The Principle of Relativity

From (59), it follows that the system of equations

is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (60) contains essentially four equations, which can be thus written:—

This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put

Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis:—

The first Maxwellian system is obtained by a generalisation of the form given by Minkowski.

We introduce the contra-variant six-vector F_αβ by the equation

and also a contra-variant four-vector J^μ, which is the electrical current-density in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates).

If we put

which quantities become equal to H_x ... E_x in the case of the special relativity theory, and besides

we get instead of (63)

The equations (60), (62) and (63) give thus a generalisation of Maxwell’s field-equations in vacuum, which remains true in our chosen system of co-ordinates.

The energy-components of the electro-magnetic field.

Let us form the inner-product

According to (61) its components can be written down in the three-dimensional notation.

K_σ is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time, and per unit of volume, by the electrical masses. If the electrical masses be free, that is, under the influence of the electro-magnetic field only, then the covariant four-vector K_σ will vanish.

In order to get the energy components T_σ^ν of the electro-magnetic field, we require only to give to the equation K_σ = 0, the form of the equation (57).

From (63) and (65) we get first,

On account of (60) the second member on the right-hand side admits of the transformation—

Owing to symmetry, this expression can also be written in the form

which can also be put in the form

The first of these terms can be written shortly as

and the second after differentiation can be transformed in the form

If we take all the three terms together, we get the relation

where

On account of (30) the equation (66) becomes equivalent to (57) and (57a) when K_σ vanishes. Thus τ_σ^ν’s are the energy-components of the electro-magnetic field. With the help of (61) and (64) we can easily show that the energy-components of the electro-magnetic field, in the case of the special relativity theory, give rise to the well-known Maxwell-Poynting expressions.

We have now deduced the most general laws which the gravitation-field and matter satisfy when we use a co-ordinate system for which √(-g) = 1. Thereby we achieve an important simplification in all our formulas and calculations, without renouncing the conditions of general covariance, as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations. Still the question is not without formal interest, whether, when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates, the laws of conservation have the form of the equation (56), and the field-equations of gravitation hold in the form (52) or (52a); such that on the left-hand side, we have a divergence in the usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation. I have found out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive work on this subject will not pay, for nothing essentially new comes out of it.

E. §21. Newton’s theory as a first approximation.

We have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general, in which g_μν’s have constant values (4). This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get one important approximation if we consider the case when g_μν’s differ from (4) only by small magnitudes (compared to 1) where we can neglect small quantities of the second and higher orders (first aspect of the approximation.)

Further it should be assumed that within the space-time region considered, g_μν’s at infinite distances (using the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); i.e., we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions.

We can assume that this approximation should lead to Newton’s theory. For it however, it is necessary to treat the fundamental equations from another point of view. Let us consider the motion of a particle according to the equation (46). In the case of the special relativity theory, the components

can take any values. This signifies that any velocity

can appear which is less than the velocity of light in vacuum (v < 1). If we finally limit ourselves to the consideration of the case when v is small compared to the velocity of light, it signifies that the components

can be treated as small quantities, whereas dx₄/ds is equal to 1, up to the second-order magnitudes (the second point of view for approximation).

Now we see that, according to the first view of approximation, the magnitudes γ_μν^τ’s are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which μ = ν = 4.

By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations:—

or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,

If we further assume that the gravitation-field is quasi-static, i.e., it is limited only to the case when the matter producing the gravitation-field is moving slowly (relative to the velocity of light) we can neglect the differentiations of the positional co-ordinates on the right-hand side with respect to time, so that we get

This is the equation of motion of a material point according to Newton’s theory, where g₄₄/₂ plays the part of gravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the material point, only the component g₄₄ of the fundamental tensor appears.

