Conference: Symplectic Techniques in Topology and Dynamics 2015

Symplectic Techniques in Topology and Dynamics 2015During the week from the 24.08 to the 28.08.2015 I participated at a conference Symplectic Techniques in Topology and Dynamics 2015 at the University of Cologne. The areas the speakers have talked about were widely spread in the field of symplectic geometry starting for example  with the n-body problem by Andreas Knauf which is very closely related to application and going to Counterexample by Tolman, which is more on the pure side of math. Over all the whole conference was very inspiring and very good organized. Much thanks for this to Prof. Dr. H. Geiges, Prof. Dr. S. Sabatini and Dr. Kunze. Since I am not an expert in these topics and completely new to the field I could not follow my initial thought of writing a little sum up for every talk. Instead I will just present the titles of all the talkes.

Participating in this conference was very fruitful for me since I realized that this is an area which I want to focus more on! Also I could get in touch with famous persons like Susan Tolman, which are and have been very successful in the area of symplectic geometry.


Day 1 - 24.08.2015

The Flashka transformation as a moment map - Tudor S. Ratiu - EPF Lausanne

New techniques in the n-body problem - Andreas Knauf - Universität Erlangen-Nürnberg

Contact open books with exotic pages - Burak Özbağcı - Koç University Istanbul

Non-loose transverse knots - Sinem Onaran - Hacettepe Üniversitesi Ankara


Day 2 - 25.08.2015

Polygons, Hyperpolygons and beyond - Leonor Godino - IST Lisboa

Integrable systems in the semiclassical limit - Álvaro Pelayo - UC San Diego

Point vortex dynamics - Thomas Bartsch - Universität Gießen

A variational approach to colliding solutions in celestial mechanics - Susanna Terracini - Università di Torino


Day 3 - 26.08.2015

Non-existence of intermediate symplectic capacities and shadows - Felix Schlenk - Université de Neuchâtel

The homotopy type of the space of contact structures on certain 3-manifolds - Yuri Chekanov - Independent University Moskau


Day 4 - 27.08.2015

Dynamics of a ping-pong model - Rafael Ortega - Universidad de Granada

Conformal symplectic systems - Rafael de la Llave - Georgia Institute of Technology

On unbounded motions of the N-body problem - Jacques Féjoz - Université Paris-Dauphine

The most classical way of being Hamiltonian in two different ways - Alain Albouy - IMCCE Paris


Day 5 - 28.08.2015

Magnetic two-spheres - Kai Zehmisch - WWU Münster

Non-Hamiltonian action with isolated fixed points - Susan Tolman - University of Illinois at Urbana-Champaign

Non-avoided crossings for n-body balanced configuartion in \mathbb{R}^3 near a central configuration - Alain Chenciner - IMCCE Paris & Université Paris 7

An introduction to Symplectic Geometry

During the last two weeks I gave in the seminar series of our work group two talks about Symplectic Geometry and its relations to physics. The first part consists of a basic introduction to the language of Symplectic Geometry and fixes the used notation. Since I had only one talk for the introduction I refer to the given literature for further details.
The second part is dedicated to the applications and results of Sympletic Geometry to Physics. Most importantly we talked about Gromov's theorem from 1985 which can be interpreted as the mathematical analog of the uncertain principle of quantum mechanics

As always I have prepared a little handout of everything that I have presented. You can find it here. In case you find any mistake of do have a suggestion, don't hesitate to write me an email.

Book review: The story of collapsing stars

9780199686766In the business of writing my master thesis about the possible treatment of naked singularities with canonical quantization I encountered a lot of papers of Pankaj S. Joshi, who is a theoretical physics at the Tata Institute of Fundamental Research in Mumbai, India. He is an expert in the field of general relativity and cosmology with a big interest in the fascinating topic of dynamical collapse scenarios.

