By Prof. J. S. Ames
Johns Hopkins University
... In the treatment of Maxwell's equations of the electromagnetic
field, several investigators realized the importance of deducing the
form of the equations when applied to a system moving with a uniform
velocity. One object of such an investigation would be to determine
such a set of transformation formulæ as would leave the mathematical
form of the equations unaltered. The necessary relations between
the new space-coordinates, those applying to the moving system,
and the original set were of course obvious; and elementary methods
led to the deduction of a new variable which should replace the time
coordinate. This step was taken by Lorentz and also, I believe, by
Larmor and by Voigt. The mathematical deductions and applications
in the hands of these men were extremely beautiful, and are probably
well known to you all.
Lorentz' paper on this subject appeared in the Proceedings of the
Amsterdam Academy in 1904. In the following year there was published
in the Annalen der Physik a paper by Einstein, written without any
knowledge of the work of Lorentz, in which he arrived at the same
transformation equations as did the latter, but with an entirely
different and fundamentally new interpretation. Einstein called
attention in his paper to the lack of definiteness in the concepts
of time and space, as ordinarily stated and used. He analyzed clearly
the definitions and postulates which were necessary before one could
speak with exactness of a length or of an interval of time. He disposed
forever of the propriety of speaking of the "true" length of a rod or
of the "true" duration of time, showing, in fact, that the numerical
values which we attach to lengths or intervals of time depend upon the
definitions and postulates which we adopt. The words "absolute" space
or time intervals are devoid of meaning. As an illustration of what
is meant Einstein discussed two possible ways of measuring the length
of a rod when it is moving in the direction of its own length with
a uniform velocity, that is, after having adopted a scale of length,
two ways of assigning a number to the length of the rod concerned. One
method is to imagine the observer moving with the rod, applying along
its length the measuring scale, and reading off the positions of the
ends of the rod. Another method would be to have two observers at rest
on the body with reference to which the rod has the uniform velocity,
so stationed along the line of motion of the rod that as the rod
moves past them they can note simultaneously on a stationary measuring
scale the positions of the two ends of the rod. Einstein showed that,
accepting two postulates which need no defense at this time, the two
methods of measurements would lead to different numerical values, and,
further, that the divergence of the two results would increase as the
velocity of the rod was increased. In assigning a number, therefore,
to the length of a moving rod, one must make a choice of the method to
be used in measuring it. Obviously the preferable method is to agree
that the observer shall move with the rod, carrying his measuring
instrument with him. This disposes of the problem of measuring space
relations. The observed fact that, if we measure the length of the rod
on different days, or when the rod is lying in different positions,
we always obtain the same value offers no information concerning the
"real" length of the rod. It may have changed, or it may not. It
must always be remembered that measurement of the length of a
rod is simply a process of comparison between it and an arbitrary
standard, e.g., a meter-rod or yard-stick. In regard to the problem
of assigning numbers to intervals of time, it must be borne in mind
that, strictly speaking, we do not "measure" such intervals, i.e.,
that we do not select a unit interval of time and find how many times
it is contained in the interval in question. (Similarly, we do not
"measure" the pitch of a sound or the temperature of a room.) Our
practical instruments for assigning numbers to time-intervals depend
in the main upon our agreeing to believe that a pendulum swings in
a perfectly uniform manner, each vibration taking the same time as
the next one. Of course we cannot prove that this is true, it is,
strictly speaking, a definition of what we mean by equal intervals
of time; and it is not a particularly good definition at that. Its
limitations are sufficiently obvious. The best way to proceed is
to consider the concept of uniform velocity, and then, using the
idea of some entity having such a uniform velocity, to define equal
intervals of time as such intervals as are required for the entity
to traverse equal lengths. These last we have already defined. What
is required in addition is to adopt some moving entity as giving our
definition of uniform velocity. Considering our known universe it
is self-evident that we should choose in our definition of uniform
velocity the velocity of light, since this selection could be made by
an observer anywhere in our universe. Having agreed then to illustrate
by the words "uniform velocity" that of light, our definition of equal
intervals of time is complete. This implies, of course, that there is
no uncertainty on our part as to the fact that the velocity of light
always has the same value at any one point in the universe to any
observer, quite regardless of the source of light. In other words,
the postulate that this is true underlies our definition. Following
this method Einstein developed a system of measuring both space and
