Jekyll2020-11-08T23:15:39+00:00https://www.briancseymour.com/feed.xmlBrian C. Seymour The personal website and blog of Brian Seymour. I am a physics PhD student at Caltech. I study general relativity, tests of modified gravity theories, and gravitational waves. Brian C. SeymourAutomated Natural Unit Conversions in Mathematica2019-06-17T00:00:00+00:002019-06-17T00:00:00+00:00https://www.briancseymour.com/Natural-Units<h3 id="introduction-and-review">Introduction and Review</h3>
<p>Natural units are a way to simplify repetitive fundamental constants in theoretical physics equations. In my research, I used three particular unit systems: geometric ($c=G=1$), natural ($c=\hbar=1$), and Planck units ($c=\hbar=G=1$). However, for me it is time consuming to convert to and from natural units. I decided that I would write an extension to Mathematica’s units package to remedy this.</p>
<p>A great summary of natural units goes in more detail in an excellent document <a href="https://www.seas.upenn.edu/~amyers/NaturalUnits.pdf">here</a>. Given a SI quantity $Q_{\text{SI}}$, the quantity in natural units is given by $Q_{\text{geometric}} = Q_{\text{SI}}/A$ for some factor $A$. For the rest of this post, I will use geometric units. Suppose that the quantity has SI units of the form</p>
\[[Q_{\text{SI}}] = M^\alpha \times L^\beta \times T^\gamma \, ,\]
<p>then, one can find $Q_{\text{geometric}}$ with the following factor</p>
\[A = G^{-\alpha} c^{2\alpha-\gamma} \, .\]
<p>So all we need to do to convert between the units is to find $A$.</p>
<h3 id="mathmatica-implementation">Mathmatica Implementation</h3>
<p>The first thing that we need to do is write a helper function that will return the $\alpha$, $\beta$, and $\gamma$ from our previous equation. All I need to do was write a wrapper class for the UnitDimensions[] fuction that Mathematica has.</p>
<figure class="highlight"><pre><code class="language-mathematica" data-lang="mathematica"><span class="nv">GetUnitVal</span><span class="p">[</span><span class="nv">qu</span><span class="o">_,</span><span class="w"> </span><span class="nv">unit</span><span class="o">_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="bp">Module</span><span class="p">[{</span><span class="nv">result</span><span class="o">,</span><span class="w"> </span><span class="nv">dims</span><span class="p">}</span><span class="o">,</span><span class="w">
</span><span class="nv">dims</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">UnitDimensions</span><span class="p">[</span><span class="nv">qu</span><span class="p">]</span><span class="o">;</span><span class="w">
</span><span class="nb">If</span><span class="p">[</span><span class="nb">Position</span><span class="p">[</span><span class="nv">dims</span><span class="o">,</span><span class="w"> </span><span class="nv">unit</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="p">{}</span><span class="o">,</span><span class="w"> </span><span class="nv">result</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="o">,</span><span class="w">
</span><span class="nv">result</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nv">dims</span><span class="p">[[</span><span class="nb">Position</span><span class="p">[</span><span class="nv">dims</span><span class="o">,</span><span class="w"> </span><span class="nv">unit</span><span class="p">][[</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="p">]]]</span><span class="o">;</span><span class="w">
</span><span class="nv">result</span><span class="w">
</span><span class="p">]</span></code></pre></figure>
<p>Next, I wrote a function to calculate the constant $A$.</p>
<figure class="highlight"><pre><code class="language-mathematica" data-lang="mathematica"><span class="nv">SItoGeometricDivideFactor</span><span class="p">[</span><span class="nv">qu</span><span class="o">_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="bp">Module</span><span class="p">[{</span><span class="nv">\[Alpha]</span><span class="o">,</span><span class="w"> </span><span class="nv">\[Beta]</span><span class="o">,</span><span class="w"> </span><span class="nv">\[Gamma]</span><span class="p">}</span><span class="o">,</span><span class="w">
</span><span class="nv">\[Alpha]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nv">GetUnitVal</span><span class="p">[</span><span class="nv">qu</span><span class="o">,</span><span class="w"> </span><span class="s">"MassUnit"</span><span class="p">]</span><span class="o">;</span><span class="w"> </span><span class="nv">\[Beta]</span><span class="w"> </span><span class="o">=</span><span class="w">
</span><span class="nv">GetUnitVal</span><span class="p">[</span><span class="nv">qu</span><span class="o">,</span><span class="w"> </span><span class="s">"LengthUnit"</span><span class="p">]</span><span class="o">;</span><span class="w"> </span><span class="nv">\[Gamma]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nv">GetUnitVal</span><span class="p">[</span><span class="nv">qu</span><span class="o">,</span><span class="w"> </span><span class="s">"TimeUnit"</span><span class="p">]</span><span class="o">;</span><span class="w">
