Jekyll2018-03-07T16:59:06-08:00https://duetosymmetry.com/Leo C. SteinPostdoctoral researcher @ Caltech. Specializing in gravity and general relativity.Leo C. Steinleostein@tapir.caltech.eduBlack-hole kicks from numerical-relativity surrogate models2018-02-13T00:00:00-08:002018-02-13T00:00:00-08:00https://duetosymmetry.com/pubs/kick-surr<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/explore.png" alt="" /></p>
<blockquote>
<p>Binary black holes radiate linear momentum in gravitational waves as
they merge. Recoils imparted to the black-hole remnant can reach
thousands of km/s, thus ejecting black holes from their host
galaxies. We exploit recent advances in gravitational waveform
modeling to quickly and reliably extract recoils imparted to
generic, precessing, black hole binaries. Our procedure uses a
numerical- relativity surrogate model to obtain the gravitational
waveform given a set of binary parameters, then from this waveform
we directly integrate the gravitational-wave linear momentum
flux. This entirely bypasses the need of fitting formulae which are
typically used to model black-hole recoils in astrophysical
contexts. We provide a thorough exploration of the black-hole kick
phenomenology in the parameter space, summarizing and extending
previous numerical results on the topic. Our extraction procedure is
made publicly available as a module for the Python programming
language named surrkick. Kick evaluations take ∼ 0.1 s on a standard
off-the-shelf machine, thus making our code ideal to be ported to
large-scale astrophysical studies.</p>
</blockquote>Leo C. Steinleostein@tapir.caltech.eduModeling black holes' kicks directly from numerical relativityDeformation of extremal black holes from stringy interactions2018-02-08T00:00:00-08:002018-02-08T00:00:00-08:00https://duetosymmetry.com/pubs/NHEK-stringy<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/DeltaV-u.png" alt="" /></p>
<blockquote>
<p>Black holes are a powerful setting for studying general relativity
and theories beyond GR. However, analytical solutions for rotating
black holes in beyond-GR theories are difficult to find because of
the complexity of such theories. In this paper, we solve for the
deformation to the near-horizon extremal Kerr metric due to two
example string-inspired beyond-GR theories:
Einstein-dilaton-Gauss-Bonnet, and dynamical Chern-Simons theory. We
accomplish this by making use of the enhanced symmetry group of NHEK
and the weak-coupling limit of EdGB and dCS. We find that the EdGB
metric deformation has a curvature singularity, while the dCS metric
is regular. From these solutions we compute orbital frequencies,
horizon areas, and entropies. This sets the stage for analytically
understanding the microscopic origin of black hole entropy in
beyond-GR theories.</p>
</blockquote>Leo C. Steinleostein@tapir.caltech.eduFinding the shape of extremal black holes in beyond-GR theoriesComplex polynomial roots toy2017-12-15T00:00:00-08:002017-12-15T00:00:00-08:00https://duetosymmetry.com/tool/polynomial-roots-toy<!------------------------------------------------------------>
<style>
.mybox {
width: 360px;
height: 360px;
margin-bottom: 1em;
display: inline-block;
}
.myDegInput {
width: 4em;
}
.myLabel {
display: inline-block;
}
.myBoxTitle {
padding: 10px;
text-decoration: underline;
}
</style>
<p>Grab the red dots and play around! Or jump to the
<a href="#explanation">explanation</a>, or <a href="#things-to-try">try this</a>.</p>
<div id="coeffbox" class="jxgbox mybox" style="">
</div>
<div id="rootbox" class="jxgbox mybox" style="">
</div>
<form onsubmit="return false;">
<label for="degView" class="myLabel">Degree of polynomial (change me!):</label>
<input type="number" name="degView" id="degView" class="myDegInput" min="1" max="7" step="1" value="4" />
</form>
<!------------------------------------------------------------>
<!-- CODE -->
<script type="text/javascript" src="https://duetosymmetry.com/assets/js/fraction.min.js"></script>
<script type="text/javascript" src="https://duetosymmetry.com/assets/js/complex.min.js"></script>
<script type="text/javascript" src="https://duetosymmetry.com/assets/js/quaternion.min.js"></script>
<script type="text/javascript" src="https://duetosymmetry.com/assets/js/polynomial.min.js"></script>
<script type="text/javascript" src="https://duetosymmetry.com/assets/js/poly-root-toy.js"></script>
<script type="text/javascript">
var controller = new PolyRootController("rootbox","coeffbox", "degView");
</script>
<h2 id="explanation">Explanation</h2>
<p>A <a href="https://en.wikipedia.org/wiki/Polynomial">polynomial</a> in <em>x</em> of
<em>degree n</em> has the form</p>
<div>
\begin{align}
P(x) = \sum_{i=0}^n a_i x^i .
