_dx₄_ = 1/√(_g_₄₄) = 1/√(1 + (_g_₄₄ - 1)) = 1 - (_g_₄₄ - 1)/2
or _dx₄_ = 1 + _k_/8π ∫ ρ_d_τ/_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