Jekyll2023-11-27T00:29:29+00:00https://duetosymmetry.com/Leo C. SteinAssistant Professor @ U of MS. Specializing in gravity and general relativity.Leo C. Steinlcstein@olemiss.eduNotes: Magnetostatic multipole expansion using STF tensors2023-08-27T06:00:00+00:002023-08-27T06:00:00+00:00https://duetosymmetry.com/notes/magnetostatics-stf-mpoles<script type="math/tex">
\newcommand{\pd}{\partial}
\newcommand{\cd}{\nabla}
\newcommand{\bs}{\boldsymbol}
\newcommand{\nn}{\nonumber}
</script>
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<li><a href="#refresher-electrostatic-stf-multipole-expansion" id="markdown-toc-refresher-electrostatic-stf-multipole-expansion">Refresher: Electrostatic STF multipole expansion</a></li>
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<p>These notes are intended for students (or profs) aware of the
multipole expansion for electrostatics in terms of symmetric tracefree
(STF) tensors. Standard texts on electrodynamics (like Jackson)
hardly mention the STF version, though it is extremely well-known to
researchers in GR. After I developed these notes, <a href="https://julioparramartinez.me/">Julio
Parra-Martinez</a> pointed me to <a href="https://arxiv.org/abs/1202.4750">a paper
by Andreas Ross</a> which implicitly
includes these results, though I want to explain a bit more slowly.</p>
<h1 id="refresher-electrostatic-stf-multipole-expansion">Refresher: Electrostatic STF multipole expansion</h1>
<p>Before getting to magnetostatics, we’ll start with electrostatics.
This is easier since we only need to solve for the scalar potential,
which satisfies</p>
<div>
\begin{align}
\cd^2 \Phi = - \frac{\rho}{\epsilon_0} .
\end{align}
</div>
<p>We’re interested in the case where <script type="math/tex">\rho</script> vanishes outside of a
compact region. The most efficient way to get to the STF version of
the multipole expansion is to start from the Green’s function
solution,</p>
<div>
\begin{align}
\Phi(\bs{x}) = \frac{1}{4\pi\epsilon_0}
\int \frac{\rho(\bs{x}')}{|\bs{x}-\bs{x}'|} d^3\bs{x}' .
\end{align}
</div>
<p>We then take the function <script type="math/tex">1/|\bs{x}-\bs{x}'|</script> and perform a multivariate
Taylor series expansion about the point <script type="math/tex">\bs{x}'=0</script>, since far away from
the source, <script type="math/tex">|\bs{x}'| \ll |\bs{x}|</script>. This expansion is</p>
<div>
\begin{align}
\frac{1}{|\bs{x}-\bs{x}'|} &= \sum_{\ell=0}^\infty \frac{(-1)^\ell}{\ell!} x^{\prime j_1} x^{\prime j_2} \cdots x^{\prime j_\ell}
\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
\,, \\
&=
\sum_{\ell=0}^\infty \frac{(-1)^\ell}{\ell!}
(r')^\ell
n'^{j_1} n'^{j_2}\cdots n'^{j_\ell}
\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
.
\end{align}
</div>
<p>In the <script type="math/tex">\ell</script> index tensor <script type="math/tex">\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell}
(1/r)</script>, all indices are obviously symmetric; they are also tracefree
away from the origin, where we get a delta function, owing to <script type="math/tex">\cd^2
(1/r) = -4\pi \delta_{(3)}(\bs{x})</script>. Since this tensor is symmetric
and tracefree (STF), we are free to take only the STF part of the
product of <script type="math/tex">\bs{x}'</script> direction vectors. We denote this with angle
brackets around the relevant indices,</p>
<div>
\begin{align}
\frac{1}{|\bs{x}-\bs{x}'|} &=
\sum_{\ell=0}^\infty \frac{(-1)^\ell}{\ell!}
x'^{\langle j_1} x'^{j_2}\cdots x'^{j_\ell\rangle}
\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
.
\end{align}
</div>
<p>Plugging this in to the Green’s function integral, we get</p>
<div>
\begin{align}
\Phi(\bs{x}) &= \frac{1}{4\pi\epsilon_{0}} \sum_{\ell=0}^{\infty}
\frac{(-1)^\ell}{\ell!}
\left(
\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
\right) M^{j_{1}j_{2}\cdots j_{\ell}} ,
\end{align}
</div>
<p>where we have defined the <script type="math/tex">\ell</script>th STF multipole tensor of the
source as</p>
<div>
\begin{align}
M^{j_{1}j_{2}\cdots j_{\ell}} \equiv \int
\rho(\bs{x}) x^{\langle j_1} x^{j_2} \cdots x^{j_\ell \rangle}
\ d^{3} \bs{x}
.
\end{align}
</div>
<h1 id="magnetostatic-multipole-expansion">Magnetostatic multipole expansion</h1>
<p>We can apply our results from the electrostatic multipole expansion to
magnetostatics, using the potential formulation. In magnetostatics,
we are trying to find a magnetic field <script type="math/tex">\bs{B}(\bs{x})</script> satisfying</p>
<div>
\begin{align}
\cd\times\bs{B} &= \mu_{0} \bs{J}\,, & \text{(static)}
\end{align}
</div>
<p>where as usual <script type="math/tex">\cd\cdot\bs{B}=0</script>, and in statics, conservation of
charge demands that <script type="math/tex">\cd\cdot\bs{J}=0</script>. Now we go to the potential
formulation, <script type="math/tex">\bs{B}=\cd\times\bs{A}</script>, and use our gauge freedom to go
to Coulomb gauge, <script type="math/tex">\cd\cdot\bs{A}=0</script>. Plugging in, we are now trying
to solve</p>
<div>
\begin{align}
\cd^{2} A^{i} = - \mu_{0} J^{i} \,.