\end{align}
</div>
<p>From the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra">fundamental theorem of
algebra</a>,
there are exactly <em>n</em> roots <script type="math/tex">z_i \in \mathbb{C}</script> in the complex
plane.<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup> The same polynomial can be written as</p>
<div>
\begin{align}
\label{eq:factored}
P(x) = a_n (x-z_1)(x-z_2)\cdots(x-z_n) .
\end{align}
</div>
<p>Let’s set <script type="math/tex">a_n=1</script> for simplicity (this is called a monic
polynomial).</p>
<p>Now, if you have the roots, finding the values of the coefficients
<script type="math/tex">a_i</script> is straightforward: just expand out the product in Eq. \eqref{eq:factored}.
The functions <script type="math/tex">a_i(z_1, z_2, \ldots, z_n)</script> are known as <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial">elementary
symmetric
polynomials</a>
(up to a sign). If you move a root, you can see how all the
coefficients change.</p>
<p>Solving for the <script type="math/tex">z_i</script>’s from the <script type="math/tex">a_i</script>’s with some algebraic
formula is possible for <script type="math/tex">n\le 4</script> but generally impossible for higher
degrees.<sup id="fnref:2"><a href="#fn:2" class="footnote">2</a></sup> Nonetheless, we can numerically solve for roots
with <a href="https://en.wikipedia.org/wiki/Category:Root-finding_algorithms">various numerical
algorithms</a>.<sup id="fnref:3"><a href="#fn:3" class="footnote">3</a></sup>
If you move a coefficient, your computer will solve for the
new locations of the roots, and you can see how they respond.</p>
<h2 id="things-to-try">Things to try</h2>
<p>Grab a coefficient <script type="math/tex">a_i</script> and move it around in a closed loop. If it
comes back to where it started, then the <em>set</em> of roots <script type="math/tex">\{ z_j \}</script>
have to return to the starting set.</p>
<p>But we can also talk about each individual root’s trajectory as
<script type="math/tex">a_i</script> is varied. If <script type="math/tex">a_i</script> moves in a very small loop, so does
each <script type="math/tex">z_j</script>.</p>
<p>Now try to find a larger loop for some <script type="math/tex">a_i</script> so that some <script type="math/tex">z_j</script>’s
swap places!</p>
<p>Hint (spoiler): a really simple choice is to move <script type="math/tex">a_0</script> around the
unit circle, if all the other <script type="math/tex">a_i</script>’s are close to the origin. Then
you should see the <em>n</em> roots <script type="math/tex">z_j</script> each shift one spot to the
left/right around their unit circle. This is an n-cycle.</p>
<p>Try to find a 2-cycle (two roots swap places) or other more
complicated types of <em>permutations</em>.</p>
<p>What we have here is a map from closed loops in <em>a</em>-space to
<a href="https://en.wikipedia.org/wiki/Permutation_group">permutations</a> of the
<em>n</em> roots.</p>
<p>Question: What determines the type of permutation (cycle structure or
conjugacy class)? Does it have anything to do with the zeros of the
<a href="https://en.wikipedia.org/wiki/Discriminant">discriminant</a>?</p>
<h2 id="acknowledgments">Acknowledgments</h2>
<p>This toy was somewhat inspired by <a href="https://plus.google.com/+johncbaez999/posts/81M1B5TCmhb">John Baez’s
post</a>, which
in turn was discussing <a href="http://twocubes.tumblr.com/post/140680223428/same-polynomials-but-this-time-im-letting-t-vary">this tumblr
post</a>.</p>
<p>This toy makes use of <a href="http://jsxgraph.uni-bayreuth.de/wp/">JSXGraph</a>
and my extended version of
<a href="https://github.com/infusion/Polynomial.js">Polynomial.js</a>.<sup id="fnref:3:1"><a href="#fn:3" class="footnote">3</a></sup></p>
<p>Suggestions welcome!</p>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>I’m only considering <script type="math/tex">a_i \in \mathbb{C}</script>; things like
polynomials over finite fields are trickier! <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>Proved by <a href="https://en.wikipedia.org/wiki/%C3%89variste_Galois">Évariste
Galois</a> before
