Space Time and Gravitation/Appendix

by Arthur Stanley Eddington

APPENDIX

Mathematical Notes

The references marked "Report" are to the writer's "Report on the Relativity Theory of Gravitation" for the Physical Society of London (Fleetway Press), where fuller mathematical details are given.

Probably the most complete treatise on the mathematical theory of the subject is H. Weyl's Raum, Zeit, Materie (Julius Springer, Berlin).

Note 1 (p. 20).

It is not possible to predict the contraction rigorously from the universally accepted electromagnetic equations, because these do not cover the whole ground. There must be other forces or conditions which govern the form and size of an electron; under electromagnetic forces alone it would expand indefinitely. The old electrodynamics is entirely vague as to these forces.

The theory of Larmor and Lorentz shows that if any system at rest in the aether is in equilibrium, a similar system in uniform motion through the aether, but with all lengths in the direction of motion diminished in FitzGerald's ratio, will also be in equilibrium so far as the differential equations of the electromagnetic field are concerned. There is thus a general theoretical agreement with the observed contraction, provided the boundary conditions at the surface of an electron behave in the same way. The latter suggestion is confirmed by experiments on isolated electrons in rapid motion (Kaufmann's experiment). It turns out that this requires an electron to suffer the same kind of contraction as a material rod; and thus, although the theory throws light on the adjustments involved in material contraction, it can scarcely be said to give an explanation of the occurrence of contraction generally.

Note 2 (p. 47).

Suppose a particle moves from $(x_{1},y_{1},z_{1},t_{1})$ to $(x_{2},y_{2},z_{2},t_{2})$, its velocity $u$ is given by $u^{2}={\frac {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}{(t_{2}-t_{1})^{2}}}$ Hence from the formula for $s^{2}$$s^{2}=(t_{2}-t_{1}){\sqrt {(1-u^{2})}}$. (We omit a ${\sqrt {-1}}$, as the sign of $s^{2}$ is changed later in the chapter.)

If we take $t_{1}$ and $t_{2}$ to be the start and finish of the aviator's cigar (Chapter i), then as judged by a terrestrial observer, $t_{2}-t_{1}={\text{60 minutes,}}\quad {\sqrt {(1-u^{2})}}={\text{FitzGerald contraction}}={\tfrac {1}{2}}$.

As judged by the aviator, $t_{2}-t_{1}={\text{30 minutes,}}\quad {\sqrt {(1-u^{2})}}=1$.

Thus for both observers $s$ = 30 minutes, verifying that it is an absolute quantity independent of the observer.

Note 3 (p. 48).

The formulae of transformation to axes with a different orientation are $x=x^{\prime }\cos \theta -\tau ^{\prime }\sin \theta {\text{,}}\quad y=y^{\prime }{\text{,}}\quad z=z^{\prime }{\text{,}}\quad \tau =x^{\prime }\sin \theta +\tau ^{\prime }\cos \theta {\text{,}}$ where $\theta$ is the angle turned through in the plane $x\tau$.

Let $u=i\tan \theta$, so that $\cos \theta =(1-u^{2})^{-{\tfrac {1}{2}}}=\beta$, say. The formulae become $x=\beta \,(x^{\prime }-iu\tau ^{\prime }){\text{,}}\quad y=y^{\prime }{\text{,}}\quad z=z^{\prime }{\text{,}}\quad \tau =\beta \,(\tau ^{\prime }-iux^{\prime }){\text{,}}$ or, reverting to real time by setting $i\tau =t$, $x=\beta \,(x^{\prime }-ut^{\prime }){\text{,}}\quad y=y^{\prime }{\text{,}}\quad z=z^{\prime }{\text{,}}\quad t=\beta \,(t^{\prime }-ux^{\prime }){\text{,}}$ which gives the relation between the estimates of space and time by two different observers.

The factor $\beta$ gives in the first equation the FitzGerald contraction, and in the fourth equation the retardation of time. The terms $ut^{\prime }$ and $ux^{\prime }$ correspond to the changed conventions as to rest and simultaneity.

