Stellar Collapse in Semi-classical Gravity

Core-collapse simulation image (Philipp Mösta)

Supernova explosion (artist: Mehau Kulyk)

In the simplest stellar collapse model of classical General Relativity developed by Oppenheimer & Snyder 1939, the collapsing star is idealized as a uniform ball of dust that contracts under the pull of gravity. The dust particles that make up the star are assumed to be classical and thus infinitely small, infinitely light, and interact only gravitationally with other matter. This approximation is not perfect since infinitesimally small particles would be infinitely large in size, and could never be localized within the stellar horizon.  

The inclusion of quantum effects is unavoidable both when determining whether a black hole will form and when understanding the final stages of the stellar collapse. When we include quantum effects (arXiv:1501.04250, submitted to Physical Review D),  some of the assumptions above are lifted. A particle on the surface of the star is no longer localized, but is instead represented by its wavefunction. Every particle now has a finite probability of escaping the gravitational pull of the star. This allows for the possibility that some configurations will not collapse to black holes, but will instead disperse or even form stable new configurations. The smaller the mass of the constituent particles, the more significant the quantum mechanical effects become.

Classical paths with all initial velocities

To simulate the stellar collapse using a path integral formulation, we have to integrate over all possible paths towards and away from the center of the star. This includes classical paths with all initial velocities. In the special case of the dust ball collapse these paths can be computed analytically.  We first derive analytic solutions to all classical paths (space-like, time-like, and light-like) in Schwarzschild (Table I and II in arXiv:1501.04250) and Kruskal coordinates (see Table V and VI).

The evolution of the wavefunction

In Schwarzschild coordinates, we can only study the collapse outside r=2M. The motion of the particle on the surface of the star is analogous to the vertical motion of a ball moving under gravity, which can go from an initial point to a final point directly or reach its highest and then come down or escape. Each of these paths is unique taking a different amount of time to complete.

In Kruskal coordinates,  we can model the behavior of a particle on the surface of the collapsing star up to the physical singularity at r=0.  We find that classical time-like paths are unique. A path between an initial and final point can be either direct or indirect (turns back in space). Thus some particles that initially move away from the star can return and contribute to the collapse. Space-like paths can turn back in time, but cannot turn back in space. They are also no longer unique when the final point lies inside r=2M. Classically, no information can exit the black hole. However, by integrating around the classical paths one might be able to extract information from inside the horizon. We only compute the paths in the Kruskal case, and leave the computation of the wavefunction and full exploration of the quantum collapse to future work.

The minisuperspace approximation

Models that apply quantization procedures to general relativity operate in superspace, where a 4-geometry space is represented as a trajectory within space-like 3-geometries. A minisuperspace is a symmetry reduced space where these trajectories are limited to a finite number of parameters describing the constant t slices. We assume that the radius is the only spatial degree of freedom that is not frozen. Such restrictive approximations make the problem tractable analytically. Most physicists reading this have likely worked in minisuperspace unknowingly.

The wavefunction of a particle on the surface of the collapsing star

WKB & Schrödinger comparison

t=5 M, multiple masses

The initial wavefunction is taken to be a Gaussian centered far away from r=2M.  In Schwarzschild coordinates, we then compute closed form solutions to the propagator in the WKB approximation where an expansion is performed around the classical paths and in the Schrödinger approximation, and compare the resulting wavefunctions. We find that the two solutions converge towards each other at intermediate times. They are out of step at early and late times with the Schrödinger solution being more exact at early times, and the WKB approximation being more accurate at late times. For lower particle mass, the star resists collapse longer and the probability that it disperses increases. Further work is needed (including the addition of higher order corrections to the Schrödinger solution) to determine the mass limit at which stellar configurations no longer collapse.

M=0, dotted (WKB), solid (Schrödinger)

In the limit when the mass of the star is zero, the WKB approximation converges to the Schrödinger solution at all times. This checks that the WKB approximation is accurate. The Wheeler-DeWitt equation converges to Schrödinger equation in this limit making the Schrödinger solution the exact solution of the free particle problem (Redmount and Suen 1992). Note that Redmount and Suen did not find a good agreement between their WKB approximation and the exact Schrödinger solution due to numerical issues.

