The Schwarzschild Solution 

As might be expected, the calculation of the Schwarzschild metric is a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!

In Schwarzschild Coordinates, (tr, θ, φ), with an origin at the centre of an isolated spherically symmetric, non-rotating, gravitating body, such as a planet or a star, ignoring the gravity of other stellar objects, the metric is
GravitationSub-2
Put k = eλ and κ = eμ which will simplify calculation,
GravitationSub-5
Using GravitationSub-6
GravitationSub-7
Using prime to denote differentiation with respect to r = x1, the non-vanishing partial derivatives of the metric are
GravitationSub-8
GravitationSub-9
GravitationSub-10
GravitationSub-11
GravitationSub-12
The non-vanishing Christoffel symbols,
GravitationSub-13
are
GravitationSub-14GravitationSub-15
GravitationSub-16
GravitationSub-17 GravitationSub-18
GravitationSub-19GravitationSub-20
GravitationSub-21GravitationSub-22

On raising the first index,
GravitationSub-23GravitationSub-24
GravitationSub-25
GravitationSub-26GravitationSub-27
GravitationSub-28GravitationSub-29
GravitationSub-30GravitationSub-31

The Ricci tensor is
GravitationSub-33
If one calculates all the terms, one finds a diagonal matrix. We have
GravitationSub-34
Calculate the terms individually
GravitationSub-35
GravitationSub-36
GravitationSub-37
GravitationSub-38
GravitationSub-39
Substitute into R00 = 0 as required by Einstein’s equation,
GravitationSub-40
GravitationSub-41

Also
GravitationSub-42
Calculate the terms individually
GravitationSub-43
GravitationSub-44
GravitationSub-45
GravitationSub-46
GravitationSub-47
GravitationSub-48
Substitute into R11 = 0 as required by Einstein’s equation,
GravitationSub-49
GravitationSub-50

We also have
GravitationSub-51
Calculate the terms individually
GravitationSub-52
GravitationSub-53
GravitationSub-54
GravitationSub-55
GravitationSub-56
GravitationSub-57
GravitationSub-58
Substitute into R22 = 0,
GravitationSub-59
GravitationSub-60

The other angular component, R33, contains the same information as R22 and does not require calculation. Using the equations for R00 and R11
GravitationSub-61
The integration constant determines units and is set to zero by choosing unit light speed. Thus
μ = −λ and κ = k−1.
So, the equation for R22 is
GravitationSub-63
GravitationSub-64
GravitationSub-65
where C is a constant of integration. Thus,
GravitationSub-66
In the weak field limit, Newton’s law of gravity states,
GravitationSub-67
So, the constant of integration is C = −2GM. Thus the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass M is
GravitationSub-68
This is the Schwarzschild metric. Return

The Newtonian Orbit

In a Newtonian potential we have
GravitationSub-69
GravitationSub-70
So,
GravitationSub-71
This can be integrated with respect to r.
GravitationSub-72
GravitationSub-73
Substituting GravitationSub-74,
GravitationSub-75
Let u = r−1.
GravitationSub-77
This can be integrated by writing it as a first order linear equation, separating the variables, and making a substitution. However, it is easier to differentiate with respect to u. Let p = h2 ⁄ μ. Then,
GravitationSub-79
This has general solution
GravitationSub-80
where e and f are constants of integration. Thus,
GravitationSub-81
If the axis is chosen with r a minimum for φ = 0, the orbit is recognised as an ellipse with eccentricity e and semi-latus rectum p,
GravitationSub-82
Return