\end{align}
</div>
<p>In Cartesian coordinates, this is just three independent copies of the
Poisson equation, one for each Cartesian component <script type="math/tex">A^{i}</script>. Therefore
we can use the Green’s function for the scalar Laplacian’s for each
component,</p>
<div>
\begin{align}
A^{i}(\bs{x}) = \frac{\mu_{0}}{4\pi} \int
\frac{J^{i}(\bs{x}')}{|\bs{x}-\bs{x}'|} d^{3}\bs{x}'
\,.
\end{align}
</div>
<p>Just like in the electrostatic case, we Taylor expand
<script type="math/tex">\tfrac{1}{|\bs{x}-\bs{x}'|}</script>, pull things out of the integrals, etc.
Essentially, we are just making a replacement in the electrostatic
case: <script type="math/tex">\Phi \to A^{i}, \tfrac{\rho}{\epsilon_{0}} \to \mu_{0} J^{i}</script>.
This means our multipole moments get an extra index that does not
participate in the STF operation. As it stands, our solution is</p>
<div>
\begin{align}
\label{eq:A-mpole-external}
A^{i}(\bs{x}) &= \frac{\mu_{0}}{4\pi}
\sum_{\ell=0}^{\infty}
\frac{(-1)^\ell}{\ell!}
\left(
\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
\right) \mathcal{M}^{i;j_{1}j_{2}\cdots j_{\ell}} \,,
\end{align}
</div>
<p>where the magnetic multipole moments are defined as</p>
<div>
\begin{align}
\label{eq:Bstatic-mpole-tensor-def}
\mathcal{M}^{i;j_{1}j_{2}\cdots j_{\ell}}
\equiv
\int
J^{i}(\bs{x})
x^{\langle j_1} x^{j_2} \cdots x^{j_\ell \rangle}
\ d^{3} \bs{x}
\,,
\end{align}
</div>
<p>which are STF only on the <script type="math/tex">j_{1}\cdots j_{\ell}</script> indices after the
semicolon.</p>
<p>Notice that the <script type="math/tex">\ell=0</script> term vanishes – no magnetic monopoles! – by
conservation of charge. Integrate <script type="math/tex">(\cd\cdot\bs{J})x^{j}</script> and use
integration by parts:</p>
<div>
\begin{align}
\int (\pd_{i}J^{i})x^{j} \ d^{3} \bs{x}
=
-\int J^{i}\pd_{i}x^{j} \ d^{3} \bs{x}
=
-\int J^{i}\delta_{i}^{j} \ d^{3} \bs{x}
=
- \mathcal{M}^{i}
\,.
\end{align}
</div>
<p>The left hand side vanishes since in magnetostatics,
<script type="math/tex">\cd\cdot\bs{J}=0</script>. Therefore the <script type="math/tex">\ell=0</script> magnetic monopole moment
vanishes, <script type="math/tex">\mathcal{M}^{i}=0</script>.</p>
<p>Before handling the arbitrary <script type="math/tex">\ell</script> term, let’s write the dipole in
the traditional form seen in e.g. Griffiths. The traditional form for
a magnetic dipole is</p>
<div>
\begin{align}
\bs{A}_{\text{dip}} &= \frac{\mu_{0}}{4\pi} \frac{\bs{m}\times\bs{n}}{r^{2}} \,,\\
A^{i}_{\text{dip}} &= \frac{\mu_{0}}{4\pi} \epsilon^{ijk} m_{j} \pd_{k} \frac{-1}{r} \,.
\end{align}
</div>
<p>Here the magnetic dipole pseudo-vector is related to the 2-index
magnetic dipole tensor,</p>
<div>
\begin{align}
m^{i} &= \frac{1}{2} \epsilon^{ijk} \mathcal{M}_{k;j} \,, &
\mathcal{M}^{k;j} &= \epsilon^{jki} m_{i} \,,\\
\bs{m} &= \frac{1}{2} \int \bs{x} \times \bs{J}(\bs{x}) \ d^{3}\bs{x}
\,.
\end{align}
</div>
<p>This gives the ideal dipole magnetic field</p>
<div>
\begin{align}
\bs{B}_{\text{dip}} &= \cd\times\bs{A}_{\text{dip}} \,, \\
B^{i}_{\text{dip}} &= \frac{\mu_{0}}{4\pi} m^{j} \pd_{i}\pd_{j}\frac{1}{r}
\,,
\end{align}
</div>
<p>except that we have dropped the singular
<script type="math/tex">\mu_{0}m^{i}\delta_{(3)}(\bs{x})</script> term.</p>
<p>It seems like we’ve discarded some information — only the
antisymmetric part of <script type="math/tex">\mathcal{M}^{i;j}</script> contributed to <script type="math/tex">m^{i}</script>.
