How can Killing Vector Equations help find Christoffel symbols?

[TerryW]

I'm working my way through D'Inverno's Understanding GR and have reached Chapter 7 with no real problems. I'm stuck though on p101 (and the corresponding problem 7.7). D'inverno sets out two forms of the geodesic equations, one of which is the Lagrangian with the Killing Vectors and then has a throwaway line (paraphrased as) '..it is possible ...using these two equations to read off the components of the Christoffel symbols..'

I have managed all the problems so far but I just cannot see this one. Am I missing something blindingly obvious?

 

[George Jones]

I'm working my way through D'Inverno's Understanding GR and have reached Chapter 7 with no real problems. I'm stuck though on p101 (and the corresponding problem 7.7). D'inverno sets out two forms of the geodesic equations, one of which is the Lagrangian with the Killing Vectors and then has a throwaway line (paraphrased as) '..it is possible ...using these two equations to read off the components of the Christoffel symbols..'

I have managed all the problems so far but I just cannot see this one. Am I missing something blindingly obvious?

Welcome to Physics Forums!

Let's work through this. Can you find

\frac{\partial K}{\partial x^a}?

for x^a equal to t, r, \theta, and \phi

If you can, post what you get; if you can't, ask more questions.

 

[TerryW]

Hi George, and thanks for helping.

For a=0, I reckon I'll get 1/2(g_00,0 + g_11,0 +g_22,0 + g_33,0).

I worked out that
d_a (dx/ds^bdx/ds^c = 0 when I was verifying the workings on the previous page.

I already know what the g_ααs are.

How am I doing?


Terry

 

[George Jones]

For a=0, I reckon I'll get 1/2(g_00,0 + g_11,0 +g_22,0 + g_33,0).

For a diagonal metric, this is almost, but not quite, correct.

Since K = g_{ab} \dot{x}^a \dot{x}^b /2,

\frac{\partial K}{\partial x^0} = \frac{1}{2} \left( \frac{\partial g_{ab}}{\partial x^0} \right) \dot{x}^a \dot{x}^b = \frac{1}{2} \left[ \left( \frac{\partial g_{00}}{\partial x^0} \right) \left( \dot{x}^0 \right)^2 + \left( \frac{\partial g_{11}}{\partial x^0} \right) \left( \dot{x}^1 \right)^2 + \left( \frac{\partial g_{22}}{\partial x^0} \right) \left( \dot{x}^2 \right)^2 + \left( \frac{\partial g_{33}}{\partial x^0} \right) \left( \dot{x}^3 \right)^2 \right]

for a diagonal metric.

I already know what the g_ααs are.

How am I doing?

Very well.

Now, what about

\frac{\partial K}{\partial x^0}?

You can learn about entering mathematical expression in the thread

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This is a time consuming process that is sometimes (and sometimes not) worth the effort. For now, don't worry too much about learning how to do this.

 

[Altabeh]

I'm working my way through D'Inverno's Understanding GR and have reached Chapter 7 with no real problems.

Sorry for asking an irrelevant question: Is this book really great for a self-study, George, or are there any other books which can be more helpful and interesting than D'Inverno's book?

 

[TerryW]

Thanks George,

Oh yes, I forgot about the (dx^a/ds)² terms. It's the next part of the Lagrangian which is the problem though and it seems to be a bit of a circular argument which brings you back to the first equation for the geodesics with d²x⁰/ds² and the Christoffel symbols.

Do you use Latex and then cut/paste the result into your posts?

I'm off to bed now. Hope to hear from you tomorrow.

Regards


Terry

 

[TerryW]

Hi Altabeh,

I'm finding it OK but I did work my way all the way through Schaum's Tensor Calculus (doing all the problems!) before I started. I think I might have struggled with D'Inverno's chapters on Tensor Calculus if I hadn't done this.

RegardsTerry

 

[George Jones]

Now, what about

\frac{\partial K}{\partial x^0}?

