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 = _g₄₄__dx₄²_
_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
B = ∫_{-∞}^∞ ∂γ/∂[_x_]₁ _dx₂_
where (73) and (70) gives
γ = √(-_g₄₄_/_g₂₂_) = 1 - α/2_r_ (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″.
Public-domain text, read in full here on John Shaqi.
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