Jekyll2019-03-25T17:38:57+00:00https://duetosymmetry.com/Leo C. SteinAssistant Professor @ Ole Miss. Specializing in gravity and general relativity.Leo C. Steinlcstein@olemiss.eduOn the Starts with a Bang Podcast!2019-03-25T06:00:00+00:002019-03-25T06:00:00+00:00https://duetosymmetry.com/news/on-the-starts-with-a-bang-podcast<iframe width="100%" height="300" scrolling="no" frameborder="no" allow="autoplay" src="https://w.soundcloud.com/player/?url=https%3A//api.soundcloud.com/tracks/595631034&color=%23ff5500&auto_play=false&hide_related=false&show_comments=true&show_user=true&show_reposts=false&show_teaser=true&visual=true"></iframe>
<p><a href="https://twitter.com/StartsWithABang">Ethan Siegel</a> is an
astrophysicist and science communicator who regular writes <a href="https://www.forbes.com/sites/ethansiegel/">a column
for Forbes</a>. When he’s not
busy writing, he also records the
<a href="https://soundcloud.com/ethan-siegel-172073460/tracks">Starts with a Bang
Podcast</a> which
has been running for 3 years. Ethan was kind enough to invite me on as
a guest for
<a href="https://soundcloud.com/ethan-siegel-172073460/starts-with-a-bang-42-black-holes-and-gravitation">Episode 42: Black Holes and
Gravitation</a>. We
talked about what you would see falling into a black hole, and what
the near future might hold for black hole astrophysics! Enjoy :)</p>Leo C. Steinlcstein@olemiss.eduI'm on Episode 42 of the Starts with a Bang PodcastTalk on the Latin American Webinar on Physics2019-03-13T06:00:00+00:002019-03-13T06:00:00+00:00https://duetosymmetry.com/news/lawphysics-webinar<iframe width="560" height="315" src="https://www.youtube.com/embed/7HO07-QtvMI" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
<p>Muchas gracias to <a href="https://acardenasavendano.org">Alejandro
Cárdenas-Avendaño</a> for inviting me to
give a webinar for the <a href="https://lawphysics.wordpress.com">Lating American Webinar on Physics (lawphysics)</a>.
Want to hear me talk about numerical relativity and theories beyond
GR? Well, you’re in luck, because the transmission is on YouTube for
you to watch! It was really fun, and viewers had great questions.</p>
<p>Listen above or click through to the lawphysics site. Talk
information:</p>
<p><em>Title</em>: Testing Einstein with numerical relativity: the precision
frontier, and theories beyond general relativity</p>
<p><em>Abstract</em>: Advanced LIGO and Virgo have already detected black holes
crashing into each other ten times. With their upgrades we anticipate
a rate of about 1 gravitational-wave detection per week. More signals
and higher precision will take the dream of testing Einstein’s theory
of gravity, general relativity, and make it a reality. But would we
know a correction to Einstein’s theory if we saw it? How do we make
predictions from theories beyond GR? And do current numerical
relativity simulations have enough precision that we could be
confident in any potential discrepancy between observations and
predictions?</p>Leo C. Steinlcstein@olemiss.eduWant to hear me talk about numerical relativity and theories beyond GR?The binary black hole explorer: on-the-fly visualizations of precessing binary black holes2018-11-18T00:00:00+00:002018-11-18T00:00:00+00:00https://duetosymmetry.com/pubs/binaryBHexp<p>Check out the <a href="https://vijayvarma392.github.io/binaryBHexp/">project demo page</a>!</p>
<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/precessing_m2000.png" alt="" /></p>
<blockquote>
<p>Binary black hole mergers are of great interest to the astrophysics
community, not least because of their promise to test general
relativity in the highly dynamic, strong field regime. Detections
of gravitational waves from these sources by LIGO and Virgo have
garnered widespread media and public attention. Among these sources,
precessing systems (with misaligned black-hole spin/orbital angular
momentum) are of particular interest because of the rich dynamics
they offer. However, these systems are, in turn, more complex
compared to nonprecessing systems, making them harder to model or
develop intuition about. Visualizations of numerical simulations of
precessing systems provide a means to understand and gain insights
about these systems. However, since these simulations are very
expensive, they can only be performed at a small number of points in
parameter space. We present <em>binaryBHexp</em>, a tool that makes use of
surrogate models of numerical simulations to generate on-the-fly
interactive visualizations of precessing binary black holes. These
visualizations can be generated in a few seconds, and at any point
in the 7-dimensional parameter space of the underlying surrogate
models. With illustrative examples, we demonstrate how this tool can
be used to learn about precessing binary black hole systems.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduReal-time interactive visualizations of merging black holes in seconds!High-accuracy mass, spin, and recoil predictions of generic black-hole merger remnants2018-09-26T00:00:00+00:002018-09-26T00:00:00+00:00https://duetosymmetry.com/pubs/surfinBH<p>This work was featured in <a href="http://www.caltech.edu/news/when-black-holes-collide-85110">Caltech
News</a>, <a href="https://news.olemiss.edu/making-better-predictions-black-hole-smash-ups/">U
of MS
News</a>,
and was picked up by
<a href="https://phys.org/news/2019-01-physicists-supercomputers-ai-accurate-black.html">phys.org</a>
and <a href="https://aps.altmetric.com/details/53748662">other news outlets</a>.</p>
<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/interaction.png" alt="" /></p>
<blockquote>
<p>We present accurate fits for the remnant properties of generically
precessing binary black holes, trained on large banks of
