Jekyll2020-02-18T06:13:12+00:00https://duetosymmetry.com/Leo C. SteinAssistant Professor @ U of MS. Specializing in gravity and general relativity.Leo C. Steinlcstein@olemiss.eduThe location of the last stable orbit in Kerr spacetime2019-12-17T00:00:00+00:002019-12-17T00:00:00+00:00https://duetosymmetry.com/pubs/kerr-separatrix<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/SepPlotZoomedOut.png" alt="" /></p>
<blockquote>
<p>Black hole spacetimes, like the Kerr spacetime, admit both stable
and plunging orbits, separated in parameter space by the
separatrix. Determining the location of the separatrix is of
fundamental interest in understanding black holes, and is of crucial
importance for modeling extreme mass-ratio inspirals. Previous
numerical approaches to locating the Kerr separatrix were not always
efficient or stable across all of parameter space. In this paper we
show that the Kerr separatrix is the zero set of a single polynomial
in parameter space. This gives two main results. First, we
thoroughly analyze special cases (extreme Kerr, polar orbits, etc.),
finding strict bounds on the limits of roots, and unifying a number
of results in the literature. Second, we pose a stable numerical
method which is guaranteed to quickly and robustly converge to the
separatrix. This new approach is implemented in the Black Hole
Perturbation Toolkit, and results in a ~45x speedup over the prior
robust approach.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduThe dividing line between bound and plunging orbits is an algebraic varietyNumerical relativity simulation of GW150914 beyond general relativity2019-11-11T00:00:00+00:002019-11-11T00:00:00+00:00https://duetosymmetry.com/pubs/DCS-merger<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/DCS-merger-waveform.png" alt="" /></p>
<blockquote>
<p>We produce the first astrophysically-relevant numerical binary black
hole gravitational waveform in a higher-curvature theory of gravity
beyond general relativity. We simulate a system with parameters
consistent with GW150914, the first LIGO detection, in order-reduced
dynamical Chern-Simons gravity, a theory with motivations in string
theory and loop quantum gravity. We present results for the
leading-order corrections to the merger and ringdown waveforms, as
well as the ringdown quasi-normal mode spectrum. We estimate that
such corrections may be discriminated in detections with signal to
noise ratio ≳ 24.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduThe first astrophysically-relevant numerical simulation of merging black holes in a higher-curvature theory beyond GR.qnm: A Python package for calculating Kerr quasinormal modes, separation constants, and spherical-spheroidal mixing coefficients2019-08-28T00:00:00+00:002019-08-28T00:00:00+00:00https://duetosymmetry.com/pubs/qnm-package<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/example_22n.png" alt="" /></p>
<blockquote>
<p><code class="highlighter-rouge">qnm</code> is an open-source Python package for computing the Kerr
quasinormal mode frequencies, angular separation constants, and
spherical-spheroidal mixing coefficients. The <code class="highlighter-rouge">qnm</code> package includes
a Leaver solver with the Cook-Zalutskiy spectral approach to the
angular sector, and a caching mechanism to avoid repeating
calculations. We provide a large cache of low <script type="math/tex">\ell, m, n</script> modes,
which can be downloaded and installed with a single function call,
and interpolated to provide good initial guess for root-polishing at
new values of spin.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduqnm is an open-source Python package for computing the Kerr quasinormal mode frequencies, angular separation constants, and spherical-spheroidal mixing coefficients.Titanium Physicists podcast episode 802019-08-21T08:00:00+00:002019-08-21T08:00:00+00:00https://duetosymmetry.com/news/Titanium-Physicists-podcast-episode-80<audio src="http://traffic.libsyn.com/titaniumphysics/Ep_80_Picturing_The_Bach_Hole.mp3" preload="auto" controls=""></audio>
<p class="align-right" style="width: 300px; margin-left: 1em; margin-bottom: 1em;"><img src="https://duetosymmetry.com/images/M87-800x466.png" alt="" /></p>
<p>A few months ago you might have heard about the ground-breaking
observations made by the <a href="https://eventhorizontelescope.org">Event Horizon Telescope
collaboration</a>. If you want an
explainer in podcast form, we’ve got a treat for you! Ben Tippett had
me back for the third time on his
<a href="http://titaniumphysicists.brachiolopemedia.com/">Titanium Physicists Podcast</a>.
This time it’s
<a href="http://titaniumphysicists.brachiolopemedia.com/2019/08/21/episode-80-picturing-the-bach-hole-with-adal-rifai/">Episode 80: Picturing the Bach Hole</a>
along with Ben,
<a href="https://twitter.com/adalrifai">Adal Rifai</a> (a.k.a. Chunt, the
shapeshifting king of the badgers),
and the incomparable
<a href="https://people.csail.mit.edu/klbouman/">Dr. Katie Bouman</a>! I met
Katie when she was a grad student at MIT, working on the imaging
algorithms to take radio data and turn them into what’s now one of the
most famous images in the history of astrophysics. Now she’s an
assistant professor in computing and mathematical sciences at Caltech!