A point at rest, $x$ = const., for the first observer corresponds to a point moving with velocity $u$, $x^{\prime }-ut^{\prime }$ = const., for the second observer. Hence their relative velocity is $u$.

Note 4 (p. 81).

The condition for flat space in two dimensions is

${\frac {\partial }{\partial x_{1}}}\left({\frac {g_{12}}{g_{11}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}}{\frac {\partial g_{11}}{\partial x_{2}}}-{\frac {1}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{22}}{\delta x_{1}}}\right)$

$+{\frac {\partial }{\partial x_{2}}}\left({\frac {2}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{12}}{\partial x_{1}}}-{\frac {1}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{11}}{\partial x_{2}}}\right.$

⁠⁠⁠⁠$\left.-{\frac {g_{12}}{g_{11}{\sqrt {\left(g_{12}g_{22}-{g_{12}}^{2}\right)}}}}{\frac {\partial g_{11}}{\partial x_{1}}}\right)=0$.

Note 5 (p. 89).

Let $g$ be the determinant of four rows and columns formed with the elements $g_{\mu \nu }$.

Let $g^{\mu \nu }$ be the minor of $g_{\mu \nu }$, divided by $g$.

Let the "3-index symbol" {$\mu \nu$, $\lambda$} denote ${\tfrac {1}{2}}g^{\lambda a}\left({\frac {\partial g_{\mu a}}{\partial x_{\nu }}}+{\frac {\partial g_{\nu a}}{\partial x_{\mu }}}-{\frac {\partial g_{\mu \nu }}{\partial x_{a}}}\right)$ summed for values of $a$ from 1 to 4. There will be 40 different 3-index symbols.

Then the Riemann-Christoffel tensor is $B_{\mu \nu \sigma }^{\rho }=\{\mu \sigma ,\epsilon \}\{\epsilon \nu ,\rho \}-\{\mu \nu ,\epsilon \}\{\epsilon \sigma ,\rho \}+{\frac {\partial }{\partial x_{\nu }}}\{\mu \sigma ,\rho \}-{\frac {\partial }{\partial x_{\sigma }}}\{\mu \nu ,\rho \}$, the terms containing $\epsilon$ being summed for values of $\epsilon$ from 1 to 4.

The "contracted" Riemann-Christoffel tensor $G_{\mu \nu }$ can be reduced to

$G_{\mu \nu }=-{\frac {\partial }{\partial x_{a}}}\{\mu \nu ,a\}+\{\mu a,\beta \}\{\nu \beta ,a\}$

$+{\frac {\partial ^{2}}{\partial x_{\mu }\partial x_{\nu }}}\log {\sqrt {-g}}-\{\mu \nu ,a\}{\frac {\partial }{\partial x_{a}}}\log {\sqrt {-g}}$,

where in accordance with a general convention in this subject, each term containing a suffix twice over ($a$ and $\beta$) must be summed for the values 1, 2, 3, 4 of that suffix.

The curvature $G=g^{\mu \nu }G_{\mu \nu }$, summed in accordance with the foregoing convention.

Note 6 (p. 94).

The electric potential due to a charge $e$ is $\phi ={\frac {e}{\left[r\left(1-v_{r}/C\right)\right]}}$, where $v_{r}$ is the velocity of the charge in the direction of $r$, $C$ the velocity of light, and the square bracket signifies antedated values. To the first order of $v_{r}/C$, the denominator is equal to the present distance $r$, so the expression reduces to $e/r$ in spite of the time of propagation. The foregoing formula for the potential was found by Liénard and Wiechert.

Note 7 (p. 97).