numerical-relativity simulations. We use Gaussian process regression
to interpolate the remnant mass, spin, and recoil velocity in the
seven-dimensional parameter space of precessing black-hole binaries
with mass ratios q≤2, and spin magnitudes χ₁, χ₂≤0.8. For precessing
systems, our errors in estimating the remnant mass, spin magnitude,
and kick magnitude are lower than those of existing fitting formulae
by at least an order of magnitude (improvement is also reported in
the extrapolated region at high mass ratios and spins). In addition,
we also model the remnant spin and kick directions. Being trained
directly on precessing simulations, our fits are free from
ambiguities regarding the initial frequency at which precessing
quantities are defined. We also construct a model for remnant
properties of aligned-spin systems with mass ratios q≤8, and spin
magnitudes χ₁, χ₂≤0.8. As a byproduct, we also provide error
estimates for all fitted quantities, which can be consistently
incorporated into current and future gravitational-wave
parameter-estimation analyses. Our model(s) are made publicly
available through a fast and easy-to-use Python module called
<a href="https://pypi.org/project/surfinBH/">surfinBH</a>.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduModeling black holes remnants directly from numerical relativityMeasuring stochastic gravitational-wave energy beyond general relativity2018-07-09T00:00:00+00:002018-07-09T00:00:00+00:00https://duetosymmetry.com/pubs/stochastic-GWs-beyond-GR<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/stochastic-relations.png" alt="" /></p>
<blockquote>
<p>Gravity theories beyond general relativity (GR) can change the
properties of gravitational waves: their polarizations, dispersion,
speed, and, importantly, energy content are all heavily theory
dependent. All these corrections can potentially be probed by
measuring the stochastic gravitational-wave background. However,
most existing treatments of this background beyond GR overlook
modifications to the energy carried by gravitational waves, or rely
on GR assumptions that are invalid in other theories. This may lead
to mistranslation between the observable cross-correlation of
detector outputs and gravitational-wave energy density, and thus to
errors when deriving observational constraints on theories. In this
article, we lay out a generic formalism for stochastic
gravitational-wave searches, applicable to a large family of
theories beyond GR. We explicitly state the (often tacit)
assumptions that go into these searches, evaluating their generic
applicability, or lack thereof. Examples of problematic assumptions
are as follows: statistical independence of linear polarization
amplitudes; which polarizations satisfy equipartition; and which
polarizations have well-defined phase velocities. We also show how
to correctly infer the value of the stochastic energy density in the
context of any given theory. We demonstrate with specific theories
in which some of the traditional assumptions break down:
Chern-Simons gravity, scalar-tensor theory, and Fierz-Pauli massive
gravity. In each theory, we show how to properly include the
beyond-GR corrections, and how to interpret observational results.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduGravity theories beyond general relativity (GR) can change the properties of gravitational waves: their polarizations, dispersion, speed, and, importantly, energy content are all heavily theory-dependent. All these corrections can potentially be probed by measuring the stochastic gravitational-wave backgroundBlack hole scalar charge from a topological horizon integral in Einstein-dilaton-Gauss-Bonnet gravity2018-05-08T00:00:00+00:002018-05-08T00:00:00+00:00https://duetosymmetry.com/pubs/EDGB-BH-charge<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/spacetime-standalone.png" alt="" /></p>
<blockquote>
<p>In theories of gravity that include a scalar field, a compact
object’s scalar charge is a crucial quantity since it controls
dipole radiation, which can be strongly constrained by pulsar timing
and gravitational wave observations. However in most such theories,
computing the scalar charge requires simultaneously solving the
coupled, nonlinear metric and scalar field equations of motion. In
this article we prove that in linearly-coupled
Einstein-dilaton-Gauss-Bonnet gravity (which admits a shift symmetry
of the dilaton), a black hole’s scalar charge is completely
determined by the horizon surface gravity times the Euler
characteristic of the bifurcation surface, without solving any
equations of motion. Within this theory, black holes announce their
horizon topology and surface gravity to the rest of the universe
through the dilaton field. In our proof, a 4-dimensional topological
density descends to a 2-dimensional topological density on the
bifurcation surface of a Killing horizon. We also comment on how our
proof can be generalised to other topological densities on general
G-bundles.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduIn theories of gravity that include a scalar field, a compact object’s scalar charge is a crucial quantity since it controls dipole radiation, which can be strongly constrained by pulsar timing and gravitational wave observations(Notes) When is a metric conformal to *Ricci-flat*?2018-04-04T23:53:00+00:002018-04-04T23:53:00+00:00https://duetosymmetry.com/notes/when-is-metric-conformal-to-Ricci-flat<p>This question arose when I was discussing with <a href="http://kacabradonjic.com/">Kaća
Bradonjić</a>, who has done work on gravity
theories where conformal geometry plays a different role than in
general relativity.</p>
<script type="math/tex">
\newcommand{\cd}{\nabla}
</script>
<p>Just to be precise, here’s the question again.