time intervals. As a matter of fact his system is identically that
which we use in daily life with reference to events here on the
earth. He further showed that if a man were to measure the length
of a rod, for instance, on the earth and then were able to carry the
rod and his measuring apparatus to Mars, the sun, or to Arcturus he
would obtain the same numerical value for the length in all places
and at all times. This doesn't mean that any statement is implied
as to whether the length of the rod has remained unchanged or not;
such words do not have any meaning--remember that we can not speak of
true length. It is thus clear that an observer living on the earth
would have a definite system of units in terms of which to express
space and time intervals, i.e., he would have a definite system
of space coordinates (x, y, z) and a definite time coordinate (t);
and similarly an observer living on Mars would have his system of
coordinates (x', y', z', t'). Provided that one observer has a definite
uniform velocity with reference to the other, it is a comparatively
simple matter to deduce the mathematical relations between the two
sets of coordinates. When Einstein did this, he arrived at the same
transformation formulæ as those used by Lorentz in his development of
Maxwell's equations. The latter had shown that, using these formulæ,
the form of the laws for all electromagnetic phenomena maintained
the same form; so Einstein's method proves that using his system of
measurement an observer, anywhere in the universe, would as the result
of his own investigation of electromagnetic phenomena arrive at the
same mathematical statement of them as any other observer, provided
only that the relative-velocity of the two observers was uniform.
Einstein discussed many other most important questions at this time;
but it is not necessary to refer to them in connection with the
present subject. So far as this is concerned, the next important
step to note is that taken in the famous address of Minkowski, in
1908, on the subject of "Space and Time." It would be difficult to
overstate the importance of the concepts advanced by Minkowski. They
marked the beginning of a new period in the philosophy of physics. I
shall not attempt to explain his ideas in detail, but shall confine
myself to a few general statements. His point of view and his line of
development of the theme are absolutely different from those of Lorentz
or of Einstein; but in the end he makes use of the same transformation
formulæ. His great contribution consists in giving us a new geometrical
picture of their meaning. It is scarcely fair to call Minkowski's
development a picture; for to us a picture can never have more than
three dimensions, our senses limit us; while his picture calls for
perception of four dimensions. It is this fact that renders any even
semi-popular discussion of Minkowski's work so impossible. We can all
see that for us to describe any event a knowledge of four coordinates
is necessary, three for the space specification and one for the time. A
complete picture could be given then by a point in four dimensions. All
four coordinates are necessary: we never observe an event except at
a certain time, and we never observe an instant of time except with
reference to space. Discussing the laws of electromagnetic phenomena,
Minkowski showed how in a space of four dimensions, by a suitable
definition of axes, the mathematical transformation of Lorentz and
Einstein could be described by a rotation of the set of axes. We are
all accustomed to a rotation of our ordinary cartesian set of axes
describing the position of a point. We ordinarily choose our axes at
any location on the earth as follows: one vertical, one east and west,
one north and south. So if we move from any one laboratory to another,
we change our axes; they are always orthogonal, but in moving from
place to place there is a rotation. Similarly, Minkowski showed that
if we choose four orthogonal axes at any point on the earth, according
to his method, to represent a space-time point using the method of
measuring space and time intervals as outlined by Einstein; and, if
an observer on Arcturus used a similar set of axes and the method of
measurement which he naturally would, the set of axes of the latter
could be obtained from those of the observer on the earth by a pure
rotation (and naturally a transfer of the origin). This is a beautiful
geometrical result. To complete my statement of the method, I must
add that instead of using as his fourth axis one along which numerical
values of time are laid off, Minkowski defined his fourth coordinate
as the product of time and the imaginary constant, the square root of
minus one. This introduction of imaginary quantities might be expected,
possibly, to introduce difficulties; but, in reality, it is the very
essence of the simplicity of the geometrical description just given
of the rotation of the sets of axes. It thus appears that different
observers situated at different points in the universe would each have
their own set of axes, all different, yet all connected by the fact
that any one can be rotated so as to coincide with any other. This
means that there is no one direction in the four-dimensional space
that corresponds to time for all observers. Just as with reference to
the earth there is no direction which can be called vertical for all
observers living on the earth. In the sense of an absolute meaning
the words "up and down," "before and after," "sooner or later,"
are entirely meaningless.