It is found that the following scheme of potentials rigorously satisfies the equations $G_{\mu \nu }=0$, according to the values of $G_{\mu \nu }$ in Note 5, ${\begin{matrix}-1/\gamma &0&0&0\\&-{x_{1}}^{2}&0&0\\&&-{x_{1}}^{2}\sin ^{2}{x_{2}}^{2}&0\\&&&\gamma \end{matrix}}$ where $\gamma =1-\kappa /x_{1}$ and $\kappa$ is any constant (see Report, § 28). Hence these potentials describe a kind of space-time which can occur in nature referred to a possible mesh-system. If $\kappa =0$, the potentials reduce to those for flat space-time referred to polar coordinates; and, since in the applications required $\kappa$ will always be extremely small, our coordinates can scarcely be distinguished from polar coordinates. We can therefore use the familiar symbols $r$, $\theta$, $\phi$, $t$, instead of $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. It must, however, be remembered that the identification with polar coordinates is only approximate; and, for example, an equally good approximation is obtained if we write $x_{1}=r+{\tfrac {1}{2}}\kappa$, a substitution often used instead of $x_{1}=r$ since it has the advantage of making the coordinate-velocity of light more symmetrical.

We next work out analytically all the mechanical and optical properties of this kind of space-time, and find that they agree observationally with those existing round a particle at rest at the origin with gravitational mass ${\tfrac {1}{2}}\kappa$. The conclusion is that the gravitational field here described is produced by a particle of mass ${\tfrac {1}{2}}\kappa$—or, if preferred, a particle of matter at rest is produced by the kind of space-time here described.

Note 8 (p. 98).

Setting the gravitational constant equal to unity, we have for a circular orbit $m/r^{2}=v^{2}/r$, so that $m=v^{2}r$. The earth's speed, $v$, is approximately 30 km. per sec., or  1/10000 in terms of the velocity of light. The radius of its orbit, $r$, is about 1.5 . 10⁸ km. Hence, $m$, the gravitational mass of the sun is approximately 1.5 km.

The radius of the sun is 697,000 kms., so that the quantity $2m/r$ occurring in the formulae is, for the sun's surface, .00000424 or 0″.87.

Note 9 (p. 123).

See Report, §§ 29, 30. The general equations of a geodesic are ${\frac {d^{2}x_{\mu }}{ds^{2}}}+\{a\beta ,\mu \}{\frac {dx_{a}}{ds}}{\frac {dx_{\beta }}{ds}}=0\quad {\text{(}}\mu ={\text{1, 2, 3, 4)}}$.

From the formula for the line-element $ds^{2}=-\gamma ^{-1}dr^{2}-r^{2}\,d\theta ^{2}+\gamma \,dt^{2}\quad {\text{............ (1),}}$ we calculate the three-index symbols and it is found that two of the equations of the geodesic take the rather simple form ${\begin{aligned}{\frac {d^{2}\theta }{ds^{2}}}+{\frac {2}{r}}\cdot {\frac {dr}{ds}}{\frac {d\theta }{ds}}=0{\text{,}}\\{\frac {d^{2}t}{ds^{2}}}+{\frac {d\left(\log \gamma \right)}{dr}}\cdot {\frac {dr}{ds}}{\frac {dt}{ds}}=0{\text{,}}\end{aligned}}$ which can be integrated giving ${\begin{aligned}r^{2}{\frac {d\theta }{ds}}&=h\quad {\text{...........................(2),}}\\{\frac {dt}{ds}}&={\frac {c}{\gamma }}\quad {\text{...........................(3),}}\end{aligned}}$ where $h$ and $c$ are constants of integration.

Eliminating $dt$ and $ds$ from (1), (2) and (3), we have $\left({\frac {h}{r^{2}}}{\frac {dr}{d\theta }}\right)^{2}+{\frac {h^{2}}{r^{2}}}=c^{2}-1+{\frac {2m}{r}}+{\frac {2mh^{2}}{r^{2}}}$, or writing $\left({\frac {du}{d\theta }}\right)^{2}+u^{2}={\frac {c^{2}-1}{h^{2}}}+{\frac {2mu}{h^{2}}}+2mu^{3}$.

Differentiating with respect to $\theta$ ${\frac {d^{2}u}{d\theta ^{2}}}+u={\frac {m}{h^{2}}}+3mu^{2}$, which gives the equation of the orbit in the usual form in particle dynamics. It differs from the equation of the Newtonian orbit by the small term $3mu^{2}$ , which is easily shown to give the motion of perihelion.

