Jekyll2020-05-27T04:25:58+00:00https://dnichols1.github.io/feed.xmlDavid A. NicholsMy professional websiteDavid A. NicholsHiring a Postdoc for the Fall 2020 Semester2019-10-10T00:00:00+00:002019-10-10T00:00:00+00:00https://dnichols1.github.io/news/hiring-a-postdoc<p>I have recently begun a search for a postdoc in gravitational physics who
will join my new group in the larger gravity group in the Department of
Physics at University of Virginia (UVA) starting in the fall semester of 2020
(the precise date is flexible).
The details about the position can be found in the job ad on
<a href="https://uva.wd1.myworkdayjobs.com/en-US/UVAJobs/job/Charlottesville-VA/Research-Associate-in-Gravitational-Physics_R0011062">UVA Workday website</a>.
I will begin reviewing applications on December 1, 2019.
If you know of promising candidates, then please tell them about this job
opportunity.</p>David A. NicholsI have recently begun a search for a postdoc in gravitational physics who will join my new group in the larger gravity group in the Department of Physics at University of Virginia (UVA) starting in the fall semester of 2020 (the precise date is flexible). The details about the position can be found in the job ad on UVA Workday website. I will begin reviewing applications on December 1, 2019. If you know of promising candidates, then please tell them about this job opportunity.New Position and New Website2019-08-09T00:00:00+00:002019-08-09T00:00:00+00:00https://dnichols1.github.io/news/new-position-and-new-website<p>On July 25, 2019, I moved from the University of Amsterdam (UvA) in the Netherlands to begin a new position as an Assistant Professor of Physics at the University of Virginia (UVA) in Charlottesville.
I am excited to begin teaching and starting my research group at UVA!</p>
<p>I also took this as an opportunity to update my website to a more modern format.
I have joined the many people who use <a href="https://jekyllrb.com/">Jekyll</a> and the <a href="https://mademistakes.com/work/minimal-mistakes-jekyll-theme/">Minimal Mistakes</a> theme.</p>David A. NicholsOn July 25, 2019, I moved from the University of Amsterdam (UvA) in the Netherlands to begin a new position as an Assistant Professor of Physics at the University of Virginia (UVA) in Charlottesville. I am excited to begin teaching and starting my research group at UVA!Paper: Persistent Gravitational-Wave Observables: General Framework2019-01-23T00:00:00+00:002019-01-23T00:00:00+00:00https://dnichols1.github.io/papers/persistent-observables<p>This post summarizes a paper about three new procedures by which one can measure generalized notions of gravitational-wave memory.
The paper was an Editors’ Suggestion in Phys. Rev. D, and it had a
<a href="https://physics.aps.org/synopsis-for/10.1103/PhysRevD.99.084044">synopsis</a>
published about it in <a href="https://physics.aps.org/">Physics</a>.</p>
<h2 id="paper-highlighted">Paper Highlighted</h2>
<ul>
<li>E. E. Flanagan, A. M. Grant, A. I. Harte, <strong>D. A. Nichols</strong>.
“Persistent gravitational-wave observables: General framework.”
<a href="https://dx.doi.org/10.1103/PhysRevD.99.084044">Phys. Rev. D 99, 084004 (2019)</a>,
<a href="https://arxiv.org/abs/1901.00021">arXiv:1901.00021</a>.</li>
</ul>
<h2 id="summary-of-the-paper">Summary of the Paper</h2>
<p><img src="/images/SpinObservable.png" alt="" class="align-left" />
The gravitational-wave memory effect is often characterized by the lasting displacement it would cause between freely falling observers after a burst of gravitational waves pass by their locations.
Subsequently, it was realized that there other types of memory effects that freely falling observers could measure, including lasting relative velocities, changes in proper time elapsed, and relative rotation of inertial gyroscopes.
We were interested in determining procedures that observers could, in principle implement, by which observers could meaure all these memory effects (and potentially other new effects).
We in fact found three such types of procedures, which encompass the known memory effects, and potentially other new ones.
The first procedure involved measuring a type of deviation vector between two neary accelerating observers.
The second involves transporting a certain kind of linear and angular momentum around a closed curve in spacetime.
The third is based on measuring the location, linear momentum, and intrinsic angular momentum of a nearby spinning point particle.