</span><span class="nb">Quantity</span><span class="p">[</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="s">"GravitationalConstant"</span><span class="p">]</span><span class="o">^-</span><span class="nv">\[Alpha]</span><span class="o">*</span><span class="w">
</span><span class="nb">Quantity</span><span class="p">[</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="s">"SpeedOfLight"</span><span class="p">]</span><span class="o">^</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="nv">\[Alpha]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nv">\[Gamma]</span><span class="p">)</span><span class="w">
</span><span class="p">]</span></code></pre></figure>
<p>Finally, I used a function that will find $Q_{\text{geometric}}$ by dividing $Q_{\text{SI}}$ by $A$. Afterwards, it converts the result into meters to the $n$ power as is used in geometric units.</p>
<figure class="highlight"><pre><code class="language-mathematica" data-lang="mathematica"><span class="nv">SItoGeometricUnits</span><span class="p">[</span><span class="nv">qu</span><span class="o">_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="bp">Module</span><span class="p">[{</span><span class="nv">factor</span><span class="o">,</span><span class="w"> </span><span class="nv">geoqu</span><span class="p">}</span><span class="o">,</span><span class="w">
</span><span class="nv">factor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nv">SItoGeometricDivideFactor</span><span class="p">[</span><span class="nv">qu</span><span class="p">]</span><span class="o">;</span><span class="w"> </span><span class="nv">geoqu</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nv">qu</span><span class="o">/</span><span class="nv">factor</span><span class="o">;</span><span class="w">
</span><span class="nb">UnitConvert</span><span class="p">[</span><span class="nv">geoqu</span><span class="o">,</span><span class="w"> </span><span class="p">(</span><span class="s">"Meters"</span><span class="p">)</span><span class="o">^</span><span class="nv">GetUnitVal</span><span class="p">[</span><span class="nv">geoqu</span><span class="o">,</span><span class="w"> </span><span class="s">"LengthUnit"</span><span class="p">]]</span><span class="w">
</span><span class="p">]</span></code></pre></figure>
<p>The full version of these functions which includes geometric, natural, and Planck units is <a href="https://github.com/BrianCSeymour/natural-units-mathematica-conversion">publicly available</a> on my Github. Feel free to use it and I hope that it will save you time in your research!</p>Brian C. SeymourIntroduction and ReviewSigma Pi Sigma Presentation2018-11-09T00:00:00+00:002018-11-09T00:00:00+00:00https://www.briancseymour.com/Sigma-Pi-Sigma-Presentation<p>I gave a presentation to a conference organized at Sigma Pi Sigma at University of Virginia.</p>
<p>You can <a href="/assets/documents/Presentations/11-9-18-SigmaPiSigma.pdf">view the slides here</a> directly.</p>Brian C. SeymourI gave a presentation to a conference organized at Sigma Pi Sigma at University of Virginia.Gravity Group Presentation2018-11-01T00:00:00+00:002018-11-01T00:00:00+00:00https://www.briancseymour.com/Gravity-Group-Meeting<p>I gave an hour long presentation to the gravity research group at University of Virginia. It specializes on content from my upcoming paper.</p>
<p>You can <a href="/assets/documents/Presentations/11-1-18-Gravity-Group-Presentation.pdf">view the slides here</a> directly.</p>Brian C. SeymourI gave an hour long presentation to the gravity research group at University of Virginia. It specializes on content from my upcoming paper.Society of Physics Students Research Introduction2018-10-19T00:00:00+00:002018-10-19T00:00:00+00:00https://www.briancseymour.com/SPS-Presentation<p>I gave a five minute presentation to Society of Physics Students about my research. This was to encourage first year students to get involved in research.</p>
<p>You can <a href="/assets/documents/Presentations/SPS/SPS-Research-Presentation.pdf">view the slides here</a> directly.</p>Brian C. SeymourI gave a five minute presentation to Society of Physics Students about my research. This was to encourage first year students to get involved in research.College Science Scholar Presentation2018-10-17T00:00:00+00:002018-10-17T00:00:00+00:00https://www.briancseymour.com/CSS-Poster<p>I had the pleasure of presenting my research to a group of first year students who are interested in scientific research. I gave an introduction to my research with a poster describing black hole-pulsar tests of general relativity. I used the example of a theory of gravity with varying gravitational constant to show the process of bounding the time derivative of G.</p>
<p>You can <a href="/assets/documents/Presentations/CSS/CSS-Poster-BrianSeymour.pdf">view the poster here</a> directly.</p>Brian C. SeymourI had the pleasure of presenting my research to a group of first year students who are interested in scientific research. I gave an introduction to my research with a poster describing black hole-pulsar tests of general relativity. I used the example of a theory of gravity with varying gravitational constant to show the process of bounding the time derivative of G.Testing General Relativity with Black Hole-Pulsar Binaries2018-08-01T00:00:00+00:002018-08-01T00:00:00+00:00https://www.briancseymour.com/Testing-GR-with-BH-PSR-Binaries<p>Here is my upcoming paper describing tests of general relativity using black hole-pulsar binaries if one is found with next generation telescopes.</p>