Since I started my master thesis only some month ago and I am completely new in this field I looked around for some references that treat gravitational collapses and their outcomes, i.e. black holes and naked singularities, at a very broad sense. In January of this year Prof. Joshi published his book 'the story of collapsing stars' via the Oxford University Press. In order to get more knowledge about my kind of research I decided to read his book and now, finally, I finished an want to give a little review.

Before betting into detail I want to sum up my impression about the book: Overall his book quite nice to read and most notably very comprehensible. For everyone who is interested in topic concerning the formation of stars I would really recommend this book to read.

The first chapter of the book has the title 'our universe' and concerns the role of gravity in the evolution of our universe. In particular this is were the reader meets the word 'singularity' for the first time. In the following three chapters we makes these concepts more concrete and talks also about problems that arise from these concepts. Most importantly he points at the problem of the so-called cosmic censorship conjecture(CCC) which states that every gravitational collapse of reasonable initial conditions should result in a black hole, i.e. a singularity that is covered by an event horizon that forbids and interaction the to far away part of the universe. In other words, this conjecture would rule out the existence of so-called naked singularities. The mathematical problems of defining such a theorem or conjecture are pointed out and explained. The conclusion was made that only the study of different types of collapsing scenarios can give us a hint whether the CCC is right has to be thrown away.
He also explains that a star under reasonable initial conditions will more likely end up as a naked singularity instead of a black hole. After this statement has been made he explains various other types and models of gravitational collapses and points out that naked singularities are as common as black holes. This makes it, of course, difficult to write down an explicit form of the CCC.
In the last parts of his book he explains how naked singularities affect the structure of spacetime and discusses some phenomena that might be observed to distinguish between a naked singularity and a black hole.

BCGS Weekend seminar

Today I was invited to give a tiny talk about my current research in my master’s thesis. In 15 minutes I had the opportunity to explain to the audience the concepts a black holes, naked singularities and to tell them something about my ultimate goal of quantizing solutions of Einstein’s field equations that have naked singularities as solutions.

In the first part I talked a little bit about the so-called Schwarzschild solution. There I have shown in a hand-waving way that there are two types of singularities. The first type is a real singularity and the other type was a so-called event horizon which is a surface in spacetime that has the property, that if you pass it, you are doomed to fall into the black hole and there is no way to escape this fate. Summarized, there is no way to escape from a singularity since there is an event horizon surrounding it, i.e. no communication to the outside is possible.

A naked singularity is now the case where a real singularity is not hidden behind an event horizon. In several calculations people have shown that this are ‘real’ solutions for some choices of initial data. In particular in the case of the so-called LTB-model it is possible to write down analytical solutions that have a naked singularity in the center. This is the sage of my current work.

Also I have shown that the existence of naked singularities violates the notion of causality, i.e. there are closed curves in spacetime itself. For example I could travel through a naked singularity back in time to kill my father. In order to avoid these weird phenomena Roger Penrosé imposed in 1969 the so-called cosmic censorship hypotheses, which tells that under certain initial conditions a formation of a naked singularity is not possible. But since naked singularities predict somehow predict the breakdown of general relativity itself one tries to solve this:

One idea is, that naked singularities or singularities at all are only one artifact of general relativity which result from the fact of neglecting quantum mechanical effects. Therefor in the last part of my master thesis I will try to ally to procedure of canonical quantization to a solution of Einstein’s field equations that admit a naked singularity. If everything works properly I will study the properties of my solution. I am really looking forward to this because the result is completely unknown and even wage predictions are not possible at all.

You can find the slides to my talk here.

A Short Introduction to Morse Theory

These are the notes of my talk about Morse theory in the seminar same-named seminar organized by Prof. Dr. S. Sabatini at the university of Cologne. Morse theory is the study of the relations between functions on a space and the shape of the space. In this short introduction we will follow the excellent book of Yukio Matsumoto[1]. Since this is just a short introduction it covers only the first part of the book which deals only with the case of two dimensional spaces, i.e. surfaces.

You can find the here.