Oops, I forgot the latex \dot. This should read

Now, what about

\frac{\partial K}{\partial \dot{x}^0}?

Do you use Latex and then cut/paste the result into your posts?

I usually type the latex commands as I type my posts.

 

[Altabeh]

Hi Altabeh,

I'm finding it OK but I did work my way all the way through Schaum's Tensor Calculus (doing all the problems!) before I started. I think I might have struggled with D'Inverno's chapters on Tensor Calculus if I hadn't done this.

Regards


Terry

And was Schaum's Tensor Calculus useful and easygoing?

 

[TerryW]

Hi George,

My first venture into Latex:

<br /> \frac{\partial K}{\partial \dot{x}^a} = g_{ab}\dot{x}^b \ \ \ \ (2K = g_{ab}\dot{x}^a\dot{x}^b\ \ \ \frac{\partial g_{ab}}{\partial \dot{x}^a}\ =\ \frac{\partial g_{ab}}{\partial {x}^a}\frac{\partial{x}}{\partial \dot{x}^a} \rightarrow \0\ as\ \ x\rightarrow0)<br />

Now you are going to ask me what happens when I take the derivative of

<br /> \frac{\partial K}{\partial \dot{x}^a}<br />

with respect to the affine parameter represented by the dot over the x

 

[TerryW]

Schaum's Tensor Calculus was very useful, but you have to do the problems. There are quite a few printing errors however which leave you pondering for a while. Whether or not it is easy going depends on you!

 

[Altabeh]

Hi George,

My first venture into Latex:

<br /> \frac{\partial K}{\partial \dot{x}^a} = g_{ab}\dot{x}^b \ \ \ \ (2K = g_{ab}\dot{x}^a\dot{x}^b\ \ \ \frac{\partial g_{ab}}{\partial \dot{x}^a}\ =\ \frac{\partial g_{ab}}{\partial {x}^a}\frac{\partial{x}}{\partial \dot{x}^a} \rightarrow \0\ as\ \ x\rightarrow0)<br />

Now you are going to ask me what happens when I take the derivative of

<br /> \frac{\partial K}{\partial \dot{x}^a}<br />

with respect to the affine parameter represented by the dot over the x

See that you are getting professional in using Latex...

<br /> \frac{\partial K}{\partial \dot{x}^a} = \frac{\partial}{\partial \dot{x}^a}(\frac{1}{2}g_{bc}\dot{x}^b\dot{x^c}) = ...

Now you continue this calculation and use the Kronecker's delta to manage combining two terms appearing in the operation. (Remember that g_ab is symmetric).

AB

 

[TerryW]

Hi Altabeh,

I thought I'd done that already

\frac{\partial K}{\partial \dot{x}^a} = g_{ab}\dot{x}^b}

 

[George Jones]

Hi George,

My first venture into Latex:

<br /> \frac{\partial K}{\partial \dot{x}^a} = g_{ab}\dot{x}^b<br />

Yes, this is correct. Did you arrive at this by using Kronecker deltas, as Altabeh suggested?

(2K = g_{ab}\dot{x}^a\dot{x}^b\ \ \ \frac{\partial g_{ab}}{\partial \dot{x}^a}\ =\ \frac{\partial g_{ab}}{\partial {x}^a}\frac{\partial{x}}{\partial \dot{x}^a} \rightarrow \0\ as\ \ x\rightarrow0)

I'm not sure what this means.

Now you are going to ask me what happens when I take the derivative of

<br /> \frac{\partial K}{\partial \dot{x}^a}<br />

with respect to the affine parameter represented by the dot over the x

Right!

 

[TerryW]

Hi George,

Yes I did arrive at my answer by Kronecker deltas. The other bit was just an explanation of why the partial derivative of g_ab disappears.

So for the next bit;

\frac{d}{du}(g_{ab}\dot{x}^b) = \partial_{c}g_{ab}\dot{x}^b\dot{x}^c+g_{ab}\ddot{x}^b

Where now?