<p>You can view the paper at <a href="https://arxiv.org/abs/1808.00080">1808.00080</a>.</p>
<p>The abstract is as follows:</p>
<p>Binary pulsars allow us to carry out precision tests of gravity and have placed stringent bounds on a broad class of theories beyond general relativity. Current and future radio telescopes, such as FAST, SKA, and MeerKAT, may find a new astrophysical system, a pulsar orbiting around a black hole, which will provide us a new source for probing gravity. In this paper, we systematically study the prospects of testing general relativity with such black hole-pulsar binaries. We begin by finding a mapping between generic non-Einsteinian parameters in the orbital decay rate and theoretical constants in various modified theories of gravity and then summarize this mapping with a ready-to-use list. Theories we study here include scalar-tensor theories, varying G theories, massive gravity theories, generic screening gravity and quadratic curvature-corrected theories. We next use simulated measurement accuracy of the orbital decay rate for black hole-pulsar binaries with FAST/SKA and derive projected upper bounds on the above generic non-Einsteinian parameters. We find that such bounds from black hole-pulsars can be stronger than those from neutron star-pulsar and neutron star-white dwarf binaries by a few orders of magnitude when the correction enters at negative post-Newtonian orders. By mapping such bounds on generic parameters to those on various modified theories of gravity, we find that one can constrain the amount of time variation in Newton’s constant G to be comparable to or slightly weaker than than the current strongest bound from solar system experiments, though the former bounds are complementary to the latter since they probe different regime of gravity. We also study how well one can probe quadratic gravity from black hole quadrupole moment measurements of black hole-pulsars. We find that bounds on the parity-violating sector of quadratic gravity can be stronger than current bounds by six orders of magnitude. These results suggest that a new discovery of black hole-pulsars in the future will provide powerful ways to probe gravity further.</p>Brian C. SeymourHere is my upcoming paper describing tests of general relativity using black hole-pulsar binaries if one is found with next generation telescopes.LIGO SURF Final Report2017-09-23T00:00:00+00:002017-09-23T00:00:00+00:00https://www.briancseymour.com/LIGO-Final-Report<p>I spent the 2017 summer working with Marie Kasprazack, Arnaud Pele, and Adam Mullavey at LIGO Livingston through the Caltech SURF Program. I examined nonlinear angular noise coupling into differential arm length of the LIGO Livingston detector. This could be summarized in the following way: I modeled the change in optical path length in the interferometer cavity arms due to misalignment of the cavity mirrors. I did this through an geometric analysis of the change in path length of a Fabry-Perot cavity (such as LIGO uses).</p>
<p>You can <a href="/assets/documents/Papers/LIGO-SURF/LIGOSURF_FinalReport_BrianSeymour.pdf">view the final report here</a> directly.</p>
<p>Alternatively, the full directory containing my work that summer can be found <a href="https://dcc.ligo.org/LIGO-T1700343/public">here</a>.</p>Brian C. SeymourI spent the 2017 summer working with Marie Kasprazack, Arnaud Pele, and Adam Mullavey at LIGO Livingston through the Caltech SURF Program. I examined nonlinear angular noise coupling into differential arm length of the LIGO Livingston detector. This could be summarized in the following way: I modeled the change in optical path length in the interferometer cavity arms due to misalignment of the cavity mirrors. I did this through an geometric analysis of the change in path length of a Fabry-Perot cavity (such as LIGO uses).LIGO SURF Final Presentation2017-08-24T00:00:00+00:002017-08-24T00:00:00+00:00https://www.briancseymour.com/LIGO-Final-Presentation<p>I presented the results of my research at LIGO from the summer of 2017 at Caltech for the SURF final presentation.</p>
<p>You can <a href="/assets/documents/Presentations/LIGO-SURF/BrianSeymour_FinalPresentation_LIGO_SURF.pdf">view the final report here</a> directly.</p>
<p>Alternatively, the full directory containing my work that summer can be found <a href="https://dcc.ligo.org/LIGO-T1700343/public">here</a>.</p>Brian C. SeymourI presented the results of my research at LIGO from the summer of 2017 at Caltech for the SURF final presentation.Hello!2000-01-01T00:00:00+00:002000-01-01T00:00:00+00:00https://www.briancseymour.com/First-Post<p>I made this website on 10/21/2018 (though I have retroactively added some materials). I look forward to adding to it in the future!</p>Brian C. SeymourI made this website on 10/21/2018 (though I have retroactively added some materials). I look forward to adding to it in the future!