This recording was super fun, thanks for having me on again, Ben :)</p>Leo C. Steinlcstein@olemiss.eduHear me, Katie Bouman, and friends talk about how to snap a picture of a black hole with the Event Horizon Telescope!Numerical binary black hole collisions in dynamical Chern-Simons gravity2019-06-24T00:00:00+00:002019-06-24T00:00:00+00:00https://duetosymmetry.com/pubs/DCS-head-on<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/DCS-head-on-degeneracy.png" alt="" /></p>
<blockquote>
<p>We produce the first numerical relativity binary black hole
gravitational waveforms in a higher-curvature theory beyond general
relativity. In particular, we study head-on collisions of binary
black holes in order-reduced dynamical Chern-Simons gravity. This is
a precursor to producing beyond-general-relativity waveforms for
inspiraling binary black hole systems that are useful for
gravitational wave detection. Head-on collisions are interesting in
their own right, however, as they cleanly probe the quasi-normal
mode spectrum of the final black hole. We thus compute the
leading-order dynamical Chern-Simons modifications to the complex
frequencies of the post-merger gravitational radiation. We consider
equal-mass systems, with equal spins oriented along the axis of
collision, resulting in remnant black holes with spin. We find
modifications to the complex frequencies of the quasi-normal mode
spectrum that behave as a power law with the spin of the remnant,
and that are not degenerate with the frequencies associated with a
Kerr black hole of any mass and spin. We discuss these results in
the context of testing general relativity with gravitational wave
observations.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduThe first numerical beyond-GR binary black hole merger simulation.Notes: A family of ramp functions and the Beta function2019-06-13T06:00:00+00:002019-06-13T06:00:00+00:00https://duetosymmetry.com/notes/a-family-of-ramp-functions-and-the-Beta-function<div id="box" class="jxgbox" style="width:500px; height:500px; margin-bottom:1em;"></div>
<div id="out"></div>
<p>Sometimes in a numerical method, you need to be able to continuously
turn a calculation on or off in space or time (here I will pretend
it’s in time). This can be easily accomplished if you have a function
that starts at a value of 0 before some first time <script type="math/tex">t_0</script>, and rises
up to a value of 1 by time <script type="math/tex">t_1</script>. Through an affine transformation
you can always map <script type="math/tex">[t_0, t_1] \to [0, 1]</script>. An example “ramp”
function is plotted above.</p>
<p>Now if this function appears in a differential equation, and you are
integrating it with an <script type="math/tex">n^{\textrm{th}}</script> order method, then it’s not
enough for the function to be continuous: you probably want the first
<em>n</em> derivatives to match (and thus vanish) at each endpoint.</p>
<p>Let’s go for a piecewise ramp function,</p>
<div>
\begin{align}
R_n(t) = \begin{cases}
0 & t < 0 \\
p_n(t) & 0 \le t \le 1 \\
1 & 1 < t
\end{cases}
\end{align}
</div>
<p>where <script type="math/tex">p_n(t)</script> is some polynomial in <em>t</em>.
Some counting tells you that these 2 endpoint values and <em>2n</em>
derivative conditions can be satisfied with a polynomial of degree
<em>2n+1</em>. Try changing the value of <em>n</em> above and see how the
smoothness changes.</p>
<p>Now for any value of <em>n</em>, it’s a straightforward algebra problem to
set up the polynomial and solve for the coefficients. You probably
want to know the answer for the general case, and a simple approach is
to do a few examples and look for the pattern. Here are the first
few:</p>
<div>
\begin{align}
p_0(t) &= t \\
p_1(t) &= t^2 (3-2t) \\
p_2(t) &= t^3 (10 - 15 t + 6 t^2) \\
p_3(t) &= t^4 (35 - 84 t + 70 t^2 - 20 t^3) \\
p_4(t) &= t^5 (126 - 420 t + 540 t^2 - 315 t^3 + 70 t^4)
\end{align}
</div>
<p>Can you spot the pattern? Don’t feel bad if you can’t, that’s why we
have the <a href="https://oeis.org/">OEIS</a>. If you search for the above
integers as a sequence, you’ll find
<a href="https://oeis.org/A091811">A091811</a>.</p>
<p>With this newfound knowledge, we can now write down the closed form
for the polynomial,</p>
<div>
\begin{align}
\label{eq:def}
p_n(t) = t^{n+1} \sum_{k=0}^n (-1)^k \binom{n+k}{k} \binom{2n+1}{n-k} t^k
\,.
\end{align}
</div>
<p>But was there a better way to find this than relying on the OEIS to
already have the result? But of course!</p>
<p>Rather than thinking about <script type="math/tex">p_n(t)</script> itself, let’s think about
<script type="math/tex">p_n^\prime(t)</script>. Since <script type="math/tex">p_n</script> is strictly increasing,
<script type="math/tex">p_n^\prime</script> is positive, while going to zero at the endpoints. In
fact it goes to zero like <script type="math/tex">t^n</script> at one endpoint, and <script type="math/tex">(1-t)^n</script> at
the other endpoint, because we wanted <em>n</em> derivatives to vanish at
each endpoint. Therefore we know the proportionality</p>
<div>
\begin{align}
p_n^\prime(t) \propto t^n (1-t)^n \,.