We called the outcome of these measurement procedures “persistent gravitational-wave observables,” and we are currently investigating their properties in specific gravitational-wave spacetimes.</p>David A. NicholsThis post summarizes a paper about three new procedures by which one can measure generalized notions of gravitational-wave memory. The paper was an Editors’ Suggestion in Phys. Rev. D, and it had a synopsis published about it in Physics.Paper: Constraining the Properties of the Progenitor of GW1708172018-08-11T00:00:00+00:002018-08-11T00:00:00+00:00https://dnichols1.github.io/papers/gw170817-bns-vs-nsbh<p>This post summarizes a paper about how well one can distinguish whether the progenitor of the GW170817 event was a binary neutron star or a neutron-star black-hole binary.</p>
<h2 id="paper-highlighted">Paper Highlighted</h2>
<ul>
<li>T. Hinderer, S. Nissanke, F. Foucart, K. Hotokezaka, T. Vincent, M. Kasliwal, P. Schmidt, A. R. Williamson, <strong>D. A. Nichols</strong>, M. Duez, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel.
“Discerning the binary neutron star or neutron star-black hole nature of GW170817 with Gravitational Wave and Electromagnetic Measurements.”
<a href="https://arxiv.org/abs/1808.03836">arXiv:1808.03836</a>.</li>
</ul>
<h2 id="summary-of-the-paper">Summary of the Paper</h2>
<p><img src="/images/MremQplot.png" alt="" class="align-left" />
The gravitational-wave (GW) event GW170817 was followed by electromagnetic (EM) observations of the system over a broad range of the EM spectrum from gamma rays to radio waves.
The GWs were consistent with the merger of two objects with masses typical of neutron stars, and the EM observations were those expected if at least one of the objects were a neutron star.
Thus, while a binary neutron star is the favored progenitor for the system, it has not been definitively shown that this is the only possible binary progenitor.
In this paper, we investigated whether a binary composed of a low mass black hole and a neutron star could explain the observed GW and EM signals.
Using new numerical relativity simulations of black-hole neutron-star binaries, we showed that both the GWs and kilonova emission from an unequal-mass binary could be consistent with the observed emission from GW170817.
When we combined information from the GW measurements with predictions of the remaining mass around a black hole following its merger with a neutron star, we could put conservative constraints on the regions of parameter space in which the system could have been a black-hole neutron-star binary.
Surprisingly, there was still a substantial region that could be consistent with a neutron-star black-hole system.</p>David A. NicholsThis post summarizes a paper about how well one can distinguish whether the progenitor of the GW170817 event was a binary neutron star or a neutron-star black-hole binary.Research Theme: Extended BMS Algebra, Spin and Center-of-Mass Memory Effects2018-07-23T00:00:00+00:002018-07-23T00:00:00+00:00https://dnichols1.github.io/papers/bms-spin-cm-memories<p>This post summarizes three papers related to the conserved quantities of an extension of the Bondi-Metzner-Sachs (BMS) group and two related new memory effects.</p>
<h2 id="papers-highlighted">Papers Highlighted</h2>
<ul>
<li>
<p><strong>D. A. Nichols</strong>.
“Center-of-mass angular momentum and memory effect in asymptotically flat spacetimes.”
<a href="https://doi.org/10.1103/PhysRevD.98.064032">Phys. Rev. D 98, 064032 (2018)</a>,
<a href="https://arxiv.org/abs/1807.08767">arXiv:1807.08767</a>.</p>
</li>
<li>
<p><strong>D. A. Nichols</strong>.
“Spin memory effect for compact binaries in the post-Newtonian approximation.”
<a href="https://doi.org/10.1103/PhysRevD.95.084048">Phys. Rev. D 95, 084048 (2017)</a>,
<a href="http://arxiv.org/abs/1702.03300">arxiv:1702.03300</a>.</p>
</li>
<li>
<p>E. E. Flanagan and <strong>D. A. Nichols</strong>.
“Conserved charges of the extended Bondi-Metzner-Sachs algebra.”
<a href="http://dx.doi.org/10.1103/PhysRevD.95.044002">Phys. Rev. D 95, 044002 (2017)</a>,
<a href="http://arxiv.org/abs/1510.03386">arxiv:1510.03386</a>.</p>
</li>
</ul>
<h2 id="summary-of-the-papers">Summary of the Papers</h2>
<p><img src="/images/SpinMemMode.png" alt="" class="align-left" />
The Bondi-Metzner-Sachs (BMS) group is the symmetry group of asymptotically flat spacetimes, and it consists of the Lorentz transformations and the supertranslations (an infinite-dimensional commutative group that has the spacetime translations as a subgroup).
