Mathematics - 2020-03-26 (page 2 of 2)

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11:00 PM

And answer my context question. It looks like you're using the expression for the Riemannian metric in normal polar coordinates. Warning: Because of the virus, I am not home, so I have neither my computer nor my library.

eigenvalue

The context is that I have a 2D metric in the following general form $g=t(dt)^{2}+G(t,x)(dx)^{2}$ and I want to figure out how the metric component( function of (t,x)) $G(t,x)$ has to look like to avoid a curvature singularity (re Gauss curvature).

Alessandro Codenotti

How badly hit is the area of the US you live in? @Ted

Ted Shifrin

Not too bad yet, Alessandro. I've been essentially indoors for about two weeks.

eigenvalue

@TedShifrin Here is the question again:

I have calculated the Gauss curvature for $g=t(dt)^{2}+G(t,x)(dx)^{2}$:

$K=-\frac{1}{2}\frac{\frac{\partial^{2}G}{\partial t^{2}}}{tG}+\frac{1}{4}\left(\frac{\frac{\partial G}{\partial t}}{t^{2}G}+\frac{(\frac{\partial G}{\partial t})^{2}}{tG^{2}}\right)=-\frac{2Gt\cdot\frac{\partial^{2}G}{\partial t^{2}}-\frac{\partial G}{\partial t}(\frac{\partial G}{\partial t}\cdot t+G)}{4G^{2}t^{2}}$, and apparently it diverges for $G^{2}t^{2}=0$. I am trying to figure out how $G:=G(t,x)$ (which is defined for all $t,x\in\mathbb{R}$) has to look like in order t…

Ted Shifrin

@eigenvalue: i'm troubled by the $t$ coefficient. So we're on $t\ge 0$ only.

Alessandro Codenotti

11:04 PM

Good, I hope you remain safe and healthy

Ted Shifrin

You wrote $t\in\Bbb R$.

Me too, Alessandro. I went to a doctor more than 2 wks ago who, it turned out, subsequently tested positive. But no symptoms for me.

eigenvalue

It's a signature changing metric again. so it's valid vor all t. The Gauss curvature diverges at t=0 (= hyper surface of signature change)

Ted Shifrin

Oh, right. I never do that stuff :)

But we have a singular metric when $t=0$.

eigenvalue

I am actually trying to figure out how G should look like to avoid the divergence. How can I use the Taylor expansion to figure that out?

@TedShifrin You tested positive? Did the doctor tell you whether you are immune now (for this and future outbreaks)?

Ted Shifrin

I would have to compute myself, using moving frames.

No, the doctor tested positive. With no symptoms I can't be tested (yet).

eigenvalue

11:09 PM

Of what use is the Taylor series in this context? I have heard that it could be useful but don't see how or why?

Oh, I see. It's a shame that getting tested is such a problem. There should be at home test kits for everyone.

Ted Shifrin

You don't know the zero in the denominator isn't canceled by the numerator without computing Taylor expansion (in $t$), yes.

I will do my own computation. Give me a few minutes.

eigenvalue

Thanks! :-)
I am so happy to be able to talk with someone about this. Nothing worse that being stuck at home with nobody around to discuss your stuff with

Ted Shifrin

Is the metric with $G$ or $G^2$?

eigenvalue

The metric is: $g=t(dt)^{2}+G(t,x)(dx)^{2}$

Ted Shifrin

OK, I agree with your formula.

Are we expecting $G\ne 0$ at dt=0$?

eigenvalue

11:23 PM

You mean the formula for the Taylor series?

Ted Shifrin

No, I worked out $K$.

eigenvalue

I assume $G(t,x)\geq0$ for all $t$.

Because only the two metric signs (-,+) and (+,+) occur, I only need to consider the non-negative case $G(t,x)\geq0$.

But looking at the formula $K=-\frac{2Gt\cdot\frac{\partial^{2}G}{\partial t^{2}}-\frac{\partial G}{\partial t}(\frac{\partial G}{\partial t}\cdot t+G)}{4G^{2}t^{2}}$ I thought that we have to have $t^{2}G^{2}(t,x)\neq0$ to avoid a curvature singularity at $t=0$

Ted Shifrin

Write $G = G_0 + tG_1+ t^2G_2+ \dots$. Assume $G_0 >0$. If $G_0=0$, then your sign condition says $G_1=0$ too, so factor out the $t^2$ and proceed similarly.

I would write partials as subscripts for convenience.

No, you're wrong. Obviously it vanishes at $t=0$. You have to figure out how to cancel with the numerator and get a removable singularity.

The $G_i$ in my notation are functions of $x$, so computing $t$ derivatives is easy.

eigenvalue

ok, so in your notation the $G = G_0 + tG_1+ t^2G_2+ \dots$ is the Taylor series. ... - I see, that's why the Taylor series becomes useful?
... It makes more sense now. I am going to work on that now.... and allow myself to bug you again if I get suck somewhere... hope that's fine with you?

Ted Shifrin

Sure. I'm not around as much as usual because I'm not at home with my computer. But sure.

eigenvalue

11:34 PM

Sounds good!
Make sure you stay healthy (or as healthy as you can) and be safe!!

Ted Shifrin

Of course :) You too.