- #1
Rasalhague
- 1,387
- 2
Find the coefficients in coordinate bases, of the metric tensors for Minkowski space, for "rotating coordinates" defined by
[tex]t' = t,[/tex]
[tex]x' = (x^2+y^2)^{-1/2} \cos(\phi - \omega t),[/tex]
[tex]y' = (x^2+y^2)^{-1/2} \sin(\phi - \omega t),[/tex]
[tex]z' = z.[/tex]
I think the inverse coordinate transformation should be
[tex]t = t',[/tex]
[tex]x = (x'^2+y'^2)^{-1/2} \cos(-\phi + \omega t),[/tex]
[tex]y = (x^2+y^2)^{-1/2} \sin(-\phi + \omega t),[/tex]
[tex]z = z'.[/tex]
(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?
[tex]t' = t,[/tex]
[tex]x' = (x^2+y^2)^{-1/2} \cos(\phi - \omega t),[/tex]
[tex]y' = (x^2+y^2)^{-1/2} \sin(\phi - \omega t),[/tex]
[tex]z' = z.[/tex]
I think the inverse coordinate transformation should be
[tex]t = t',[/tex]
[tex]x = (x'^2+y'^2)^{-1/2} \cos(-\phi + \omega t),[/tex]
[tex]y = (x^2+y^2)^{-1/2} \sin(-\phi + \omega t),[/tex]
[tex]z = z'.[/tex]
(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?