This concept of Minkowski's may be made clearer, perhaps, by the
following process of thought. If we take a section through our
three-dimensional space, we have a plane, i.e., a two-dimensional
space. Similarly, if a section is made through a four-dimensional
space, one of three dimensions is obtained. Thus, for an observer on
the earth a definite section of Minkowski's four-dimensional space will
give us our ordinary three-dimensional one; so that this section will,
as it were, break up Minkowski's space into our space and give us our
ordinary time. Similarly, a different section would have to be used
to the observer on Arcturus; but by a suitable selection he would
get his own familiar three-dimensional space and his own time. Thus
the space defined by Minkowski is completely isotropic in reference
to measured lengths and times, there is absolutely no difference
between any two directions in an absolute sense; for any particular
observer, of course, a particular section will cause the space to
fall apart so as to suit his habits of measurement; any section,
however, taken at random will do the same thing for some observer
somewhere. From another point of view, that of Lorentz and Einstein,
it is obvious that, since this four-dimensional space is isotropic,
the expression of the laws of electromagnetic phenomena take identical
mathematical forms when expressed by any observer.
The question of course must be raised as to what can be said in regard
to phenomena which so far as we know do not have an electromagnetic
origin. In particular what can be done with respect to gravitational
phenomena? Before, however, showing how this problem was attacked by
Einstein; and the fact that the subject of my address is Einstein's
work on gravitation shows that ultimately I shall explain this, I
must emphasize another feature of Minkowski's geometry. To describe
the space-time characteristics of any event a point, defined by its
four coordinates, is sufficient; so, if one observes the life-history
of any entity, e.g., a particle of matter, a light-wave, etc., he
observes a sequence of points in the space-time continuum; that is,
the life-history of any entity is described fully by a line in this
space. Such a line was called by Minkowski a "world-line." Further,
from a different point of view, all of our observations of nature
are in reality observations of coincidences, e.g., if one reads
a thermometer, what he does is to note the coincidence of the
end of the column of mercury with a certain scale division on
the thermometer tube. In other words, thinking of the world-line
of the end of the mercury column and the world-line of the scale
division, what we have observed was the intersection or crossing of
these lines. In a similar manner any observation may be analyzed;
and remembering that light rays, a point on the retina of the eye,
etc., all have their world-lines, it will be recognized that it is
a perfectly accurate statement to say that every observation is the
perception of the intersection of world-lines. Further, since all we
know of a world-line is the result of observations, it is evident
that we do not know a world-line as a continuous series of points,
but simply as a series of discontinuous points, each point being where
the particular world-line in question is crossed by another world-line.
It is clear, moreover, that for the description of a world-line
we are not limited to the particular set of four orthogonal axes
adopted by Minkowski. We can choose any set of four-dimensional
axes we wish. It is further evident that the mathematical expression
for the coincidence of two points is absolutely independent of our
selection of reference axes. If we change our axes, we will change
the coordinates of both points simultaneously, so that the question
of axes ceases to be of interest. But our so-called laws of nature are
nothing but descriptions in mathematical language of our observations;
we observe only coincidences; a sequence of coincidences when put in
mathematical terms takes a form which is independent of the selection
of reference axes; therefore the mathematical expression of our laws
of nature, of every character, must be such that their form does
not change if we make a transformation of axes. This is a simple but
far-reaching deduction.