What about the symmetric part? This exactly vanishes, and that
generalizes to all higher <script type="math/tex">\ell</script>. The proof follows similarly to why
<script type="math/tex">\mathcal{M}^{i}</script> vanished above. Use that <script type="math/tex">\cd\cdot\bs{J}=0</script>, and
integrate this divergence against <script type="math/tex">x^{j_{1}}x^{j_{2}}\cdots
x^{j_{\ell}}</script>,</p>
<div>
\begin{align}
0 &= -\int (\pd_{i}J^{i})x^{j_{1}}x^{j_{2}}\cdots x^{j_{\ell}}
\ d^{3}\bs{x} \\
&= +\int J^{i} \pd_{i}
\left(
x^{j_{1}}x^{j_{2}}\cdots x^{j_{\ell}}
\right) \ d^{3}\bs{x} \\
&=
\int J^{i}
\left[
\delta_{i}^{j_{1}}x^{j_{2}}\cdots x^{j_{\ell}}
+
x^{j_{1}}\delta_{i}^{j_{2}}x^{j_{3}}\cdots x^{j_{\ell}}
\right. \nn\\
&\qquad\qquad
\left.
+ \ldots + x^{j_{1}}x^{j_{2}}\cdots x^{j_{\ell-1}} \delta_{i}^{j_{\ell}}
\right]\ d^{3}\bs{x} \\
&= \ell \int J^{(j_{1}} x^{j_{2}}\cdots x^{j_{\ell})}
\ d^{3}\bs{x} \,.
\end{align}
</div>
<p>What we found is that the completely symmetric part of
<script type="math/tex">\mathcal{M}^{i;j_{1}\cdots j_{\ell-1}}</script> vanishes (in fact it vanishes
even before removing the traces on the indices after the semicolon).</p>
<p>The next step for understanding these magnetic multipole tensors
requires a little knowledge of how Young diagrams classify the index
symmetries of tensors (ok, maybe not strictly necessary, but this was
how I first realized what to do). We know that the tensor
<script type="math/tex">x^{\langle j_{1}}x^{j_{2}}\cdots x^{j_{\ell}\rangle}</script> lives in the
representation labeled by the diagram of shape <script type="math/tex">(\ell)</script>,</p>
<p><img src="https://duetosymmetry.com/images/yt-j1-jl.png" alt="Young tableau of shape (ell)" class="align-center" style="width: 250px" /></p>
<p>Now recall that when we tensor-product a vector with some tensor in a
diagram with shape <script type="math/tex">\lambda</script>, we generate tensors in irreps related
by adding one box at the end of any allowed row or as a new row
underneath (the decomposition of tensor products into irreps is
determined by the <a href="https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule">Littlewood–Richardson
rule</a>;
adding one box where allowed is the simplest case. This is
encapsulated in a <em>Hasse diagram</em> called <a href="https://en.wikipedia.org/wiki/Young%27s_lattice">Young’s
lattice</a>, which gives
a partial order on Young diagrams, seen in here:</p>
<p><img src="https://duetosymmetry.com/images/Young-lattice.png" alt="Young lattice" class="align-center" /></p>
<p>Now, since <script type="math/tex">x^{\langle j_{1}}x^{j_{2}}\cdots x^{j_{\ell}\rangle}</script>
lives in the <script type="math/tex">(\ell)</script> representation, we know that tensoring with
<script type="math/tex">J^{i}</script> can produce content in exactly two representations: the
<script type="math/tex">(\ell+1)</script> diagram, and the <script type="math/tex">(\ell,1)</script> diagram, having shapes</p>
<p><img src="https://duetosymmetry.com/images/yt-lp1-and-l1.png" alt="Two Young tableaux, one of shape (ell+1), one of shape (ell,1)" class="align-center" /></p>
<p>However, above we showed that the completely symmetric part, labeled
by <script type="math/tex">(\ell+1)</script>, vanishes. Therefore, we have shown that each
<script type="math/tex">\mathcal{M}^{i;j_{1}\cdots j_{\ell}}</script> lives in the <script type="math/tex">(\ell,1)</script>
diagram, and this means the <script type="math/tex">i</script> index is antisymmetric with each <script type="math/tex">j</script>
index.</p>
<p>Because of the antisymmetry between <script type="math/tex">i</script> and any one of the <script type="math/tex">j</script>’s, we
are free to insert a projector in the space of 2-forms,</p>
<div>
\begin{align}
\mathcal{M}^{i;j_{1}j_{2}\cdots j_{\ell}}
= \delta^{i}_{[k}\delta^{j_{1}}_{p]} \mathcal{M}^{k;pj_{2}\cdots j_{\ell}}
= \tfrac{1}{2}\epsilon^{ij_{1}q}\epsilon_{qkp} \mathcal{M}^{k;pj_{2}\cdots j_{\ell}}
\,.
\end{align}
</div>
<p>This motivates defining an auxiliary tensor <script type="math/tex">m</script>, like in the dipole
case,</p>
<div>
\begin{align}
\label{eq:mstatic-mpole-and-dual-rels}
m^{k j_{2}j_{3}\cdots j_{\ell}} &\equiv -\tfrac{1}{2} \epsilon^{k}{}_{i j_{1}} \mathcal{M}^{i;j_{1}j_{2}\cdots j_{\ell}}
\,, &
\mathcal{M}^{i;j_{1}j_{2}\cdots j_{\ell}} &=
-\epsilon^{ij_{1}}{}_{k} m^{k j_{2}j_{3}\cdots j_{\ell}}
\,.
\end{align}
</div>
<p>The two minus signs (which cancel) are here to agree with the
traditional notation for the magnetic dipole vector. We can insert
the integral expression,</p>
<div>
\begin{align}
m^{k j_{2}j_{3}\cdots j_{\ell}}
=
\int -\tfrac{1}{2} \epsilon^{k}{}_{i j_{1}} J^{i}
x^{\langle j_{1}} x^{j_{2}} \cdots x^{j_{\ell}\rangle} d^{3}x
\,.
