*Reference: Einstein’s 1920 Book*

*This paper presents Part II, Chapter 5 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).*

*The paragraphs of the original material (in black) are accompanied
by brief comments (in color) based on the present understanding. Feedback
on these comments is appreciated.*

*The heading below is linked to the original materials.*

.

## A Few Inferences from the General Theory of Relativity

The considerations of Section XX show
that the general theory of relativity puts us in a position to derive
properties of the gravitational field in a purely theoretical manner. Let us
suppose, for instance, that we know the space-time “course” for any natural
process whatsoever, as regards the manner in which it takes place in the
Galileian domain relative to a Galileian body of reference ** K.** By means of purely
theoretical operations (i.e. simply by calculation) we are then able to find how
this known natural process appears, as seen from a reference-body

**which is accelerated relatively to**

*K’***. But since a gravitational field exists with respect to this new body of reference**

*K***, our consideration also teaches us how the gravitational field influences the process studied.**

*K’**A non-accelerating mass is
always at rest when viewed from surrounding space. An accelerating mass is also
at rest when viewed from surrounding field. *

For example, we learn that a body
which is in a state of uniform rectilinear motion with respect to ** K**
(in accordance with the law of Galilei) is executing an accelerated and in
general curvilinear motion with respect to the accelerated reference-body

**(chest). This acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to**

*K’***It is known that a gravitational field influences the movement of bodies in this way, so that our consideration supplies us with nothing essentially new.**

*K’.**Space becomes field when
it contains acceleration due to curvilinear motion. In other words, a field is
a curved space.*

*A field may be viewed as
a rotating space like a whirlpool. Everything in this field is accelerating
toward a center. All substance ends up at the center where it collects and
condenses.*

However, we obtain a new result of
fundamental importance when we carry out the analogous consideration for a ray
of light. With respect to the Galileian reference-body ** K**, such a ray of light is
transmitted rectilinearly with the velocity c. It can easily be shown that the
path of the same ray of light is no longer a straight line when we consider it
with reference to the accelerated chest (reference-body

**). From this we conclude,**

*K’**that, in general, rays of light are propagated curvilinearly in gravitational fields.*In two respects this result is of great importance.

*A ray of light is a very
small amount of substance moving very fast. When this ray enters a
gravitational field it accelerates toward the center of that field before
exiting it. Therefore, the path of the ray of light becomes slightly curved as
it passed through a gravitational field.*

In the first place, it can be compared
with the reality. Although a detailed examination of the question shows that
the curvature of light rays required by the general theory of relativity is
only exceedingly small for the gravitational fields at our disposal in
practice, its estimated magnitude for light rays passing the sun at grazing
incidence is nevertheless 1.7 seconds of arc. This ought to manifest itself in
the following way. As seen from the earth, certain fixed stars appear to be in
the neighbourhood of the sun, and are thus capable of observation during a
total eclipse of the sun. At such times, these stars ought to appear to be
displaced outwards from the sun by an amount indicated above, as compared with
their apparent position in the sky when the sun is situated at another part of
the heavens. The examination of the correctness or otherwise of this deduction
is a problem of the greatest importance, the early solution of which is to be
expected of astronomers. *[By means of the
star photographs of two expeditions equipped by a Joint Committee of the Royal
and Royal Astronomical Societies, the existence of the deflection of light
demanded by theory was confirmed during the solar eclipse of 29th May, 1919.
(Cf. Appendix III.)]*

*The increased curvature of
light in a gravitational field, as demanded by the general theory of relativity
has been confirmed experimentally. *

In the second place our result shows
that, according to the general theory of relativity, the law of the constancy
of the velocity of light *in vacuo*,
which constitutes one of the two fundamental assumptions in the special theory
of relativity and to which we have already frequently referred, cannot claim
any unlimited validity. A curvature of rays of light can only take place when
the velocity of propagation of light varies with position. Now we might think
that as a consequence of this, the special theory of relativity and with it the
whole theory of relativity would be laid in the dust. But in reality this is
not the case. We can only conclude that the special theory of relativity cannot
claim an unlimited domain of validity; its result hold only so long as we are
able to disregard the influences of gravitational fields on the phenomena (e.g.
of light).

*The velocity of light also
must change as it passes through a gravitational field. But this change is so
negligible that it does not affect the validity of the special theory of
relativity.*

Since it has often been contended by opponents of the theory of relativity that the special theory of relativity is overthrown by the general theory of relativity, it is perhaps advisable to make the facts of the case clearer by means of an appropriate comparison. Before the development of electrodynamics the laws of electrostatics and the laws of electricity were regarded indiscriminately. At the present time we know that electric fields can be derived correctly from electrostatic considerations only for the case, which is never strictly realised, in which the electrical masses are quite at rest relatively to each other, and to the co-ordinate system. Should we be justified in saying that for this reason electrostatics is overthrown by the field-equations of Maxwell in electrodynamics? Not in the least. Electrostatics is contained in electrodynamics as a limiting case; the laws of the latter lead directly to those of the former for the case in which the fields are invariable with regard to time. No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.

*The special theory of
relativity is contained within the general theory of relativity as a limiting
case of zero acceleration.*

In the example of the transmission of light just dealt with, we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes, the laws of which are already known when a gravitational field is absent. But the most attractive problem, to the solution of which the general theory of relativity supplies the key, concerns the investigation of the laws satisfied by the gravitational field itself. Let us consider this for a moment.

*The special theory considers
the natural processes in the absence of a gravitational field. The general theory
enables us to derive theoretically the influence of a gravitational field on
these processes. It also tells us about the laws satisfied by the gravitational
field itself.*

We are acquainted with space-time
domains which behave (approximately) in a “Galileian” fashion under suitable
choice of reference-body, i.e. domains in which gravitational fields are
absent. If we now refer such a domain to a reference-body ** K’** possessing any kind of
motion, then relative to

**there exists a gravitational field which is variable with respect to space and time.**

*K’**[This follows from a generalisation of the discussion in Section XX.]*The character of this field will of course depend on the motion chosen for

**. According to the general theory of relativity, the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. Even though by no means all gravitational fields can be produced in this way, yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. This hope has been realised in the most beautiful manner. But between the clear vision of this goal and its actual realisation it was necessary to surmount a serious difficulty, and as this lies deep at the root of things, I dare not withhold it from the reader. We require to extend our ideas of the space-time continuum still farther.**

*K’**The Galileian system
considers space-time in the absence of a gravitation field. The general theory
of relativity considers space-time in the presence of a gravitation field. This
requires that we extend our ideas of space-time continuum further.*

.

## FINAL COMMENTS

*The general theory of
relativity forces us to consider the space-time continuum in the presence of a
gravitational field.*

*The space-time continuum
is all about the extent (space) and duration (time) of a very fine form of
substance (force). When this fine form of substance is not there then the
space-time is not there either.*

*Therefore, the
space-time continuum represents the attributes of force as a very fine
substance. As this substance condenses it produces the electromagnetic spectrum.
At the upper end of the spectrum it produces the fundamental particles and
finally, matter.*

*A gravitational field is, therefore, made up of forces in motion similar to those in a whirlpool.*

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