Line element to polar coordinates
$begingroup$
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 FerrándezÁlvaro Ferrández
64
$endgroup$
add a comment |
$begingroup$
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 FerrándezÁlvaro Ferrández
64
$endgroup$
add a comment |
$begingroup$
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 FerrándezÁlvaro Ferrández
64
$endgroup$
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 FerrándezÁlvaro Ferrández
64
asked Dec 2 '18 at 19:02
Álvaro FerrándezÁlvaro Ferrández
64
edited Dec 2 '18 at 19:44
Álvaro Ferrández
asked Dec 2 '18 at 19:02
Álvaro FerrándezÁlvaro Ferrández
64
asked Dec 2 '18 at 19:02
Álvaro FerrándezÁlvaro Ferrández
64
asked Dec 2 '18 at 19:02
Álvaro FerrándezÁlvaro Ferrández
64
64
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023049%2fline-element-to-polar-coordinates%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023049%2fline-element-to-polar-coordinates%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
