Wald: Ch. 2, Problem 8.b

[Rasalhague]

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?

 

[PAllen]

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.

 

[Rasalhague]

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.

 

[Bill_K]

Several typos in your transformation as given. (x² + y²) should have an exponent + 1/2. Also in the sin, cos argument you want to change the sign of just t, not both φ and t. In other words, you want the (x,y) coords to rotate in the opposite (time-reversed) sense wrt (x',y'). Since cos is an even function and sin is odd, what you have written amounts to a reflection y' = - y.

 

[Rasalhague]

Thanks, Bill. I never spotted the minus sign in the exponent in the LaTeX till you pointed it out; that was just a typo, and didn't enter my calculations. I did evenually realize my mistake with inserting a minus sign before the phi. Also, the argument of sine and cosine in the inverse transformation should be phi'+omega*t', where phi' = arctan(y'/x'), shouldn't it?