In order to understand how objects move in Schwarzschild spacetime, we therefore require the geodesic equations defined by the Schwarzschild metric.
We have shown that those equations are in the form of parameterised curves 
Replacing α by the four variables t, r, φ and θ gives us four complicated-looking differential Schwarzschild equations
For α = t, looking at the values of Γtμν from our previous article, we see that only Γttr and Γtrt are not null, leading to
Looking at α = r, we know that the four diagonal elements of the matrix Γrμν are not null
so that we can write
If we replace now α by θ, and reminding us the Γθμν matrix, we can write
Finally, we retrieve the last geodesic equation by replacing α by Φ
Simplify the equations by choosing Θ = Π/2
We can be clever and try to describe the motion of a free falling particle of mass m in the constant plane θ=π/2 (because the field is spherically symmetric, we can understand all the orbits by studying only these equatorial oribts, and there is no loss of generality), so we are left with only three equations (the third equation reduces to 0 + 0 = 0...)
Let's consider the first equation and start by multiplying each member by A to finally find out that this expression reduces to the derivative of d(A(dt/dλ))/dλ=0 so that A(dt/dλ) =cste
If we consider the equation with Φ, we realize equivalently that it can be expressed as a derivative:
We are now left with the third equation with r, which should be equivalent to a derivative as well:
which has the advantage to be dependant on r variable only
As before, we have to find out the expression from which this equation derives.
For that, we recall the spacetime interval expression
At this point, we have to consider two cases.
If we consider a massive particule, we know that the proper time τ between two distinct events is defined and in terms of the spacetime line element is given by
ds2 = c2 x dτ2
which is the time measured by a stationary clock at the same position as the two events.
and replacing again dt/dτ and dφ/dτ by their respective values
Deriving this equation with respect to proper times gives
which is exactly the third equation with respect to r that we have deduced aboce.
In case of a photon (without mass), we have ds2 = 0, so that finally our final equation can be written as
We can interpret this expression as being strictly equivalent to the conservation of energy, with the radial cinetic energy per mass unit equals to Ec = 1/2(dr/dλ)2, E the total energy per mass unity and a potential energy per mass unity V(r) in the form of:
Comparison with Newtonian mechanics
We can compare this expression with what can be found in newtonian mechanics for a radial potential energy per mass, which only depends on r:
We notice that the two expressions are almost equivalent, except for the presence of the extra term in (1/h3) in the context of the Schwarzschild equation.
Newtonian approximation in case of radial free fall
Radial free fall implies the object is moving 'straight down', ie Φ is constant, therefore dΦ/dλ = 0 and so h = r2(dΦ/dλ) = 0
And the last equation thus reduces to
We can differentiate this equation with respect to τ, giving
which should remind you something very familiar in Newtonian mechanics ;-)
 To be more precise, time-like geodesics (where ds2 > 0 and proper time dτ ≠ 0) describe the paths of massive objects and can use proper time as a affine parameter. Null geodesics (where ds2 = 0 and proper time dτ = 0) describe the paths of (massless) photons and need another parameter. In the rest of the article, we use λ for either.