Let us now turn to the field-equation (53). In this case, we have to remember that the energy-tensor of matter is exclusively defined in a narrow sense by the density ρ of matter, i.e., by the second member on the right-hand side of 58 [(58a, or 58b)]. If we make the necessary approximations, then all component vanish except

On the left-hand side of (53) the second term is an infinitesimal of the second order, so that the first leads to the following terms in the approximation, which are rather interesting for us:

By neglecting all differentiations with regard to time, this leads, when μ = ν =4, to the expression

The last of the equations (53) thus leads to

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

For the gravitation-potential we get from (67) and (68) the exp.

whereas the Newtonian theory for the chosen unit of time gives

where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating them we get

§22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature of light-rays. Perihelion-motion of the paths of the Planets.

In order to obtain Newton’s theory as a first approximation we had to calculate only g₄₄, out of the 10 components g_μν of the gravitation-potential, for that is the only component which comes in the first approximate equations of motion of a material point in a gravitational field.

We see however, that the other components of g_μν should also differ from the values given in (4) as required by the condition g = -1.

For a heavy particle at the origin of co-ordinates and generating the gravitational field, we get as a first approximation the symmetrical solution of the equation:—

δ_ρσ is 1 or 0, according as ρ = σ or not and r is the quantity

On account of (68a) we have

where M denotes the mass generating the field. It is easy to verify that this solution satisfies approximately the field-equation outside the mass M.

Let us now investigate the influences which the field of mass M will have upon the metrical properties of the field. Between the lengths and times measured locally on the one hand, and the differences in co-ordinates dx_ν on the other, we have the relation

For a unit measuring rod, for example, placed parallel to the x axis, we have to put

If the unit measuring rod lies on the x axis, the first of the equations (70) gives

From both these relations it follows as a first approximation that

The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational field, when we place it radially in the field.

Similarly we can get its co-ordinate-length in a tangential position, if we put for example

we then get

The gravitational field has no influence upon the length of the rod, when we put it tangentially in the field.

Thus Euclidean geometry does not hold in the gravitational field even in the first approximation, if we conceive that one and the same rod independent of its position and its orientation can serve as the measure of the same extension. But a glance at (70a) and (69) shows that the expected difference is much too small to be noticeable in the measurement of earth’s surface.

We would further investigate the rate of going of a unit-clock which is placed in a statical gravitational field. Here we have for a period of the clock

then we have

Therefore the clock goes slowly what it is placed in the neighbourhood of ponderable masses. It follows from this that the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the red end of the spectrum.

Let us further investigate the path of light-rays in a statical gravitational field. According to the special relativity theory, the velocity of light is given by the equation

thus also according to the generalised relativity theory it is given by the equation

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

and with it the velocity,

in the sense of the Euclidean Geometry. We can easily see that, with reference to the co-ordinate system, the rays of light must appear curved in case g_μν’s are not constants. If n be the direction perpendicular to the direction of propagation, we have, from Huygen’s principle, that light-rays (taken in the plane (γ, n)] must suffer a curvature ∂λ/∂n.

Let us find out the curvature which a light-ray suffers when it goes by a mass M at a distance Δ from it. If we use the co-ordinate system according to the above scheme, then the total bending B of light-rays (reckoned positive when it is concave to the origin) is given as a sufficient approximation by

where (73) and (70) gives

The calculation gives

A ray of light just grazing the sun would suffer a bending of 1·7″, whereas one coming by Jupiter would have a deviation of about ·02″.

If we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of a material particle of a relatively small (infinitesimal) mass we get a deviation of the following kind from the Kepler-Newtonian Laws of Planetary motion. The Ellipse of Planetary motion suffers a slow rotation in the direction of motion, of amount

In this Formula ‘a’ signifies the semi-major axis, c, the velocity of light, measured in the usual way, e, the eccentricity, τ, the time of revolution in seconds.

The calculation gives for the planet Mercury, a rotation of path of amount 43″ per century, corresponding sufficiently to what has been found by astronomers (Leverrier). They found a residual perihelion motion of this planet of the given magnitude which can not be explained by the perturbation of the other planets.