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  which quantities are equal to the quantities Hx . . . Ez in the special case of the restricted theory of relativity; and in addition

  we obtain in place of (63)

  The equations (60), (62), and (63) thus form the generalization of Maxwell’s field equations for free space, with the convention which we have established with respect to the choice of coordinates.

  The Energy-components of the Electromagnetic Field.—We form the inner product

  By (61) its components, written in the three-dimensional manner, are

  κσ is a covariant vector the components of which are equal to the negative momentum, or, respectively, the energy, which is transferred from the electric masses to the electromagnetic field per unit of time and volume. If the electric masses are free, that is, under the sole influence of the electromagnetic field, the covariant vector κσ will vanish.

  To obtain the energy-components of the electromagnetic field, we need only give to equation κσ = 0 the form of equation (57). From (63) and (65) we have in the first place

  The second term of the right-hand side, by reason of (60), permits the transformation

  which latter expression may, for reasons of symmetry, also be written in the form

  But for this we may set

  The first of these terms is written more briefly

  the second, after the differentiation is carried out, and after some reduction, results in

  Taking all three terms together we obtain the relation

  where

  Equation (66), if κσ vanishes, is, on account of (30), equivalent to (57) or (57a) respectively. Therefore the are the energy-components of the electromagnetic field. With the help of (61) and (64), it is easy to show that these energy-components of the electromagnetic field in the case of the special theory of relativity give the well-known Maxwell-Poynting expressions.

  We have now deduced the general laws which are satisfied by the gravitational field and matter, by consistently using a system of coordinates for which We have thereby achieved a considerable simplification of formulae and calculations, without failing to comply with the requirement of general covariance; for we have drawn our equations from generally covariant equations by specializing the system of co-ordinates.

  Still the question is not without a formal interest, whether with a correspondingly generalized definition of the energy-components of gravitational field and matter, even without specializing the system of co-ordinates, it is possible to formulate laws of conservation in the form of equation (56), and field equations of gravitation of the same nature as (52) or (52a), in such a manner that on the left we have a divergence (in the ordinary sense), and on the right the sum of the energy-components of matter and gravitation. I have found that in both cases this is actually so. But I do not think that the communication of my somewhat extensive reflexions on this subject would be worth while, because after all they do not give us anything that is materially new.

  E

  § 21. NEWTON’S THEORY AS A FIRST APPROXIMATION

  As has already been mentioned more than once, the special theory of relativity as a special case of the general theory is characterized by the gμv having the constant values (4). From what has already been said, this means complete neglect of the effects of gravitation. We arrive at a closer approximation to reality by considering the case where the gμv differ from the values of (4) by quantities which are small compared with 1, and neglecting small quantities of second and higher order. (First point of view of approximation.)

  It is further to be assumed that in the space-time territory under consideration the gμv at spatial infinity, with a suitable choice of coordinates, tend toward the values (4); i.e. we are considering gravitational fields which may be regarded as generated exclusively by matter in the finite region.

  It might be thought that these approximations must lead us to Newton’s theory. But to that end we still need to approximate the fundamental equations from a second point of view. We give our attention to the motion of a material point in accordance with the equations (16). In the case of the special theory of relativity the components

  may take on any values. This signifies that any velocity

  may occur, which is less than the velocity of light in vacuo. If we restrict ourselves to the case which almost exclusively offers itself to our experience, of v being small as compared with the velocity of light, this denotes that the components

  are to be treated as small quantities, while dx4/ds, to the second order of small quantities, is equal to one. (Second point of view of approximation.)

  Now we remark that from the first point of view of approximation the magnitudes are all small magnitudes of at least the first order. A glance at (46) thus shows that in this equation, from the second point of view of approximation, we have to consider only terms for which μ = v = 4. Restricting ourselves to terms of lowest order we first obtain in place of (46) the equations

  where we have set ds = dx4 = dt; or with restriction to terms which from the first point of view of approximation are of first order:—

  If in addition we suppose the gravitational field to be a quasi-static field, by confining ourselves to the case where the motion of the matter generating the gravitational field is but slow (in comparison with the velocity of the propagation of light), we may neglect on the right-hand side differentiations with respect to the time in comparison with those with respect to the space co-ordinates, so that we have

  This is the equation of motion of the material point according to Newton’s theory, in which g44 plays the part of the gravitational potential. What is remarkable in this result is that the component g44 of the fundamental tensor alone defines, to a first approximation, the motion of the material point.

  We now turn to the field equations (53). Here we have to take into consideration that the energy-tensor of “matter” is almost exclusively defined by the density of matter in the narrower sense, i.e. by the second term of the right-hand side of (58) [or, respectively, (58a) or (58b)]. If we form the approximation in question, all the components vanish with the one exception of T44 = ρ = T. On the left-hand side of (53) the second term is a small quantity of second order; the first yields, to the approximation in question,

  For μ = v = 4, this gives, with the omission of terms differentiated with respect to time,

  The last of equations (53) thus yields

  The equations (67) and (68) together are equivalent to Newton’s law of gravitation.

