter is allowed to fall together from any configuration
to a closer one, the work done by the gravitational to reive is expressed
by the increase made in the quantity This
is identically the same as the quantity summed
through all space. If, for example, the matter be given initially in a
state of infinitely fine division, infinitely widely separated, then the
work done by the gravitational forcive in passing to any other configuration
is or ,
which therefore expresses the "exhaustion of potential energy." We may
therefore assume that expresses
the exhaustion of potential energy per unit volume of the medium. The equivalent
of the exhaustion of potential energy is, of course, the gain of kinetic
energy, if no other forces have been in action.**

**We can now express the flux of energy. We may compare the present
problem with that of the motion of electrification. If moved about slowly
in a dielectric, the electric force is appreciably the static distribution.
Nevertheless, the flux of energy depends upon the magnetic force as well.
It may, indeed, be represented in another way, without introducing the
magnetic force, but then the formula would not be sufficiently comprehensive
to suit other cases. Now what is there analogous to magnetic force in the
gravitational case? And if it have its analogue, what is there to correspond
with electric current? At first glance it might seem that the whole of
the magnetic side of elctromagnetism was absent in the gravitational I
analogy. But this is not true.**

**Thus, if u is the velocity of ,
then is the density of a
current (or flux) of matter. It is analogous to a convective current of
electrification. Also, when the matter enters any region through its boundary,
there is a simultaneous convergence of gravitational force into that region
proportional to. This is
expressed by saying that if**

**,
(4)**

**then is a circuital
flux. It is the analogue of Maxwell's true current; for although Maxwell
did not include the convective term ,
yet it would be against his principles to ignore it. Being a circuital
flux, it is the curl of a vector, say**

**.
(5)**

**This defines except
as regards its divergence, which is arbitrary, and may be made zero. Then is
the analogue of magnetic force, for it bears the same relation to flux
of matter as magnetic force does to convective current. We have**

**
(6)**

**if . But, since instantaneous
action is here involved, we may equally well take**

**,
(7)**

**and its curl will be .
Thus, whilst the ordinary potential is
the potential of the matter, the new potential is
that of its flux.**

**Now if we multiply (5) by,
we obtain**

**,
(8)**

**or, which is the same,**

**,
(9)**

**if . But represents
the rate of exhaustion potential energy, so -represents
its rate of increase, whiles represents
the activity of the force on ,
increasing its kinetic energy. Consequently, the vector expresses
the flux of gravitational energy. More strictly, any circuital flux whatever
may be added. This is analogous
to the electromagnetic found
by Poynting and myself. But there is a reversal of direction. Thus, comparing
a single moving particle of matter with a similarly-moving electric charge,
describe a sphere round each. Let the direction of motion be the axis,
the positive pole being at the forward end. Then in the electrical case
the magnetic force follows the lines of latitude with positive rotation
about the axis, and the flux of energy coincides with the lines of longitude
from the negative pole to the positive. But in the gravitational case,
although h still follows the lines of latitude positively, yet since the
radial e is directed to instead of from the centre, the flux of energy
is along the lines of longitude from the positive pole to the negative.
This reversal arises from all matter being alike and attractive, whereas
like electrifications repel one another.**

**The electromagnetic analogy may be pushed further. It is as incredible
now as it was in Newton's time that gravitative influence can be exerted
without a medium; and, granting a medium, we may as well consider that
it propagates in time, although immensely fast. Suppose, then, instead
of instantaneous action, which involves**

**
(10)**

**we assert that the gravitational force in
ether is propagated at a single finite speed .
This requires that**

**so in space free from matter we have**

**.
(12)**

**But we also have, by (5),**

**
(13)**

**away from matter. This gives a second value to ,
when we differentiate (13) to the time, say**

**.
(14)**

**So, by (12) and (14), and remembering that we have already chosen circuital,
we derive**

**.
(15)**

**Or, if is a new constant,
such that**

**,
(16)**

**then (15) may be written in the form**

**.**

**leads to a second one, namely (17), if we introduce the hypothesis
propagation at finite speed. This, of course, might be inferred from the
electromagnetic case.**

**In order that the speed should
be not less than any value that may be settled upon as the least possible,
we have merely to make be
of the necessary smallness. The equation of activity becomes, instead of
(9),**

**,
(3)**

**if . The negative
sign before the time-increase of this quantity points to exhaustion of
energy, as before. If so, we should still represent the flux of energy
by . But, of course, an
almost vanishing quantity when is
small enough, or big enough.
Note that is not a negligible
quantity, though the product is.
Thus results will be sensibly as in the common theory of instantaneous
action, although expressed in terms of wave-propagation. Results showing
signs of wave-propagation would require an inordinately large velocity
of matter through the ether. It may be worth while to point out that the
lines of gravitational force connected with a particle of matter will no
longer converge to it uniformly from all directions when the velocity is
finite, but will show a tendency to lateral concentration, though only
to a sensible extent when the velocity of the matter is not an insensible
fraction of .**