\end{align}
</div>
<p>What are the symmetries of <script type="math/tex">m^{kj_{2}\cdots j_{\ell}}</script>? It is
obviously symmetric and tracefree on the <script type="math/tex">j</script>’s. It is also easy to
see that tracing <script type="math/tex">k</script> with any of the <script type="math/tex">j</script>’s would result in a symmetric
pair of indices contracting with the <script type="math/tex">\epsilon</script> tensor in the
definition of <script type="math/tex">m</script>, so by symmetry-antisymmetry,
<script type="math/tex">m^{kj_{2}\cdots j_{\ell}}</script> is tracefree on all indices.</p>
<p>Now we will show that <script type="math/tex">m^{kj_{2}\cdots j_{\ell}}</script> is symmetric on
<script type="math/tex">(k,j_{2})</script> and thus on <script type="math/tex">k</script> with any of the <script type="math/tex">j</script>’s. Suppose we split
the tensor into parts that are symmetric and antisymmetric on these
two indices,
<script type="math/tex">m^{kj_{2}\cdots j_{\ell}} = m^{(kj_{2})\cdots j_{\ell}} +
m^{[kj_{2}]\cdots j_{\ell}}</script>. For the antisymmetric part, we could
again insert a projector in the space of 2-forms.
While evaluating this projector, we have the dual on <script type="math/tex">[k j_{2}]</script>.
But this is simply a trace of <script type="math/tex">\mathcal{M}</script>: from
Eq. \eqref{eq:mstatic-mpole-and-dual-rels},</p>
<div>
\begin{align}
\epsilon^{p}{}_{kj_{2}}m^{kj_{2}\cdots j_{\ell}} =
\mathcal{M}^{p;j_{1}j_{2}j_{3}\cdots j_{\ell}}\delta_{j_{1}j_{2}} = 0
\,,
\end{align}
</div>
<p>which vanishes since <script type="math/tex">\mathcal{M}</script> is tracefree on all the <script type="math/tex">j</script>’s.
Since this antisymmetric part of <script type="math/tex">m^{kj_{2}\cdots j_{\ell}}</script> vanished,
we found that <script type="math/tex">m</script> is STF on all indices.</p>
<p>We can finally restate <script type="math/tex">A^{k}</script> and <script type="math/tex">B^{i}</script> in terms of these magnetic
STF moments, after a bit of algebra:</p>
<div>
\begin{align}
A^{k}(\bs{x}) &= \frac{\mu_{0}}{4\pi}
\sum_{\ell=0}^{\infty}
\frac{(-1)^{\ell}}{\ell!}
\left(
\pd_{j_1} \pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
\right) \epsilon^{kp j_{1}}m^{p j_{2}\cdots j_{\ell}} \,,
\\
B^{i} = \epsilon^{ijk}\pd_{j}A_{k} &=
\frac{\mu_{0}}{4\pi}
\sum_{\ell=0}^{\infty}
\frac{(-1)^{\ell+1}}{\ell!}
\left(
\pd_{i} \pd_{j_{1}}\pd_{j_2} \cdots \pd_{j_\ell} \frac{1}{r}
\right)
m^{j_{1} j_{2}\cdots j_{\ell}} \,.
\end{align}
</div>Leo C. Steinlcstein@olemiss.eduHow to do the STF multipole expansion of the magnetic potential and field (it's been on my TODO list for a while)Numerical relativity surrogate model with memory effects and post-Newtonian hybridization2023-06-05T00:00:00+00:002023-06-05T00:00:00+00:00https://duetosymmetry.com/pubs/CCE-surrogate<p class="align-right" style="width: 350px; margin: 2em 0 0 1em;"><img src="https://duetosymmetry.com/images/CCE-surrogate.png" alt="" /></p>
<blockquote>
<p>Numerical relativity simulations provide the most precise templates
for the gravitational waves produced by binary black hole
mergers. However, many of these simulations use an incomplete
waveform extraction technique – extrapolation – that fails to
capture important physics, such as gravitational memory
effects. Cauchy-characteristic evolution (CCE), by contrast, is a
much more physically accurate extraction procedure that fully
evolves Einstein’s equations to future null infinity and accurately
captures the expected physics. In this work, we present a new
surrogate model, <tt>NRHybSur3dq8_CCE</tt>, built from CCE waveforms
that have been mapped to the post-Newtonian (PN) BMS frame and then
hybridized with PN and effective one-body (EOB) waveforms. This
model is trained on 102 waveforms with mass ratios q≤8 and aligned
spins χ<sub>1z</sub>,χ<sub>2z</sub> ∈ [−0.8,0.8]. The model spans
the entire LIGO-Virgo-KAGRA (LVK) frequency band (with
f<sub>low</sub>=20Hz) for total masses M ≳ 2.25M<sub>⊙</sub> and
includes the ℓ≤4 and (ℓ,m)=(5,5) spin-weight −2 spherical harmonic
modes, but not the (3,1), (4,2) or (4,1) modes. We find that
<tt>NRHybSur3dq8_CCE</tt> can accurately reproduce the training
waveforms with mismatches ≲ 2×10<sup>−4</sup> for total masses
2.25M<sub>⊙</sub> ≤ M ≤ 300M<sub>⊙</sub> and can, for a modest
degree of extrapolation, capably model outside of its training
region. Most importantly, unlike previous waveform models, the new