 

Black holes, black stars or boson stars?

The dust ball collapse is a testbed for relativistic corrections to the Schrödinger equation. Recent work by Dvali and Gomez argue that black holes are a collection of Bose-Einstein condensates. Quantum effects are very important in this situation, and could add potentially new semi-classical corrections that can be investigated.

Other authors argue that there are no black holes and that nature has black stars or boson stars instead. Neither have an event horizon. 

Classical boson stars are formed from dark matter particles that are spin zero bosons and Bose-condense creating a macroscopic quantum object that depending on the particle size can either fit in your pocket or be the size of a galaxy. Since no spin zero particles have been discovered other than the perhaps the Higgs boson, it is difficult to predict which theory is correct. Supermassive boson stars could lie in the centers of galaxies. Upcoming instruments sensitive to light may detect supermassive boson stars through lensing (e.g., see Boson stars as Gravitational lenses and Method for detecting a boson star at Sgr A* through gravitational lensing). In the future, it may also be possible to detect gravitational waves from a pair of co-orbiting boson stars. Additionally, detectors beyond LISA may see gravitational waves from perturbed supermassive boson stars, which would have quasinormal modes that damp slower than if the object had been a black hole.

Macroscopic dark matter particles?

Feeding on dark matter?

Dark matter particles come close to the attributes of classical dust. If somehow black holes can feed on dark matter particles,  the paucity of super-massive black holes immediately above a certain mass M could be linked to the presence of an ultra-light particle halos that black holes heavier than M can feed on. Black holes lighter than M would be unable to capture such dark matter particles in the same way mini black holes produced at the LHC cannot be grown on atomic matter and pose no danger to the Earth. As soon as a black hole reaches mass M, it starts feeding on dark matter and grows rapidly, perhaps swallowing the entire halo. It is important to note that unlike virialized dust halos or stars, the super-fluid dark matter particles would have negligible momentum and thus accrete easily.

Why should we care about black holes?

Supermassive black hole. Artist conception.

In order to understand the universe, we have to understand black holes. They are the most permanent objects in our universe. All life revolves around them. Every galaxy has a black hole at its center. In the early universe, black holes acted as the seeds around which material collected, and eventually galaxies as complex as our own formed.  Black holes are also believed to be the only objects that will be left in the very far future as our universe continues to expand growing cold and empty. They do evaporate, but the evaporation of black holes that are stellar mass or higher happens on timescales longer than the age of the universe and thus is not observable. To truly understand black holes we have to understand their quantum nature, which is not only important in understanding the final stages of stellar collapse, but also in understanding whether they will form at all. The lighter the constituent particles, the more important the quantum mechanical effects become.

This post is based on arXiv:1501.04250, which is work done in collaboration with Prof. Jayashree Balakrishna (Harris Stowe State U.) and Dr. Christine Corbett-Moran (Caltech), and has been submitted to Physical Review D.

This is the first article I have written where all the authors are women. It is also coincidentally my most technical article to date. Note that my co-authors and I do not discriminate and believe in equal rights for everyone. Our collaboration just happens to be 100% female this time.

Should clothes define us?

My answer to this question is a strong "No!". I like to wear pretty clothes, but I think that I am the one with the personality and not my shirt or skirt.

When I was growing up, my shirts did not have slogans on them. They generally had one color and were made of cotton. Sometimes they did have flowers or lines drawn on them or some other equally non-defining drawing.  However, as small children, my brother and I did receive compliments of the kind "you are smart like your father, and handsome/pretty like your mother".  Our natural response to this was "No! no! I am smart like my mother". We did not care about looking pretty/beautiful/handsome at the time.