There recently was a proposal that a larger symmetry algebra than the BMS algebra could be relevant.
This new algebra was called the extended BMS algebra, and it included the infinite number of conformal Killing vectors on the two-sphere, not just the smooth six-parameter family of Lorentz transformations.
We computed the conserved quantities corresponding to these new symmetries in a somewhat more general context than before, and we found that they are finite and that they have an electric- and a magnetic-parity part.
These two parts are generalizations of the spin angular momentum and the center-of-mass parts of the relativistic angular momentum (the superspin and super-center-of-mass charges, respectively).
The change in the superspin charges can be a source of a new kind of memory effect, which had been proposed earlier, called the spin memory effect.</p>
<p>Another source of the spin memory effect is the effective angular momentum per solid angle that is radiated in gravitational waves.
I recently computed the expansion of this effective angular momentum flux in terms of radiative moments of the gravitational-wave strain, thereby deriving a quadrupole-like formula for the spin memory.
I then specialized to compact binary sources in the post-Newtonian approximation.
In this limit, I found that the spin memory has a substantial slow secular growth during the inspiral stage of a compact binary.
In addition, the rate of accumulation of the spin memory is related to a non-hereditary, nonlinear, and non-oscillatory contribution to the gravitational waveform that appears at a high order in the post-Newtonian expansion.
This rate of accumulation of the spin memory might be detectable by third-generation gravitational-wave detectors, such as the Einstein Telescope, by coherently adding the spin memory signals from hundreds of gravitational-wave observations of black-hole-binary mergers.</p>
<p>I also showed that changes in the super-center-of-mass charges can also produce a new type of memory effect, which I called the center-of-mass memory effect.
Like the spin memory effect, it produces a lasting change in a quantity related to the time integral of the gravitational-wave strain.
There are terms in the gravitational waves related to the center-of-mass memory effect.
These gravitational-wave terms, however, will be weaker than those related to the spin memory effect.
As a result, it is less likely that the next generation of gravitational-wave detectors will be able to find evidence for the effect.
It should, however, appear in the post-Newtonian expansion of the gravitational waveform, when the gravtiational waveform is computed at fourth post-Newtonian order.</p>David A. NicholsThis post summarizes three papers related to the conserved quantities of an extension of the Bondi-Metzner-Sachs (BMS) group and two related new memory effects.Paper: Testing General Relativity Using Golden Black-Hole Binaries2016-02-08T00:00:00+00:002016-02-08T00:00:00+00:00https://dnichols1.github.io/papers/testing-gr-golden-binaries<p>This post describes a paper outlining the methodology for testing general relativity with binary black holes, which was later applied to the LIGO and Virgo detections of these systems.</p>
<h2 id="paper-highlighted">Paper Highlighted</h2>
<ul>
<li>Ab. Ghosh, Ar. Ghosh, N. K. Johnson-McDaniel, C. K. Mitra, P. Ajith, W. Del Pozzo, <strong>D. A. Nichols</strong>, Y. Chen, A. B. Nielsen, C. P. L. Berry, and L. London.
“Testing general relativity using golden black-hole binaries.”
<a href="https://doi.org/10.1103/PhysRevD.94.021101">Phys. Rev. D 94, 201101 (2016)</a>,
<a href="http://arxiv.org/abs/1602.02453">arXiv:1602.02453</a>.</li>
</ul>
<h2 id="summary-of-the-paper">Summary of the Paper</h2>
<p><img src="/images/posteriorIMR.png" alt="" class="align-left" />
The merger of a black-hole binary releases an enormous amount of energy that is carried off by gravitaitonal waves.
Before the detections of binary black holes and neutron stars by the LIGO and Virgo Collaborations, general relativity had not been well tested for this aspect of the theory.
Einstein’s theory turns out to describe the merger of these black holes well, but it is important to be able to quantitatively test whether any deviations might occur.</p>
<p>We propsed a test of general relativity that was similar to one proposed by Hughes and Menou for so-called golden binaries: supermassive black-hole binaries that could be observed with very high signal-to-noise ratio by the proposed space-based detector, LISA.