surrogate model successfully captures memory effects.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduNumerical relativity simulations provide the most precise templates for the gravitational waves produced by binary black hole mergers. However, many of these simulations use an incomplete waveform extraction technique – extrapolation – that fails to capture important physics, such as gravitational memory effects. Cauchy-characteristic evolution (CCE), by contrast, is a much more physically accurate extraction procedure that fully evolves Einstein’s equations to future null infinity and accurately captures the expected physics. In this work, we present a new surrogate model, NRHybSur3dq8_CCE, built from CCE waveforms that have been mapped to the post-Newtonian (PN) BMS frame and then hybridized with PN and effective one-body (EOB) waveforms. This model is trained on 102 waveforms with mass ratios q≤8 and aligned spins χ1z,χ2z ∈ [−0.8,0.8]. The model spans the entire LIGO-Virgo-KAGRA (LVK) frequency band (with flow=20Hz) for total masses M ≳ 2.25M⊙ and includes the ℓ≤4 and (ℓ,m)=(5,5) spin-weight −2 spherical harmonic modes, but not the (3,1), (4,2) or (4,1) modes. We find that NRHybSur3dq8_CCE can accurately reproduce the training waveforms with mismatches ≲ 2×10−4 for total masses 2.25M⊙ ≤ M ≤ 300M⊙ and can, for a modest degree of extrapolation, capably model outside of its training region. Most importantly, unlike previous waveform models, the new surrogate model successfully captures memory effects.Numerical simulations of black hole-neutron star mergers in scalar-tensor gravity2023-04-25T00:00:00+00:002023-04-25T00:00:00+00:00https://duetosymmetry.com/pubs/NSBH-in-ST<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/GR_ST_edited.png" alt="" /></p>
<blockquote>
<p>We present a numerical-relativity simulation of a black hole -
neutron star merger in scalar-tensor (ST) gravity with binary
parameters consistent with the gravitational wave event GW200115. In
this exploratory simulation, we consider the Damour-Esposito-Farese
extension to Brans-Dicke theory, and maximize the effect of
spontaneous scalarization by choosing a soft equation of state and
ST theory parameters at the edge of known constraints. We
extrapolate the gravitational waves, including tensor and scalar
(breathing) modes, to future null-infinity. The numerical waveforms
undergo ~ 22 wave cycles before the merger, and are in good
agreement with predictions from post-Newtonian theory during the
inspiral. We find the ST system evolves faster than its
general-relativity (GR) counterpart due to dipole radiation, merging
a full gravitational-wave cycle before the GR counterpart. This
enables easy differentiation between the ST waveforms and GR in the
context of parameter estimation. However, we find that dipole
radiation’s effect may be partially degenerate with the NS tidal
deformability during the late inspiral stage, and a full Bayesian
analysis is necessary to fully understand the degeneracies between
ST and binary parameters in GR.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduWe present a numerical-relativity simulation of a black hole - neutron star merger in scalar-tensor (ST) gravity with binary parameters consistent with the gravitational wave event GW200115. In this exploratory simulation, we consider the Damour-Esposito-Farese extension to Brans-Dicke theory, and maximize the effect of spontaneous scalarization by choosing a soft equation of state and ST theory parameters at the edge of known constraints. We extrapolate the gravitational waves, including tensor and scalar (breathing) modes, to future null-infinity. The numerical waveforms undergo ~ 22 wave cycles before the merger, and are in good agreement with predictions from post-Newtonian theory during the inspiral. We find the ST system evolves faster than its general-relativity (GR) counterpart due to dipole radiation, merging a full gravitational-wave cycle before the GR counterpart. This enables easy differentiation between the ST waveforms and GR in the context of parameter estimation. However, we find that dipole radiation’s effect may be partially degenerate with the NS tidal deformability during the late inspiral stage, and a full Bayesian analysis is necessary to fully understand the degeneracies between ST and binary parameters in GR.Nonlinear ringdown in the news2023-02-22T00:00:00+00:002023-02-22T00:00:00+00:00https://duetosymmetry.com/news/Nonlinear-paper-press<p class="align-right" style="width: 400px"><img src="https://duetosymmetry.com/images/nonlinear-qnm-cartoon.png" alt="" /></p>
<p>Our latest paper, <a href="/pubs/Nonlinear-BH-ringdown/">Nonlinearities in black hole ringdowns</a>, was just <a href="https://doi.org/10.1103/PhysRevLett.130.081402">published in
Physical Review
Letters</a>!
This article was selected as an ❦ Editors’ Suggestion, and <a href="https://physics.aps.org/articles/v16/29">Featured in
APS’s Physics magazine</a>.
