Reason for Disapearance...

Hello all,

I was terribly bad at updating my blog posts... not a post since Christmas. But I assure you, I was busy. I was busy doing this... And also studying for that test thing.... called the prelim or something like that. :)

Hardly working... I mean, hard at work. Yeah.

Merging Black Holes Pt. 3: Simulating Relativity

This is the third installment of the merging black holes posts (if you're just coming in, I suggest you read the intro post). This part will focus on some of the numerical techniques used to simulate black hole mergers. So, if you've ever been interested in how people actually simulate relativity, and what are the current limitations/problems we face, then read on! If not, well, this may be particularly boring for you, ha ha! Just skip the text and look at the pretty pictures (psst... it's what I do most of the time). Just know that when it comes to black holes, especially coalescing binary black holes, numerical relativity is complicated... much like a 12 year old girl's love life.

Bad joke. Anyway, moving on...

Black hole dynamics can be described reasonably well for most of its orbit. For example, analytic post-Newtonian expansions can simulate the early inspiral phase, and black hole perturbation methods can treat the late ringdown phase. However, sophisticated computational techniques are required for the late inspiral and merger phases, where, unfortunately (or fortunately?) gravitational waves are at their highest amplitudes.

There are three major hurtles in simulating these highly relativistic systems, which are:

1. Scale: The gravitational field is incredibly large around the black holes, but gravitational waves themselves are only small perturbations in the gravitational field. Simulations that cover the entire dynamic range from itty bitty waves to huge spacetime warping are extremely hard to write. In addition, simulations cannot deal with singularities themselves, which, unfortunately, is what black holes are. So it makes it challenging to find a numerical technique that can ignore those regions while still simulating their effects.

2. Non-linearity: Einstein’s equations are non-linear. This means that an element of chaos is introduced into the system. In a chaotic system, vastly different outcomes will occur from similar initial conditions - this is simply what results from non-linear equations of motion. Therefore, if the initial conditions are not exactly right, the black hole merger may display vastly different characteristics than it would in reality.

3. Wonky spacetime: current computational models need to decompose Einstein’s equations of General Relativity into equations computers can handle. Einstein’s equations very elegantly combine space and time into one quantity: spacetime. Unfortunately, computers decide to poop out on us. Why? To simulate a system that starts at a specific initial time, computational solutions demand that space and time have to be separated - to be decoupled. There is a way, however, to work around this. Einstein’s equations can be re-written using the 3+1 decomposition technique. In this technique, the equations are split up into two sets. One set describes the gravitational field at all points within a specific time, and the other set then evolves the gravitational field in time. In this way, once the initial conditions have been set, the code can evolve the merger.

Briefly, there are a couple of different numerical techniques that I know of that work around all these problems: finite difference mesh, spectral or pseudo-spectral, and adaptive mesh refinement. Finite difference involves equally spaced grids along the system. The partial differential equations are solved in the middle of each grid, and then extrapolated across the boundary. The picture below is an example of this kind of technique.

The grid spacing in this rendition is constant, which is the hallmark of a finite difference simulation.

The spectral and pseudo-spectral techniques expand the differential equations in a Fourier series, and the terms that pop out are used as the basis states of the simulation. This simplifies spatial and time derivatives, but also requires the extra step of actually Fourier transforming the field variables.

The last technique (adaptive mesh refinement), uses grids that change size in order to make the simulation more computationally efficient. The simulation essentially determines the number of grid points needed at a specific point in the domain such that the simulation can guarantee accuracy and efficiency. It can also change grid shape or orientation as needed. This can be particularly useful for General Relativity, which is the description of how space becomes warped in the presence of matter (in a gravitational field). This technique is the most commonly used one, although to simulate the full range of the merger, still higher level techniques must be developed.

Yes, the Simpsons got it right! Notice how the grid spacing changes size near the singularity. This is why it's called adaptive mesh refinement.

It should be noted that adding all spacial dimensions or throwing away symmetry arguments developed instabilities after only short integration times!

Next time, I'll talk about what happens when these simulations are implemented. There was a surprise when black holes merged, recoil, which is strictly a consequence of Relativity (and thus reinforces my conviction that Relativity is just plain weird). Stay tuned for more weird black hole fun... I promise to write about it in a timely fashion this time.

Merry Christmas!

Many years ago on this day, an amazing child was born. This child forever changed the way we understood the world, and his ideas opened up an unprecedented era of enlightenment and discovery.

Happy birthday, Isaac Newton! 

* Thank you, Sam Collopy, for pointing this out. *

 

 

Merging Black Holes Pt. 2: Ripples in Spacetime

The above Youtube video shows a merging event generating gravitational waves. Now, this video shows two merging white dwarfs, but two merging black holes will do the same thing - only on a much larger scale. It seems crazy to think that this sort of event happens in the universe, but is currently invisible to us. However, in the coming years, that should change.

Gravitational waves are a hot topic in modern astrophysics. They are analogous to most waves, such as water waves or electromagnetic waves (light), except that they are ripples on the background curvature of spacetime itself.