his death in a duel at age 20. <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>I implemented the <a href="https://en.wikipedia.org/wiki/Aberth_method">Aberth-Ehrlich
method</a> into the
javascript package
<a href="https://github.com/infusion/Polynomial.js">Polynomial.js</a>,
following <a href="https://doi.org/10.1007/BF02207694">Dario Bini’s paper</a>
and his <a href="http://www.netlib.org/numeralgo/na10">FORTRAN
implementation</a>. <a href="#fnref:3" class="reversefootnote">↩</a> <a href="#fnref:3:1" class="reversefootnote">↩<sup>2</sup></a></p>
</li>
</ol>
</div>Leo C. Steinleostein@tapir.caltech.eduInteractive toy for visualizing relationship between polynomial roots and coefficientsSimple slow-rotation neutron star structure solver2017-09-08T14:24:09-07:002017-09-08T14:24:09-07:00https://duetosymmetry.com/code/simple-slow-rot-NS-solver<p>I’m releasing into the wild <a href="https://github.com/duetosymmetry/simple-slow-rot-NS-solver">a simple code for computing neutron star
structure in the slow-rotation
expansion</a>
to first and second order. This code was originally from Nico Yunes,
with a bunch of development by Kent Yagi. I rewrote huge chunks of it
to give it a command line/config file interface, turned some magic
numbers into configurable parameters, C++ified some important bits,
etc. Kent gave me his blessing to release it into the wild. For IP
reasons, I first had to remove the dependence on code from Numerical
Recipes, which is why it looks like the commit history starts in
Sept. 2017.</p>
<p>Of course there are already codes like
<a href="http://www.gravity.phys.uwm.edu/rns/">RNS</a> and
<a href="http://www.lorene.obspm.fr/">LORENE</a>, so who needs another NS code?
This code is useful for two reasons:</p>
<ol>
<li>There is a modular implementation of the piecewise-polytropic model
from Read, Lackey, Owen, and Friedman (2009)
[<a href="https://arxiv.org/abs/0812.2163">arXiv:0812.2163</a>] along with their
fits for named EOSs; and</li>
<li>The slow-rotation expansion allows to
accurately extract the moment of inertia and quadrupole moment.</li>
</ol>
<p>Feel free to improve the code in any way you see fit and send me a
pull request, or open a new issue, anything you want under the MIT
license.</p>Leo C. Steinleostein@tapir.caltech.eduI’m releasing into the wild a simple code for computing neutron star structure in the slow-rotation expansion to first and second order. This code was originally from Nico Yunes, with a bunch of development by Kent Yagi. I rewrote huge chunks of it to give it a command line/config file interface, turned some magic numbers into configurable parameters, C++ified some important bits, etc. Kent gave me his blessing to release it into the wild. For IP reasons, I first had to remove the dependence on code from Numerical Recipes, which is why it looks like the commit history starts in Sept. 2017.Living Reviews in Relativity author index2017-08-28T21:10:00-07:002017-08-28T21:10:00-07:00https://duetosymmetry.com/rebuilt-lrr-index<p>Some time ago I hit the ‘old man shaking his fist at a cloud’ age.
Then Springer took over Living Reviews in Relativity and moved all the
papers into their infrastructure. Along with this move was the demise
of the LRR server and the old LRR author index (snapshot seen
<a href="https://web.archive.org/web/20161125104822/http://relativity.livingreviews.org/Articles/author.html">here</a>
from the Internet Archive).</p>
<p>Thanks to the magic of INSPIRE’s xml and writing
<a href="https://github.com/duetosymmetry/lrr-index">a bit of python code</a>,
I was able to resurrect my own copy of the author index. I don’t know
if anybody else will find it useful, but you can browse it here:
<a href="https://duetosymmetry.com/lrr-index/">LRR author index</a>.</p>Leo C. Steinleostein@tapir.caltech.eduI missed the author index from LRR, so I rebuilt one.Note on simple(r) equations for Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons theories2017-08-06T22:43:00-07:002017-08-06T22:43:00-07:00https://duetosymmetry.com/notes/note-on-simple-eoms-for-edgb-dcs<p>Special thanks to Helvi Witek, who originally showed me one simplified
form of the “C-tensor” in Einstein-dilaton-Gauss-Bonnet (EDGB).