There were a number of other news stories covering this work:</p>
<ul>
<li>University of Mississippi: <a href="https://news.olemiss.edu/black-hole-researchers-make-progress-in-gravitational-wave-research/">Black Hole Researchers Make Progress in
Gravitational Wave
Research</a></li>
<li>Caltech: <a href="https://www.caltech.edu/about/news/physicists-create-new-model-of-ringing-black-holes">Physicists Create New Model of Ringing Black
Holes</a></li>
<li>Columbia University: <a href="https://news.columbia.edu/news/new-model-better-understand-whats-inside-colliding-black-holes">A New Model to Better Understand What’s Inside
Colliding Black
Holes</a></li>
<li>Johns Hopkins University: <a href="https://hub.jhu.edu/2023/02/22/hopkins-scientists-simulate-black-hole-collision/">Simulations show aftermath of black hole
collision</a></li>
<li>Keefe’s <a href="https://www.youtube.com/watch?v=4IJAf4UTwbA">60 Second Science: Keefe Mitman on Black Hole
Mergers</a></li>
</ul>Leo C. Steinlcstein@olemiss.eduOur latest paper, Nonlinearities in black hole ringdowns, was just published in Physical Review Letters! This article was selected as an ❦ Editors’ Suggestion, and Featured in APS’s Physics magazine. There were a number of other news stories covering this work: University of Mississippi: Black Hole Researchers Make Progress in Gravitational Wave Research Caltech: Physicists Create New Model of Ringing Black Holes Columbia University: A New Model to Better Understand What’s Inside Colliding Black Holes Johns Hopkins University: Simulations show aftermath of black hole collision Keefe’s 60 Second Science: Keefe Mitman on Black Hole MergersSloan Research Fellowship2023-02-15T15:00:00+00:002023-02-15T15:00:00+00:00https://duetosymmetry.com/news/Sloan-fellowship<p class="align-right" style="width: 250px"><img src="https://duetosymmetry.com/images/sloan-research-fellowships-2023-facebook.png" alt="" /></p>
<p>I am honored to have been selected as one of this year’s Sloan
Research Fellows! The Alfred P. Sloan Foundation awards these
fellowship annually (since 1955) through a highly competitive
application process. <a href="https://sloan.org/fellowships/2023-Fellows">This year, 126 early-career researchers were
named Sloan Fellows</a>. You
can see in Sloan’s <a href="https://sloan.org/fellows-database">fellows
database</a> the good company I’m in
(including undergrad friend <a href="https://chemistry.northwestern.edu/people/core-faculty/profiles/todd-gingrich.html">Todd
Gingrich</a>,
and grad school friend <a href="https://live-sas-physics.pantheon.sas.upenn.edu/people/standing-faculty/robyn-sanderson">Robyn
Sanderson</a>).
The Foundation writes (emphasis mine):</p>
<blockquote>
<p>[T]he Sloan Research Fellowships are one of the most competitive and
prestigious awards available to early-career researchers. They are
also often seen as a marker of the quality of an institution’s
science faculty and proof of an institution’s success in attracting
the most promising junior researchers to its ranks. <strong>This year
marks the first time a faculty member from University of Mississippi
has received a Sloan Research Fellowship</strong>—we want to extend our
congratulations and hope you’re as excited as we are!</p>
</blockquote>
<p>Thanks to the Sloan Foundation, my nominator, those who wrote letters
of support, and to all my colleagues who supported me over the years!</p>Leo C. Steinlcstein@olemiss.eduI am honored to have been selected as one of this year's Sloan Research Fellows!Nonlinearities in black hole ringdowns2022-08-17T00:00:00+00:002022-08-17T00:00:00+00:00https://duetosymmetry.com/pubs/Nonlinear-BH-ringdown<p>This article was selected as an ❦ Editors’ Suggestion, and <a href="https://physics.aps.org/articles/v16/29">Featured in
APS’s Physics magazine</a>.
<a href="/news/Nonlinear-paper-press/">More press coverage links here</a>.</p>
<p class="align-right" style="width: 400px"><img src="https://duetosymmetry.com/images/amp_vs_amp_both_sets_4panels.png" alt="" /></p>
<blockquote>
<p>The gravitational wave strain emitted by a perturbed black hole (BH)
ringing down is typically modeled analytically using first-order BH
perturbation theory. In this Letter, we show that second-order
effects are necessary for modeling ringdowns from BH merger
simulations. Focusing on the strain’s (ℓ,m)=(4,4) angular harmonic,
we show the presence of a quadratic effect across a range of binary
BH mass ratios that agrees with theoretical expectations. We find
that the quadratic (4, 4) mode’s amplitude exhibits quadratic
scaling with the fundamental (2, 2) mode—its parent mode. The
nonlinear mode’s amplitude is comparable to or even larger than that
of the linear (4, 4) mode. Therefore, correctly modeling the
ringdown of higher harmonics—improving mode mismatches by up to 2
orders of magnitude—requires the inclusion of nonlinear effects.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduThis article was selected as an ❦ Editors’ Suggestion, and Featured in APS’s Physics magazine. More press coverage links here.Fixing the BMS Frame of Numerical Relativity Waveforms with BMS Charges2022-08-10T00:00:00+00:002022-08-10T00:00:00+00:00https://duetosymmetry.com/pubs/BMS-fixing-charges<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/Comparing_EXT_CCE_errs.png" alt="" /></p>
<blockquote>
<p>The Bondi-van der Burg-Metzner-Sachs (BMS) group, which uniquely
describes the symmetries of asymptotic infinity and therefore of the
gravitational waves that propagate there, has become increasingly
important for accurate modeling of waveforms. In particular,
waveform models, such as post-Newtonian (PN) expressions, numerical
relativity (NR), and black hole perturbation theory, produce results
that are in different BMS frames. Consequently, to build a model for
the waveforms produced during the merging of compact objects, which
ideally would be a hybridization of PN, NR, and black hole
perturbation theory, one needs a fast and robust method for fixing
the BMS freedoms. In this work, we present the first means of fixing
the entire BMS freedom of NR waveforms to match the frame of either
PN waveforms or black hole perturbation theory. We achieve this by
finding the BMS transformations that change certain charges in a
prescribed way — e.g., finding the center-of-mass transformation
that maps the center-of-mass charge to a mean of zero. We find that
this new method is 20 times faster, and more correct when mapping to
the superrest frame, than previous methods that relied on