There is a geometrical method of picturing the effect of a change
of axes of reference, i.e., of a mathematical transformation. To a
man in a railway coach the path of a drop of water does not appear
vertical, i.e., it is not parallel to the edge of the window; still
less so does it appear vertical to a man performing manoeuvres in
an airplane. This means that whereas with reference to axes fixed
to the earth the path of the drop is vertical; with reference to
other axes, the path is not. Or, stating the conclusion in general
language, changing the axes of reference (or effecting a mathematical
transformation) in general changes the shape of any line. If one
imagines the line forming a part of the space, it is evident that
if the space is deformed by compression or expansion the shape of
the line is changed, and if sufficient care is taken it is clearly
possible, by deforming the space, to make the line take any shape
desired, or better stated, any shape specified by the previous change
of axes. It is thus possible to picture a mathematical transformation
as a deformation of space. Thus I can draw a line on a sheet of paper
or of rubber and by bending and stretching the sheet, I can make the
line assume a great variety of shapes; each of these new shapes is
a picture of a suitable transformation.
Now, consider world-lines in our four-dimensional space. The
complete record of all our knowledge is a series of sequences of
intersections of such lines. By analogy I can draw in ordinary space
a great number of intersecting lines on a sheet of rubber; I can
then bend and deform the sheet to please myself; by so doing I do
not introduce any new intersections nor do I alter in the least the
sequence of intersections. So in the space of our world-lines, the
space may be deformed in any imaginable manner without introducing
any new intersections or changing the sequence of the existing
intersections. It is this sequence which gives us the mathematical
expression of our so-called experimental laws; a deformation of
our space is equivalent mathematically to a transformation of axes,
consequently we see why it is that the form of our laws must be the
same when referred to any and all sets of axes, that is, must remain
unaltered by any mathematical transformation.
Now, at last we come to gravitation. We can not imagine any world-line
simpler than that of a particle of matter left to itself; we shall
therefore call it a "straight" line. Our experience is that two
particles of matter attract one another. Expressed in terms of
world-lines, this means that, if the world-lines of two isolated
particles come near each other, the lines, instead of being straight,
will be deflected or bent in towards each other. The world-line of
any one particle is therefore deformed; and we have just seen that a
deformation is the equivalent of a mathematical transformation. In
other words, for any one particle it is possible to replace the
effect of a gravitational field at any instant by a mathematical
transformation of axes. The statement that this is always possible
for any particle at any instant is Einstein's famous "Principle
of Equivalence."
Let us rest for a moment, while I call attention to a most interesting
coincidence, not to be thought of as an intersection of world-lines. It
is said that Newton's thoughts were directed to the observation of
gravitational phenomena by an apple falling on his head; from this
striking event he passed by natural steps to a consideration of the
universality of gravitation. Einstein in describing his mental process
in the evolution of his law of gravitation says that his attention
was called to a new point of view by discussing his experiences with
a man whose fall from a high building he had just witnessed. The man
fortunately suffered no serious injuries and assured Einstein that in
the course of his fall he had not been conscious in the least of any
pull downward on his body. In mathematical language, with reference to
axes moving with the man the force of gravity had disappeared. This is
a case where by the transfer of the axes from the earth itself to the
man, the force of the gravitational field is annulled. The converse
change of axes from the falling man to a point on the earth could be
considered as introducing the force of gravity into the equations of
motion. Another illustration of the introduction into our equations of
a force by a means of a change of axes is furnished by the ordinary
treatment of a body in uniform rotation about an axis. For instance,
in the case of a so-called conical pendulum, that is, the motion of a
bob suspended from a fixed point by string, which is so set in motion
that the bob describes a horizontal circle and the string therefore
describes a circular cone, if we transfer our axes from the earth and
have them rotate around the vertical line through the fixed point with
the same angular velocity as the bob, it is necessary to introduce into
our equations of motion a fictitious "force" called the centrifugal
force. No one ever thinks of this force other than as a mathematical
quantity introduced into the equations for the sake of simplicity of
treatment; no physical meaning is attached to it. Why should there
be to any other so-called "force," which like centrifugal force,
is independent of the nature of the matter? Again, here on the earth
our sensation of weight is interpreted mathematically by combining
expressions for centrifugal force and gravity; we have no distinct
sensation for either separately. Why then is there any difference in
the essence of the two? Why not consider them both as brought into
our equations by the agency of mathematical transformations? This is
Einstein's point of view.
