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RELATIVITY: THE SPECIAL AND GENERAL THEORY
BY ALBERT EINSTEIN
Written: 1916 (this revised edition: 1924)
Source: Relativity: The Special and General Theory (1920)
Publisher: Methuen & Co Ltd
First Published: December, 1916
Translated: Robert W. Lawson (Authorised translation)
Transcription/Markup: Brian Basgen <brian@marxists.org>
Transcription to text: Gregory B. Newby <gbnewby@petascale.org>
Thanks to: Einstein Reference Archive (marxists.org)
The Einstein Reference Archive is online at:
http://www.marxists.org/reference/archive/einstein/index.htm
Transcriber note: This file is a plain text rendition of HTML.
Because many equations cannot be presented effectively in plain text,
images are supplied for many equations and for all figures and tables.
CONTENTS
Preface
Part I: The Special Theory of Relativity
01. Physical Meaning of Geometrical Propositions
02. The System of Co-ordinates
03. Space and Time in Classical Mechanics
04. The Galileian System of Co-ordinates
05. The Principle of Relativity (in the Restricted Sense)
06. The Theorem of the Addition of Velocities employed in
Classical Mechanics
07. The Apparent Incompatability of the Law of Propagation of
Light with the Principle of Relativity
08. On the Idea of Time in Physics
09. The Relativity of Simultaneity
10. On the Relativity of the Conception of Distance
11. The Lorentz Transformation
12. The Behaviour of Measuring-Rods and Clocks in Motion
13. Theorem of the Addition of Velocities. The Experiment of Fizeau
14. The Hueristic Value of the Theory of Relativity
15. General Results of the Theory
16. Expereince and the Special Theory of Relativity
17. Minkowski's Four-dimensial Space
Part II: The General Theory of Relativity
18. Special and General Principle of Relativity
19. The Gravitational Field
20. The Equality of Inertial and Gravitational Mass as an Argument
for the General Postulate of Relativity
21. In What Respects are the Foundations of Classical Mechanics
and of the Special Theory of Relativity Unsatisfactory?
22. A Few Inferences from the General Principle of Relativity
23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of
Reference
24. Euclidean and non-Euclidean Continuum
25. Gaussian Co-ordinates
26. The Space-Time Continuum of the Speical Theory of Relativity
Considered as a Euclidean Continuum
27. The Space-Time Continuum of the General Theory of Relativity
is Not a Eculidean Continuum
28. Exact Formulation of the General Principle of Relativity
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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 unlinlited domain of validity ;
its results hold only so long as we are able to disregard the
influences of gravitational fields on the phenomena (e.g. of light).
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
were looked upon as the laws of electricity. 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.
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 Iaws 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.
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 K1 possessing any kind of motion,
then relative to K1 there exists a gravitational field which is
variable with respect to space and time.[3]** The character of this
field will of course depend on the motion chosen for K1. 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 gravitationial 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.
Notes
*) 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 first
confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
III.)
**) This follows from a generalisation of the discussion in
Section 20
BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
Hitherto I have purposely refrained from speaking about the physical
interpretation of space- and time-data in the case of the general
theory of relativity. As a consequence, I am guilty of a certain
slovenliness of treatment, which, as we know from the special theory
of relativity, is far from being unimportant and pardonable. It is now
high time that we remedy this defect; but I would mention at the
outset, that this matter lays no small claims on the patience and on
the power of abstraction of the reader.
We start off again from quite special cases, which we have frequently
used before. Let us consider a space time domain in which no
gravitational field exists relative to a reference-body K whose state
of motion has been suitably chosen. K is then a Galileian
reference-body as regards the domain considered, and the results of
the special theory of relativity hold relative to K. Let us supposse
the same domain referred to a second body of reference K1, which is
rotating uniformly with respect to K. In order to fix our ideas, we
shall imagine K1 to be in the form of a plane circular disc, which
rotates uniformly in its own plane about its centre. An observer who
is sitting eccentrically on the disc K1 is sensible of a force which
acts outwards in a radial direction, and which would be interpreted as
an effect of inertia (centrifugal force) by an observer who was at
rest with respect to the original reference-body K. But the observer
on the disc may regard his disc as a reference-body which is " at rest
" ; on the basis of the general principle of relativity he is
justified in doing this. The force acting on himself, and in fact on
all other bodies which are at rest relative to the disc, he regards as
the effect of a gravitational field. Nevertheless, the
space-distribution of this gravitational field is of a kind that would
not be possible on Newton's theory of gravitation.* But since the
observer believes in the general theory of relativity, this does not
disturb him; he is quite in the right when he believes that a general
law of gravitation can be formulated- a law which not only explains
the motion of the stars correctly, but also the field of force
experienced by himself.
