What Are the Values of \(a\) and \(b\) in the Schwarzschild 4-Velocity?

[Omikron123]

Homework Statement

I have an object orbiting in free fall with constant radius $r$ in the plane $\theta = \frac{\pi}{2}$.
I am supposed to prove that the 4-velocity $U = a\partial _t + b\partial _\phi$ and find the values of $a$ & $b$ for a free falling object in the plane $\theta = \pi/2$

Relevant Equations

The Schwarzchild line element is given by $$ ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^-1dr^2 - r^2d\Omega ^2$$

So the line element is given by $$ ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^{-1}dr^2 - r^2d\Omega ^2$$
The object is orbiting at constant radius $r$ in the plane $\theta = \frac{\pi}{2}$. I am supposed to find the values of $a$ and $b$ in the 4-velocity given by: $$U = a\partial _t + b\partial _\phi$$.
Im pretty new the general relativity and Schwarzschild geodesics but here is my attempted solution:

For a massive particle the squared 4-velocity $U^2 = -1$, space-like which i can expand with the Schwarzschild metric, which is diagonal:
$$ U^2 = U^tU_t + U^\phi U_\phi = (g^{ta}U_a)U_t + (g^{\phi a}U_\phi) = -1$$ Since the metric is diagonal only $g^{tt}, g^{\phi \phi}$ are non-zero with values $$g^{\phi \phi} = \frac{1}{g_{\phi \phi}} = -\frac{1}{r^2}, g^{tt} = ... = \frac{1}{1-\frac{R_s}{r}}$$ At this point I am not sure how to continue, because I am not sure if $U_\phi = \partial _\phi$ etc. One idea was to compare the following:
$$U^2 = g^{tt}U_tU_t + g^{\phi \phi}U_\phi U_\phi = -\frac{1}{r^2}\partial _t^2 + \frac{1}{1-\frac{R_s}{r}}\partial _\phi ^2$$ and $$U^2 = (a\partial _t + b\partial _\phi)^2 = a^2\partial _t ^2 + b^2\partial _\phi^2$$ give $a = \sqrt{\frac{1}{1-\frac{R_s}{r}}}, b = \sqrt{-\frac{1}{r^2}}$ (since $2ab\partial _t\partial _\phi = 0$ due to diagonal metric??) As I said I don't really know what I am doing here so there might be some major errors in my thinking..

 

[PeroK]

Posting advanced homework at this level without Latex is problematic:

https://www.physicsforums.com/help/latexhelp/

Also, you need to show us your best attempt. What do you know about Schwarzschild geodesics?

 

[Omikron123]

Oh sorry of course, thank you I couldn´ t figure out how to use it!
So the line element is given by $$ ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^-1dr^2 - r^2d\Omega ^2$$
The object is orbiting at constant radius $r$ in the plane $\theta = \frac{\pi}{2}$. I am supposed to find the values of $a$ and $b$ in the 4-velocity given by: $$U = a\partial _t + b\partial _\phi$$.
Im pretty new the general relativity and Schwarzschild geodesics but here is my attempted solution:

For a massive particle the squared 4-velocity $U^2 = -1$, space-like which i can expand with the Schwarzschild metric, which is diagonal:
$$ U^2 = U^tU_t + U^\phi U_\phi = (g^{ta}U_a)U_t + (g^{\phi a}U_\phi) = -1$$ Since the metric is diagonal only $g^{tt}, g^{\phi \phi}$ are non-zero with values $$g^{\phi \phi} = \frac{1}{g_{\phi \phi}} = -\frac{1}{r^2}, g^{tt} = ... = \frac{1}{1-\frac{R_s}{r}}$$ At this point I am not sure how to continue, because I am not sure if $U_\phi = \partial _\phi$ etc. One idea was to compare the following:
$$U^2 = g^{tt}U_tU_t + g^{\phi \phi}U_\phi U_\phi = -\frac{1}{r^2}\partial _t^2 + \frac{1}{1-\frac{R_s}{r}}\partial _\phi ^2$$ and $$U^2 = (a\partial _t + b\partial _\phi)^2 = a^2\partial _t ^2 + b^2\partial _\phi^2$$ give $a = \sqrt{\frac{1}{1-\frac{R_s}{r}}}, b = \sqrt{-\frac{1}{r^2}}$ (since $2ab\partial _t\partial _\phi = 0$ due to diagonal metric??) As I said I don't really know what I am doing here so there might be some major errors in my thinking..

 

[PeroK]

First, $\partial_t$ and $\partial_{\phi}$ are the Schwarzschild coordinate unit vectors. So, $U = a \partial_t + b\partial_{\phi}$ is an alternative way of writing: $U = (\frac{dt}{d\tau}, 0, 0, \frac{d \phi}{d\tau})$. Where $a = \frac{dt}{d\tau}$ etc.

Your main problem is that you are only looking at timelike paths: $U \cdot U = -1$. What you also need are the conditions for a timelike geodesic. Have you heard of Killing vectors?

 

[Omikron123]

Okay thank you. Yes that was another trail I tried to follow. From what I have learned both $\partial _t$ and $\partial _\phi$ are killing fields due to the spherical symmetri in Schwarzschild spacetime? So the derivative with respect to proper time $\tau$ of a scalarproduct between one of the killing fields "$K$" should be zero:

$$\frac{d}{d\tau}(g(K,U)) = \nabla_Ug(K,U) = g(\nabla_UK,U) + g(K,\nabla _UU) = U^\mu U^\nu \nabla _\mu K_\nu + 0 = 0$$ since $U^\mu U^\nu$ is symmetric and $\nabla _\mu K_\nu$ antisymmetric, which mean that $g(K,U)$ is constant along the geodesic. However i don't know how I can decide the 4-velocity components from this.