My mom & Mihai

My mom and dad

While both my parents are very intelligent and were good looking, my mother had always been the practical one. And, no, not in terms of fixing things around the house (my father did that and he was very much the man in the house), but in the overall picture my mom was the one who hired people and got things done.    She had been not only exceedingly smart and very beautiful, but also generous and with a talent in interacting with people of all ages. She was an outstanding doctor (my father was a doctor, too, but in the military; my mom was a gynecologist). She had the ability to fix people from all walks of life with very few resources. Whenever we had visits from neighbors, family, friends or strangers, they would all tell her their health problems, and unless they had lung cancer or something equally terminal, she managed to help them get well. She also loved us very much and has always supported us unconditionally.

 Most women I know are both strong and talented. However, our culture does define men and women differently and this is reflected in the writing on the clothes in the stores and in the popular Hollywood movies. Yes, we are different, and we want the men in our lives to be kind to us, but we are neither stupid nor incompetent, and do not like to be insulted and treated as such. This latter fact should not be difficult to understand, and yet when I mention I graduated from Cornell, most people assume I cheated in some way. They make statements of the form "I know many men who are way smarter than you, and did not achieve half of the things you did" and assume that people do me favors. Such beliefs seem to be ingrained at a fundamental level and cannot be uprooted at 20+ or 30+ years of age.

Men's T-shirt

Label: "Women's geek shirt"

I started writing this post after reading an article about sexist clothes and their impact on young children and particularly on girls, which said that the paucity of women in science is related to how we dress girls. It is hard to disagree.

 A simple google on "rocket scientist shirt" shows the two images I included here and a few others. All are very ugly. The one for women here has the crazy old man scientist stereotype and the label "women's geek shirt", which implies that women who are interested in rockets and wear this shirt are geeks.  The one for men is worse, and so I will simply abstain from comments on that.  It cannot be so hard to design some shirts that promote learning, and are decent looking.

Mihai and me at the beach. Hint: I am standing.

Personally, I still prefer clothes that are simple and beautiful without silly statements on them. I am raising boys at the moment, and I would not buy them shirts with "Be a hero", "Training to be Batman", etc. I don't want them to be someone else. I want them to be happy being themselves on all days other than perhaps Halloween.

Conclusion: In principle, clothes should either (1) hold messages that encourage children towards learning something (e.g., math, science, music, art) and increase self-esteem & creativity,  (2) contain some message that is generally positive or (3) simply be cute with no message at all. I generally vote for (3) in the clothes I buy, but that is a personal preference. Anything that is sexist or demeaning should just not be manufactured, and especially not for children who have no say in what they wear.

Note: Thanks to Ruth for pointing out that math & science are not the only subjects that are important ...  I tend to forget that.

Belated end of year summary

Sledding

2014 was beautiful and hard in its own way.  It was a good year.  See pictures below. David is in second grade. He has been going to a new school. The move has been quite stressful, but he is starting to like it now. Edward is 4 and not yet allowed to go to kindergarten because he was born after July 31st, and the law in Switzerland is quite strict in that respect. I also live close to work now, which is quite convenient, and close to the veterinary hospital, which has been fun. We have watched the autopsy of a sheep, and how they put all its organs in bags with liquid nitrogen. Cows that come here are often very thin and have IV lines with fluids. We've seen the ambulance for animals and were disappointed that it does not make loud sounds.

Edward and David are both growing and I learn about as many things from them as they learn from me. 

The sheep

David and a goat

Me: “Why do I have to ask you and David so many times before you do something I ask?”

Edward: “Because you don’t always understand us.”
Me: "Just like you do not understand me...but you have to try a little harder. I will try, too".

Just now:

Me: “Are you going to continue to have temper tantrums?”

Edward: “I am not sure yet. You will have to see.”

They often stand up for each other and do most things together. When David was not allowed to board the plane to Israel because his passport was valid for another five months instead of six, Edward chose to stay home, too. I am proud of how reasonable both of them were. 

In Fallanden

Above the parliament in Neuchatel

We have been walking through parks, meadows and forests, and when on vacation on beaches. The children have had extended conversations with swans, ducks, goats, frogs and tad poles. In and around Zurich we have visited various museums, gardens, the zoo and parks with family and friends whom we hope will come again. We’ve watched baby turtles hatch in Greece, seen the Swiss parliament meeting room in Neuchatel (it’s in a castle), and grown quite attached to three sheep in Romania, and to a little dog. 