The observer performs experiments on his circular disc with clocks and
measuring-rods. In doing so, it is his intention to arrive at exact
definitions for the signification of time- and space-data with
reference to the circular disc K1, these definitions being based on
his observations. What will be his experience in this enterprise ?
To start with, he places one of two identically constructed clocks at
the centre of the circular disc, and the other on the edge of the
disc, so that they are at rest relative to it. We now ask ourselves
whether both clocks go at the same rate from the standpoint of the
non-rotating Galileian reference-body K. As judged from this body, the
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A four-dimensional continuum described by the "co-ordinates" x[1],
x[2], x[3], x[4], was called "world" by Minkowski, who also termed a
point-event a " world-point." From a "happening" in three-dimensional
space, physics becomes, as it were, an " existence " in the
four-dimensional " world."
This four-dimensional " world " bears a close similarity to the
three-dimensional " space " of (Euclidean) analytical geometry. If we
introduce into the latter a new Cartesian co-ordinate system (x'[1],
x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are
linear homogeneous functions of x[1], x[2], x[3] which identically
satisfy the equation
x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2
The analogy with (12) is a complete one. We can regard Minkowski's "
world " in a formal manner as a four-dimensional Euclidean space (with
an imaginary time coordinate) ; the Lorentz transformation corresponds
to a " rotation " of the co-ordinate system in the fourdimensional "
world."
APPENDIX III
THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
From a systematic theoretical point of view, we may imagine the
process of evolution of an empirical science to be a continuous
process of induction. Theories are evolved and are expressed in short
compass as statements of a large number of individual observations in
the form of empirical laws, from which the general laws can be
ascertained by comparison. Regarded in this way, the development of a
science bears some resemblance to the compilation of a classified
catalogue. It is, as it were, a purely empirical enterprise.
But this point of view by no means embraces the whole of the actual
process ; for it slurs over the important part played by intuition and
deductive thought in the development of an exact science. As soon as a
science has emerged from its initial stages, theoretical advances are
no longer achieved merely by a process of arrangement. Guided by
empirical data, the investigator rather develops a system of thought
which, in general, is built up logically from a small number of
fundamental assumptions, the so-called axioms. We call such a system
of thought a theory. The theory finds the justification for its
existence in the fact that it correlates a large number of single
observations, and it is just here that the " truth " of the theory
lies.
Corresponding to the same complex of empirical data, there may be
several theories, which differ from one another to a considerable
extent. But as regards the deductions from the theories which are
capable of being tested, the agreement between the theories may be so
complete that it becomes difficult to find any deductions in which the
two theories differ from each other. As an example, a case of general
interest is available in the province of biology, in the Darwinian
theory of the development of species by selection in the struggle for
existence, and in the theory of development which is based on the
hypothesis of the hereditary transmission of acquired characters.
We have another instance of far-reaching agreement between the
deductions from two theories in Newtonian mechanics on the one hand,
and the general theory of relativity on the other. This agreement goes
so far, that up to the preseat we have been able to find only a few
deductions from the general theory of relativity which are capable of
investigation, and to which the physics of pre-relativity days does
not also lead, and this despite the profound difference in the
fundamental assumptions of the two theories. In what follows, we shall
again consider these important deductions, and we shall also discuss
the empirical evidence appertaining to them which has hitherto been
obtained.
(a) Motion of the Perihelion of Mercury
According to Newtonian mechanics and Newton's law of gravitation, a
planet which is revolving round the sun would describe an ellipse
round the latter, or, more correctly, round the common centre of
gravity of the sun and the planet. In such a system, the sun, or the
common centre of gravity, lies in one of the foci of the orbital
ellipse in such a manner that, in the course of a planet-year, the
distance sun-planet grows from a minimum to a maximum, and then
decreases again to a minimum. If instead of Newton's law we insert a
somewhat different law of attraction into the calculation, we find
that, according to this new law, the motion would still take place in
such a manner that the distance sun-planet exhibits periodic
variations; but in this case the angle described by the line joining
sun and planet during such a period (from perihelion--closest
proximity to the sun--to perihelion) would differ from 360^0. The line
of the orbit would not then be a closed one but in the course of time
it would fill up an annular part of the orbital plane, viz. between
the circle of least and the circle of greatest distance of the planet
from the sun.
According also to the general theory of relativity, which differs of
course from the theory of Newton, a small variation from the
Newton-Kepler motion of a planet in its orbit should take place, and
in such away, that the angle described by the radius sun-planet
between one perhelion and the next should exceed that corresponding to
one complete revolution by an amount given by
eq. 41: file eq41.gif
(N.B. -- One complete revolution corresponds to the angle 2p in the
absolute angular measure customary in physics, and the above
expression giver the amount by which the radius sun-planet exceeds
this angle during the interval between one perihelion and the next.)
