From Oleg D. Jefimenko, Causality, Electromagnetic Induction, and
Gravitation: A Different
Approach to the Theory of Electromagnetic and Gravitational
Fields, 2nd ed.,
(Electret Scientific, Star City, 2000)
This reproduction of Heaviside's article is an unedited copy of the
original. Oleg D. Jefimenko have converted some formulas and all vector equations
appearing in the article to modern mathematical notation. A GRAVITATIONAL AND ELECTROMAGNETIC
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. (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, 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 be the density
of matter, and e the intensity of force, or the force per unit matter,
Now the force
F, or the intensity e, 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 e now be the intensity of gravitational force, and F 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 distribution of matter by
(3)where c 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 forcive 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 ce2/2 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
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 electromagnetism was absent in the gravitational analogy. But
this is not true.
Thus, if u is the velocity
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
This is expressed by saying that if
(4)then C 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
defines h except as regards its divergence, which is arbitrary, and may
be made zero. Then h is the analogue of magnetic force, for it bears the
same relation to flux of matter as magnetic force does to convective current. We
(6)if A = Pot C.
But, since instantaneous action is here involved, we may equally well
(7)and its curl will be h.
Thus, whilst the ordinary potential P is the potential of the matter, the
new potential A is that of its flux.
Now if we
multiply (5) by e, we obtain
(8)or, which is the same,
(9)if U =
ce2/2. But represents the rate of
exhaustion of potential energy, so - represents its rate of
increase, whilst 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 e in ether is propagated at a single finite speed
v. This requires that
(11)for this is the general
characteristic of undissipated propagation at finite speed. Now,
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 h circuital, we derive
(15)Or, if is a new constant, such
(16)then (15) may be
written in the form
(17)To sum up, the first
circuital law (5), or
(18)leads to a second one, namely
(17), if we introduce the hypothesis of propagation at finite speed.
of course, might be inferred from the electromagnetic case.
In order that the speed v 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, T is an almost vanishing quantity when is small enough, or
v big enough. Note that h 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 v 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 e by the medium, with the concomitant convection of
h, requires us to turn the circuital laws (17), (18) to
(20)where q 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 e and the other upon h, analogous to the electric
and magnetic stresses. The one depending upon h is, of course,
insignificant. The other consists of a pressure parallel to e combined
with a lateral tension all round it, both of magnitude ce2/2.
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
Now there is a magnetic problem in which we
have a kind of similarity of behaviour, viz., when currents in material circuits
are 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 is represented by , where is the
inductivity and H 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 ce2/2 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 indeterminateness, 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
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, p.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(1)
and of Prof. Lodge(2)
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 aberrational effect, but only a slight alteration in the
intensity of force in different directions round a moving body considered as an
attractor. (1)using rational units in order
to harmonise with the electromagnetic laws when rationally expressed. Also, let
F be the modified force when the Sun is in motion at speed u
through the ether. Then(3)
(2)where s is the small
quantity u2/v2, and is the angle between
r and the line of motion. ("Electrical Papers," Vol. II., pp. 495,
Thus, take the case of a big Sun and small
Earth, of masses S and E, at distance r apart. Let f
be the unmodified force of S on E, thus
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
u = 3 107 centim. per sec.
v = 3 1010 centim. per sec.
This value of u is not very different from the speed attributed to
fast stars, and the value of v is the speed of light itself.
(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.
So we have
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 S on E of one two-millionth part, without
alteration of direction or variation in the orbit.(4)
But when the common motion of the Sun and Earth is in
their plane, varies from 0 to in a revolution, so
that the attraction on E, whilst towards the Sun's centre, always
undergoes a periodic variation from
(6)when = 0, to
(7)when . The extreme
variation is, therefore, 3sf/2, 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
(8)where F is
as before, in (2) above, q is the actual speed of the Earth (not the same
as u), and in the third vectorial factor q1,
u1, and r1 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 F from f before considered, because
u and q will be of the same order, at least when u is big.
As for x, it is simply a numerical factor, which cannot exceed 1, and is
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 outwards, so that the result
is merely to reduce the size of' the previous correction, viz., the difference
of F from f in the same motional circumstances.
But when the line of motion of Sun is in the plane of the
orbit, the case is much more complicated. The force G 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.
1. The Electrician, July 14, p. 277, and July 23, p.
2. The Electrician, July 28, p. 347.
3. This is the case of steady motion. There is no
simple formula when the motion is unsteady.
4. But Prof. Lodge tells me that our own particular
Sun is considered to move only 109 miles per second. This is stupendously slow.
The size of s is reduced to about 1/360 part of that in the text, and the
same applies to the corrections depending upon it.
5. Heaviside uses the word "tensor" for the magnitude
of a force vector (O. D. J.).
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