Presenting our work

in the conference room

Mihai and I have discussed volcanoes, general relativity & atomic clocks at the ICNFP conference under a picture of Jesus and the Orthodox patriarch. The meeting was organised with the help of the Orthodox Academy. 

I finished my first set of job applications for professor positions.  I’ve applied to about 20 positions world-wide.  I have done the first set of seminars and interviews in Israel over the Christmas period. There should be more in 2015.

Tree climbing in the park

With a swan and duck close by

On a cliff

Overall, I am extremely thankful for my family and for the rest of the people and things in my life. My mother told me that she is efficient in everything she does because in her life she has had a lot of practice in controlling her own feelings and emotions, and that in order to take charge of difficult situation one has to first control their own fears and then have the common sense to take the steps that solve the problem. As a doctor of humans she has had plenty of practice with difficult situations where lives were in balance. As a doctor of stars, planets and clocks, it is still often hard to take decisions. It’s also hard to move, and hard to never be certain where I will be next. The search for jobs in previously unknown countries has its own challenges, and I sometimes want to drop everything and run somewhere far away, but I will stick to what I do a bit longer and see what comes out. After all new experiences lead to personal growth. 

On roller-blades

Learning how to fly a hellicopter

Edward at the Petting Zoo

Kite flying

Baby frogs

Changing tires with uncle Mihai

Xmas tree with handmade decorations

Ballons

Trying to get carnivorous plants to close

Our former house in Fallanden

Living in Ashkelon

Ashkelon is a little city in Israel just off the Mediterranean coast where we had rented an apartment. It lies some 10 km from the Gaza border. It is cheap to park your yacht there and so, people who own yachts stop in Ashkelon, and then spend their time traveling through Israel.

Mihai being cold

The giraffe cannot keep its head up

In the direction of the Gaza border, there are two towers that look like nuclear power plants, but they are actually coal based with a cooling system that uses sea water.

Ashkelon has ruins from the neolithic period, which contain the remains from many fire-places but only one wall. This we took to mean that the weather was always so nice that people did not need to build houses back then.

Since buried objects last so long in Israel, we were tempted to bury something to test how we find it on return. We decided against it since it is an area where people sometimes worry about bombs, and we did not want to look suspect by burying random jars into the ground. Burying stuff at night made me think of Tom Sawyer & Huck with warts carrying a dead cat - only they were not 30+ years old in the story. There is also Bridget Jones' diaries, but there she has a Mr. Darcy-extremely capable and good looking guy - to get her out of trouble, and we had had enough trouble already.

Beyond offices of interesting people, I saw Israel's various highways and parking lots. The highways were well built. The country felt like California with Deutsche-Bahn trains. All trains in Israel were made in Germany - they even had the same color as in Germany and the architecture of the train stations was similar as well. 

Ashkelon parking lot

Some of the parking lots had interesting topology. One was below sea level and had water in it. Another had similar uneven ground, and was on a hill. It was fun to see how people drive their cars and park with a lot less care than in other parts of the world. My theory is that because the army is mandatory in Israel for both men and women, they get used to driving tanks and flying planes. So, when they have a personal car, they drive it the same way as they drove their tanks. 

There are plenty of stray cats in Israel, and some stray dogs. Along the corridors of the Weizmann Institute there is an array of happy looking cats. I was glad for their company. Some of the cats must have been female and un-spayed and so the female line was not that poorly represented after all.  I was told that people let them come in because the cats are fast enough to catch vipers and other poisonous snakes without being poisoned. Cats are not immune to the venom. Slowing them down by feeding them dry cat food, which retains water and makes them fat, or by tying a bell around their necks like in the US or Western Europe could be cat-lethal in Israel if this story is true. I was glad I did not see any poisonous snakes in my office or any where else I stepped for that matter.