Let’s say we have an <em>n</em>-dimensional manifold <em>M</em>, and somebody
hands you the metric <script type="math/tex">g_{ab}</script>. In all generality, the Riemann
curvature of <script type="math/tex">\cd</script> (the Levi-Civita connection of <em>g</em>) can have
nonzero Ricci curvature and nonzero Weyl part (details below). Given
this metric, can there exist a conformal factor <script type="math/tex">\Omega(x)>0</script> such
that the conformally transformed metric, <script type="math/tex">\tilde{g}_{ab} = \Omega^2
g_{ab}</script>, has zero Ricci curvature for its connection <script type="math/tex">\tilde{\cd}</script>?</p>
<p>First let’s recall the
Ricci/Weyl decomposition of the Riemann tensor,</p>
<div>
\begin{align}
\label{eq:R-decomp-proj}
R_{abcd} = S_{r[abcd]} \left[
C_{abcd}
+ \frac{4}{n-2} R_{ac}g_{bd}
- \frac{2}{(n-1)(n-2)} R g_{ac}g_{bd}
\right] \,,
\end{align}
</div>
<p>where the symbol <script type="math/tex">S_{r[abcd]}</script> was one I used in <a href="https://duetosymmetry.com/notes/notes-on-the-E-B-and-3+1-decomp-of-Riem/">this post</a> to mean
“project onto the algebraic symmetries of the Riemann tensor” by
antisymmetrizing on <em>ab</em>, antisymmetrizing on <em>cd</em>, then symmetrizing
on the two pairs. In <a href="https://en.wikipedia.org/wiki/Young_tableau">Young
tableau</a> language, that
means projecting with the Young symmetrizer that looks like
<img src="https://duetosymmetry.com/images/youngabcd.png" alt="" class="align-center" style="height: 3em" /></p>
<p>Next we have to crack open a textbook and look up how tensors that
depend on <script type="math/tex">\tilde{g}_{ab}</script> are determined by their counterparts from
<script type="math/tex">g_{ab}</script> and various derivatives of <script type="math/tex">\Omega</script>. You can find these
in e.g. Appendix D of Wald.<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup> The easy ones are</p>
<div>
\begin{align}
\tilde{g}^{ab} &= \Omega^{-2} g^{ab} \nonumber\\
\label{eq:confC}
\tilde{C}_{abc}{}^d &= C_{abc}{}^d \,.
\end{align}
</div>
<p>From Eq. \eqref{eq:confC} we immediately see that you can never get
rid of Weyl curvature by conformal transformations (why the Weyl
tensor is sometimes called the conformal tensor). Hence the most you can
hope for is to be able to get rid of Ricci curvature through a
conformal transformation.</p>
<p>The relationship between the Ricci tensors <script type="math/tex">\tilde{R}_{ab}</script> and
<script type="math/tex">R_{ab}</script> is nastier. It’s slightly nicer to focus on a
“trace-adjusted” version of Ricci called the <a href="https://en.wikipedia.org/wiki/Schouten_tensor">Schouten
tensor</a>, which in
components is</p>
<div>
\begin{align}
\label{eq:schout}
P_{ab} = \frac{1}{n-2} \left( R_{ab} - \frac{1}{2(n-1)} g_{ab} R
\right) \,.