Here, we plan to determine the masses and spins of the stellar-mass black holes seen by LIGO and Virgo during their inspiral (prior to merger) and independently determine the final mass and spin of the black hole using only the merger and ringdown part of the waveform after the inspiral.
Numerical relativity simulations predict a relationship between the initial and final masses and spins.
Thus, we can use these relationships to determine if our predictions of the final masses and spins from the inspiral and the merger and ringdown, respectively, give rise to estimates that are consistent the the results of general relativity.</p>
<p>The figure shows the results from a simulation of a binary black hole with equal masses and total mass equal to one-hundred solar masses.
The predictions from the inspiral, the merger and ringdown, and the complete gravitational waveform are all consistent within their errors.
Because the posterior distributions of the parameters are spread over a large range of the parameter space, a single measurement will not give very strong constraints on general relativity.
However, if we chose parameters that are the fractional deviations from the results of general relativity, we can combine the posteriors from multiple different events to get a much stronger constraint, once many black-hole binary mergers are discovered.
This test has been applied to several of the LIGO and Virgo detections to show that the observed waveforms agree with the predictions of general relativity to within around ten percent.</p>David A. NicholsThis post describes a paper outlining the methodology for testing general relativity with binary black holes, which was later applied to the LIGO and Virgo detections of these systems.Research Theme: Gravitational-Wave Memory and Relativistic Angular Momentum2016-02-04T00:00:00+00:002016-02-04T00:00:00+00:00https://dnichols1.github.io/papers/memory-and-angular-momentum<p>This post describes three papers related to defining a locally measured angular momentum, comparing this angular momentum measured by different observers, and understanding the relation of the observer dependence of these measurements to the gravitational-wave memory effect.</p>
<h2 id="papers-highlighted">Papers Highlighted</h2>
<ul>
<li>
<p>E. E. Flanagan, <strong>D. A. Nichols</strong>, L. C. Stein, and J. Vines.
“Prescriptions for measuring and transporting local angular momenta in general relativity.”
<a href="http://dx.doi.org/10.1103/PhysRevD.93.104007">Phys. Rev. D 93, 104007 (2016)</a>,
<a href="http://arxiv.org/abs/1602.01847">arxiv:1602.01847</a>.</p>
</li>
<li>
<p>J. Vines and <strong>D. A. Nichols</strong>.
“Properties of an affine transport equation and its holonomy.”
<a href="http://dx.doi.org/10.1007/s10714-016-2118-2">Gen. Relativ. Gravit. 48(10), 1 (2016)</a>,
<a href="http://arxiv.org/abs/1412.4077">arXiv:1412.4077</a>.</p>
</li>
<li>
<p>E. E. Flanagan and <strong>D. A. Nichols</strong>.
“Observer dependence of angular momentum in general relativity and its relationship to the gravitational-wave memory effect.”
<a href="http://dx.doi.org/10.1103/PhysRevD.92.084057">Phys. Rev. D 92, 084057 (2015)</a>,
<a href="http://arxiv.org/abs/1411.4599">arXiv:1411.4599</a>.</p>
</li>
</ul>
<h2 id="summary-of-the-papers">Summary of the Papers</h2>
<p><img src="/images/PJtransport.png" alt="" class="align-left" />
Bondi, Metzner, and Sachs (BMS) discovered that asymptotically flat spacetimes have an infinite-dimensional symmetry group, now called the BMS group, instead of the ten-dimensional symmetry group of flat spacetimes, the Poincare group.
The additional symmetries not present in Minkowski spacetime are the so-called supertranslations, which are a type of angle-dependent translation around an isolated source.
One important consequence is that in asymptotically flat spacetimes, there is not, in general, a preferred Poincare group with which one can define a special-relativistic angular momentum (except when the spacetime is stationary).
Thus, observers who feign ignorance of the BMS group will notice that if they try to measure a type of special-relativistic angular momentum locally and compare their values with other observers, then their results will disagree.
Thus, they would conclude that their notion of a special-relativistic angular momentum is observer dependent.
But what would cause this observer dependence?</p>
<p>To investigate this question, we first defined a local procedure by which observers can measure special-relativistic angular momentum in stationary spacetimes.
The algorithm uses just local measurements of the spacetime curvature and its gradients, and it gives the expected result in stationary spacetimes when measured by observers at large distances from the source.
