Jekyll2024-07-09T03:35:35+00:00https://duetosymmetry.com/Leo C. SteinPhysics Professor @ U of MS. Specializing in gravity, general relativity, black holes, gravitational waves, numerical relativityLeo C. Steinlcstein@olemiss.eduA Review of Gravitational Memory and BMS Frame Fixing in Numerical Relativity2024-05-16T00:00:00+00:002024-05-16T00:00:00+00:00https://duetosymmetry.com/pubs/memory-BMS-review<p class="align-right" style="width: 350px; margin: 2em 0 0 1em;"><img src="https://duetosymmetry.com/images/ScriCylinderST.jpg" alt="" /></p>
<blockquote>
<p>Gravitational memory effects and the BMS freedoms exhibited at
future null infinity have recently been resolved and utilized in
numerical relativity simulations. With this, gravitational wave
models and our understanding of the fundamental nature of general
relativity have been vastly improved. In this paper, we review the
history and intuition behind memory effects and BMS symmetries, how
they manifest in gravitational waves, and how controlling the
infinite number of BMS freedoms of numerical relativity simulations
can crucially improve the waveform models that are used by
gravitational wave detectors. We reiterate the fact that, with
memory effects and BMS symmetries, not only can these
next-generation numerical waveforms be used to observe
never-before-seen physics, but they can also be used to test GR and
learn new astrophysical information about our universe.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduGravitational memory effects and the BMS freedoms exhibited at future null infinity have recently been resolved and utilized in numerical relativity simulations. With this, gravitational wave models and our understanding of the fundamental nature of general relativity have been vastly improved. In this paper, we review the history and intuition behind memory effects and BMS symmetries, how they manifest in gravitational waves, and how controlling the infinite number of BMS freedoms of numerical relativity simulations can crucially improve the waveform models that are used by gravitational wave detectors. We reiterate the fact that, with memory effects and BMS symmetries, not only can these next-generation numerical waveforms be used to observe never-before-seen physics, but they can also be used to test GR and learn new astrophysical information about our universe.Imprints of Changing Mass and Spin on Black Hole Ringdown2024-04-22T00:00:00+00:002024-04-22T00:00:00+00:00https://duetosymmetry.com/pubs/changing-mass-spin-ringdown<p class="align-right" style="width: 350px; margin: 2em 0 0 1em;"><img src="https://duetosymmetry.com/images/nonlinear-phase-diff.png" alt="" /></p>
<blockquote>
<p>We numerically investigate the imprints of gravitational
radiation-reaction driven changes to a black hole’s mass and spin on
the corresponding ringdown waveform. We do so by comparing the
dynamics of a perturbed black hole evolved with the full (nonlinear)
versus linearized Einstein equations. As expected, we find that the
quasinormal mode amplitudes extracted from nonlinear evolution
deviate from their linear counterparts at third order in initial
perturbation amplitude. For perturbations leading to a change in the
black hole mass and spin of ∼5%, which is reasonable for a remnant
formed in an astrophysical merger, we find that nonlinear
distortions to the complex amplitudes of some quasinormal modes can
be as large as ∼50% at the peak of the waveform. Furthermore, the
change in the mass and spin results in a drift in the quasinormal
mode frequencies, which for large amplitude perturbations causes the
nonlinear waveform to rapidly dephase with respect to its linear
counterpart. These two nonlinear effects together create a large
distortion in both the amplitude and phase of the ringdown
gravitational waveform. Surprisingly, despite these nonlinear
effects creating significant deviations in the nonlinear waveform,
we show that a linear quasinormal mode model still performs quite
well from close to the peak amplitude onwards.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduWe numerically investigate the imprints of gravitational radiation-reaction driven changes to a black hole’s mass and spin on the corresponding ringdown waveform. We do so by comparing the dynamics of a perturbed black hole evolved with the full (nonlinear) versus linearized Einstein equations. As expected, we find that the quasinormal mode amplitudes extracted from nonlinear evolution deviate from their linear counterparts at third order in initial perturbation amplitude. For perturbations leading to a change in the black hole mass and spin of ∼5%, which is reasonable for a remnant formed in an astrophysical merger, we find that nonlinear distortions to the complex amplitudes of some quasinormal modes can be as large as ∼50% at the peak of the waveform. Furthermore, the change in the mass and spin results in a drift in the quasinormal mode frequencies, which for large amplitude perturbations causes the nonlinear waveform to rapidly dephase with respect to its linear counterpart. These two nonlinear effects together create a large distortion in both the amplitude and phase of the ringdown gravitational waveform. Surprisingly, despite these nonlinear effects creating significant deviations in the nonlinear waveform, we show that a linear quasinormal mode model still performs quite well from close to the peak amplitude onwards.Can a radiation gauge be horizon-locking?2024-04-16T00:00:00+00:002024-04-16T00:00:00+00:00https://duetosymmetry.com/pubs/IRG-horizon-locking<blockquote>
