Schwarzschild’s Christoffel Symbols

Begin by defining the Christoffel symbols through the
parallel transport of a contravariant vector A^{i} along a coordinate
line x^{j}. Parallel transport does not change the magnitude or the
direction of the vector, so relative to a non-Cartesian coordinate frame the
vector changes in accordance with

(Eq’n 1)

The upper-case gamma represents the Christoffel symbol and the delta indicates a four-vector derivative.

Analysis of the algebraic expression describing a geodesic path, the path of shortest invariant distance between two points, gives us the means to calculate the Christoffel symbols associated with the coordinate frame to which we have referred the path. Written in terms of the metric tensor associated with the coordinate frame, the calculation is

(Eq’n 2)

In that equation we have ∂_{i}= ∂/∂x_{i}.

The semi-classical Schwarzschild metric

Presented in 1916 by Karl Schwarzschild, this solution comes from the metric equation

(Eq’n 3)

From that equation we extract the covariant metric tensor

(Eq’n 4)

That tensor has the determinant

(Eq’n 5)

which enables us to calculate the contravariant metric tensor by dividing the cofactor of each element by the determinant

(Eq’n 6)

Now we can calculate the Christoffel symbols in accordance with Equation 2.
To show more explicitly how it’s done, let’s look at the calculation of the
element Γ^{1}_{44};

(Eq’n 7)

In that equation I have defined for convenience
κ=(1-2MG/rc^{2}).
Also note that the Christoffel symbols are symmetric with respect to an
interchange of the indices i and j; that is, if we represent the Christoffel
symbols as a 4x4x4 cube, the elements are symmetrical about the plane that runs
from upper left to lower right. So now we have the Christoffel symbols of the
semi-classical Schwarzschild solution in four matrices as

(Eq’ns 8)

Thus we have the Christoffel symbols for the semi-classical Schwarzschild solution.

The fully-relativistic Schwarzschild metric

The fully-relativistic version of Schwarzschild’s solution of Einstein’s equation has the metric equation

(Eq’n 9)

For convenience in that equation I have defined
μ=(1+MG/rc^{2}).
From that equation we extract the covariant metric tensor

(Eq’n 10)

That tensor has the determinant

(Eq’n 11)

which enables us to calculate the contravariant metric tensor

(Eq’n 12)

Now we can calculate the Christoffel symbols in accordance with Equation 2:

(Eq’ns 13)

Thus we have the Christoffel symbols that correspond to both the semi-classical and the fully-relativistic Schwarzschild solutions. Now we want to look at one application of those symbols.

The Geodesic Path

The primary application of the Christoffel symbols in General Relativity is the calculation of geodesic paths through warped spacetime. For this essay it will be sufficient to calculate the acceleration imposed on small bodies by the fully-relativistic Schwarzschild metric. In this case we have two accelerations to consider – the four-acceleration (which we infer) and the coordinate acceleration (which we observe).

We calculate four-acceleration by dividing a differential
change in four-velocity by an interval of proper time, noting that the change in
the four-vector equals the observed change minus the change due to parallel
transport. Making the substitution A^{i}=dβ^{i}
in Equation 1 thus gives us

(Eq’n 14)

If we have a body in free fall in a warped spacetime (that is, the body follows a geodesic path), then the four-acceleration must equal zero, so we have the geodesic equation

(Eq’n 15)

Now we want to convert that equation into one that expresses the coordinate
acceleration in terms of the coordinate velocity v^{i}=dx^{i}/dt.
Because proper time relates to coordinate time through the Lorentz factor (dt=γdτ;
γ^{2}=1/(1-β^{2}))
and the four-velocity relates to the coordinate velocity as **
β**=γ(**v**/c,1),
we have Equation 15 as

(Eq’n 16)

Differentiating the Lorentz factor is unnecessary in this case; we need only
look at the component of that equation for which k=4. Because v^{4} is a
constant we get an equation that gives us a simple substitution for the product
of the Lorentz factor and its time derivative. Making that substitution,
rearranging a bit, and dividing out the square of the Lorentz factor and the
speed of light transforms Equation 15 into

(Eq’n 17)

In that equation the index k takes only the values 1, 2, 3 and the other indices take all four values.

Now we take our coordinate velocity as v^{i}=(v_{r},
ω_{θ},
ω_{ϕ},
c), in which v represents a displacement velocity and omega represents an
angular velocity. Theta represents colatitude and phi represents longitude in
our spherical coordinate system. Using that velocity with Equations 13 in
Equation 17 gives us three equations representing the coordinate acceleration in
3-space;

(Eq’ns 18)

In the first of those equations the second and third terms represent the centrifugal force acting on the small body. The first term represents a kind of antigravity, though the cee-squared term in the denominator makes it too small to notice. And the fourth term represents the gravitational attraction associated with the warped spacetime.

In the second and third equations the first terms represent the Coriolis force acting on the small body and the second terms appear to be related to the precession of rotating bodies.

The last terms in all three equations represent a component of acceleration due to changes in the gravitational time dilation, the source of the gravitational redshift, in the system. As the small body moves in the radial direction, time dilation increases or decreases and thereby augments or decrements all three components of the body’s motion.

And there we have one straightforward aspect of the theory of General Relativity.

eabf