The track of a ray of light is also obtained from this formula, since by the principle of equivalence it agrees with that of a material particle moving with the speed of light. This case is given by $ds=0$, and therefore $h=\infty$. The differentia] equation for the path of a light-ray is thus ${\frac {d^{2}u}{d\theta ^{2}}}+u=3mu^{2}$.

An approximate solution is $u={\frac {\cos \theta }{R}}+{\frac {m}{R^{2}}}\left(\cos ^{2}\theta +2\sin ^{2}\theta \right)$, neglecting the very small quantity $m^{2}/R^{2}$. Converting to Cartesian coordinates, this becomes $x=R-{\frac {m}{R}}{\frac {x^{2}+2y^{2}}{\sqrt {\left(x^{2}+y^{2}\right)}}}$.

The asymptotes of the light-track are found by taking $y$ very large compared with $x$, giving $x=R\pm {\frac {2m}{R}}y$ so that the angle between them is $4m/R$.

Note 10 (p. 126).

Writing the line element in the form $ds^{2}=-\left(1+a{\frac {m}{r}}+\ldots \right)dr^{2}-r^{2}\,d\theta ^{2}+\left(1+b{\frac {m}{r}}+c{\frac {m^{2}}{r^{2}}}+\ldots \right)dt^{2}$, the approximate Newtonian attraction fixes $b$ equal to + 2; then the observed deflection of light fixes $a$ equal to − 2; and with these values the observed motion of Mercury fixes $c$ equal to 0.

To insert an arbitrary coefficient of $r^{2}d\theta ^{2}$ would merely vary the coordinate system. We cannot arrive at any intrinsically different kind of space-time in that way. Hence, within the limits of accuracy mentioned, the expression found by Einstein is completely determinable by observation.

It may be mentioned that the line-element $ds^{2}=-dr^{2}-r^{2}d\theta ^{2}+\left(1-2m/r\right)dt^{2}$, gives one-half the observed deflection of light, and one-third the motion of perihelion of Mercury. As both these can be obtained on older theories, taking account of the variation of mass with velocity, the coefficient $\gamma ^{-1}$ of $dr^{2}$ is the essentially novel point in Einstein's theory.

Note 11 (p. 131).

It is often supposed that by the Principle of Equivalence any invariant property which holds outside a gravitational field also holds in a gravitational field; but there is necessarily some limitation on this equivalence. Consider for instance the two invariant equations $ds^{2}=1$, $ds^{2}\left(1+k^{4}B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }\right)=1$, where $k$ is some constant having the dimensions of a length. Since $B_{\mu \nu \sigma }^{\rho }$ vanishes outside a gravitational field, if one of these equations is true the other will be. But they cannot both hold in a gravitational field, since there $B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }$ does not vanish, and is in fact equal to $24m^{2}/r^{6}$. (I believe that the numerical factor 24 is correct; but there are 65,536 terms in the expression, and the terms which do not vanish have to be picked out.

This ambiguity of the Principle of Equivalence is referred to in Report, §§ 14, 27; and an enunciation is given which makes it definite. The enunciation however is merely an explicit statement, and not a defence, of the assumptions commonly made in applying the principle.

So far as general reasoning goes there seems no ground for choosing $ds^{2}$ rather than $ds^{2}(1+24k^{4}m^{2}/r^{6})$, or any similar expression, as the constant character in the vibration of an atom.

Note 12 (p. 134).

Let two rays diverging from a point at a distance $R$ pass at distances $r$ and $r+dr$ from a star of mass $m$. The deflection being $4m/r$, their divergence will be increased by $4mdr/r^{2}$. This increase will be equal to the original divergence $dr/R$ if $r={\sqrt {4mR}}$. Take for instance $4m=10$ km., $R=10^{1}5$ km., then $r=10^{8}$ km. So that the divergence of the light will be doubled, when the actual deflection of the ray is only 10^-7, or 0″.02. In the case of a star seen behind the sun the added divergence has no time to take effect; but when the light has to travel a stellar distance after the divergence is produced, it becomes weakened by it. Generally in stellar phenomena the weakening of the light should be more prominent than the actual deflection.