Granting, then, the principle of equivalence, we can so choose axes at
any point at any instant that the gravitational field will disappear;
these axes are therefore of what Eddington calls the "Galilean"
type, the simplest possible. Consider, that is, an observer in a
box, or compartment, which is falling with the acceleration of the
gravitational field at that point. He would not be conscious of the
field. If there were a projectile fired off in this compartment,
the observer would describe its path as being straight. In this space
the infinitesimal interval between two space-time points would then
be given by the formula
$$ds^2 = dx^2_1 + dx2_2 + dx^2_3 + dx2_4,$$
where ds is the interval and $x_1, x_2, x_3, x_4$ are coordinates. If
we make a mathematical transformation, i.e., use another set of axes,
this interval would obviously take the form
$$ds^2 = g_{11}dx^2_{33} + g_{22}dx^2_2 + g_{33}dx^2_3 +
g_{44}dx2_4 + 2g_{12}dx_1dx_2 + \rm{etc.},$$
where $x_1, x_2, x_3$ and $x_4$ are now coordinates referring to the
new axes. This relation involves ten coefficients, the coefficients
defining the transformation.
But of course a certain dynamical value is also attached to the
g's, because by the transfer of our axes from the Galilean type
we have made a change which is equivalent to the introduction of
a gravitational field; and the g's must specify the field. That
is, these g's are the expressions of our experiences, and hence
their values can not depend upon the use of any special axes; the
values must be the same for all selections. In other words, whatever
function of the coordinates any one g is for one set of axes, if other
axes are chosen, this g must still be the same function of the new
coordinates. There are ten g's defined by differential equations;
so we have ten covariant equations. Einstein showed how these g's
could be regarded as generalized potentials of the field. Our own
experiments and observations upon gravitation have given us a certain
knowledge concerning its potential; that is, we know a value for it
which must be so near the truth that we can properly call it at least
a first approximation. Or, stated differently, if Einstein succeeds in
deducing the rigid value for the gravitational potential in any field,
it must degenerate to the Newtonian value for the great majority of
cases with which we have actual experience. Einstein's method, then,
was to investigate the functions (or equations) which would satisfy
the mathematical conditions just described. A transformation from
the axes used by the observer in the following box may be made so as
to introduce into the equations the gravitational field recognized
by an observer on the earth near the box; but this, obviously, would
not be the general gravitational field, because the field changes as
one moves over the surface of the earth. A solution found, therefore,
as just indicated, would not be the one sought for the general field;
and another must be found which is less stringent than the former
but reduces to it as a special case. He found himself at liberty to
make a selection from among several possibilities, and for several
reasons chose the simplest solution. He then tested this decision
by seeing if his formulæ would degenerate to Newton's law for the
limiting case of velocities small when compared with that of light,
because this condition is satisfied in those cases to which Newton's
law applies. His formulæ satisfied this test, and he therefore was
able to announce a "law of gravitation," of which Newton's was a
special form for a simple case.
To the ordinary scholar the difficulties surmounted by Einstein in
his investigations appear stupendous. It is not improbable that the
statement which he is alleged to have made to his editor, that only
ten men in the world could understand his treatment of the subject,
is true. I am fully prepared to believe it, and wish to add that I
certainly am not one of the ten. But I can also say that, after a
careful and serious study of his papers, I feel confident that there
is nothing in them which I can not understand, given the time to become
familiar with the special mathematical processes used. The more I work
over Einstein's papers, the more impressed I am, not simply by his
genius in viewing the problem, but also by his great technical skill.
Following the path outlined, Einstein, as just said, arrived at certain
mathematical laws for a gravitational field, laws which reduced
to Newton's form in most cases where observations are possible,
but which led to different conclusions in a few cases, knowledge
concerning which we might obtain by careful observations. I shall
mention a few deductions from Einstein's formulæ.
1. If a heavy particle is put at the center of a circle, and, if
the length of the circumference and the length of the diameter
are measured, it will be found that their ratio is not [pi]
(3.14159). In other words the geometrical properties of space in such
a gravitational field are not those discussed by Euclid; the space is,
then, non-Euclidean. There is no way by which this deduction can be
verified, the difference between the predicted ratio and [pi] is
too minute for us to hope to make our measurements with sufficient
exactness to determine the difference.