<script type="math/tex">
\newcommand{\cd}{\nabla}
\newcommand{\dR}{ {}^{*}\!R}
\newcommand{\ddR}{ {}^{*}\!R^{*}{} }
</script></p>
<hr />
<p>If you’ve ever looked into theories beyond general relativity, you’re
already aware that their field equations can be very complicated. For
example, here’s a clip from <a href="https://arxiv.org/abs/1511.05513">one paper on EDGB</a>:</p>
<p><img src="https://duetosymmetry.com/images/EDGB-scary.png" alt="Scary equations for EDGB" /></p>
<p>Ack! That’s pretty unwieldy. But don’t despair, it turns out that
the above mess can be written much more compactly.</p>
<p>Just to set conventions, let’s work with the action</p>
<div>
\begin{align}
\label{eq:action1}
S = \int d^4x \sqrt{-g} \left[
\frac{1}{2}m_{pl}^2 R
- \frac{1}{2} (\cd^a \vartheta) (\cd_a \vartheta)
\right]
+ S_{int}
\end{align}
</div>
<p>where <script type="math/tex">\vartheta</script> is a scalar (dilaton or axion) and <script type="math/tex">S_{int}</script> is
a non-minimal interaction term between the scalar and curvature.</p>
<h2 id="einstein-dilaton-gauss-bonnet">Einstein-dilaton-Gauss-Bonnet</h2>
<p>For EDGB, let’s take</p>
<div>
\begin{align}
\label{eq:SEDGB}
S_{int}^{EDGB} = \frac{1}{8} \varepsilon
\int d^4x \sqrt{-g}
F(\vartheta)
\left[
R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd}
\right]
\end{align}
</div>
<p>with some arbitrary coupling function F, and some dimensionless perturbation
parameter <script type="math/tex">\varepsilon</script>. Now this above curvature combination might seem
arbitrary, but it’s actually the 4-dimensional Euler density (see
e.g. <a href="http://jacobi.luc.edu/Useful.html#EulerDensities">Bob McNees’s notes</a>).
It’s more natural to write that as</p>
<div>
\begin{align}
\label{eq:euler4}
\ddR_{abcd}R^{abdc} = R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd}.
\end{align}
</div>
<p>Here we’ve defined the <em>double-dual</em> <script type="math/tex">\ddR</script> of the Riemann tensor.
First, we dualize on the left two antisymmetric indices to define the
left-dual,</p>
<div>
\begin{align}
\label{eq:leftdual}
\dR^{abcd} \equiv \frac{1}{2} \epsilon^{abef} R_{ef}{}^{cd},
\end{align}
</div>
<p>and then we further dualize on the right two antisymmetric indices to
get the double-dual,</p>
<div>
\begin{align}
\label{eq:doubledual}
\ddR^{abcd} \equiv \dR^{ab}{}_{gh} \frac{1}{2} \epsilon^{ghcd}
= \frac{1}{2} \epsilon^{abef} R_{efgh} \frac{1}{2} \epsilon^{ghcd}.
\end{align}
</div>
<p>Now, it’s an exercise in algebraic manipulation to show that the
equation for the metric from the action defined in
Eqs. \eqref{eq:action1}, \eqref{eq:SEDGB} is given simply by</p>
<div>
\begin{align}
\label{eq:eom-EDGB}
\boxed{
m_{pl}^2 G_{ab} + \varepsilon \cd^c \cd^d
\left[
\ddR_{cabd} F(\vartheta)
\right] = T_{ab}
}
\end{align}
</div>
<p>where <script type="math/tex">T_{ab}</script> is the stress-energy tensor for matter plus the
stress-energy tensor for the scalar field. That’s quite a
bit simpler than the image above, isn’t it! If you want to get
xTensor to verify this for you, grab
<code class="highlighter-rouge">EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb</code> from the
<a href="https://github.com/xAct-contrib/examples">xAct examples collection</a>.</p>
<p>I wrote the above in a certain way to make it look very similar to the
case of dynamical Chern-Simons (DCS, below), but before moving on—recall
that one reason people like EDGB is that the equations of motion are
only second order in the metric. That’s not obvious from the way I
wrote it, because it looks like you might get third and fourth
derivatives of the metric. However, one nice property of the
double-dual of Riemann is that it’s divergence free (see MTW
Eq. (13.51) and exercise 13.11). This means we can rewrite</p>
<div>
\begin{align}
\label{eq:C-tensor-EDGB-idents}
\cd^c \cd^d
\left[
\ddR_{cabd} F(\vartheta)
\right]
=
\cd^c
\left[
\ddR_{cabd} \cd^d F(\vartheta)
\right]
=
\ddR_{cabd} \cd^c \cd^d F(\vartheta).