\end{align}
</div>
<p>This happens to make the Ricci/Weyl decomposition
\eqref{eq:R-decomp-proj} slightly simpler,</p>
<div>
\begin{align}
\label{eq:R-decomp-proj-schout}
R_{abcd} = S_{r[abcd]} \left[
C_{abcd}
+ 4 P_{ac}g_{bd}
\right] \,.
\end{align}
</div>
<p>It <em>also</em> has a somewhat straightforward relationship under conformal
transformations,</p>
<div>
\begin{align}
\label{eq:schoutenConf}
\tilde{P}_{ab} = P_{ab} &- \cd_b \cd_a \ln \Omega +
(\cd_a \ln\Omega)(\cd_b \ln \Omega) \nonumber \\
&- \frac{1}{2} g_{ab} g^{cd}(\cd_c \ln\Omega)(\cd_d \ln \Omega)
\,.
\end{align}
</div>
<p>Notice that the trace of Schouten is determined by the trace of Ricci,
<script type="math/tex">P = R/(2(n-1))</script>, so the vanishing of the Schouten tensor is
equivalent to the vanishing of the Ricci tensor. Thus we can restate
the original question to ask: when does there exist an <script type="math/tex">\Omega</script> such
that <script type="math/tex">\tilde{P}_{ab}=0</script>? And this immediately becomes a question of
“integrability” of the system of equations</p>
<div>
\begin{align}
0 = P_{ab} &- \cd_b \cd_a \ln \Omega +
(\cd_a \ln\Omega)(\cd_b \ln \Omega) \nonumber \\
&- \frac{1}{2} g_{ab} g^{cd}(\cd_c \ln\Omega)(\cd_d \ln \Omega)
\,,
\label{eq:integ0}
\end{align}
</div>
<p>which are <em>n(n+1)/2</em> partial differential equations for the single
scalar field <script type="math/tex">\Omega</script>. For a gentle introduction to the theory of
integrability from a geometric viewpoint, I recommend the sections on
<a href="https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology)">Frobenius’
theorem</a>
from Schutz’s little geometry book.<sup id="fnref:2"><a href="#fn:2" class="footnote">2</a></sup></p>
<p>To start with, we recast the second-order system \eqref{eq:integ0} in
first order form by adjoining an auxiliary one-form</p>
<div>
\begin{align}
\label{eq:omega}
\omega_a \equiv \cd_a \ln \Omega
\end{align}
</div>
<p>which must satisfy the integrability condition</p>
<div>
\begin{align}
\label{eq:integomega}
\cd_{[a}\omega_{b]} = 0 \,,
\end{align}
</div>
<p>by the commutativity of (torsion-free) covariant derivatives on a
scalar field (i.e. the one-form <script type="math/tex">\omega</script> is closed). In terms of
this new one-form, Eq. \eqref{eq:integ0} becomes the first-order
system of PDEs</p>
<div>
\begin{align}
\label{eq:integomega1}
\cd_b \omega_a = P_{ab} & +
\omega_a \omega_b - \frac{1}{2} g_{ab} g^{cd}\omega_c \omega_d
\,.
\end{align}
</div>
<p>If we can solve \eqref{eq:integomega1} for <script type="math/tex">\omega_a</script> which
satisfies \eqref{eq:integomega}, and there is no topological
obstruction (i.e. <script type="math/tex">\omega</script> is not just closed but exact; a question
in <a href="https://en.wikipedia.org/wiki/De_Rham_cohomology">de Rham
cohomology</a>), then
it can immediately be integrated up into a solution for <script type="math/tex">\ln\Omega.</script></p>
<p>Now to find the integrability of this first-order system, let’s take a
further derivative <script type="math/tex">\cd_c</script> of Eq. \eqref{eq:integomega1} and
antisymmetrize over the <em>[cb]</em> index pair. On the left-hand side, this
gives a commutator of covariant derivatives, which means we can
convert it into a curvature tensor. The price is that on the
right-hand side, we will be taking derivatives of the <script type="math/tex">\omega</script>’s. But if
solutions exist, then we can just reuse \eqref{eq:integomega1} to
replace <script type="math/tex">\cd\omega</script> with <em>P</em> and products of <script type="math/tex">\omega</script> without
derivatives. I encourage you to do these manipulations with
<a href="http://www.xact.es/">xAct/xTensor</a>, of course.</p>
<p>The resulting integrability condition is the amazingly simple</p>
<div>
\begin{align}
\label{eq:intCondWeyl}
\boxed{C_{abc}{}^d\omega_d = 2\cd_{[a}P_{b]c}} \,.
