The equation for geodesic line is introduced by means of "Ricci" package for tensor analysis in Mathematica

<<Ricci`

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    Copyright 1992 - 2000 John M. Lee
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In the beginning we make a basic definition of the 4-dimensional bundle (Gravity) and metric g:

DefineBundle[Gravity,4,g,{i,j,k,l}, MetricType->Riemannian]

Index i associated with Gravity
Index j associated with Gravity
Index k associated with Gravity
Index l associated with Gravity
Tensor g defined.
  Rank = 2  Symmetries = Symmetric
  Type = {Real}  Bundle = Gravity  Variance = Covariant
Tensor Rm defined.
  Rank = 4  Symmetries = RiemannSymmetries
  Type = {Real}  Bundle = Gravity  Variance = Covariant
Tensor Rc defined.
  Rank = 2  Symmetries = Symmetric
  Type = {Real}  Bundle = Gravity  Variance = Covariant
Tensor Sc defined.
  Rank = 0  Symmetries = NoSymmetries
  Type = {Real}  Bundle = Gravity  Variance = Covariant
Bundle Gravity defined.
   Metric = g   Dimension = 4   Indices = {i, j, k, l}
   Bundle Type = Real   Metric Type = Riemannian
   Tangent Bundle = {Gravity}
   Connection is torsion free.

Gravity

Definition of vectors u, n, x and scalar l will be used, too:

DefineTensor[u, 1, Variance->Contravariant]
DefineTensor[n, 1, Variance->Contravariant]
DefineTensor[x, 1, Variance->Contravariant]
DefineTensor[λ, 0]

Tensor u defined.
  Rank = 1  Symmetries = NoSymmetries
  Type = {Real}  Bundle = {Gravity}  Variance = Contravariant
Tensor n defined.
  Rank = 1  Symmetries = NoSymmetries
  Type = {Real}  Bundle = {Gravity}  Variance = Contravariant
Tensor x defined.
  Rank = 1  Symmetries = NoSymmetries
  Type = {Real}  Bundle = {Gravity}  Variance = Contravariant
Tensor λ defined.
  Rank = 0  Symmetries = NoSymmetries
  Type = {Real}  Bundle = {Gravity}  Variance = Covariant

l;

The trajectory of the point P corresponding to the parallel translation along the tangent vector u is the geodesic line. The derivative, which is resulted from the changes of tensor field due to infinitesimal parallel translation, is the covariant derivative. If

(l is the parameter on trajectory, i. e. affine parameter) is the tangent vector, then the following equation e1 is equal to zero by definition of the geodesic line:

e1 = Del[u, u]

Del_u[u]

The infinitesimal difference between neighboring geodesic lines is (

e2 = Del[n, e1]

Del_n[Del_u[u]]

In the arbitrary basis this can be overwritten as

e3 = BasisExpand[e2]

n^l2u^l1_{;l3 l2}u^l3Basis_l1+ n^l2u^l1_;l3u^l3_;l2Basis _l1

e4 = TensorSimplify[e3]

n^juⁱ_{;k j}u^kBasis_i+ n^juⁱ_;ku^k_;jBasis _i

Let exchange the covariant n and u derivatives , that results in

e5 = CommuteCovD[e4,L[k], L[j]]

n^juⁱ_;ku^k_;jBasis _i+ n^ju^k(uⁱ_{;j k}-Rmⁱ_{l4 j k} u^l4)Basis_i

e6 = TensorSimplify[e5]

n^juⁱ_{;j k}u^kBasis_i+ n^juⁱ_;ku^k_;jBasis _i- n^jRmⁱ_{k j l}u^ku^lBasis_i

But this expression is

Del[u,Del[n, u]]+Rm == 0

Del_u[Del_n[u]]+Rm==0

and to be equal to zero by definition. The appearance of the Riemann's tensor is the result of the noncommutative character of the operator

Now we are to take into account the symmetry of the covariant derivative:

that causes the zero value of

Del[u,Del[u, n]]+Rm == 0

Del_u[Del_u[n]]+Rm==0

Now let suppose, that g is the linear correction for Lorenzian metrics. We will consider the slowly moving particles (v<<c). Then the affine parameter l=t --> t (t is the local time, t is the global time) and the equation of the geodesic line is

Del[u,u]
TensorSimplify[BasisExpand[%]]==0

uⁱ_;ju^jBasis_i==0

But u-derivative can be rewritten as

u [U[i]] [L[j]]
TensorSimplify[CovDExpand[%]]

Del_j[uⁱ]+Connⁱ_{k j}u^k

As the first term is gradient of u, we have for equation of the geodesic line

Grad[u[U[i]]] u [U[j]]+Conn [L[k], U[i], L[j]] u [U[k]] u[U[j]]==0

u^jGrad[uⁱ]+Connⁱ_{k j}u^ju^k==0

Now if coordinates of P are x^j consequently

But

consequently

as result we have the definition of the geodesic line through points coordinates:

Connⁱ_{k j}(x^j)'[l](x^k)'[l]+ (xⁱ)''[l]==0

or in natural mathematical form:

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