In this expression a represents the major semi-axis of the ellipse, e
its eccentricity, c the velocity of light, and T the period of
revolution of the planet. Our result may also be stated as follows :
According to the general theory of relativity, the major axis of the
ellipse rotates round the sun in the same sense as the orbital motion
of the planet. Theory requires that this rotation should amount to 43
seconds of arc per century for the planet Mercury, but for the other
Planets of our solar system its magnitude should be so small that it
would necessarily escape detection. *
In point of fact, astronomers have found that the theory of Newton
does not suffice to calculate the observed motion of Mercury with an
exactness corresponding to that of the delicacy of observation
attainable at the present time. After taking account of all the
disturbing influences exerted on Mercury by the remaining planets, it
was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
perihelial movement of the orbit of Mercury remained over, the amount
of which does not differ sensibly from the above mentioned +43 seconds
of arc per century. The uncertainty of the empirical result amounts to
a few seconds only.
(b) Deflection of Light by a Gravitational Field
In Section 22 it has been already mentioned that according to the
general theory of relativity, a ray of light will experience a
curvature of its path when passing through a gravitational field, this
curvature being similar to that experienced by the path of a body
which is projected through a gravitational field. As a result of this
theory, we should expect that a ray of light which is passing close to
a heavenly body would be deviated towards the latter. For a ray of
light which passes the sun at a distance of D sun-radii from its
centre, the angle of deflection (a) should amount to
eq. 42: file eq42.gif
It may be added that, according to the theory, half of Figure 05 this
deflection is produced by the Newtonian field of attraction of the
sun, and the other half by the geometrical modification (" curvature
") of space caused by the sun.
This result admits of an experimental test by means of the
photographic registration of stars during a total eclipse of the sun.
The only reason why we must wait for a total eclipse is because at
every other time the atmosphere is so strongly illuminated by the
light from the sun that the stars situated near the sun's disc are
invisible. The predicted effect can be seen clearly from the
accompanying diagram. If the sun (S) were not present, a star which is
practically infinitely distant would be seen in the direction D[1], as
observed front the earth. But as a consequence of the deflection of
light from the star by the sun, the star will be seen in the direction
D[2], i.e. at a somewhat greater distance from the centre of the sun
than corresponds to its real position.
In practice, the question is tested in the following way. The stars in
the neighbourhood of the sun are photographed during a solar eclipse.
In addition, a second photograph of the same stars is taken when the
sun is situated at another position in the sky, i.e. a few months
earlier or later. As compared whh the standard photograph, the
positions of the stars on the eclipse-photograph ought to appear
displaced radially outwards (away from the centre of the sun) by an
amount corresponding to the angle a.
We are indebted to the [British] Royal Society and to the Royal
Astronomical Society for the investigation of this important
deduction. Undaunted by the [first world] war and by difficulties of
both a material and a psychological nature aroused by the war, these
societies equipped two expeditions -- to Sobral (Brazil), and to the
island of Principe (West Africa) -- and sent several of Britain's most
celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),
in order to obtain photographs of the solar eclipse of 29th May, 1919.
The relative discrepancies to be expected between the stellar
photographs obtained during the eclipse and the comparison photographs
amounted to a few hundredths of a millimetre only. Thus great accuracy
was necessary in making the adjustments required for the taking of the
photographs, and in their subsequent measurement.
The results of the measurements confirmed the theory in a thoroughly
satisfactory manner. The rectangular components of the observed and of
the calculated deviations of the stars (in seconds of arc) are set
forth in the following table of results :
Table 01: file table01.gif
(c) Displacement of Spectral Lines Towards the Red
In Section 23 it has been shown that in a system K1 which is in
rotation with regard to a Galileian system K, clocks of identical
construction, and which are considered at rest with respect to the
rotating reference-body, go at rates which are dependent on the
positions of the clocks. We shall now examine this dependence
quantitatively. A clock, which is situated at a distance r from the
centre of the disc, has a velocity relative to K which is given by
V = wr
where w represents the angular velocity of rotation of the disc K1
with respect to K. If v[0], represents the number of ticks of the
clock per unit time (" rate " of the clock) relative to K when the
clock is at rest, then the " rate " of the clock (v) when it is moving
relative to K with a velocity V, but at rest with respect to the disc,
will, in accordance with Section 12, be given by
eq. 43: file eq43.gif
or with sufficient accuracy by
eq. 44: file eq44.gif
This expression may also be stated in the following form:
eq. 45: file eq45.gif
If we represent the difference of potential of the centrifugal force
between the position of the clock and the centre of the disc by f,
i.e. the work, considered negatively, which must be performed on the
unit of mass against the centrifugal force in order to transport it
from the position of the clock on the rotating disc to the centre of
the disc, then we have
eq. 46: file eq46.gif
From this it follows that
eq. 47: file eq47.gif
In the first place, we see from this expression that two clocks of
identical construction will go at different rates when situated at
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at the surface of the earth, the amount of this displacement being
eq. 48: file eq48.gif
For the sun, the displacement towards the red predicted by theory
amounts to about two millionths of the wave-length. A trustworthy
calculation is not possible in the case of the stars, because in
general neither the mass M nor the radius r are known.
