Page:The principle of relativity (1920).djvu/236

 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 [part]λ/[part]n.

Χ_{2}

Light-ray

Χ_{1} Δ

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 = [integral]_{-[infinity]}^{[infinity]} [part]γ/[part][x]_{1} dx_{2}

where (73) and (70) gives

γ = [sqrt](-g_{4 4}/g_{2 2}) = 1 - α/2r (1 + x_{2}^2/r^2).

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´´.