Note 13 (p. 141).

The relations are (Report, § 39) $G_{\mu \nu }^{\nu }={\tfrac {1}{2}}{\frac {\partial G}{\partial x_{\mu }}}$⁠($\mu$ = 1, 2, 3, 4),</math> where $G_{\mu \nu }^{\nu }$ is the (contracted) covariant derivative of $G_{\mu }^{\nu }$, or $g^{\nu a}G_{\mu a}$.

I doubt whether anyone has performed the laborious task of verifying these identities by straightforward algebra.

Note 14 (p. 158).

The modified law for spherical space-time is in empty space $G_{\mu \nu }=\lambda g_{\mu \nu }$.

In cylindrical space-time, matter is essential. The law in space occupied by matter is $G_{\mu \nu }-{\tfrac {1}{2}}g_{\mu \nu }\left(G-2\lambda \right)=-8\pi T_{\mu \nu }$, the term $2\lambda$ being the only modification. Spherical space-time of radius $R$ is given by $\lambda =3/R^{2}$; cylindrical space-time by $\lambda =1/R^{2}$ provided matter of average density $\rho =1/4\pi R^{2}$ is present. (See Report, §§ 50, 51.) The total mass of matter in the cylindrical world is ${\tfrac {1}{2}}\pi R$. This must be enormous, seeing that the sun s mass is only 1 1/2 kilometres.

Note 15 (p. 174).

Weyl's theory is given in Berlin. Sitzungsberichte, 30 May, 1918; Annalen der Physik, Bd. 59 (1919), p. 101.

Note 16 (p. 177).

The argument is rather more complicated than appears in the text, where the distinction between action-density and action in a region, curvature and total curvature in a region, has not been elaborated. Taking a definitely marked out region in space and time, its measured volume will be increased 16-fold by halving the gauge. Therefore for action-density we must take an expression which will be diminished 16-fold by halving the gauge. Now $G$ is proportional to $1/^{2}$, where $R$ is the radius of curvature, and so is diminished 4-fold. The invariant $B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }$ has the same gauge-dimensions as $G^{2}$; and hence when integrated through a volume gives a pure number independent of the gauge. In Weyl's theory this is only the gravitational part of the complete invariant $\left(B_{\mu \nu \sigma }^{\rho }-{\tfrac {1}{2}}g_{\mu }^{\rho }F_{\nu \sigma }\right)\left(B_{\rho }^{\mu \nu \sigma }-{\tfrac {1}{2}}g_{\rho }^{\mu }F^{\nu \sigma }\right)$, which reduces to $B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }+F_{\nu \sigma }F^{\nu \sigma }$.

The second term gives actually the well-known expression for the action-density of the electromagnetic field, and this evidently strengthens the identification of this invariant with action-density.

Einstein's theory, on the other hand, creates a difficulty here, because although there may be action in an electromagnetic field without electrons, the curvature is zero.

Historical Note

Before the Michelson-Morley experiment the question had been widely discussed whether the aether in and near the earth was carried along by the earth in its motion, or whether it slipped through the interstices between the atoms. Astronomical aberration pointed decidedly to a stagnant aether; but the experiments of Arago and Fizeau on the effect of motion of transparent media on the velocity of light in those media, suggested a partial convection of the aether in such cases. These experiments were first-order experiments, i.e they depended on the ratio of the velocity of the transparent body to the velocity of light. The Michelson-Morley experiment is the first example of an experiment delicate enough to detect second-order effects, depending on the square of the above ratio; the result, that no current of aether past terrestrial objects could be detected, appeared favourable to the view that the aether must be convected by the earth. The difficulty of reconciling this with astronomical aberration was recognised.

An attempt was made by Stokes to reconcile mathematically a convection of aether by the earth with the accurately verified facts of astronomical aberration; but his theory cannot be regarded as tenable. Lodge investigated experimentally the question whether smaller bodies carried the aether with them in their motion, and showed that the aether between two whirling steel discs was undisturbed.