It is an open question whether or not this effect exists, and at the
present time (1920) astronomers are working with great zeal towards
the solution. Owing to the smallness of the effect in the case of the
sun, it is difficult to form an opinion as to its existence. Whereas
Grebe and Bachem (Bonn), as a result of their own measurements and
those of Evershed and Schwarzschild on the cyanogen bands, have placed
the existence of the effect almost beyond doubt, while other
investigators, particularly St. John, have been led to the opposite
opinion in consequence of their measurements.
Mean displacements of lines towards the less refrangible end of the
spectrum are certainly revealed by statistical investigations of the
fixed stars ; but up to the present the examination of the available
data does not allow of any definite decision being arrived at, as to
whether or not these displacements are to be referred in reality to
the effect of gravitation. The results of observation have been
collected together, and discussed in detail from the standpoint of the
question which has been engaging our attention here, in a paper by E.
Freundlich entitled "Zur Prüfung der allgemeinen
Relativit¨aut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35,
p. 520: Julius Springer, Berlin).
At all events, a definite decision will be reached during the next few
years. If the displacement of spectral lines towards the red by the
gravitational potential does not exist, then the general theory of
relativity will be untenable. On the other hand, if the cause of the
displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will
furnish us with important information as to the mass of the heavenly
bodies. [5][A]
Notes
*) Especially since the next planet Venus has an orbit that is
almost an exact circle, which makes it more difficult to locate the
perihelion with precision.
The displacentent of spectral lines towards the red end of the
spectrum was definitely established by Adams in 1924, by observations
on the dense companion of Sirius, for which the effect is about thirty
times greater than for the Sun. R.W.L. -- translator
APPENDIX IV
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
(SUPPLEMENTARY TO SECTION 32)
Since the publication of the first edition of this little book, our
knowledge about the structure of space in the large (" cosmological
problem ") has had an important development, which ought to be
mentioned even in a popular presentation of the subject.
My original considerations on the subject were based on two
hypotheses:
(1) There exists an average density of matter in the whole of space
which is everywhere the same and different from zero.
(2) The magnitude (" radius ") of space is independent of time.
Both these hypotheses proved to be consistent, according to the
general theory of relativity, but only after a hypothetical term was
added to the field equations, a term which was not required by the
theory as such nor did it seem natural from a theoretical point of
view (" cosmological term of the field equations ").
Hypothesis (2) appeared unavoidable to me at the time, since I thought
that one would get into bottomless speculations if one departed from
it.
However, already in the 'twenties, the Russian mathematician Friedman
showed that a different hypothesis was natural from a purely
theoretical point of view. He realized that it was possible to
preserve hypothesis (1) without introducing the less natural
cosmological term into the field equations of gravitation, if one was
ready to drop hypothesis (2). Namely, the original field equations
admit a solution in which the " world radius " depends on time
(expanding space). In that sense one can say, according to Friedman,
that the theory demands an expansion of space.
A few years later Hubble showed, by a special investigation of the
extra-galactic nebulae (" milky ways "), that the spectral lines
emitted showed a red shift which increased regularly with the distance
of the nebulae. This can be interpreted in regard to our present
knowledge only in the sense of Doppler's principle, as an expansive
motion of the system of stars in the large -- as required, according
to Friedman, by the field equations of gravitation. Hubble's discovery
can, therefore, be considered to some extent as a confirmation of the
theory.
There does arise, however, a strange difficulty. The interpretation of
the galactic line-shift discovered by Hubble as an expansion (which
can hardly be doubted from a theoretical point of view), leads to an
origin of this expansion which lies " only " about 10^9 years ago,
while physical astronomy makes it appear likely that the development
of individual stars and systems of stars takes considerably longer. It
is in no way known how this incongruity is to be overcome.
I further want to rernark that the theory of expanding space, together
with the empirical data of astronomy, permit no decision to be reached
about the finite or infinite character of (three-dimensional) space,
while the original " static " hypothesis of space yielded the closure
(finiteness) of space.
K = co-ordinate system
x, y = two-dimensional co-ordinates
x, y, z = three-dimensional co-ordinates
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