\end{align}
</div>
<p>The only thing to get right is the normalization, which we enforce by
asking that the integral of <script type="math/tex">p_n^\prime</script> is 1 at <script type="math/tex">t=1</script>. If you’ve
spent enough time on probability and statistics, then you’ll recognize
<script type="math/tex">p_n^\prime(t)</script> as a special case of the <a href="https://en.wikipedia.org/wiki/Beta_distribution">Beta
distribution</a>, with
shape parameters <script type="math/tex">\alpha = \beta = n+1</script>. So we know the
normalization,</p>
<div>
\begin{align}
p_n^\prime(t) = \frac{1}{B(n+1, n+1)} t^n (1-t)^n \,,
\end{align}
</div>
<p>where <script type="math/tex">B(a,b)</script> is the <a href="https://en.wikipedia.org/wiki/Beta_function">beta
function</a>, and we can
now call <script type="math/tex">p_n(t)</script> the <a href="https://dlmf.nist.gov/8.17#i">regularized incomplete beta
function</a>,
<script type="math/tex">p_n(t) = I_t(n+1,n+1)</script>.</p>
<p>The incomplete beta function has a <a href="https://dlmf.nist.gov/8.17#ii">representation in terms of the
Gauss hypergeometric function</a>,</p>
<div>
\begin{align}
B_x(a,b) = \frac{x^a}{a} F(a, 1-b; a+1; x) \,.
\end{align}
</div>
<p>The important fact for us is that we’re interested in <script type="math/tex">a=b=n+1</script>, in
which case one of the first two arguments is a non-positive integer,
and therefore <a href="https://dlmf.nist.gov/15.2#E4">the series will
terminate</a> as a finite-degree
polynomial. This way, you can prove Eq. \eqref{eq:def}!</p>
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</script>Leo C. Steinlcstein@olemiss.eduSometimes in a numerical method, you need to be able to continuously turn a calculation on or off in space or time.Surrogate models for precessing binary black hole simulations with unequal masses2019-05-22T00:00:00+00:002019-05-22T00:00:00+00:00https://duetosymmetry.com/pubs/NRSur7dq4<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/NRSur7dq4-mismatch.png" alt="" /></p>
<blockquote>
<p>Only numerical relativity simulations can capture the full
complexities of binary black hole mergers. These simulations,
however, are prohibitively expensive for direct data analysis
applications such as parameter estimation. We present two new fast
and accurate surrogate models for the outputs of these simulations:
the first model, NRSur7dq4, predicts the gravitational waveform and
the second model, surfinBH7dq4, predicts the properties of the
remnant black hole. These models extend previous 7-dimensional,
non-eccentric precessing models to higher mass ratios, and have been
trained against 1528 simulations with mass ratios q≤4 and spin
magnitudes χ₁,χ₂≤0.8, with generic spin directions. The waveform
model, NRSur7dq4, which begins about 20 orbits before merger,
includes all ℓ≤4 spin-weighted spherical harmonic modes, as well as
the precession frame dynamics and spin evolution of the black
holes. The final black hole model, surfinBH7dq4, models the mass,
spin, and recoil kick velocity of the remnant black hole. In their
regime of validity, both models are shown to be more accurate than
existing models by at least an order of magnitude, with errors
comparable to the estimated errors in the numerical relativity
simulations.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduA surrogate model extending the parameter space range of fully precessing quasicircular inspiralsThe SXS Collaboration catalog of binary black hole simulations2019-04-10T00:00:00+00:002019-04-10T00:00:00+00:00https://duetosymmetry.com/pubs/SXS-catalog<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/ParamsCorner_qchi.png" alt="" /></p>
<blockquote>
<p>Accurate models of gravitational waves from merging black holes are
necessary for detectors to observe as many events as possible while
extracting the maximum science. Near the time of merger, the
gravitational waves from merging black holes can be computed only
using numerical relativity. In this paper, we present a major update
of the Simulating eXtreme Spacetimes (SXS) Collaboration catalog of
numerical simulations for merging black holes. The catalog contains
2,018 distinct configurations (a factor of 11 increase compared to
the 2013 SXS catalog), including 1426 spin-precessing
configurations, with mass ratios between 1 and 10, and spin
magnitudes up to 0.998. The median length of a waveform in the
catalog is 39 cycles of the dominant <script type="math/tex">\ell=m=2</script> gravitational-wave
mode, with the shortest waveform containing 7.0 cycles and the
longest 351.3 cycles. We discuss improvements such as correcting for
moving centers of mass and extended coverage of the parameter
space. We also present a thorough analysis of numerical errors,
finding typical truncation errors corresponding to a waveform
mismatch of <script type="math/tex">\sim 10^{-4}</script>. The simulations provide remnant masses
and spins with uncertainties of 0.03% and 0.1% (90<script type="math/tex">^{\text{th}}</script>
percentile), about an order of magnitude better than analytical
models for remnant properties. The full catalog is publicly
available at <a href="https://www.black-holes.org/waveforms">https://www.black-holes.org/waveforms</a>.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduAccurate models of gravitational waves from merging black holes are necessary for detectors to observe as many events as possible while extracting the maximum scienceOn 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?