Jekyll2018-05-10T18:01:40-07:00https://duetosymmetry.com/Leo C. SteinPostdoctoral researcher @ Caltech. Specializing in gravity and general relativity.Leo C. Steinleostein@tapir.caltech.eduBlack hole scalar charge from a topological horizon integral in Einstein-dilaton-Gauss-Bonnet gravity2018-05-08T00:00:00-07:002018-05-08T00:00:00-07:00https://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. Steinleostein@tapir.caltech.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*?2018-04-04T16:53:00-07:002018-04-04T16:53:00-07:00https://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. Steinleostein@tapir.caltech.eduIntegrability conditions when trying to solve for a conformal factorBlack-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!