So, what's the big deal with gravitational waves, aside from the fact that they're a cool physics buzzword?

Well, the most exciting prospect of gravitational waves is their potential in the study of different astrophysical phenomena. For most of astronomy, electromagnetic radiation has been the sole source of probing the universe (or, photons for short... it's amazing what we can learn from light rays). Electromagnetic radiation comes from the superposition of (mostly) randomly oriented photons being emitted or absorbed from electrons, atoms, or molecules. Gravitational waves, on the other hand, are generated most strongly from large bulk motions of mass. And, even more incredibly, gravitational waves are not dampened by intervening matter. As a result, the detection of gravitational waves should open an entirely new area of astrophysics previously unseen.  Perhaps, like the advent of infrared and radio astronomy, gravitational wave astronomy will usher in a new era of unprecedented scientific discovery and advancement.

Interferometers like the Laser Interferometer Space Antenna (LISA) and the Laser Interferometer Gravitational-Wave Observatory (LIGO) are currently the most promising ways to detect gravitational waves. Simply put, like most interferometers, a laser is shot at a beamsplitter, which splits the laser beam into two directions (usually perpendicular to each other). These beams hit mirrors, and are then reflected back to a detector, where they are recombined. *Note: I have simplified the process quite dramatically - I have never been very good with instrumentation, so I'm not going to try to go into any specifics.*

(Is it bad that I can't ever say laser without thinking of Dr. Evil's "laser")?

Very simple interferometer design, except for gravitational wave detectors, the movable mirror would not be moved by anything other than gravitational waves. The screen has a great example of what fringe patterns look like.

Since gravitational waves are ripples in the fabric of spacetime itself, a gravitational wave passing through a point in space causes the gravitational field to increase or decrease in magnitude. As a result, if a gravitational wave hits one of the mirrors, the oscillation in gravitational strength should shift the mirror back and forth. This causes the path length for one of the mirrors to deviate from the original value, so that when the beams are later recombined at the detector, the observed fringe patterns should change in accordance with the mirror oscillations. These fringe patterns can then be used to deduce the strength of the passing gravitational wave.

Gravitational wave schematic. The top portion shows the three phases of coalescence: the quasi-circular inspiral phase, the plunge and merger, and the ringdown. The bottom portion shows the amplitude of the emitted gravitational waves that correspond to the above merger process. (Image credit: Baumgarte and Shapiro 2011).

Black hole mergers, incidentally, are especially interesting in the study of gravitational waves. This is because gravitational waves are theorized to carry away most of the angular momentum and energy from these systems, which causes the merging black holes to coalesce. The above figure (that I scanned, sorry for the poor quality, ha ha!) shows a schematic of the phases of a black hole merger, and how those phases relate to gravitational wave strength. The top portion shows a cartoon of the merger, including the early quasi-circular inspiral phase, the plunge and merger phase, and the final ringdown stage. The bottom part shows the wave amplitude (or, strength) versus time. For two black holes approaching each other in the early inspiral phase, the emission of gravitational radiation causes the orbits to circularize and decay. Gravitational waves at this point will be at relatively low levels. But during the late inspiral phase, when the binary is in tight, circular orbits, and during the actual merger, gravitational waves should be at their strongest. This is where LIGO and LISA will be able to detect gravitational waves.

Unfortunately, even though the very early and late stages of coalescence has been simulated fairly easily, the actual late inspiral and merging phases are not as simple. For that part of the merger, post-Newtonian and perturbation methods break down. Therefore, providing templates for the signatures of merging black holes requires advancements in numerical relativity, which will push computational techniques to their limits. 

For more info, please see:
Abramovici, A. et. al 1992, Science, Vol. 256, pp. 325-333
Baumgarte, T. W. & Shapiro, S. L. 2011, Physics Today, Vol. 64 No. 10, pp. 32-3
Berti, E., Cardoso, V. & Will, C. M. 2006, Phys. Rev., 73, 6

Merging Black Holes Pt. 1: An Introduction

There are monsters in the night sky. If you were unfortunate enough to pass by one, there would be nothing to stop the inexorable pull toward destruction. But for such behemoths, you wouldn't be able to see them. You could see the effects of their hunger (if they happened to be hungry) in bright, energetic X-rays. But that's all. That's all you could see of one of the mysteries of astrophysics today - supermassive black holes.

However, funny enough, these monsters may see and remember you. According to Stephen Hawking, black holes can dissipate, because the huge gravitational field may occasionally pop out particles (it's all about E=mc², where energy is converted to mass). Over time, as more and more particles are ejected from the black hole, the mass that originally fed the black hole will once again return to the universe, making the black hole no more.

Artist rendition of a black hole. All images stolen shamlessly from the internet.

Massive black holes are everywhere in the universe, yet their formation is still largely unknown. Cosmological models suggest that massive black holes form from the mergers of smaller seed black holes, but can we reproduce this behaviour in simulations? Do these simulations generate surprises about relativity we previously didn't know about? In the next 5 posts, I'll discuss recent simulations of black hole mergers, some of the main techniques used for handling computational relativity, and possible observational signatures of these merging monster systems. Oh, and spoiler alert - there will be a surprise from these simulations, which suggests that there are some things about General Relativity that can continue to surprise us.