optimization algorithms. Furthermore, in the course of developing
this charge-based frame fixing method, we compute the PN expression
for the Moreschi supermomentum to 3PN order without spins and 2PN
order with spins. This Moreschi supermomentum is effectively
equivalent to the energy flux or the null memory contribution at
future null infinity ℐ⁺. From this PN calculation, we also compute
oscillatory (m≠0 modes) and spin-dependent memory terms that have
not been identified previously or have been missing from strain
expressions in the post-Newtonian literature.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduThe Bondi-van der Burg-Metzner-Sachs (BMS) group, which uniquely describes the symmetries of asymptotic infinity and therefore of the gravitational waves that propagate there, has become increasingly important for accurate modeling of waveforms. In particular, waveform models, such as post-Newtonian (PN) expressions, numerical relativity (NR), and black hole perturbation theory, produce results that are in different BMS frames. Consequently, to build a model for the waveforms produced during the merging of compact objects, which ideally would be a hybridization of PN, NR, and black hole perturbation theory, one needs a fast and robust method for fixing the BMS freedoms. In this work, we present the first means of fixing the entire BMS freedom of NR waveforms to match the frame of either PN waveforms or black hole perturbation theory. We achieve this by finding the BMS transformations that change certain charges in a prescribed way — e.g., finding the center-of-mass transformation that maps the center-of-mass charge to a mean of zero. We find that this new method is 20 times faster, and more correct when mapping to the superrest frame, than previous methods that relied on optimization algorithms. Furthermore, in the course of developing this charge-based frame fixing method, we compute the PN expression for the Moreschi supermomentum to 3PN order without spins and 2PN order with spins. This Moreschi supermomentum is effectively equivalent to the energy flux or the null memory contribution at future null infinity ℐ⁺. From this PN calculation, we also compute oscillatory (m≠0 modes) and spin-dependent memory terms that have not been identified previously or have been missing from strain expressions in the post-Newtonian literature.Gravitational-wave energy and other fluxes in ghost-free bigravity2022-08-03T00:00:00+00:002022-08-03T00:00:00+00:00https://duetosymmetry.com/pubs/bigravity-energy<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/radiation-cylinder.png" alt="" /></p>
<blockquote>
<p>One of the key ingredients for making binary waveform predictions in
a beyond-GR theory of gravity is understanding the energy and
angular momentum carried by gravitational waves and any other
radiated fields. Identifying the appropriate energy functional is
unclear in Hassan-Rosen bigravity, a ghost-free theory with one
massive and one massless graviton. The difficulty arises from the
new degrees of freedom and length scales which are not present in
GR, rendering an Isaacson-style averaging calculation ambiguous. In
this article we compute the energy carried by gravitational waves in
bigravity starting from the action, using the canonical current
formalism. The canonical current agrees with other common energy
calculations in GR, and is unambiguous (modulo boundary terms),
making it a convenient choice for quantifying the energy of
gravitational waves in bigravity or any diffeomorphism-invariant
theories of gravity. This calculation opens the door for future
waveform modeling in bigravity to correctly include backreaction due
to emission of gravitational waves.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduOne of the key ingredients for making binary waveform predictions in a beyond-GR theory of gravity is understanding the energy and angular momentum carried by gravitational waves and any other radiated fields. Identifying the appropriate energy functional is unclear in Hassan-Rosen bigravity, a ghost-free theory with one massive and one massless graviton. The difficulty arises from the new degrees of freedom and length scales which are not present in GR, rendering an Isaacson-style averaging calculation ambiguous. In this article we compute the energy carried by gravitational waves in bigravity starting from the action, using the canonical current formalism. The canonical current agrees with other common energy calculations in GR, and is unambiguous (modulo boundary terms), making it a convenient choice for quantifying the energy of gravitational waves in bigravity or any diffeomorphism-invariant theories of gravity. This calculation opens the door for future waveform modeling in bigravity to correctly include backreaction due to emission of gravitational waves.Tidally-induced nonlinear resonances in EMRIs with an analogue model2022-03-18T00:00:00+00:002022-03-18T00:00:00+00:00https://duetosymmetry.com/pubs/EMRI-tidal-resonance<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/poincare_4d.png" alt="" /></p>
<blockquote>
<p>One of the important targets for the future space-based
gravitational wave observatory LISA is extreme mass ratio inspirals
(EMRIs), where long and accurate waveform modeling is necessary for
detection and characterization. Modeling EMRI dynamics requires
accounting for effects such as the ones induced by an external tidal
field, which can break integrability at resonances and cause
significant dephasing. In this paper, we use a Newtonian analogue of
a Kerr black hole to study the effect of an external tidal field on
the dynamics and the gravitational waveform. We have developed a
numerical framework that takes advantage of the integrability of the
background system to evolve it with a symplectic splitting
integrator, and compute approximate gravitational waveforms to
estimate the timescale over which the perturbation affects the
dynamics. Comparing this timescale with the characteristic time
under radiation reaction at resonance, we introduce a tool for
quantifying the regime in which tidal effects might be included when
modeling EMRI gravitational waves. As an application of this
framework, we perform a detailed analysis of the dynamics at one
resonance to show how different entry points into the resonance in
phase-space can produce substantially different dynamics, and how
one can estimate bounds for the parameter space where tidal effects
may become dominant. Such bounds will scale as <script type="math/tex">\varepsilon \gtrsim