Naked Singularities

In preparation for my master thesis I give at the 20th of January a talk in the seminar 'Advanced Seminar on Relativity and Cosmology' about naked singularities. Here is a brief outline of the topics I want to talk about:

Outline

  • Gravitational collapse and naked singularities
  • Cosmic censorship
  • Charging a black hole
  • Summary and the Third law of Black-Hole-Dynamics

 


 Gravitational collapse and naked singularities

The topic of naked singularities arises from the question: How does a Black Hole form? This leads us to the topic of so-called gravitational collapse. There exist a plenty of models to describe matter, that collapse into itself to form a black hole. One of these many models is the so-called Laimaître-Tolmann-Bondi model (1933) with the line element:

 \text{d}s^2 = - \text{d} t^2 + \frac{(\partial_r R)^2}{1+2 \cdot E(r)} \text{d}r^2+R(r)^2\text{d} \Omega^2.

This is a solution to Einstein's field equations for a spherical shell of dust under the influence of gravity, that is expanding or collapsing. Under certain initial conditions and assumptions one recovers the standard Schwarzschild solution or the Friedmann-Laimaître equations. But also so-called naked singularities can occur (these were found the first time numerically by Eardley et. al. in 1979). Now I want to specify what a naked singularity is:

Definition
A naked singularity is a gravitational singularity, i.e. a point in spacetime which is infinitely large curvature, which is not hidden behind an event horizon (black hole horizon).

Now one could ask: Where is the problem with these 'objects'? The thing is, that the occurrence of such a non-hidden singularity would break down the predictability of general relativity itself. We could not say anything about any trajectory of particles that move within this spacetime. Since this inconsistency is really annoying and seems to be contradictionary to our real would, Penrose developed in 1969 the idea of the so-called cosmic censorship:


 Cosmic censorship

There are indeed two different statements of the cosmic censorship:

Weak case
There can be no singularity visible from future null infinity.

Strong case
General relativity is a deterministic theory, i.e. the classical fate of all observers should be predictable from initial date.

Apart from the more or less general agreement in this conjecture, there are, up to now, no proves for this two conjectures.


 Charging a black hole

In order to 'check' if the above statements are correct we want to consider the case of a non-rotating charged black hole, so-called Reissner-Nordström-black hole. It is given via the following line element

 \text{d}s^2 = - \left( \frac{\Delta}{r^2} \right) \text{d}t^2 + \frac{r^2}{\Delta} \text{d} r^2 + r^2 \text{d}\Omega^2,


where \Delta = r^2 - 2 M r + Q^2, M is the mass of the black hole and Q denotes its charge. From this we can calculate, that the so-called event horizon is given by

 r_{\text{H}} = M + \sqrt{M^2 - Q^2} := M \left( 1 + \sqrt{1 - \lambda^2}\right),


where we introduced the so-called charge-to-mass parameter \lambda = \frac{Q}{M}. This question I want to consider now is:
Can we enlarge Q (resp. \lambda) such that \lambda > 1 and therefore r_{\text{H}} \in \mathbb{C}?  This would correspond to a vanishing event horizon and a naked singularity.

The simplest idea is to trough into the black hole charges particles, i.e. we search for suitable values of the charge q, the mass m and the energy E of a particle such that:

  1. The particle falls into the black hole from infinity and is not reflected
  2. The resulting charge-to-mass parameter \lambda is greater than before

In order to calculate this we will use the principle of stationary action and the Lorentz-force on a charged particle. Then the Lagrangian is given by

 \mathcal{L} = - m \sqrt{ - g_{\mu \nu} \frac{\text{d}x^\mu}{\text{d}s} \frac{\text{d}x^\nu}{\text{d}s}}+ q \frac{\text{d}x^\mu}{\text{d}s} A_\mu(x(s)),


where x(s) = \{ x^\mu(s)\} is the trajectory of the particle, parametrized by an affine parameter s. Also we require that the trajectory of the particle is timelike, i.e. the condition  g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = -1 has to be full filled. We will now calculate all equations of motion to get conditions for the parameters of the particle. Due to radial symmetry of the problem we will use spherical coordinates (t , r , \phi ,\theta).

