This is to explore these ideas.

### Plotting geodesics in the space with metric: ds2 = (dx2 − dt2)/t2

Coordinates are xi = <t, x> (i →). The space is of such points where t > 0.
gij =
 −1/t2 0 0 1/t2
(i →; j ↓)
gij =
 −t2 0 0 t2
(i →; j ↓)
kgij =
 2/t3 0 0 −2/t3
|
 0 0 0 0
(i →; j ↓; k →→)

### About the (i →; j ↓; k →→) Notation

Tkji =
 Tttt Tttx Ttxt Ttxx
|
 Txtt Txtx Txxt Txxx
(i →; j ↓; k →→)

### Christoffel symbols

igjk + ∂jgik − ∂kgij =
 2/t3 0 0 2/t3
|
 0 −2/t3 −2/t3 0
(i →; j ↓; k →→)
gkn( ∂igjn + ∂jgin − ∂ngij ) =
 −2/t 0 0 −2/t
|
 0 −2/t −2/t 0
(i →; j ↓; k →→)
Γijk = 1/2 gkn( ∂igjn + ∂jgin − ∂ngij ) =
 −1/t 0 0 −1/t
|
 0 −1/t −1/t 0
(i →; j ↓; k →→)

### The Geodesic Equation

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:
d2xk/dλ2 = −Γijk(dxi/dλ)(dxj/dλ)

d2t/dλ2 = ((dt/dλ)2 + (dx/dλ)2)/t
d2x/dλ2 = 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 gij = δij/y2 geodesics are semi-circles 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 X-axis 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
gij =
 1/t2 0 0 1/t2
we get
Γijk =
 −1/t 0 0 −1/t
|
 0 −1/t −1/t 0
(i →; j ↓; k →→)
and
d2t/dλ2 = (−(dt/dλ)2 − (dx/dλ)2)/t
d2x/dλ2 = 2(dt/dλ) (dx/dλ)/t

Making the relevant change to the java program (‘td*td+xd*xd’ → ‘td*td-xd*xd’) yields the familiar semi-circles. 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?