## Wald: Ch. 2, Problem 8.b

Find the coefficients in coordinate bases, of the metric tensors for Minkowski space, for "rotating coordinates" defined by

$$t' = t,$$

$$x' = (x^2+y^2)^{-1/2} \cos(\phi - \omega t),$$

$$y' = (x^2+y^2)^{-1/2} \sin(\phi - \omega t),$$

$$z' = z.$$

I think the inverse coordinate transformation should be

$$t = t',$$

$$x = (x'^2+y'^2)^{-1/2} \cos(-\phi + \omega t),$$

$$y = (x^2+y^2)^{-1/2} \sin(-\phi + \omega t),$$

$$z = z'.$$

(EDIT: Insert prime symbols on x and y inside the brackets in the 3rd line of the inverse transformation.)

In Mathematica, I calculated the Jacobian matrix of this inverse transformation, using doubled letters for primed ones:

In[1]:= q = {tt, Sqrt[xx^2 + yy^2]*Cos[-phi + omega*tt],
Sqrt[xx^2 + yy^2]*Sin[-phi + omega*tt], zz}; J =
D[q, {{tt, xx, yy, zz}}]

Out[1]:= {{1, 0, 0, 0}, {omega Sqrt[xx^2 + yy^2] Sin[phi - omega tt], (
xx Cos[phi - omega tt])/Sqrt[xx^2 + yy^2], (yy Cos[phi - omega tt])/
Sqrt[xx^2 + yy^2],
0}, {omega Sqrt[xx^2 + yy^2] Cos[phi - omega tt], -((
xx Sin[phi - omega tt])/Sqrt[xx^2 + yy^2]), -((
yy Sin[phi - omega tt])/Sqrt[xx^2 + yy^2]), 0}, {0, 0, 0, 1}}

Then I calculated the new coefficients of the metric tensors thus:

In[2]:= g = DiagonalMatrix[{-1, 1, 1, 1}]; gg =
Transpose[J].g.J

Out[2]: {{-1 + omega^2 (xx^2 + yy^2) Cos[phi - omega tt]^2 +
omega^2 (xx^2 + yy^2) Sin[phi - omega tt]^2, 0, 0,
0}, {0, (xx^2 Cos[phi - omega tt]^2)/(xx^2 + yy^2) + (
xx^2 Sin[phi - omega tt]^2)/(xx^2 + yy^2), (
xx yy Cos[phi - omega tt]^2)/(xx^2 + yy^2) + (
xx yy Sin[phi - omega tt]^2)/(xx^2 + yy^2),
0}, {0, (xx yy Cos[phi - omega tt]^2)/(xx^2 + yy^2) + (
xx yy Sin[phi - omega tt]^2)/(xx^2 + yy^2), (
yy^2 Cos[phi - omega tt]^2)/(xx^2 + yy^2) + (
yy^2 Sin[phi - omega tt]^2)/(xx^2 + yy^2), 0}, {0, 0, 0, 1}}

The result is symmetric, but both the Jacobian matrix and the new coefficient matrix of the metric tensor field have determinant zero. I guess this means I'm doing something wrong, since the determinant of the latter matrix is used to measure spacetime volumes, but volume wouldn't be well defined if a particular volume could be zero when measured in one chart, and nonzero in another. Any suggestions?
 PhysOrg.com science news on PhysOrg.com >> Ants and carnivorous plants conspire for mutualistic feeding>> Forecast for Titan: Wild weather could be ahead>> Researchers stitch defects into the world's thinnest semiconductor
 Blog Entries: 1 Recognitions: Gold Member Science Advisor It took me a moment to notice this issue, but there is something fishy about the transform. For given z and t, any x, y on the circle (x^2+Y^2)=k produce the same x', y' values. Thus, a circle in (x,y,z,t) gets mapped to a point in (x',y',z',t'). That's not a valid coordinate transform.
 Ah, thanks, I see what's amiss now. I omitted his final condition: tan(phi) = y/x. Setting phi = ArcTan[y/x] gives a Jacobian matrix with determinant 1, and a metric matrix with determinant -1, as expected.

Blog Entries: 1
Recognitions: