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 →→)
|
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 ) =
|
| |
|
| (i →; j ↓; k →→)
|
| Γijk =
1/2 gkn(
∂igjn +
∂jgin –
∂ngij ) =
|
| |
|
| (i →; j ↓; k →→) |
If <t(λ), x(λ)> are points along a geodesic curve, where λ is the parameter 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 look like hyperbolae with centers on 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.
For we get
| Γijk =
|
| |
|
| (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 yields the familiar semi-circles.
This forms a check of sorts for our calculations.