\end{align}
</div>
<p>As an easy consequence, when the Weyl tensor vanishes, <script type="math/tex">g_{ab}</script>
is conformal to Ricci-flat (and indeed conformal to flat) if and only
if <script type="math/tex">\cd_{[a}P_{b]c} = 0</script>. This result was already known to Schouten
back in 1920,<sup id="fnref:3"><a href="#fn:3" class="footnote">3</a></sup> but I’m interested in the more general case when
the Weyl tensor is non-vanishing.</p>
<p>So, we have succeeded in turning integrability of \eqref{eq:integ0}
into an algebraic test. In words, this condition is: Treat the
Weyl tensor as a linear map from <script type="math/tex">T^*M</script> to <script type="math/tex">T^*M^{\otimes 3}</script>;
then a necessary (but not yet sufficient) condition for
\eqref{eq:integ0} to be integrable is that <script type="math/tex">\cd_{[a}P_{b]c}</script> must be
in the image of this linear map.</p>
<p>At this point I would be remiss in my duties if I neglected to tell
you that <script type="math/tex">\cd_{[a}P_{b]c}</script> has a perhaps more familiar expression.
Start from the Bianchi identity <script type="math/tex">\cd_{[a}R_{bc]de}=0</script> and decompose
Riemann into the Weyl and Schouten tensors. After performing a
contraction, we find the identity</p>
<div>
\begin{align}
\label{eq:divWeyl}
\cd_d C_{abc}{}^d = (3-n)2\cd_{[a}P_{b]c} \,.
\end{align}
</div>
<p>So we can see that <script type="math/tex">\cd_{[a}P_{b]c}</script> lives in the representation
labeled by the Young diagram
<img src="https://duetosymmetry.com/images/youngabc.png" alt="" class="align-center" style="height: 3em" />
In fact you could have seen this from the index symmetries without
knowing identity \eqref{eq:divWeyl}.</p>
<p>Thus it is at least algebraically possible for <script type="math/tex">\cd_{[a}P_{b]c}</script> to
live in the image of <script type="math/tex">C: T^*M \to T^*M^{\otimes 3}</script>. If it
actually does (which is the real question) then there is a space of
potential solutions <script type="math/tex">\omega_d</script> (unique up to elements of the kernel
of Weyl). We then need there to be a solution in this space which
satisfies the integrability condition <script type="math/tex">\cd_{[a}\omega_{b]}=0</script>. If
all of these conditions are satisfied, then the closed
one-form <script type="math/tex">\omega</script> exists; and if there is no topological
obstruction, <script type="math/tex">\omega</script> is also exact, so <script type="math/tex">\ln\Omega</script> may be
integrated up, and finally <script type="math/tex">\tilde{g}_{ab}</script> can be found, which will
be Ricci-flat.</p>
<h1 id="references">References</h1>
<div class="footnotes">
<ol>
<li id="fn:1">
<p><a href="http://press.uchicago.edu/ucp/books/book/chicago/G/bo5952261.html">Wald’s General Relativity</a> <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p><a href="http://www.cambridge.org/us/academic/subjects/mathematics/mathematical-physics/geometrical-methods-mathematical-physics">Schutz’s Geometrical methods of mathematical physics</a> <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>J A Schouten, <em>Über die konforme Abbildung n-dimensionaler
Mannigfaltigkeiten mit quadratischer Maßbestimmung auf eine
Mannigfaltigkeit mit euklidischer Maßbestimmung</em>, <a href="https://doi.org/10.1007/BF01203193">Math Z (1921)
11: 58</a>. Thanks to Uli
Sperhake for helping me try to understand some of the German in
this old paper! <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>Leo C. Steinlcstein@olemiss.eduIntegrability conditions when trying to solve for a conformal factorBlack-hole kicks from numerical-relativity surrogate models2018-02-13T00:00:00+00:002018-02-13T00:00:00+00:00https://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 ∼0.1s on a standard
off-the-shelf machine, thus making our code ideal to be ported to
large-scale astrophysical studies.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduModeling black holes' kicks directly from numerical relativityDeformation of extremal black holes from stringy interactions2018-02-08T00:00:00+00:002018-02-08T00:00:00+00:00https://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. Steinlcstein@olemiss.eduFinding the shape of extremal black holes in beyond-GR theoriesComplex polynomial roots toy2017-12-15T00:00:00+00:002017-12-15T00:00:00+00:00https://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. Steinlcstein@olemiss.eduInteractive toy for visualizing relationship between polynomial roots and coefficients