The controversy, stagnant versus convected aether, had now reached an intensely interesting stage. In 1895, Lorentz discussed the problem from the point of view of the electrical theory of light and matter. By his famous transformation of the electromagnetic equations, he cleared up the difficulties associated with the first-order effects, showing that they could all be reconciled with a stagnant aether. In 1900, Larmor carried the theory as far as second-order effects, and obtained an exact theoretical foundation for FitzGerald's hypothesis of contraction, which had been suggested in 1892 as an explanation of the Michelson-Morley experiment. The theory of a stagnant aether was thus reconciled with all observational results; and hence forward it held the field.

Further second-order experiments were performed by Rayleigh and Brace on double refraction (1902, 1904), Trouton and Noble on a torsional effect on a charged condenser (1903), and Trouton and Rankine on electric conductivity (1908). All showed that the earth s motion has no effect on the phenomena. On the theoretical side, Lorentz (1902) showed that the indifference of the equations of the electromagnetic field to any velocity of the axes of reference, which he had previously established to the first order, and Larmor to the second order, was exact to all orders. He was not, however, able to establish with the same exactness a corresponding transformation for bodies containing electrons.

Both Larmor and Lorentz had introduced a "local time" for the moving system. It was clear that for many phenomena this local time would replace the "real" time; but it was not suggested that the observer in the moving system would be deceived into thinking that it was the real time. Einstein, in 1905 founded the modern principle of relativity by postulating that this local time was the time for the moving observer; no real or absolute time existed, but only the local times, different for different observers. He showed that absolute simultaneity and absolute location in space are inextricably bound together, and the denial of the latter carries with it the denial of the former. By realising that an observer in the moving system would measure all velocities in terms of the local space and time of that system, Einstein removed the last discrepancies from Lorentz's transformation.

The relation between the space and time coordinates in two systems in relative motion was now obtained immediately from the principles of space and time-measurement. It must hold for all phenomena provided they do not postulate a medium which can serve as a standard for absolute location and simultaneity. The previous deduction of these formulae by lengthy transformation of the electromagnetic equations now appears as a particular case; it shows that electromagnetic phenomena have no reference to a medium with such properties.

The combination of the local spaces and times of Einstein into an absolute space-time of four dimensions is the work of Minkowski (1908). Chapter iii is largely based on his researches. Much progress was made in the four-dimensional vector-analysis of the world; but the whole problem was greatly simplified when Einstein and Grossmann introduced for this purpose the more powerful mathematical calculus of Riemann, Ricci, and Levi-Civita.

In 1911, Einstein put forward the Principle of Equivalence, thus turning the subject towards gravitation for the first time. By postulating that not only mechanical but optical and electrical phenomena in a field of gravitation and in a field produced by acceleration of the observer were equivalent, he deduced the displacement of the spectral lines on the sun and the displacement of a star during a total eclipse. In the latter case, however, he predicted only the half-deflection, since he was still working with Newton's law of gravitation. Freundlich at once examined plates obtained at previous eclipses, but failed to find sufficient data; he also prepared to observe the eclipse of 1914 in Russia with this object, but was stopped by the outbreak of war. Another attempt was made by the Lick Observatory at the not very favourable eclipse of 1918. Only preliminary results have been published; according to the information given, the probable accidental error of the mean result (reduced to the sun's limb) was about 1″.6, so that no conclusion was permissible.

The principle of equivalence opened up the possibility of a general theory of relativity not confined to uniform motion, for it pointed a way out of the objections which had been urged against such an extension from the time of Newton. At first the opening seemed a very narrow one, merely indicating that the objections could not be considered final until the possibilities of complications by gravitation had been more fully exhausted. By 1913, Einstein had surmounted the main difficulties. His theory in a complete form was published in 1915; but it was not generally accessible in England until a year or two later. As this theory forms the main subject-matter of the book, we may leave our historical survey at this point.