2. All the lines in the solar spectrum should with reference to lines
obtained by terrestrial sources be displaced slightly towards longer
wave-lengths. The amount of displacement predicted for lines in the
blue end of the spectrum is about one-hundredth of an Angstrom unit,
a quantity well within experimental limits. Unfortunately, as far
as the testing of this prediction is concerned, there are several
physical causes which are also operating to cause displacement of the
spectrum-lines; and so at present a decision can not be rendered as
to the verification. St. John and other workers at the Mount Wilson
Observatory have the question under investigation.
3. According to Newton's law an isolated planet in its motion around a
central sun would describe, period after period, the same elliptical
orbit; whereas Einstein's laws lead to the prediction that the
successive orbits traversed would not be identically the same. Each
revolution would start the planet off on an orbit very approximately
elliptical, but with the major axis of the ellipse rotated slightly in
the plane of the orbit. When calculations were made for the various
planets in our solar system, it was found that the only one which
was of interest from the standpoint of verification of Einstein's
formulæ was Mercury. It has been known for a long time that there
was actually such a change as just described in the orbit of Mercury,
amounting to 574'' of arc per century; and it has been shown that of
this a rotation of 532'' was due to the direct action of other planets,
thus leaving an unexplained rotation of 42'' per century. Einstein's
formulæ predicted a rotation of 43'', a striking agreement.
4. In accordance with Einstein's formulæ a ray of light passing close
to a heavy piece of matter, the sun, for instance, should experience
a sensible deflection in towards the sun. This might be expected from
"general" consideration of energy in motion; energy and mass are
generally considered to be identical in the sense that an amount
of energy E has the mass $E1c^2$ where c is the velocity of light;
and consequently a ray of light might fall within the province of
gravitation and the amount of deflection to be expected could be
calculated by the ordinary formula for gravitation. Another point
of view is to consider again the observer inside the compartment
falling with the acceleration of the gravitational field. To him the
path of a projectile and a ray of light would both appear straight;
so that, if the projectile had a velocity equal to that of light, it
and the light wave would travel side by side. To an observer outside
the compartment, e.g., to one on the earth, both would then appear
to have the same deflection owing to the sun. But how much would
the path of the projectile be bent? What would be the shape of its
parabola? One might apply Newton's law; but, according to Einstein's
formulæ, Newton's law should be used only for small velocities. In the
case of a ray passing close to the sun it was decided that according
to Einstein's formula there should be a deflection of 1''.75 whereas
Newton's law of gravitation predicted half this amount. Careful
plans were made by various astronomers, to investigate this question
at the solar eclipse last May, and the result announced by Dyson,
Eddington and Crommelin, the leaders of astronomy in England, was
that there was a deflection of 1''.9. Of course the detection of such
a minute deflection was an extraordinarily difficult matter, so many
corrections had to be applied to the original observations; but the
names of the men who record the conclusions are such as to inspire
confidence. Certainly any effect of refraction seems to be excluded.
It is thus seen that the formulæ deduced by Einstein have been
confirmed in a variety of ways and in a most brilliant manner. In
connection with these formulæ one question must arise in the minds of
everyone; by what process, where in the course of the mathematical
development, does the idea of mass reveal itself? It was not in
the equations at the beginning and yet here it is at the end. How
does it appear? As a matter of fact it is first seen as a constant
of integration in the discussion of the problem of the gravitational
field due to a single particle; and the identity of this constant with
mass is proved when one compares Einstein's formulæ with Newton's
law which is simply its degenerated form. This mass, though, is the
mass of which we become aware through our experiences with weight;
and Einstein proceeded to prove that this quantity which entered as
a constant of integration in his ideally simple problem also obeyed
the laws of conservation of mass and conservation of momentum when
he investigated the problems of two and more particles. Therefore
Einstein deduced from his study of gravitational fields the well-known
properties of matter which form the basis of theoretical mechanics. A
further logical consequence of Einstein's development is to show that
energy has mass, a concept with which every one nowadays is familiar.
