This is to explore these ideas.
Plotting geodesics in the space with metric: ds^{2} = (dx^{2} − dt^{2})/t^{2}
Coordinates are x^{i} = <t, x> (i →).
The space is of such points where t > 0.
∂_{k}g_{ij} =

 

 (i →; j ↓; k →→)

About the (i →; j ↓; k →→) Notation
T_{kji} =
 T_{ttt}
 T_{ttx}  T_{txt}
 T_{txx} 
 
 T_{xtt}
 T_{xtx}  T_{xxt}
 T_{xxx} 
 (i →; j ↓; k →→)

∂_{i}g_{jk} +
∂_{j}g_{ik} −
∂_{k}g_{ij} =

 

 (i →; j ↓; k →→)

g^{kn}(
∂_{i}g_{jn} +
∂_{j}g_{in} −
∂_{n}g_{ij} ) =

 

 (i →; j ↓; k →→)

Γ_{ij}^{k} =
1/2 g^{kn}(
∂_{i}g_{jn} +
∂_{j}g_{in} −
∂_{n}g_{ij} ) =

 

 (i →; j ↓; k →→) 
If <t(λ), x(λ)> are points along a geodesic curve expressed parametrically with parameter λ, then the differential equation for t and x is:
d^{2}x^{k}/dλ^{2} = −Γ_{ij}^{k}(dx^{i}/dλ)(dx^{j}/dλ)
d^{2}t/dλ^{2} =
((dt/dλ)^{2} + (dx/dλ)^{2})/t
d^{2}x/dλ^{2} =
2(dt/dλ) (dx/dλ)/t
I wish there were an easy check for these error prone calculations.
Those simple equations suggest a short Java program that plots these geodesics.
The plot is for negative y with y=0 at the top of the plot; both t and T = −log(−t) increase for higher points in the plot.
They look like hyperbolae, some with foci on the X axis, and others with foci at conjugate points about the x axis.
This would match the fact that with metric g_{ij} = δ_{ij}/y^{2} geodesics are semicircles with center on the X axis.
I don’t know how to prove either fact.
The family of geodesics plotted here all go thru a typical event in the 2D space.
A geodesic with foci on the Xaxis corresponds to a world line of a real particle.
A geodesic with foci at (x, y) and (x, −y) corresponds to a tachyon.
Null geodesics appear as 45° lines.
For we get
Γ_{ij}^{k} =

 

 (i →; j ↓; k →→) 
and
d^{2}t/dλ^{2} =
(−(dt/dλ)^{2} − (dx/dλ)^{2})/t
d^{2}x/dλ^{2} =
2(dt/dλ) (dx/dλ)/t
Making the relevant change to the java program (‘td*td+xd*xd’ → ‘td*tdxd*xd’) yields the familiar semicircles.
This forms a check of sorts for our calculations.
Sign discrepancy
The problem may be that t is positive in the code while t is negative in the math.
Why do geod.java and geodA.java (like geod on site) make different pictures since t has merely changed signs?