## LaTeX at last ! (Illustrated with a GR calculation) Sunday, Feb 18 2007

After waiting for long, Blogphysica gets $\LaTeX$ . (See the announcement at the wordpress.com Blog )

The Basic syntax is $latex <LaTeX Equation>$. You can find a list of $\LaTeX$ symbols here(pdf) . In particular, if you find the formulae are too small try $latex {\displaystyle <LaTeX Equation>}$ . It’s great !

Update(18/02/07) : See this FAQ for some more options.

Now, to illustrate LaTeX, I’ll take up a particular problem. Consider two equal masses falling towards each other, as shown below, starting from rest.

m—>—o—<—m

The question is this – How much power does this system lose as the two masses fall towards each other ?

Let $r$ be the distance from the centre of mass (which I’ve denoted by an ‘o’ above and I choose it to be the origin). Take the common axis to be z-axis.

Hence the position of two masses are respectively $(x,y,z)=(0,0,r)$ and $(0,0,-r)$

To the zeroth approximation, Newtonian mechanics tells you that

${\displaystyle \frac{d^2 r}{dt^2} = - \frac{G_Nm}{(2r)^2}}$

(We neglect the effect of gravitational wave on the masses)

The quadrupole moment $Q_{ij}$ of a mass distribution is defined by

${\displaystyle Q_{ij}(t) = \iiint d\forall\ \ \ \rho(x,t) \left[ x_i x_j - \frac{1}{3} \delta_{ij} r^2 \right]}$

Where the integral is done over the whole source($\rho$ is the mass density of the source). It is basically negative of the traceless part of the moment of inertia.

The Einstein formula for the power emitted by the source (in the form of Gravitational waves) is

${\displaystyle P = \frac{G_N}{5c^5} \sum\ \frac{d^3 Q_{ij}}{dt^3}\ \frac{d^3 Q_{ij}}{dt^3} }$

where the symbol $\sum$ denotes a sum over $i,j=1,2,3$.

Assuming that the masses are small in size, the components of quadrupole moment in this case are

${\displaystyle Q_{zz}= m\left(r^2 - \frac{1}{3}r^2\right)+ m\left((-r)^2 - \frac{1}{3}r^2\right)= \frac{4}{3}mr^2}$
${\displaystyle Q_{xx}=Q_{yy} = m(0^2 -\frac{1}{3}r^2)+ m(0^2 -\frac{1}{3}r^2)= -\frac{2}{3}mr^2 }$
${\displaystyle Q_{xy}=Q_{yx}=Q_{zx} =\ldots = 0}$

Now to calculate the third time derivative, we first use chain rule to get

${\displaystyle \frac{d^3}{dt^3} r^2 = 6 \frac{dr}{dt}\ \frac{d^2 r}{dt^2} + 2r\ \frac{d^3 r}{dt^3} }$

We can now employ Newton’s Law to get

${\displaystyle \frac{d^3}{dt^3} r^2 = - \frac{G_Nm}{2r^2}\ \frac{dr}{dt}}$

${\displaystyle \frac{d^3 Q_{zz}}{dt^3} = - \frac{2G_Nm^2}{3r^2}\ \frac{dr}{dt}}$
${\displaystyle \frac{d^3 Q_{xx}}{dt^3}=\frac{d^3 Q_{yy}}{dt^3} = \frac{G_Nm^2}{3r^2}\ \frac{dr}{dt}}$

All the other components are zero. Substituting this into the Einstein’s formula which I quoted above, the total power radiated comes out to be

${\displaystyle \mbox{Power emitted} = \frac{G_N^3m^2}{15c^5 r^4}\ \left(\frac{dr}{dt}\right)^2 }$

which is terribly small in most cases.

## Blog Theme Tuesday, Apr 18 2006

As Tom mentioned, I've changed the theme to one which automatically displays the author of each post.  I tried reading up on the stuff, but wordpress is weird and doesn't allow us to tweak the themes, because as wordpress puts it

We are definitely mindful of making everything more customizable for our users, but at the same time we don’t want people to have to look at HTML and CSS code, which is antithetical to the purpose of WordPress.com. If you want complete control over your enviroment, you’re probably better off running WordPress on a great web host of your own

Quite stupid actually, but, unless some of you can tweak something under the hood, we'll have to stick to this theme, which is a shame because I actually liked the old theme more.

Also, to all the other contributors (mainly Ajay and Venky I think), I think it shouldn't be much of a pain to create your own wordpress account, signup here: and let me know your wordpress username, so that I can add you to the blog team, and then you can post while you are logged in under your own login.  That way the confusion with everyone using the blogphysica id will end.

And I would suggest that we mail everyone to create their id's now, or whenever they feel like posting and mail one of us here about it if they want to join in as authors.  If you are all ok with it, then we could lock the blogphysica id in, say, a week's time?

P.S. : Since this post is relevant to the about page and how to post in this blog page, I am adding a trackback to those pages – Tom.

## Welcome! Friday, Mar 31 2006

Welcome to Blog Physica. This is a site where we will post interesting topics and resources for Physics.