Judgement Day (20 page)

Now

Since

And

Then

Since

Therefore

(15)

This value is
analogous to acceleration, however since we know that light neither
accelerates nor decelerates due to gravity but can only have its
path bent, this value tells us how much the path of light is bent
due to gravity. The value of this function is in the S.I. units of
m
^-1
or (radians.m
^-1
) with respect to r.
Notice also that this equation is equivalent to

Now ‘r’ is the
distance from the centre of mass ‘M’, ‘s’ is the distance along the
light path, ‘d’ is the perpendicular distance from the centre of
mass to the light path and ‘ϴ’ is the angle from the origin at M.
We need to change equation (15) to be in terms of ‘s’ rather than
‘r’ and integrate over ‘s’ for the entire path length. The angle of
deflection of the light path will be given the symbol ‘α’.

(16)

First let’s explore
the scenario where (Δt
_r
/Δt
_∞
)
²
is
close to enough to 1 as to be insignificant. In this case, close to
earth, the equation becomes

Which becomes

Which upon integration
gives the solution

(17)

Which is identical to
Einstein’s original equation from 1911 for the gravitational
deflection of light
⁹
, however Einstein later updated this
equation in 1915 after realising that this equation has assumed the
value for ‘d’ to be constant. Einstein’s final equation had
increased the value for the deflection angle by a factor of 2.

Just quickly,
integrating the full equation yields the rather complex