Part 2 — The Foundation of the Generalised Theory of Relativity By A. Einstein. From Annalen der Physik 4.49.1916. (continued 2)

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 = g_μν dx_μ dx_ν = 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 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

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.