Line element to polar coordinates
I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
begin{equation}
mathbf{v}=frac{A}{r} hat{r} + frac{B}{r}hat{theta}
end{equation}
So:
begin{equation}
mathbf{v}=boldsymbol{nabla} psi longrightarrow psi= A ~log r + B~theta
end{equation}
And I have the line element in cartesian coordinates $(t,x^1,x^2,x^3)=(t,x,y,z)$:
begin{equation}
ds^2 = dfrac{rho_0}{c_s} left[ - left( c_s^2-v_0^2right) dt^2 - v_0^i dt dx^i - v_0^j dt dx^j + delta_{ij} dx^i dx^j right]
end{equation}
I need to obtain the following line element, effective metric acoustic $(t,r,theta)$:
begin{equation}
ds^2 = - left( c_s^2-frac{A^2+B^2}{r^2}right) dt^2 +dr^2 - 2frac{A}{r}dtdr + r^2dtheta-2Bdtdtheta
end{equation}
Without $z$ because vortex is axially symmetric. I don't know how can I do it. I would appreciate some help to get started, what do I do with the terms with $i$.
cylindrical-coordinates
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
add a comment |
I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
begin{equation}
mathbf{v}=frac{A}{r} hat{r} + frac{B}{r}hat{theta}
end{equation}
So:
begin{equation}
mathbf{v}=boldsymbol{nabla} psi longrightarrow psi= A ~log r + B~theta
end{equation}
And I have the line element in cartesian coordinates $(t,x^1,x^2,x^3)=(t,x,y,z)$:
begin{equation}
ds^2 = dfrac{rho_0}{c_s} left[ - left( c_s^2-v_0^2right) dt^2 - v_0^i dt dx^i - v_0^j dt dx^j + delta_{ij} dx^i dx^j right]
end{equation}
I need to obtain the following line element, effective metric acoustic $(t,r,theta)$:
begin{equation}
ds^2 = - left( c_s^2-frac{A^2+B^2}{r^2}right) dt^2 +dr^2 - 2frac{A}{r}dtdr + r^2dtheta-2Bdtdtheta
end{equation}
Without $z$ because vortex is axially symmetric. I don't know how can I do it. I would appreciate some help to get started, what do I do with the terms with $i$.
cylindrical-coordinates
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
add a comment |
I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
begin{equation}
mathbf{v}=frac{A}{r} hat{r} + frac{B}{r}hat{theta}
end{equation}
So:
begin{equation}
mathbf{v}=boldsymbol{nabla} psi longrightarrow psi= A ~log r + B~theta
end{equation}
And I have the line element in cartesian coordinates $(t,x^1,x^2,x^3)=(t,x,y,z)$:
begin{equation}
ds^2 = dfrac{rho_0}{c_s} left[ - left( c_s^2-v_0^2right) dt^2 - v_0^i dt dx^i - v_0^j dt dx^j + delta_{ij} dx^i dx^j right]
end{equation}
I need to obtain the following line element, effective metric acoustic $(t,r,theta)$:
begin{equation}
ds^2 = - left( c_s^2-frac{A^2+B^2}{r^2}right) dt^2 +dr^2 - 2frac{A}{r}dtdr + r^2dtheta-2Bdtdtheta
end{equation}
Without $z$ because vortex is axially symmetric. I don't know how can I do it. I would appreciate some help to get started, what do I do with the terms with $i$.
cylindrical-coordinates
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
begin{equation}
mathbf{v}=frac{A}{r} hat{r} + frac{B}{r}hat{theta}
end{equation}
So:
begin{equation}
mathbf{v}=boldsymbol{nabla} psi longrightarrow psi= A ~log r + B~theta
end{equation}
And I have the line element in cartesian coordinates $(t,x^1,x^2,x^3)=(t,x,y,z)$:
begin{equation}
ds^2 = dfrac{rho_0}{c_s} left[ - left( c_s^2-v_0^2right) dt^2 - v_0^i dt dx^i - v_0^j dt dx^j + delta_{ij} dx^i dx^j right]
end{equation}
I need to obtain the following line element, effective metric acoustic $(t,r,theta)$:
begin{equation}
ds^2 = - left( c_s^2-frac{A^2+B^2}{r^2}right) dt^2 +dr^2 - 2frac{A}{r}dtdr + r^2dtheta-2Bdtdtheta
end{equation}
Without $z$ because vortex is axially symmetric. I don't know how can I do it. I would appreciate some help to get started, what do I do with the terms with $i$.
cylindrical-coordinates
cylindrical-coordinates
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
edited Dec 2 '18 at 19:44
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
asked Dec 2 '18 at 19:02
Álvaro Ferrández
64
64
add a comment |
add a comment |
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