We then developed a method by which they can compare this angular momentum between different observers that takes into account the usual origin dependence of angular momentum in flat Minkowski space.
The process of two observers comparing angular momentum at two different stationary points in a spacetime we could represent mathematically as a holonomy operation.
By performing calculations involving a short-duration burst of gravitational waves passing by these observers, we determined that the changes in angular momentum of the spacetime would differ for the two observers.
Moreover, we showed that in such a situation, the observer dependence is related to changes in the relative displacement of the observers that arise from the gravitational-wave memory effect.</p>David A. NicholsThis post describes three papers related to defining a locally measured angular momentum, comparing this angular momentum measured by different observers, and understanding the relation of the observer dependence of these measurements to the gravitational-wave memory effect.Research Theme: Short-Wavelength Quasinormal Modes2015-10-28T00:00:00+00:002015-10-28T00:00:00+00:00https://dnichols1.github.io/papers/QNMs-Kerr-eikonal<p>This post describes two papers and a comment about quasinormal modes of Kerr black holes in the short-wavelength (geometric-optics) limit.</p>
<h2 id="papers-highlighted">Papers Highlighted</h2>
<ul>
<li>
<p>A. Zimmerman, H. Yang, F. Zhang, <strong>D. A. Nichols</strong>, E. Berti, and Y. Chen.
“Reply to ‘On the branching of quasinormal resonances of near-extremal Kerr black holes’ by Shahar Hod.”
<a href="https://arxiv.org/abs/1510.08159">arXiv:1510.08159</a>.</p>
</li>
<li>
<p>H. Yang, F. Zhang, A. Zimmerman, <strong>D. A. Nichols</strong>, E. Berti, and Y. Chen.
“Branching of quasinormal modes for nearly extremal Kerr black holes.”
<a href="http://dx.doi.org/10.1103/PhysRevD.87.041502">Phys. Rev. D. 87, 041502(R) (2013)</a>,
<a href="http://arxiv.org/abs/1212.3271">arXiv:1212.3271</a>.</p>
</li>
<li>
<p>H. Yang, <strong>D. A. Nichols</strong>, F. Zhang, A. Zimmerman, Z. Zhang, and Y. Chen.
“Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation.”
<a href="http://dx.doi.org/10.1103/PhysRevD.86.104006">Phys. Rev. D 86, 104006 (2012)</a>,
<a href="http://arxiv.org/abs/1212.3271">arXiv:1207.4253</a>.</p>
</li>
</ul>
<h2 id="summary-of-the-papers">Summary of the Papers</h2>
<p><img src="/images/QNMpic.png" alt="" class="align-left" />
Perturbed black holes oscillate at characteristic frequencies and with amplitudes that decay at specific rates in what are called quasinormal modes.
For non-rotating and slowly rotating modes, it was noted that the oscillation frequency of short-wavelength modes was related to the orbital and precessional frequencies of null geodesics that remain at fixed radius (on the photon sphere).
The spherical photon orbits are unstable, and they head out or into the black hole on a time scale given by the Lyapunov exponent, which is equivalent to the rate of decay for short-wavelength quasinormal modes.
The reason for this apparent coincidence is that quasinormal modes are massless disturbances propagating on the black hole’s background, and in the short-wavelength (geometric-optics) limit, the mode follows the null rays (geodesics) of the spacetime (as would any other massless field).</p>
<p>We showed that the correspondence holds for Kerr black holes of any physical spin by explicitly comparing the equations of geometric optics and those of black-hole perturbations in the short-wavelength limit and relating the two sets of parameters in the two complementary descriptions.
Our analysis also provided a method to calculate quasinormal-mode frequencies approximately with good accuracy.
When applied to black holes with nearly maximal spin angular momentum, we were able to observe a splitting in the spectrum of the modes as the damping rate of the modes went to zero.
This is likely connected to the fact that black holes with exactly maximal spins are marginally stable.</p>David A. NicholsThis post describes two papers and a comment about quasinormal modes of Kerr black holes in the short-wavelength (geometric-optics) limit.Research Theme: Visualizing Spacetime Curvature with Vortex and Tendex Lines2012-12-19T00:00:00+00:002012-12-19T00:00:00+00:00https://dnichols1.github.io/papers/vortex-and-tendex<p>This post is a summary of six papers I wrote with collaborators during my Ph.D. about visualizing spacetime curvature around binary-black-hole mergers.</p>
<h2 id="papers-highlighted">Papers Highlighted</h2>
<ul>
<li>
<p>R. H. Price, J. W. Belcher, and <strong>D. A. Nichols</strong>.