If I remember correct, by skipping the derivative, $g(\partial _t,U)$ gives me the energy per unit mass, and $g(\partial _\phi,U)$ the angular momentum per unit mass, are these the velocity values $a$ and $b$?

 

[PeroK]

Okay thank you. Yes that was another trail I tried to follow. From what I have learned both $\partial _t$ and $\partial _\phi$ are killing fields due to the spherical symmetri in Schwarzschild spacetime? So the derivative with respect to proper time $\tau$ of a scalarproduct between one of the killing fields "$K$" should be zero:

$$\frac{d}{d\tau}(g(K,U)) = \nabla_Ug(K,U) = g(\nabla_UK,U) + g(K,\nabla _UU) = U^\mu U^\nu \nabla _\mu K_\nu + 0 = 0$$ since $U^\mu U^\nu$ is symmetric and $\nabla _\mu K_\nu$ antisymmetric, which mean that $g(K,U)$ is constant along the geodesic. However i don't know how I can decide the 4-velocity components from this.

If I remember correct, by skipping the derivative, $g(\partial _t,U)$ gives me the energy per unit mass, and $g(\partial _\phi,U)$ the angular momentum per unit mass, are these the velocity values $a$ and $b$?

That's the theory of Killing vectors right enough. In this case you know that $K_1 = (1, 0, 0, 0)$ and $K_2 = (0, 0, 0, 1)$.

And you also have a specific form for $g_{\mu \nu}$ .

 

[Omikron123]

That's the theory of Killing vectors right enough. In this case you know that $K_1 = (1, 0, 0, 0)$ and $K_2 = (0, 0, 0, 1)$.

And you also have a specific form for $g_{\mu \nu}$ .

Okay, so the only non-zero components of $g_{\mu \nu}$ are $g_{tt} = 1- \frac{R_s}{r}$ and $g_{\phi \phi} = -r^2$ which gives $$g(K_1,U) = g_{tt}\frac{dt}{d\tau} = (1-\frac{R_s}{r})\dot{t}$$ and $$g(K_4,U) = g_{\phi \phi}\frac{d\phi}{d\tau} = (-r^2)\dot{\phi}$$ which give $a = 1-\frac{R_s}{r}$ and $b = -r^2$, $$ U = (1-\frac{R_s}{r})\partial _t -r^2\partial _\phi$$ Is this correct?

 

[PeroK]

Okay, so the only non-zero components of $g_{\mu \nu}$ are $g_{tt} = 1- \frac{R_s}{r}$ and $g_{\phi \phi} = -r^2$ which gives $$g(K_1,U) = g_{tt}\frac{dt}{d\tau} = (1-\frac{R_s}{r})\dot{t}$$ and $$g(K_4,U) = g_{\phi \phi}\frac{d\phi}{d\tau} = (-r^2)\dot{\phi}$$ which give $a = 1-\frac{R_s}{r}$ and $b = -r^2$, $$ U = (1-\frac{R_s}{r})\partial _t -r^2\partial _\phi$$ Is this correct?

That's not right. You have calculated the constants of the motion: $(1-\frac{R_s}{r})\dot{t}$ and $(r^2)\dot{\phi}$.

Note that you perhaps need to try to relate the mathematical formalism to the physics. For example, generally the first of these is analagous to conservation of energy along a geodesic and we can write: $$e = (1-\frac{R_s}{r})\dot{t}$$ And the second is conservation of angular momentum: $$l = (r^2)\dot{\phi}$$
That's what the geodescic equations give us (via the short-cut of using Killing vectors).

You need to combine that with the normalisation condition for a timelike path: $U \cdot U = -1$ to get the solution.

Note that your four velocity is not normalised.

 

[PeroK]

PS You may also need to look at the condition for a stable circular orbit to give you another equation in $l$ and $r$.

Have you seen the equations for Schwarzschild orbits generally?

 

[PeroK]

PS You may also need to look at the condition for a stable circular orbit to give you another equation in $l$ and $r$.

Have you seen the equations for Schwarzschild orbits generally?

In case you haven't here's the general approach to these problems:

1) You use the Killing vectors and timelike normalisation to get a formula for $e, l, r$: $$e^2 = f(r)$$ where you have to calculate $f(r)$. Note that this equation involves $l^2$ as well.

2) Now, for a stable circular orbit $f(r)$ must be at a minimum at the orbital radius. So, we need also $f'(r) = 0$. This gives you an equation of the form $$l^2 = g(r)$$ where you have to calculate $g(r)$.

3) If you plug $g(r)$ into your equation for $e^2$ you should get $e$ in terms of $R_s$ and $r$. This gives you $dt/d\tau$.

4) Then you can also solve for $l$, which gives you $d\phi/d\tau$.

 

[Omikron123]

In case you haven't here's the general approach to these problems:

1) You use the Killing vectors and timelike normalisation to get a formula for $e, l, r$: $$e^2 = f(r)$$ where you have to calculate $f(r)$. Note that this equation involves $l^2$ as well.

2) Now, for a stable circular orbit $f(r)$ must be at a minimum at the orbital radius. So, we need also $f'(r) = 0$. This gives you an equation of the form $$l^2 = g(r)$$ where you have to calculate $g(r)$.

3) If you plug $g(r)$ into your equation for $e^2$ you should get $e$ in terms of $R_s$ and $r$. This gives you $dt/d\tau$.

4) Then you can also solve for $l$, which gives you $d\phi/d\tau$.

Thank you very much for the help, I'll give it a try!