User talk:Jaycall

Joel's First Talk Message to Jay

Jay: In the email message that you sent to me today, you indicated that you had added the 3rd component to the dx/dt equation. I don't see that modification in the current T3Coordinates page. --Tohline 15:05, 29 May 2010 (MDT)

Joel: It's at the bottom of the section entitled "Time-Derivative of Position and Velocity Vectors". Under the history tab, can you see the edits that I made? It was the very first edit and was fairly minor. I only added a couple terms. --Jaycall 20:17, 29 May 2010 (MDT)

The mediaWiki "talk" page recommends that when you reply to a query, you should not start a new subsection but, rather, just indent (using one or more colons) immediately following the query. Also, it recommends that you "sign" each "talk" message. You do this by typing 2 dashes followed by 4 tildes! I'm going to edit your "reply" (and this additional one from me) to put it in this recommended format. But you have to "sign" your own remark. --Tohline 16:23, 29 May 2010 (MDT)

By the way, I can see the edits that you made to add the 3rd component to the dx/dt equation. I did not spot the changes earlier, but via the history "diff" function, I'm able to see the edits clearly. --Tohline 16:41, 29 May 2010 (MDT)

Killing Vector Approach

Thanks for writing out all the terms in the expression for <math>d\Xi/dt</math> in your "Killing Vector Approach" discussion. On the one hand, I'm happy that the term I mentioned cancels with another one because that means we have fewer terms to integrate. On the other hand, it was one term that I thought we might actually be able to manipulate analytically. Of the two terms that remain, one is relatively simple — the one that is proportional to <math>\dot{\lambda}_2</math> — but the first term is a bear! That will take some thinking! --Tohline 10:50, 30 May 2010 (MDT)

I know it seems like the wrong direction, but unless we can express this summary integral entirely in terms of <math>\lambda_1</math>, <math>\lambda_2</math>, <math>\dot{\lambda}_1</math> and <math>\dot{\lambda}_2</math>, I think that in order to make progress we're going to have to write things out in terms of <math>\varpi</math>, <math>z</math>, <math>\dot{\varpi}</math> and <math>\dot{z}</math>. Of course, that will make the expression much messier, but I think it will be very difficult to recognize this quantity as an exact derivative unless everything's in terms of the same coordinates. --Jaycall 12:55, 30 May 2010 (MDT)

I agree. It is for similar reasons that I am planning on focusing on the case where <math>q^2=2</math>. At least in this special case we can invert the coordinate expressions to give <math>\varpi</math> and <math>z</math> in terms of the <math>\lambda</math>s. Incidentally, I have added a link to your "Killing Vector" page as well as a link to this associated "Talk" page on the page where I itemize my various "ramblings". --Tohline 16:16, 30 May 2010 (MDT)

Good point. That would be a good reason to focus on one specific case. I haven't looked closely at the inverted coordinate transformation for the <math>q^2=2</math> case for several days. Since the relation is quadratic, is it obvious which root should be used?

The proper physical root was obvious to me when I performed the coordinate inversion in the case of T1 Coordinates, so I presume it will be obvious for the T3 Coordinate system. But the inversion needs to be redone for the case of T3 Coordinates; do you want to take care of this and type up the result? In the (quadratic) case of <math>q^2=2</math>, it may be as simple as replacing <math>\chi_2</math> with <math>1/\lambda_2</math>. --Tohline 11:21, 31 May 2010 (MDT)

Sure. --Jaycall 13:34, 31 May 2010 (MDT)

Logarithmic Derivatives of T3 Scale Factors

Check out my new subsection (under T3 Coordinates) entitled Logarithmic Derivatives of Scale Factors. First, see if you agree that there is a mistake (typo) in one of your tables of partial derivatives. Specifically, I think that the correct expression is:

<math> \frac{\partial z}{\partial\lambda_2} = - (q^2-1)\frac{\varpi^2 z \ell^2}{\lambda_2} . </math>

Second, see if you agree with my derived expressions for <math>\partial\ln h_i/\partial\ln\lambda_j</math>. --Tohline 22:28, 31 May 2010 (MDT)

You're certainly right about the sign error. I have corrected the expression in the relevant table. --Jaycall 14:28, 1 June 2010 (MDT)

I have not yet been able to confirm your expressions for logarithmic derivatives of the scale factors. I'm not sure what approach you took in deriving them, but since I had already calculated partials of the scale factors (<math>\partial_i h_j</math> and so forth), I derived a little trick to help simplify the work. Do you agree that

<math> \frac{\partial \ln h_i}{\partial \ln \lambda_j} \equiv \frac{\lambda_j}{h_i} \partial_j h_i ? </math>

Yes, this is precisely how I defined the logarithmic derivatives. And I used your tables of partial derivatives (e.g., <math>\partial\varpi/\partial\lambda_j</math> and <math>\partial z/\partial\lambda_j</math>) to obtain <math>\partial_i h_j</math> and so forth. --Tohline 17:30, 1 June 2010 (MDT)

Mistake! Jay: I have found a mistake in my original derivation of the logarithmic derivatives of the <math>h_1</math> scale factor. Perhaps now they will match yours. Check it out and let me know! --Tohline 12:41, 4 June 2010 (MDT)