“Comparison of electromagnetic and gravitational radiation; what we can learn about each from the other.”
<a href="http://dx.doi.org/10.1119/1.4807853">Am. J. Phys. 81, 575 (2013)</a>,
<a href="http://arxiv.org/abs/1212.4730">arXiv:1212:4730</a>,</p>
</li>
<li>
<p><strong>D. A. Nichols</strong>, A. Zimmerman, Y. Chen, G. Lovelace, K. D. Matthews, R. Owen, F. Zhang, and K. S. Thorne.
“Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: III. Quasinormal pulsations of Schwarzschild and Kerr black holes.”
<a href="http://dx.doi.org/10.1103/PhysRevD.86.104028">Phys. Rev. D 86, 104028 (2012)</a>,
<a href="http://arxiv.org/abs/1208.3038">arXiv:1208.3038</a>.</p>
</li>
<li>
<p>F. Zhang, A. Zimmerman, <strong>D. A. Nichols</strong>, Y. Chen, G. Lovelace, K. D. Matthews, R. Owen, and K. S. Thorne.
“Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: II. Stationary black holes.”
<a href="http://dx.doi.org/10.1103/PhysRevD.86.084049">Phys. Rev. D 86, 084049 (2012)</a>,
<a href="http://arxiv.org/abs/1208.3034">arXiv:1208.3034</a>.</p>
</li>
<li>
<p><strong>D. A. Nichols</strong>, R. Owen, F. Zhang, A. Zimmerman, J. Brink, Y. Chen, J. D. Kaplan, G. Lovelace, K. D. Matthews, M. A. Scheel, and K. S. Thorne.
“Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: I. General theory and weak-gravity applications.”
<a href="http://dx.doi.org/10.1103/PhysRevD.84.124014">Phys. Rev. D 84, 124014 (2011)</a>,
<a href="http://arxiv.org/abs/1108.5486">arXiv:1108.5486</a>.</p>
</li>
<li>
<p>A. Zimmerman, <strong>D. A. Nichols</strong>, and F. Zhang.
“Classifying the isolated zeros of asymptotic gravitational radiation by tendex and vortex lines.”
<a href="http://dx.doi.org/10.1103/PhysRevD.84.044037">Phys. Rev. D 84, 044037 (2011)</a>,
<a href="http://arxiv.org/abs/1107.2959">arXiv:1107.2959</a>.</p>
</li>
<li>
<p>R. Owen, J. Brink, Y. Chen, J. D. Kaplan, G. Lovelace, K. D. Matthews, <strong>D. A. Nichols</strong>, M. A. Scheel, F. Zhang, A. Zimmerman, and K. S. Thorne.
“Frame-dragging vortexes and tidal tendexes attached to colliding black holes: Visualizing the curvature of spacetime.”
<a href="http://dx.doi.org/10.1103/PhysRevLett.106.151101">Phys. Rev. Lett. 106, 151101 (2011)</a>,
<a href="http://arxiv.org/abs/1012.4869">arXiv:1012.4869</a>.</p>
</li>
</ul>
<h2 id="summary-of-the-papers">Summary of the Papers</h2>
<p><img src="/images/BoostVortex.png" alt="" class="align-left" />
In empty space, the ten independent components of the Weyl tensor encode all the information about the spacetime curvature, but the Weyl tensor itself is difficult to visualize and to grasp its physical meaning.
The Weyl tensor in a surface of constant time splits into two symmetric and trace-free tensors called its electric and magnetic parts (in analogy to the E and B fields of electromagnetism).
The interpretation of the electric and magnetic Weyl tensors has been known for quite some time: the electric part describes the tidal acceleration of two spatially separated observers, and the magnetic part relates to the differential precession of two adjacent inertial gyroscopes.</p>
<p>To visualize the Weyl tensor’s electric and magnetic parts, we used the facts that these tensors are always diagonalizable, the sum of their eigenvalues will be zero, and the corresponding eigenvectors are orthogonal.
We formed streamlines from the eigenvectors, which we call tendex lines for the electric part and vortex lines for the magnetic part, and we plotted these lines and their corresponding eigenvalues, which we call tendicities and vorticities, respectively.
