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.
| ∂kgij =
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| |
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| (i →; j ↓; k →→)
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About the (i →; j ↓; k →→) Notation
| Tkji =
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| |
|
| (i →; j ↓; k →→)
|
|
∂igjk +
∂jgik −
∂kgij =
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| |
|
| (i →; j ↓; k →→)
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| gkn(
∂igjn +
∂jgin −
∂ngij ) =
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| |
|
| (i →; j ↓; k →→)
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| Γijk =
1/2 gkn(
∂igjn +
∂jgin −
∂ngij ) =
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| |
|
| (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:
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 we get
| Γijk =
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| |
|
| (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?