  By (67) and (68) the expression for the gravitational potential becomes

  while Newton’s theory, with the unit of time which we have chosen, gives

  in which K denotes the constant 6.7 × 10–8, usually called the constant of gravitation. By comparison we obtain

  § 22. BEHAVIOUR OF RODS AND CLOCKS IN THE STATIC GRAVITATIONAL FIELD. BENDING OF LIGHT-RAYS. MOTION OF THE PERIHELION OF A PLANETARY ORBIT

  To arrive at Newton’s theory as a first approximation we had to calculate only one component, g44, of the ten gμv of the gravitational field, since this component alone enters into the first approximation, (67), of the equation for the motion of the material point in the gravitational field. From this, however, it is already apparent that other components of the gμv must differ from the values given in (4) by small quantities of the first order. This is required by the condition g = – 1.

  For a field-producing point mass at the origin of co-ordinates, we obtain, to the first approximation, the radially symmetrical solution

  where δμσ is 1 or 0, respectively, accordingly as ρ = σ or ρ ≠ σ, and r is the quantity . On account of (68a)

  If M denotes the field-producing mass. It is easy to verify that the field equations (outside the mass) are satisfied to the first order of small quantities.

  We now examine the influence exerted by the field of the mass M upon the metrical properties of space. The relation

  always holds between the “locally” (§ 4) measured lengths and times ds on the o
ne hand, and the differences of co-ordinates dxv on the other hand.

  For a unit-measure of length laid “parallel” to the axis of x, for example, we should have to set ds2 = – 1; dx2 = dx3 dx4 = 0. Therefore – 1 = . If, in addition, the unit-measure lies on the axis of x, the first of equations (70) gives

  From these two relations it follows that, correct to a first order of small quantities,

  The unit measuring-rod thus appears a little shortened in relation to the system of co-ordinates by the presence of the gravitational field, if the rod is laid along a radius.

  In an analogous manner we obtain the length of co-ordinates in tangential direction if, for example, we set

  The result is

  With the tangential position, therefore, the gravitational field of the point of mass has no influence on the length of a rod.

  Thus Euclidean geometry does not hold even to a first approximation in the gravitational field, if we wish to take one and the same rod, independently of its place and orientation, as a realization of the same interval; although, to be sure, a glance at (70a) and (69) shows that the deviations to be expected are much too slight to be noticeable in measurements of the earth’s surface.

  Further, let us examine the rate of a unit clock, which is arranged to be at rest in a static gravitational field. Here we have for a clock period ds = 1; dx1 = dx2 = dx5 = 0

  Therefore

  or

  Thus the clock goes more slowly if set up in the neighbourhood of ponderable masses. From this it follows that the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum.*

  We now examine the course of light-rays in the static gravitational field. By the special theory of relativity the velocity of light is given by the equation

  and therefore by the general theory of relativity by the equation

  If the direction, i.e. the ratio dx1: dx2: dx3 is given, equation (73) gives the quantities

  and accordingly the velocity

  defined in the sense of Euclidean geometry. We easily recognize that the course of the light-rays must be bent with regard to the system of co-ordinates, if the gμv are not constant. If n is a direction perpendicular to the propagation of light, the Huyghens principle shows that the light-ray, envisaged in the plane (γ, n), has the curvature – ∂γ/∂n.

  We examine the curvature undergone by a ray of light passing by a mass M at the distance Δ. If we choose the system of co-ordinates in agreement with the accompanying diagram, the total bending of the ray (calculated positively if concave towards the origin) is given in sufficient approximation by

  while (73) and (70) give

  Carrying out the calculation, this gives

  FIG. 4.

  According to this, a ray of light going past the sun undergoes a deflexion of 1.7″; and a ray going past the planet Jupiter a deflexion of about .02″.

  If we calculate the gravitational field to a higher degree of approximation, and likewise with corresponding accuracy the orbital motion of a material point of relatively infinitely small mass, we find a deviation of the following kind from the Kepler-Newton laws of planetary motion. The orbital ellipse of a planet undergoes a slow rotation, in the direction of motion, of amount

  per revolution. In this formula a denotes the major semi-axis, c the velocity of light in the usual measurement, e the eccentricity, T the time of revolution in seconds.*

  Calculation gives for the planet Mercury a rotation of the orbit of 43″ per century, corresopnding exactly to astronomical observation (Leverrier); for the astronomers have discovered in the motion of the perihelion of his planet, after allowing for disturbances by other planets, an inexplicable remainder of this magnitude.

  *Of course an answer may be satisfactory from the point of view of epistemology, and yet be unsound physically, if it is in conflict with other experiences.

  *Eötvös has proved experimentally that the gravitational field has this property in great accuracy.