Extended observers are stretched along a tendex line with negative tendicity and squeezed along a tendex line of positive tendicity; similarly, observers with gyroscopes tied to their heads and feet are twisted clockwise along a positive-vorticity vortex line and counterclockwise along a negative-vorticity line.</p>
<p>We applied these vortex and tendex concepts in several directions.
To build up intuition about these new visualizations, we looked at well-known examples of stationary and radiating spacetimes.
In particular, we investigated the vortexes and tendexes of weakly-gravitating stationary spinning and non-spinning point particles, multipolar gravitational waves, and binaries made of spinning and non-spinning point particles.
We were able to understand the generation of gravitational waves in terms of vortexes and tendexes in the near zone that induce accompanying tendexes and vortexes to form those of gravitational waves.
When we studied stationary and perturbed black holes, we found similar insights about the structure of the vortexes and tendexes and their roles in generating gravitational radiation.
In addition, we found that tendex and vortex lines provide a framework to prove that gravitational radiation must vanish at isolated points far from their source.
We also studied the similarities and differences between electromagnetic field lines and vortex and tendex lines.</p>David A. NicholsThis post is a summary of six papers I wrote with collaborators during my Ph.D. about visualizing spacetime curvature around binary-black-hole mergers.Research Theme: Hybrid Post-Newtonian and Black-Hole Perturbation Model2011-09-01T00:00:00+00:002011-09-01T00:00:00+00:00https://dnichols1.github.io/papers/hybrid-method-pn-bhp<p>This post is a summary of two papers I wrote with Yanbei Chen during my Ph.D. about a semi-analytical model for explaining the properties of the gravitational waveforms from binary-black-hole mergers.</p>
<h2 id="papers-highlighted">Papers Highlighted</h2>
<ul>
<li>
<p><strong>D. A. Nichols</strong> and Y. Chen.
“Hybrid method for understanding black-hole mergers: Inspiralling case.”
<a href="http://dx.doi.org/10.1103/PhysRevD.85.044035">Phys. Rev. D 85, 044035 (2012)</a>,
<a href="http://arxiv.org/abs/1109.0081">arXiv:1109.0081</a>.</p>
</li>
<li>
<p><strong>D. A. Nichols</strong> and Y. Chen.
“Hybrid method for understanding black-hole mergers: Head-on case.”
<a href="http://dx.doi.org/10.1103/PhysRevD.82.104020">Phys. Rev. D 82, 104020 (2010)</a>,
<a href="http://arxiv.org/abs/1007.2024">arXiv:1007.2024</a>.</p>
</li>
</ul>
<h2 id="summary-of-the-papers">Summary of the Papers</h2>
<p><img src="/images/hybridUV.png" alt="" class="align-left" />
While it is now feasible to compute the merger of black holes with large-scale simulations, it is still helpful to use analytical methods to develop faster ways of calculating gravitational waveforms and to provide intuitive understanding of these processes.
We put forward a method of combining two analytical approximation schemes—Post-Newtonian (PN) and black-hole-perturbation (BHP) theories—to make approximate gravitational waveforms and to clarify the structure of a black-hole-binary merger.
The central idea of the method is to use both PN and BHP theories simultaneously at a given time, but to restrict their use to spatial regions where either they are good approximations or their errors will not effect important observables like the gravitational waveform.
Typically, PN and BHP theories are applied to the entire spatial region of distinct times in the evolution of a binary.</p>
<p>Applying both PN and BHP theory at once requires there be a region where one can match the two theories.
We found empirically that we can match the theories at the locations of the PN theory’s point particles.
We then let the matching region evolve by requiring that the point particles follow a combination of geodesic motion in the final black-hole’s background and dissipative motion from a radiation-reaction force (which was calculated self-consistently from the emitted gravitational waves).
This new set of evolution equations simultaneously computes the matching region and the gravitational waves outside of it.
The waveform that they produce is not calibrated to numerical simulations, but still agrees rather well with them for both head-on and quasi-circular binaries.
The method was also useful for exploring how black holes get large recoil velocities when they have spin vectors that are anti-aligned in the orbital plane.</p>David A. NicholsThis post is a summary of two papers I wrote with Yanbei Chen during my Ph.D. about a semi-analytical model for explaining the properties of the gravitational waveforms from binary-black-hole mergers.