 

[Altabeh]

Hi George,

Yes I did arrive at my answer by Kronecker deltas. The other bit was just an explanation of why the partial derivative of g_ab disappears.

So for the next bit;

\frac{d}{du}(g_{ab}\dot{x}^b) = \partial_{c}g_{ab}\dot{x}^b\dot{x}^c+g_{ab}\ddot{x}^b

Where now?

Nice. Now you are required to calculate \frac{\partial}{\partial x^{a}}(\frac{1}{2}g_{bc}\dot{x}^b\dot{x}^c). If you've already calculated these, then arrange all the terms obtained so far in the order given by the geodesic equation (7.46). After that just try to get rid of g_{ab} in g_{ab}\ddot{x }^b. What should you do to lead to pure terms like \ddot{x }^b? (Note: Don't take the index b of \ddot{x}^b seriously here. It must be something else if one still uses terms including \dot{x}^b\dot{x}^c.)

 

[TerryW]

Hi Altabeh,

Is all this leading towards:

\ddot{x}^a +\Gamma^a_{bc}\dot{x}^b\dot{x}^c = 0 ? (7.42 in D'Inverno)

If it is, it isn't the point of my original question which was to find out what D'Inverno means by "It is possible, by (7.42) (ie the equation above) to read off directly from

\frac{\partial{K}}{\partial{x}^a} - \frac{d}{du}\left(\frac{\partial{K}}{\partial\dot{x}^a}\right)=0 (7.46 in D'Inverno)
the components of the connection \Gamma^a_{bc}, and this proves to be a very efficient way of calculating \Gamma^a_{bc}."

I know various techniques for working out the \Gamma^a_{bc} values for a given metric, I thought that this was alluding to an even smarter way of doing it but I can't see it.

 

[Altabeh]

Hi Altabeh,

Is all this leading towards:

\ddot{x}^a +\Gamma^a_{bc}\dot{x}^b\dot{x}^c = 0 ? (7.42 in D'Inverno)

If it is, it isn't the point of my original question which was to find out what D'Inverno means by "It is possible, by (7.42) (ie the equation above) to read off directly from

\frac{\partial{K}}{\partial{x}^a} - \frac{d}{du}\left(\frac{\partial{K}}{\partial\dot{x}^a}\right)=0 (7.46 in D'Inverno)
the components of the connection \Gamma^a_{bc}, and this proves to be a very efficient way of calculating \Gamma^a_{bc}."

I know various techniques for working out the \Gamma^a_{bc} values for a given metric, I thought that this was alluding to an even smarter way of doing it but I can't see it.

Yes, it is! But the point is that if you wanted to derive the Christoffel symbols for the desired metric here directly from the Euler-Lagrange equation, you would end up leading to what D'Inverno asked for in reality. We just gave a proof of how one would extract (7.42) from (7.46) which is the Euler-Lagrange equation.

So all you need now is to start from the line element in spherical coordinates, ds², and divide both sides of it by du², so this would be your \frac{1}{2}g_{bc}\dot{x}^b\dot{x}^c. Remember that here (ds/du)^2=2K and for the sake of convenience, simply ignore that factor 2 which won't make any trouble ahead.

AB

 

[George Jones]

Sorry, I have been busy with work all day.

We have been proceeding too generally.

Yes, it is! But the point is that if you wanted to derive the Christoffel symbols for the desired metric[/B

Exactly!

Or if you wanted to write down the equations of geodesic motion for a a given metric without first calculating the \Gamma^\alpha {}_{\mu \nu} explicitly.

Let's start again. Write down the specific K for the specific metric given in exercise 6.31.

 

[TerryW]

Hi George and Altabeh,

The penny has just dropped! I've just done \Gamma^0_{00} for the Schwartzchild metric and can see my way to doing the rest.

Thanks very much both of you for helping with this.



Regards

Terry