\end{align}
</div>
<p>Now it’s obvious that there are only second derivatives of the metric.
However, the first or second forms might give more insight, because
from them you can see that this so-called “C-tensor” is itself the
divergence of some tensor. That’s the kind of thing you might want to
integrate over a region…</p>
<h2 id="dynamical-chern-simons">Dynamical Chern-Simons</h2>
<p>Anyway, on to DCS. Now we use the interaction term</p>
<div>
\begin{align}
\label{eq:SDCS}
S_{int}^{DCS} = \frac{1}{8} \varepsilon
\int d^4x \sqrt{-g}
F(\vartheta)
\dR^{abcd} R_{abdc}
\end{align}
</div>
<p>with just a single dual. Again it looks kind of arbitrary, but
when <script type="math/tex">\dR^{abcd} R_{abcd}</script> is integrated over the whole manifold,
you get a topological invariant.</p>
<p>The equation for the metric in DCS is also kind of scary looking, but
again some algebra shows that you can write it as</p>
<div>
\begin{align}
\label{eq:eom-DCS}
\boxed{
m_{pl}^2 G_{ab} + \varepsilon \cd^c \cd^d
\left[ \dR_{c(ab)d} F(\vartheta)
\right] = T_{ab}
}
\end{align}
</div>
<p>where <script type="math/tex">(ab)</script> means that we are symmetrizing (with a factor of 1/2)
on those two indices. This looks very similar to the expression for
EDGB! However, the single-dual of Riemann is only manifestly
divergence-free on the left two indices, so
this equation does have third derivatives of the metric. The
double-divergence does vanish, so there are no fourth
derivatives.</p>
<p>All of the above calculations are in my notebook
<code class="highlighter-rouge">EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb</code> in the
<a href="https://github.com/xAct-contrib/examples">xAct examples collection</a>.
Hope you learned something!</p>Leo C. Steinleostein@tapir.caltech.eduHere to save you some algebra and column-inchesRecent talks in Rome and Nottingham2017-07-24T17:00:00-07:002017-07-24T17:00:00-07:00https://duetosymmetry.com/talks/recent-talks-Rome-Nottingham<p>Just an update on talks I gave recently. Valeria Ferrari, Leonardo
Gualtieri, and Paolo Pani hosted the conference “<a href="https://agenda.infn.it/conferenceDisplay.py?ovw=True&confId=12616">New Frontiers in
Gravitational-Wave
Astrophysics</a>”
at Sapienza University in Rome, and graciously invited me to speak
about “Numerical black holes and mergers in theories beyond GR”
(<a href="https://agenda.infn.it/conferenceTimeTable.py?confId=12616#20170621.detailed">slides available from their web
site</a>:
click on the little folder next to my name).</p>
<p>Then in July, Thomas Sotiriou invited me to come visit the University
of Nottingham (home of Robin Hood). I gave two talks: first, a more
general one, “Probing strong-field gravity: black holes and mergers in
general relativity and beyond” for non-experts; and second, to
Thomas’s strong-gravity group meeting, a talk covering my <a href="https://duetosymmetry.com/pubs/NHEK-met-pert/">recent work
on separing metric perturbations in near-horizon extremal Kerr</a>.</p>Leo C. Steinleostein@tapir.caltech.eduJust an update on talks I gave recently. Valeria Ferrari, Leonardo Gualtieri, and Paolo Pani hosted the conference “New Frontiers in Gravitational-Wave Astrophysics” at Sapienza University in Rome, and graciously invited me to speak about “Numerical black holes and mergers in theories beyond GR” (slides available from their web site: click on the little folder next to my name).Separating metric perturbations in near-horizon extremal Kerr spacetimes2017-07-19T00:00:00-07:002017-07-19T00:00:00-07:00https://duetosymmetry.com/pubs/NHEK-met-pert<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/NHEK-schem.png" alt="" /></p>
<blockquote>
<p>Linear perturbation theory is a powerful toolkit for studying black
hole spacetimes. However, the perturbation equations are hard to
solve unless we can use separation of variables. In the Kerr
spacetime, metric perturbations do not separate, but curvature
perturbations do. The cost of curvature perturbations is a very
complicated metric-reconstruction procedure. This procedure can be
avoided using a symmetry-adapted choice of basis functions in highly
symmetric spacetimes, such as near-horizon extremal Kerr. In this
paper, we focus on this spacetime, and (i) construct the
symmetry-adapted basis functions; (ii) show their orthogonality; and
(iii) show that they lead to separation of variables of the scalar,
Maxwell, and metric perturbation equations. This separation turns
the system of partial differential equations into one of ordinary
differential equations over a compact domain, the polar angle.</p>