The description of Einstein's method which I have given so far is
simply the story of one success after another; and it is certainly
fair to ask if we have at last reached finality in our investigation
of nature, if we have attained to truth. Are there no outstanding
difficulties? Is there no possibility of error? Certainly, not
until all the predictions made from Einstein's formulæ have been
investigated can much be said; and further, it must be seen whether
any other lines of argument will lead to the same conclusions. But
without waiting for all this there is at least one difficulty which
is apparent at this time. We have discussed the laws of nature as
independent in their form of reference axes, a concept which appeals
strongly to our philosophy; yet it is not at all clear, at first sight,
that we can be justified in our belief. We can not imagine any way
by which we can become conscious of the translation of the earth in
space; but by means of gyroscopes we can learn a great deal about its
rotation on its axis. We could locate the positions of its two poles,
and by watching a Foucault pendulum or a gyroscope we can obtain a
number which we interpret as the angular velocity of rotation of axes
fixed in the earth; angular velocity with reference to what? Where
is the fundamental set of axes? This is a real difficulty. It can be
surmounted in several ways. Einstein himself has outlined a method
which in the end amounts to assuming the existence on the confines
of space of vast quantities of matter, a proposition which is not
attractive. deSitter has suggested a peculiar quality of the space
to which we refer our space-time coordinates. The consequences of
this are most interesting, but no decision can as yet be made as to
the justification of the hypothesis. In any case we can say that
the difficulty raised is not one that destroys the real value of
Einstein's work.
In conclusion I wish to emphasize the fact, which should be obvious,
that Einstein has not attempted any explanation of gravitation;
he has been occupied with the deduction of its laws. These laws,
together with those of electromagnetic phenomena, comprise our store
of knowledge. There is not the slightest indication of a mechanism,
meaning by that a picture in terms of our senses. In fact what we have
learned has been to realize that our desire to use such mechanisms
is futile.
THE DEFLECTION OF LIGHT BY GRAVITATION AND THE EINSTEIN THEORY OF
RELATIVITY. [16]
Sir Frank Dyson
the Astronomer Royal
The purpose of the expedition was to determine whether any displacement
is caused to a ray of light by the gravitational field of the sun,
and if so, the amount of the displacement. Einstein's theory predicted
a displacement varying inversely as the distance of the ray from
the sun's center, amounting to 1''.75 for a star seen just grazing
the sun....
A study of the conditions of the 1919 eclipse showed that the sun
would be very favorably placed among a group of bright stars--in fact,
it would be in the most favorable possible position. A study of the
conditions at various points on the path of the eclipse, in which
Mr. Hinks helped us, pointed to Sobral, in Brazil, and Principe, an
island off the west coast of Africa, as the most favorable stations....
The Greenwich party, Dr. Crommelin and Mr. Davidson, reached Brazil
in ample time to prepare for the eclipse, and the usual preliminary
focusing by photographing stellar fields was carried out. The day of
the eclipse opened cloudy, but cleared later, and the observations
were carried out with almost complete success. With the astrographic
telescope Mr. Davidson secured 15 out of 18 photographs showing the
required stellar images. Totality lasted 6 minutes, and the average
exposure of the plates was 5 to 6 seconds. Dr. Crommelin with the
other lens had 7 successful plates out of 8. The unsuccessful plates
were spoiled for this purpose by the clouds, but show the remarkable
prominence very well.
When the plates were developed the astrographic images were found to
be out of focus. This is attributed to the effect of the sun's heat
on the coelostat mirror. The images were fuzzy and quite different
from those on the check-plates secured at night before and after the
eclipse. Fortunately the mirror which fed the 4-inch lens was not
affected, and the star images secured with this lens were good and
similar to those got by the night-plates. The observers stayed on in
Brazil until July to secure the field in the night sky at the altitude
of the eclipse epoch and under identical instrumental conditions.
The plates were measured at Greenwich immediately after the observers'
return. Each plate was measured twice over by Messrs. Davidson and
Furner, and I am satisfied that such faults as lie in the results
are in the plates themselves and not in the measures. The figures
obtained may be briefly summarized as follows: The astrographic plates
gave 0''.97 for the displacement at the limb when the scale-value was
determined from the plates themselves, and 1''.40 when the scale-value
was assumed from the check plates. But the much better plates gave
for the displacement at the limb 1''.98, Einstein's predicted value
being 1''.75. Further, for these plates the agreement was all that
could be expected....