**The gravitational-electromagnetic analogy may be further extended
if we allow that the ether which supports and propagates the gravitational
influence can have a translational motion of its own, thus carrying about
and distorting the lines of force. Making allowance for this convection
of by the medium, with the
concomitant convection of ,
requires us to turn the circuital laws (17), (18) to**

**,
(19)**

**,
(20)**

**where is the velocity
of the medium itself.**

**It is needless to go into detail, because the matter may be regarded
as a special and simplified case of my investigation of the forces in the
electromagnetic field, with changed meanings of the symbols. It is sufficient
to point out that the stress in the field now becomes prominent as a working
agent. It is of two sorts, one depending upon and
the other upon , analogous
to the electric and magnetic stresses. The one depending upon is,
of course, insignificant. The other consists of a pressure parallel to combined
with a lateral tension all round it, both of magnitude .
This was equivalently suggested by Maxwell. Thus two bodies which appear
to attract are pushed together. The case of two large parallel material
planes exhibits this in a marked manner, for e is very small between them,
and relatively large on their further sides.**

**But the above analogy, though interesting in its way, and serving
to emphasise the non-necessity of the assumption of instantaneous or direct
action of matter upon matter, does not enlighten us in the least about
the ultimate nature of gravitational energy. It serves, in fact, to further
illustrate the mystery. For it must be confessed that the exhaustion of potential energy from a universal
medium is a very unintelligible and mysterious matter. When matter is infinitely
widely separated, and the forces are least, the potential energy is at
its greatest, and when the potential energy is most exhausted, the forces
are most energetic!**

**Now there is a magnetic problem in which we have a kind of similarity
of behaviour, viz., when currents in material circuits arc allowed to attract
one another. Let, for completeness, the initial state be one of infinitely
wide separation of infinitely small filamentary currents in closed circuits.
Then, on concentration to any other state, the work done by the attractive
forces it**

**represented by ,
where is the inductivity
and the magnetic force.
This has its equivalent in the energy of motion of the circuits, or may
be imagined to be so converted, or else wasted by friction, if we like.
But, over and above this energy, the same amount, ,
represents the energy of the magnetic field, which can be got out of it
in work. It was zero at the beginning. Now, as Lord Kelvin showed, this
double work is accounted for by extra work in the batteries or other sources
required to maintain the currents constant. (I have omitted reference to
the waste of energy due to electrical resistance, to avoid complications.)
In the gravitational case there is a partial analogy, but the matter is
all along assumed to be incapable of variation, and not to require any
supply of energy to keep it constant. If we asserted that was
stored energy, then its double would be the work done per unit volume by
letting bodies attract from infinity, without any apparent source. But
it is merely the exhaustion of potential energy of unknown amount and distribution.**

**Potential energy, when regarded merely as expressive of the work
that can be done by forces depending upon configuration, does not admit
of much argument. It is little more than a mathematical idea, for there
is scarcely any physics in it. It explains nothing. But in the consideration
of physics in general, it is scarcely possible to avoid the idea that potential
energy should be capable of localisation equally as well as kinetic. That
the potential energy may be itself ultimately kinetic is a separate question.
Perhaps the best definition of the former is contained in these words:
- Potential energy is energy that is not known to be kinetic. But, however
this be, there is a practical distinction between them which it is found
useful to carry out. Now, when energy can be distinctly localised, its
flux can also be traced (subject to circuital indcterminatencss, however).
Also, this flux of energy forms a useful working idea when action at a
distance is denied (even though the speed of transmission be infinitely
great, or be assumed to be so). Any distinct and practical localisation
of energy is therefore a useful step, wholly apart from the debatable question
of the identity of energy advocated by Prof. Lodge.**

**From this point of view, then, we ought to localise gravitational
energy as a preliminary to a better understanding of that mysterious agency.
It cannot be said that the theory of the potential energy of gravitation
exhausts the subject. The flux of gravitational energy in the form above
given is, perhaps, somewhat more distinct, since it considers the flux
only and the changes in the amount localised, without any statement of
the gross amount. Perhaps the above analogy may be useful, and suggest
something better.**

**[Part II, The Electrician, 31, 359 (1893)]**

**In my first article on this subject (The Electrician, July
14, 1893, р. 281), I partly assumed a knowledge on the part of the reader
of my theory of convective currents of electrification ("Electrical Papers,"
Vol. II., p. 495 and after), and only very briefly mentioned the modified
law of the inverse squares which is involved, viz., with a lateral concentration
of the lines of force. The remarks of the Editor ,
and of Prof. Lodge on gravitational
aberration, lead me to point out now some of the consequences of the modified
law which arises when we assume that the ether is the working agent in
gravitational effects, and that it propagates disturbances at speed v in
the manner supposed in my former article. There is, so far as I can see
at present, no aberrations effect, but only a slight alteration in the
intensity of force in different directions round a moving body considered
as an attractor.**