</blockquote>Leo C. Steinleostein@tapir.caltech.eduWhenever you've got symmetry, you should use it!Numerical binary black hole mergers in dynamical Chern-Simons: I. Scalar field2017-05-23T00:00:00-07:002017-05-23T00:00:00-07:00https://duetosymmetry.com/pubs/numerical-bbh-in-dcs-scalar<p>Here’s a movie from one of the simulations we ran for this paper:</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/WQH-1b_XUM4" frameborder="0" allowfullscreen=""></iframe>
<hr />
<p class="align-right" style="width: 250px"><img src="https://duetosymmetry.com/images/num-order-reduced-dcs-waveforms.png" alt="" /></p>
<blockquote>
<p>Testing general relativity in the non-linear, dynamical,
strong-field regime of gravity is one of the major goals of
gravitational wave astrophysics. Performing precision tests of
general relativity (GR) requires numerical inspiral, merger, and
ringdown waveforms for binary black hole (BBH) systems in theories
beyond GR. Currently, GR and scalar-tensor gravity are the only
theories amenable to numerical simulations. In this article, we
present a well-posed perturbation scheme for numerically integrating
beyond-GR theories that have a continuous limit to GR. We
demonstrate this scheme by simulating BBH mergers in dynamical
Chern-Simons gravity (dCS), to linear order in the perturbation
parameter. We present mode waveforms and energy fluxes of the dCS
pseudoscalar field from our numerical simulations. We find good
agreement with analytic predictions at early times, including the
absence of pseudoscalar dipole radiation. We discover new
phenomenology only accessible through numerics: a burst of dipole
radiation during merger. We also quantify the self-consistency of
the perturbation scheme. Finally, we estimate bounds that
GR-consistent LIGO detections could place on the new dCS length
scale, approximately <script type="math/tex">\ell \lesssim \mathcal{O}(10)</script> km.</p>
</blockquote>Leo C. Steinleostein@tapir.caltech.eduOne of the first numerical simulations of black hole mergers in beyond-GR effective field theoriesNote on commutation coefficients in two ways2017-05-04T13:52:00-07:002017-05-04T13:52:00-07:00https://duetosymmetry.com/notes/note-on-commutation-coefficients-two-ways<script type="math/tex">
\newcommand{\cd}{\nabla}
\newcommand{\pd}{\partial}
\newcommand{\Lie}{\mathcal{L}}
</script>
<p>Suppose somebody hands you a collection of n linearly-independent
vector fields <script type="math/tex">\{v_a\}_{a=1}^n</script> on an n-dimensional manifold, which
you can use as a frame field (not necessarily an orthonormal frame
field, because I haven’t said anything about a metric yet!). A
natural thing to compute are the <em>commutation coefficients</em> of these
vector fields,</p>
<div>
\begin{align}
\label{eq:vec-c}
[v_a, v_b] = c_{ab}{}^d v_d \,,
\end{align}
</div>
<p>where we decompose the commutators back into the basis of the vector
fields themselves. The collection of scalar fields <script type="math/tex">c_{ab}{}^d</script> are
called the commutation coefficients. Because of the antisymmetry of
the Lie bracket, the commutation coefficients are automatically
antisymmetric in the lower two indices.</p>
<p>On the other hand, let’s say somebody hands you a collection of n
linearly-independent one-forms <script type="math/tex">\{\theta^a\}_{a=1}^n</script>, which you can
use as a coframe field (again not necessarily orthonormal, because no
metric yet; and this coframe field might not be dual to the frame
field). A natural thing to compute is the exterior derivative of each
form, <script type="math/tex">d\theta^a</script>, which you could then expand in the basis of
two-forms made by wedging together the <script type="math/tex">\theta^a</script>’s. So you could
define another set of coefficients <script type="math/tex">\tilde{c}</script> from</p>
<div>
\begin{align}
\label{eq:covec-c}
d\theta^a = \tilde{c}^a{}_{bd} \frac{1}{2} \theta^b \wedge \theta^d
\end{align}
</div>
<p>where we have included a factor of 1/2 for future convenience. (This
is not to be confused with the connection 1-form <script type="math/tex">\omega^a{}_b</script>.)</p>
<p>The wedge product of two one-forms is automatically antisymmetric, so
again we have this property that the collection of scalar fields
<script type="math/tex">\tilde{c}^a{}_{bd}</script> is automatically antisymmetric in the lower
indices.</p>
<p>This should probably lead you to suspect that the two sets of
coefficients are related when the vector and covector bases are
related. So, let’s now say that the two bases are dual to each other,</p>
<div>
\begin{align}
\theta^a(v_b) = \langle \theta^a, v_b \rangle = i_{v_b} \theta^a
= \delta^a_b \,.