  *We assume the possibility of verifying “simultaneity” for events immediately proximate in space, or—to speak more precisely—for immediate proximity or coincidence in space-time, without giving a definition of this fundamental concept.

  *The unit of time is to be chosen so that the velocity of light in vacuo as measured in the “local” system of co-ordinates is to be equal to unity.

  *By outer multiplication of the vector with arbitrary components A11, A12, A13, A14 by the vector with components 1, 0, 0, 0, we produce a tensor with components

  By the addition of four tensors of this type, we obtain the tensor Aμv with any assigned components.

  *The mathematicians have proved that this is also a sufficient condition.

  *It is only between the second (and first) derivatives that, by § 12, the relations = 0 subsist.

  *Properly speaking, this can be affirmed only of the tensor

  where λ is a constant. If, however, we set this tensor = 0, we come back again to the equations Gμv, = 0.

  *The reason for the introduction of the factor – 2k will be apparent later.

  * are to be symmetrical tensors.

  *On this question cf. H. Hilbert, Nachr. d. K. Gesellsch. d. Wiss. zu Gottingen, Math.-phys. Klasse, 1915, p. 3.

  *For an observer using a system of reference in the sense of the special theory of relativity for an infinitely small region, and moving with it, the density of energy equals ρ – p. This gives the definition of ρ. Thus ρ is not constant for an incompressible fluid.

  *On the abandonment of the choice of co-ordinates with g = – 1, there remain four functions of space with liberty of choice, corresponding to the four arbitrary functions at our disposal in the choice of co-ordinates.

  *According to E. Freundlich, spectroscopical observations on fixed stars of certain types indicate the existence of an effect of this kind, but a crucial test of this consequence has not yet been made.

  *For the calculation I refer to the original papers: A. Einstein, Sitzungsber. d. Preuss. Akad. d. Wiss., 1915, p. 831; K. Schwarzschild, ibid., 1916, p. 189.

  HAMILTON’S PRINCIPLE

  AND THE GENERAL THEORY

  OF RELATIVITY

  BY

  A. Einstein

  Translated from “Hamiltonsches Princip und allgemeine Relativitätstheorie,” Sitzungsberichte der Preussischen Akad. d. Wissenschaften, 1916.

  THE general theory of relativity has recently been given in a particularly clear form by H. A. Lorentz and D. Hilbert,* who have deduced its equations from one single principle of variation. The same thing will be done in the present paper. But my purpose here is to present the fundamental connexions in as perspicuous a manner as possible, and in as general terms as is permissible from the point of view of the general theory of relativity. In particular we shall make as few specializing assumptions as possible, in marked contrast to Hilbert’s treatment of the subject. On the other hand, in antithesis to my own most recent treatment of the subject, there is to be complete liberty in the choice of the system of co-ordinates.

  § 1. THE PRINCIPLE OF VARIATION AND THE FIELD-EQUATIONS OF GRAVITATION AND MATTER

  Let the gravitational field be described as usual by the tensor† of the gμv (or the gμv); and matter, including the electromagnetic field, by any number of space-time functions q(ρ). How these functions may be characterized in the theory of invariants does not concern us. Further, let be a function of the

  The principle of variation

  then gives us as many differential equations as there are functions gμv and q(ρ) to be defined, if the gμv and q(ρ) are varied independently of one another, and in such a way that at the limits of integration the

  δq(ρ), δgμv, and all vanish.

  We will now assume that is linear in the gστ, and that the coefficients of the depend only on the gμv. We may then replace the principle of variation (1) by one which is more convenient for us. For by appropriate partial integration we obtain />
  where F denotes an integral over the boundary of the domain in question, and * depends only on the gμv, , q(ρ), q(ρ)α, and no longer on the . From (2) we obtain, for such variations as are of interest to us,

  so that we may replace our principle of variation (1) by the more convenient form

  By carrying out the variation of the gμv and the q(ρ) we obtain, as field-equations of gravitation and matter, the equations†

  § 2. SEPARATE EXISTENCE OF THE GRAVITATIONAL FIELD

  If we make no restrictive assumption as to the manner in which depends on the gμv, , , q(ρ), q(ρ)α, the energy-components cannot be divided into two parts, one belonging to the gravitational field, the other to matter. To ensure this feature of the theory, we make the following assumption

  where is to depend only on the gμv, , , and only on gμv, q(ρ), q(ρ)α. Equations (4), (4a) then assume the form

  Here * stands in the same relation to as * to .

  It is to be noted carefully that equations (8) or (5) would have to give way to others, if we were to assume or to be also dependent on derivatives of the q(ρ) of order higher than the first. Likewise it might be imaginable that the QP) would have to be taken, not as independent of one another, but as connected by conditional equations. All this is of no importance for the following developments, as these are based solely on the equations (7), which have been found by varying our integral with respect to the gμv.