*The
Electrician, July 14, p. 277, and July 23, p. 340*

*The Electrician,
July 28, p. 347*

*T*hus, take the case of a big Sun and small Earth, of masses and ,
at distance apart. Let be
the unmodified force of on ,
thus

*,
(1)*

**using rational units in order to harmonise with the electromagnetic
laws when rationally expressed. Also, let be
the modified force when the Sun is in motion at speed through
the ether. Then **

*This is the
case of steady motion. There is no simple formula*

*when the motion is unsteady*

*,
(2)*

**where is the small
quantity , and is
the angle between and the
line of motion. ("Electrical Papers," Vol. II., pp. 495, 499).**

**Therefore, if the Sun is at rest, there is no disturbance of the
Newtonian law, because its "field of force" is stationary. But if it has
a motion through space, there is a slight weakening of the force in the
line of motion, and a slight strengthening equatorially. The direction
is still radial.**

**To show the size of the effect, let**

**(3)**

**This value of is not
very different from the speed attributed to fast stars, and the value of is
the speed of light itself.**

**So we have**

**,
(4)**

*i.e.,* one millionth. All perturbing forces of the first order
are, therefore, of the order of magnitude of only one-millionth of the
full force, even when the speed of propagation is as small as that of light.

**The simplest case is when the common motion of the Sun and Earth
is perpendicular to the plane of the orbit. Then ,
all round the orbit, and**

**,
(5)**

**showing increase in the force of attraction of on of
one two-millionth part, without alteration of direction or variation in
tile orbit.**

^{4} But Prof. Lodge tells me that our own particular Sun
it considered to move only miles
per second. This 11 stupendously slow. The size *of *is
reduced to about 1/360 part of that in the text, and the same applies to
the corrections depending upon it.

**But when the common motion of the Sun and Earth is in their plane,
в varies from 0 to in a
revolution, so that the attraction on ,
whilst towards the Sun's centre, always undergoes a periodic variation
from**

**
(6)**

**when , to**

**
(7)**

**when . The extreme
variation is, therefore, ,
according to the data used. The result is a slight change in the shape
of the orbit.**

**But, to be consistent, having made v finite by certain suppositions,
we should carry out the consequences more fully, and allow not merely for
the change in the Newtonian law, as above, but for the force brought in
by the finiteness of v which is analogous to the "electromagnetic
force." This is very small truly, but so is the above change in the Newtonian
law, and since they are of the same order of magnitude, we should also
count the auxiliary force. Call it .
Then**

**,
(8)**

**where is as before,
in (2) above, is the actual
speed of the Earth (not the same as u), and in the third vectorial
factor, and ,
are unit vectors drawn parallel to the direction of the Earth's motion,
of the Sun's motion, and from the Sun to the Earth. We see at once that
the order of magnitude cannot be greater than that of the departure of from before
considered, because and will
be of the same order, at least when is
big. As for , it
is simply a numerical factor, which cannot exceed 1, and is probably 2/3**

**The simplest case is when the motion 'of the Sun is perpendicular
to the orbit of the Earth. Then**

**
(9)**

**gives the tensor ^{5} or size of the auxiliary force. It is
radial, but**

**But when the line of motion of Sun is in the plane of the orbit,
the case is much more complicated. The force is
neither constant (for the same distance) nor radial, except in four positions,
viz., two in the line of motion of the Sun, when the auxiliary force vanishes,
and two when , when it
is greatest. But this force is still in the plane of the orbit, which is
an important thing, and is, moreover, periodic, so that the tangential
component is as much one way as the other in a period.**

**All we need expect, then, so far as I can see from the above considerations,
are small perturbations due to the variation of the force of gravity in
different directions, and to the auxiliary force. Of course, there will
be numerous minor perturbations**

**If variations of the force of the size considered above are too small
to lead to observable perturbations of motion, then the striking conclusion
is that the speed of gravity may even be the same as that of light. If
they are observable, then, if existent, they should turn up, but if non-existent
then the speed of gravity should be greater. Furthermore, it is to be observed
that there may be other ways of expressing the propagation of gravity.**

**But I am mindful of the good old adage about the shoemaker and his
last, and am, therefore, reluctant to make any more remarks about perturbations.
The question of the ether in its gravitational aspect must be faced, however,
and solved sooner or later, if it be possible. Perhaps, therefore, my suggestions
may not be wholly useless.**

**________**

**Web-публикация: guts@univer.omsk.su, Omsk, December 15, 2001.**