<p>In this short Note, I answer the titular question: Yes, a radiation
gauge can be horizon-locking. Radiation gauges are very common in
black hole perturbation theory. It’s also very convenient if a gauge
choice is horizon-locking, i.e. the location of the horizon is not
moved by a linear metric perturbation. Therefore it is doubly
convenient that a radiation gauge can be horizon-locking, when some
simple criteria are satisfied. Though the calculation is
straightforward, it seemed useful enough to warrant writing this
Note. Finally I show an example: the ℓ vector of the Hartle–Hawking
tetrad in Kerr satisfies all the conditions for ingoing radiation
gauge to keep the future horizon fixed.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduIn this short Note, I answer the titular question: Yes, a radiation gauge can be horizon-locking. Radiation gauges are very common in black hole perturbation theory. It’s also very convenient if a gauge choice is horizon-locking, i.e. the location of the horizon is not moved by a linear metric perturbation. Therefore it is doubly convenient that a radiation gauge can be horizon-locking, when some simple criteria are satisfied. Though the calculation is straightforward, it seemed useful enough to warrant writing this Note. Finally I show an example: the ℓ vector of the Hartle–Hawking tetrad in Kerr satisfies all the conditions for ingoing radiation gauge to keep the future horizon fixed.Optimizing post-Newtonian parameters and fixing the BMS frame for numerical-relativity waveform hybridizations2024-03-19T00:00:00+00:002024-03-19T00:00:00+00:00https://duetosymmetry.com/pubs/Hybridize-PN-BMS<p class="align-right" style="width: 350px; margin: 2em 0 0 1em;"><img src="https://duetosymmetry.com/images/PN-NR-handshake-1920.png" alt="" /></p>
<blockquote>
<p>Numerical relativity (NR) simulations of binary black holes provide
precise waveforms, but are typically too computationally expensive
to produce waveforms with enough orbits to cover the whole frequency
band of gravitational-wave observatories. Accordingly, it is
important to be able to hybridize NR waveforms with analytic,
post-Newtonian (PN) waveforms, which are accurate during the early
inspiral phase. We show that to build such hybrids, it is crucial to
both fix the Bondi-Metzner-Sachs (BMS) frame of the NR waveforms to
match that of PN theory, and optimize over the PN parameters. We
test such a hybridization procedure including all spin-weighted
spherical harmonic modes with |m|≤ℓ for ℓ≤8, using 29 NR waveforms
with mass ratios q≤10 and spin magnitudes |χ₁|,|χ₂|≤0.8. We find
that for spin-aligned systems, the PN and NR waveforms agree very
well. The difference is limited by the small nonzero orbital
eccentricity of the NR waveforms, or equivalently by the lack of
eccentric terms in the PN waveforms. To maintain full accuracy of
the simulations, the matching window for spin-aligned systems should
be at least 5 orbits long and end at least 15 orbits before
merger. For precessing systems, the errors are larger than for
spin-aligned cases. The errors are likely limited by the absence of
precession-related spin-spin PN terms. Using 10⁵M long NR waveforms,
we find that there is no optimal choice of the matching window
within this time span, because the hybridization result for
precessing cases is always better if using earlier or longer
matching windows. We provide the mean orbital frequency of the
smallest acceptable matching window as a function of the target
error between the PN and NR waveforms and the black hole spins.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduNumerical relativity (NR) simulations of binary black holes provide precise waveforms, but are typically too computationally expensive to produce waveforms with enough orbits to cover the whole frequency band of gravitational-wave observatories. Accordingly, it is important to be able to hybridize NR waveforms with analytic, post-Newtonian (PN) waveforms, which are accurate during the early inspiral phase. We show that to build such hybrids, it is crucial to both fix the Bondi-Metzner-Sachs (BMS) frame of the NR waveforms to match that of PN theory, and optimize over the PN parameters. We test such a hybridization procedure including all spin-weighted spherical harmonic modes with |m|≤ℓ for ℓ≤8, using 29 NR waveforms with mass ratios q≤10 and spin magnitudes |χ₁|,|χ₂|≤0.8. We find that for spin-aligned systems, the PN and NR waveforms agree very well. The difference is limited by the small nonzero orbital eccentricity of the NR waveforms, or equivalently by the lack of eccentric terms in the PN waveforms. To maintain full accuracy of the simulations, the matching window for spin-aligned systems should be at least 5 orbits long and end at least 15 orbits before merger. For precessing systems, the errors are larger than for spin-aligned cases. The errors are likely limited by the absence of precession-related spin-spin PN terms. Using 10⁵M long NR waveforms, we find that there is no optimal choice of the matching window within this time span, because the hybridization result for precessing cases is always better if using earlier or longer matching windows. We provide the mean orbital frequency of the smallest acceptable matching window as a function of the target error between the PN and NR waveforms and the black hole spins.Notes: 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>
<li><a href="#magnetostatic-multipole-expansion" id="markdown-toc-magnetostatic-multipole-expansion">Magnetostatic 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.