C q</script>, where <script type="math/tex">\varepsilon</script> measures the strength of the external
tidal field, <script type="math/tex">q</script> is the mass ratio, and <script type="math/tex">C</script> is a number which
depends on the resonance and the shape of the tide. We demonstrate
how to estimate C using our framework for the 2:3 radial to polar
frequency resonance in our model system. This framework can serve as
a proxy for proper modeling of the tidal perturbation in the fully
relativistic case.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduOne of the important targets for the future space-based gravitational wave observatory LISA is extreme mass ratio inspirals (EMRIs), where long and accurate waveform modeling is necessary for detection and characterization. Modeling EMRI dynamics requires accounting for effects such as the ones induced by an external tidal field, which can break integrability at resonances and cause significant dephasing. In this paper, we use a Newtonian analogue of a Kerr black hole to study the effect of an external tidal field on the dynamics and the gravitational waveform. We have developed a numerical framework that takes advantage of the integrability of the background system to evolve it with a symplectic splitting integrator, and compute approximate gravitational waveforms to estimate the timescale over which the perturbation affects the dynamics. Comparing this timescale with the characteristic time under radiation reaction at resonance, we introduce a tool for quantifying the regime in which tidal effects might be included when modeling EMRI gravitational waves. As an application of this framework, we perform a detailed analysis of the dynamics at one resonance to show how different entry points into the resonance in phase-space can produce substantially different dynamics, and how one can estimate bounds for the parameter space where tidal effects may become dominant. Such bounds will scale as , where measures the strength of the external tidal field, is the mass ratio, and is a number which depends on the resonance and the shape of the tide. We demonstrate how to estimate C using our framework for the 2:3 radial to polar frequency resonance in our model system. This framework can serve as a proxy for proper modeling of the tidal perturbation in the fully relativistic case.High Precision Ringdown Modeling: Multimode Fits and BMS Frames2021-10-31T00:00:00+00:002021-10-31T00:00:00+00:00https://duetosymmetry.com/pubs/high-precision-ringdown<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/wrong-right-BMS-frame.png" alt="" /></p>
<blockquote>
<p>Quasi-normal mode (QNM) modeling is an invaluable tool for
characterizing remnant black holes, studying strong gravity, and
testing general relativity. Only recently have QNM studies begun to
focus on multimode fitting to numerical relativity strain waveforms.
As gravitational wave observatories become even more sensitive they
will be able to resolve higher-order modes. Consequently, multimode
QNM fits will be critically important, and in turn require a more
thorough treatment of the asymptotic frame at ℐ⁺. The first main
result of this work is a method for systematically fitting a QNM
model containing many modes to a numerical waveform produced using
Cauchy-characteristic extraction (CCE), a waveform extraction
technique which is known to resolve memory effects. We choose the
modes to model based on their power contribution to the residual
between numerical and model waveforms. We show that the all-mode
strain mismatch improves by a factor of ~10⁵ when using multimode
fitting as opposed to only fitting the (2, ±2,n) modes. Our most
significant result addresses a critical point that has been
overlooked in the QNM literature: the importance of matching the
Bondi-van der Burg-Metzner-Sachs (BMS) frame of the numerical
waveform to that of the QNM model. We show that by mapping the
numerical waveforms—which exhibit the memory effect—to a BMS frame
known as the super rest frame, there is an improvement of ~10⁵ in
the all-mode strain mismatch compared to using a strain waveform
whose BMS frame is not fixed. Furthermore, we find that by mapping
CCE waveforms to the super rest frame, we can obtain all-mode
mismatches that are, on average, a factor of ~4 better than using
the publicly-available extrapolated waveforms. We illustrate the
effectiveness of these modeling enhancements by applying them to
families of waveforms produced by numerical relativity and comparing
our results to previous QNM studies.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduQuasi-normal mode (QNM) modeling is an invaluable tool for characterizing remnant black holes, studying strong gravity, and testing general relativity. Only recently have QNM studies begun to focus on multimode fitting to numerical relativity strain waveforms. As gravitational wave observatories become even more sensitive they will be able to resolve higher-order modes. Consequently, multimode QNM fits will be critically important, and in turn require a more thorough treatment of the asymptotic frame at ℐ⁺. The first main result of this work is a method for systematically fitting a QNM model containing many modes to a numerical waveform produced using Cauchy-characteristic extraction (CCE), a waveform extraction technique which is known to resolve memory effects. We choose the modes to model based on their power contribution to the residual between numerical and model waveforms. We show that the all-mode strain mismatch improves by a factor of ~10⁵ when using multimode fitting as opposed to only fitting the (2, ±2,n) modes. Our most significant result addresses a critical point that has been overlooked in the QNM literature: the importance of matching the Bondi-van der Burg-Metzner-Sachs (BMS) frame of the numerical waveform to that of the QNM model. We show that by mapping the numerical waveforms—which exhibit the memory effect—to a BMS frame known as the super rest frame, there is an improvement of ~10⁵ in the all-mode strain mismatch compared to using a strain waveform whose BMS frame is not fixed. Furthermore, we find that by mapping CCE waveforms to the super rest frame, we can obtain all-mode mismatches that are, on average, a factor of ~4 better than using the publicly-available extrapolated waveforms. We illustrate the effectiveness of these modeling enhancements by applying them to families of waveforms produced by numerical relativity and comparing our results to previous QNM studies.