\end{align}
</div>
<p>Notice that we still haven’t needed a metric: finding a dual basis is
possible without metric (roughly, you only need to be able to do
matrix inversion).</p>
<p>Now we can extract components of each equation, Eqs. \eqref{eq:vec-c}
and \eqref{eq:covec-c}, by contracting with the right type of object.
If we contract Eq. \eqref{eq:vec-c} with <script type="math/tex">\theta^e</script>, we’ll find</p>
<div>
\begin{align}
c_{ab}{}^e = \langle \theta^e, [v_a, v_b] \rangle
= i_{[v_a, v_b]} \theta^e \,.
\end{align}
</div>
<p>Similarly, if we insert two vectors into the two slots of
Eq. \eqref{eq:covec-c}, we find</p>
<div>
\begin{align}
\tilde{c}^a{}_{ef} = (d\theta^a)(v_e, v_f)
= i_{v_f} i_{v_e} d\theta^a \,.
\end{align}
</div>
<p>So now the question is: what is the relationship, if any, between</p>
<div>
\begin{align}
i_{[v,w]} \omega && \textrm{and} && i_v i_w d\omega \ ?
\end{align}
</div>
<p>In fact, with the use of some
<a href="https://duetosymmetry.com/notes/note-on-exterior-interior-Lie-derivative-superalgebra/">differential identities</a>
(or a one-liner in my package
<a href="http://www.xact.es/xTerior/">xTerior</a>, using the function
<code class="highlighter-rouge">SortDerivations[]</code>)
you can show that for any vectors <script type="math/tex">v, w</script> and form <script type="math/tex">\omega</script>, we have</p>
<div>
\begin{align}
\label{eq:fancy}
i_v i_w d\omega = i_{[v,w]} \omega
+ d i_v i_w \omega
- \Lie_v i_w \omega + \Lie_w i_v \omega
\end{align}
</div>
<p>(there are a bunch of equivalent ways to rewrite this). Now let
<script type="math/tex">v=v_a, w = v_b, \omega=\theta^d</script>. Since
<script type="math/tex">i_{v_a}\theta^d = \delta^d_a</script>, the last three terms on the right
hand side of Eq. \eqref{eq:fancy} will vanish. In this case we’ll
find</p>
<div>
\begin{align}
i_{v_a} i_{v_b} d\theta^d = i_{[v_a, v_b]} \theta^d
\end{align}
</div>
<p>which immediately tells us that</p>
<div>
\begin{align}
c_{ab}{}^d = -\tilde{c}^d{}_{ab}
\,.
\end{align}
</div>
<p>So indeed, the same information is encoded in the commutation
coefficients of vectors, <script type="math/tex">c_{ab}{}^d</script>, and the decomposition of
<script type="math/tex">d\theta^d</script> into basis two-forms, <script type="math/tex">\tilde{c}^d{}_{ab}</script>.</p>
<p>Note again that everything has been independent of a metric.</p>
<div>
\begin{align}
\end{align}
</div>Leo C. Steinleostein@tapir.caltech.eduAn identity between vector commutation coefficients and coframe connection coefficients