After a careful study of the plates I am prepared to say that there
can be no doubt that they confirm Einstein's prediction. A very
definite result has been obtained that light is deflected according
to Einstein's law of gravitation.
Professor A. S. Eddington
Royal Observatory
Mr. Cottingham and I left the other observers at Madeira and arrived
at Principe on April 23.... We soon realized that the prospect of
a clear sky at the end of May was not very good. Not even a heavy
thunderstorm on the morning of the eclipse, three weeks after the end
of the wet season, saved the situation. The sky was completely cloudy
at first contact, but about half an hour before totality we began to
see glimpses of the sun's crescent through the clouds. We carried
out our program exactly as arranged, and the sky must have been
clearer towards the end of totality. Of the 16 plates taken during
the five minutes of totality the first ten showed no stars at all;
of the later plates two showed five stars each, from which a result
could be obtained. Comparing them with the check-plates secured at
Oxford before we went out, we obtained as the final result from the
two plates for the value of the displacement of the limb 1''.6 ±
0.3.... This result supports the figures obtained at Sobral....
I will pass now to a few words on the meaning of the result. It points
to the larger of the two possible values of the deflection. The
simplest interpretation of the bending of the ray is to consider
it as an effect of the weight of light. We know that momentum is
carried along on the path of a beam of light. Gravity in acting
creates momentum in a direction different from that of the path of
the ray and so causes it to bend. For the half-effect we have to
assume that gravity obeys Newton's law; for the full effect which has
been obtained we must assume that gravity obeys the new law proposed
by Einstein. This is one of the most crucial tests between Newton's
law and the proposed new law. Einstein's law had already indicated a
perturbation, causing the orbit of Mercury to revolve. That confirms
it for relatively small velocities. Going to the limit, where the
speed is that of light, the perturbation is increased in such a way
as to double the curvature of the path, and this is now confirmed.
This effect may be taken as proving Einstein's law rather than his
theory. It is not affected by the failure to detect the displacement
of Fraunhofer lines on the sun. If this latter failure is confirmed
it will not affect Einstein's law of gravitation, but it will affect
the views on which the law was arrived at. The law is right, though
the fundamental ideas underlying it may yet be questioned....
One further point must be touched upon. Are we to attribute the
displacement to the gravitational field and not to the refracting
matter around the sun? The refractive index required to produce the
result at a distance of 15' from the sun would be that given by gases
at a pressure of 1/60 to 1/200 of an atmosphere. This is of too great
a density considering the depth through which the light would have
to pass.
Sir J. J. Thomson
President of the Royal Society
... If the results obtained had been only that light was affected by
gravitation, it would have been of the greatest importance. Newton,
did, in fact, suggest this very point in his "Optics," and his
suggestion would presumably have led to the half-value. But this result
is not an isolated one; it is part of a whole continent of scientific
ideas affecting the most fundamental concepts of physics.... This is
the most important result obtained in connection with the theory of
gravitation since Newton's day, and it is fitting that it should be
announced at a meeting of the society so closely connected with him.
The difference between the laws of gravitation of Einstein and Newton
come only in special cases. The real interest of Einstein's theory
lies not so much in his results as in the method by which he gets
them. If his theory is right, it makes us take an entirely new view
of gravitation. If it is sustained that Einstein's reasoning holds
good--and it has survived two very severe tests in connection with the
perihelion of mercury and the present eclipse--then it is the result
of one of the highest achievements of human thought. The weak point
in the theory is the great difficulty in expressing it. It would
seem that no one can understand the new law of gravitation without
a thorough knowledge of the theory of invariants and of the calculus
of variations.
One other point of physical interest arises from the discussion. Light
is deflected in passing near huge bodies of matter. This involves
alterations in the electric and magnetic field. This, again, implies
the existence of electric and magnetic forces outside matter--forces
at present unknown, though some idea of their nature may be got from
the results of this expedition.