Feeding black hole.

Black holes are relatively simple objects compared to most physical systems. They are described by analytic solutions to Einstein's equations of General Relativity, and depend on only three parameters: charge, mass, and spin. For most astrophysical black holes, the description is further simplified because the charge is usually set to zero. This happens because in the accretion disk, the charged, ionized material rearranges itself to neutralize the black hole charge at large scales. More specifically, the gas is treated as having infinite conductivity and is therefore able to support infinite currents. These currents are generated from charge imbalances and move charges in such a way as to cancel charge imbalances, thus causing the whole system to neutralize.

Black holes also span a ridiculously large range of masses, from the predicted tiny holes from string theory to supermassive holes as large as some small galaxies. However, it is still unclear how supermassive black holes are formed. Population III stars are thought to be one of the most likely sources of the first seed black holes. These stars are the theorized first generation of stars, and consequently form from the primordial gas of pure hydrogen. A consequence of this is that the gas is unable to fragment during the formation process, meaning that these stars are extremely massive and live fairly short lives. It is predicted that the stars with masses of 25 and 140 times the mass of our sun formed the first seed black holes.

However, there is a problem with this scenario. This problem arises in the growth of these smaller seed black holes to the supermassive black holes seen today. If black hole growth proceeds purely by accretion, a liberal estimate of a growth timescale can be computed if the black hole accretes at the Eddington rate. If it is assumed that Population III stars form 100 solar mass seed black holes, then it would take roughly 0.8 Gyrs (or 800 million years) of continuous, uninterrupted accretion to form a black hole of 1 billion times the mass of our sun. But, according to observations, quasars at a redshift of z=6 have been observed. This means that massive black holes must have been in place when the Universe was only about 1 Gyr (1 billion years) old. This means that given even the best estimates for black hole growth, black holes would have had to start accretion in the universe's infancy. Therefore, there must be another mechanism that increases the black hole mass rather quickly.

The current cosmological understanding is that large black holes must then grow bottom up, where small seed black holes merge to form successively larger black holes. In this way, the formation of supermassive black holes can proceed at the quick pace observations seem to imply. But how common are black hole mergers? And if they did merge, do we know if they'll act the way we think they'll act? 

Rendition of a quasar.

In addition to the growth rate of black hole formation, supermassive black hole formation scenarios must also provide explanations for their role in galactic evolution. It is undeniable that large black hole formation is a common occurrence, since they reside in the vast majority of galaxies (in particular bulge galaxies). They are also known to be the source of quasars and active galactic nuclei. To date, though, the interaction between black hole formation and their environments are only just being explored.

But, even though this field is in it's infancy, it seems like a promising avenue to study how the universe works in conditions wholly alien to us - to push the boundaries of the physics we know into the physics of the unknown. 

* Note * This is just a compilation of knowledge I have read from people who actually do this kind of research. I haven't included references in the proper style for fear that someone may use this as a paper. But, I don't feel right about not acknowledging the people who provided us with this knowledge. If you are interested in this subject, please please please look at these sources!

Baumgarte, T. W. & Shapiro, S. L. 2011, Physics Today, Vol. 64 No. 10, pp. 32-37

Carroll, B. W. & Ostlie, D. A. 2007, An Introduction to Modern Astrophysics: 2nd ed., Pearson Education Inc.

Fan, X. et. al 2001, AJ, 121, 54-65

Madau, P. & Rees, M. J. 2001, ApJ, 551, L27-L30

Madau, P. & Quataert, E. 2004, ApJ, 606, L17-L20

Volonteri, M. 2010, Astron. Astrophys. Rev., 18, 279-315
This is a really good review paper on black holes, by the way!

Grad School Lesson #1

Well, I feel like a jerk. I wrote a post about how much schoolwork we grad students get, but then we get a week off of homework. Dang.

So, now I've decided to keep a running, improvised set of lessons from grad school. Perhaps these tidbits will be useful to you. Or... perhaps not. But, for the sake of knowledge, my first lesson is:

Lunchables are made for children, and do not make a good meal substitute.

I was hungry the entire day, and the sugar from the dang thing put me in a very silly mood. Case in point: this picture made me laugh for way longer than it should have....

Hmmm... I really should have wolfed down some caffeine, just to see the awesomeness that would have ensued. But, alas, I didn't, and I spent the rest of the day giggling in my office. I wonder what my office mates thought about that...

Also, do not touch other people's erasers.

Don't even do this by accident. Some grad students are very territorial about this.  Even if it's not their erasers.

Until next time, happy computing. I'll be kicking my computer in anticipation of the next post. :)

Advisor Meeting Analog

I'm pretty sure this is how most advisor meetings go.Thanks to Ryan Shea for finding this. Hopefully he won't find out that I shamelessly stole this from him on facebook.