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 gμν’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 gμν’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, gμν’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
dx₁/ds, dx₂/ds, dx₃/ds,
can take any values. This signifies that any velocity
v = √((dx₁/dx₄)² + (dx₂/dx₄)² + (dx₃/dx₄)²)
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
dx₁/ds, dx₂/ds, dx₃/ds,
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 γμντ’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:—
d²xτ/dt² = Γ₄₄τ, where ds = dx₄ = dt,
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
(67) d²xτ/dt² = -½ ∂g₄₄/∂xτ (τ, = 1, 2, 3)
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
τ₄₄ = ρ = τ.
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
(68) ▽² g₄₄ = κρ.
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.
(68a.) -κ/(8π) ∫ ρdτ/r
whereas the Newtonian theory for the chosen unit of time gives
-K/c² ∫ρdτ/r,
where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating them we get
(69) κ = 8πK/c² = 1.87 x 10⁻²⁷.
§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 gμν 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 gμν 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:—
{ gρσ = -δρσ - α(xρ xσ)/r³ (ρ and σ 1, 2, 3)
{
(70) { gρ4 = g4ρ = 0 (ρ 1, 2, 3)
{
{ g₄₄ = 1 - α/r.
δρσ is 1 or 0, according as ρ = σ or not and r is the quantity
+√(x₁² + x₂² + x₃²).
On account of (68a) we have
(70a) α = κM/4π
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 dxν on the other, we have the relation
ds² = gμν dxμ dxν.
For a unit measuring rod, for example, placed parallel to the x axis, we have to put
ds² = -1, dx₂ = dx₃ = dx₄ = 0
then -1 = g₁₁dx₁².
If the unit measuring rod lies on the x axis, the first of the equations (70) gives
g₁₁ = -(1 + α/r).
From both these relations it follows as a first approximation that
(71) dx = 1 - α/2r.
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
ds² = -1, dx₁ = dx₃ = dx₄ = 0, x₁ = r, x₂ = x₃ = 0
we then get
(71a) -1 = g₂₂ dx₂² = -dx₂².
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
ds = 1, dx₁ = dx₂ dx₃ = 0;
then we have
1 = g₄₄dx₄²
dx₄ = 1/√(g₄₄) = 1/√(1 + (g₄₄ - 1)) = 1 - (g₄₄ - 1)/2
or dx₄ = 1 + k/8π ∫ ρdτ/r.
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
-dx₁² - dx₂² - dx₃² + dx₄² = 0;
thus also according to the generalised relativity theory it is given by the equation
(73) ds² = gμν dxμ dxν = 0.
If the direction, i.e., the ratio dx₁ : dx₂ : dx₃ is given, the equation (73) gives the magnitudes
dx₁/dx₄, dx₂/dx₄, dx₃/dx₄,
and with it the velocity,
√((dx₁/dx₄)² + (dx₂/dx₄)² + (dx₃/dx₄)²) = γ,
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 gμν’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
B = ∫-∞∞ ∂γ/∂[x]₁ dx₂
where (73) and (70) gives
γ = √(-g₄₄/g₂₂) = 1 - α/2r (1 + x₂²/r²).
The calculation gives
B = 2α/Δ = KM/2πΔ.
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
(75) s = 24π³a²/τ²c²(1 - e²) per revolution.
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.