  • Since \mathcal{L} is cyclic in t, we know that the energy is conserved:

     \frac{\partial \mathcal{L}}{\partial \dot{t}} = - \frac{m}{\sqrt{- g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu}} \frac{\Delta}{r^2} \dot{t} - \frac{q\, Q}{r} = - \frac{m \, \Delta}{r^2} \dot{t} - \frac{q \, Q}{r} =: -E,

    with \dot{t} = \frac{\text{d}t}{\text{d}s}. Since \frac{\Delta}{r^2} \geq 0 and \dot{t} \geq 0 for all  r \geq r_{\text{G}} we get

     E - \frac{q \, Q}{r^2} = m \frac{\Delta}{r^2} \dot{t} \geq 0 \quad \forall r \geq r_{\text{H}}.

  • Also \mathcal{L} is cyclic in \phi and therefore we have conservation of angular momentum. We can set \dot{\phi} = L = 0.
  • Now we calculate the Euler-Lagrange-equations for \theta:

     \frac{\text{d}}{\text{d}s} r^2 \dot{\theta} = r^2 \sin \theta \cos \theta \dot{\phi}^2 =0.


    Set \theta = \frac{\pi}{2} and \dot{\theta} =0. Applying this to the timelike condition we get

     - \frac{\Delta}{r^2} \dot{t}^2 + \frac{r^2}{\Delta} \dot{r}^2 = -1


    and therefore

     \dot{r}^2 = \frac{\left(E - \frac{q \, Q}{r}\right)^2}{m^2} - \frac{\Delta}{r^2}.


    This is indeed the same as the Euler-Lagrange-equation for r.

Expanding this equation gives us

\frac{E}{2m} - \frac{m}{2}= \frac{1}{2} m \dot{r}^2 - \frac{m \, M}{r} + \frac{m Q^2}{2 r^2} - \frac{q^2 Q^2}{2m r^2} + \frac{E \, q \, Q}{m r} := \frac{1}{2} m \dot{r}^2 + V(r),


where we defined the effective potential V(r) = - \frac{m \, M}{r} + \frac{m Q^2}{2 r^2} - \frac{q^2 Q^2}{2m r^2} + \frac{E \, q \, Q}{m r} . Then we can deduce the conditions we need:

  • We need \dot{r}^2 > 0, so that the particle will always fall into the blackhole and will never stop. Together with the energy condition this tells us that

     E - \frac{q\, Q}{r} > m \sqrt{\frac{\Delta}{r^2}} \geq 0 \quad \forall r \geq r_{\text{H}}.

  • We also need that the energy of the particle is sufficient so overcome the potential barrier of V(r), i.e.  E > \frac{q \, Q}{r} and therefore

     E > \frac{q \, Q}{r_{\text{H}}} = \frac{q \, Q}{M + \sqrt{M^2 - Q^2}}.

  • As a last condition we need that E > m for  r \rightarrow \infty

In order to satisfy condition 2, i.e. to get an enlargement of \lambda we also need \frac{Q}{M} < \frac{Q + q}{M + E}, which implies that  E < \frac{q M}{Q}. By using the inequality from \dot{r}^2 >0, this is also equivalent to the requirment

 m < \sqrt{\frac{r^2}{\Delta}} \left( E - \frac{qQ}{r} \right) \quad \forall r \geq r_{\text{H}}.


An analytical calculation shows, that the RHS of the above equation has its minimum at

 r_{\text{m}} = Q \left( \frac{M q - Q E}{Qq - M E} \right).


Finally, if one has  M > Q the following two conditions can occur:

 \frac{qQ}{r_{\text{H}}} < E < \frac{qQ}{M} <\frac{qM}{Q} \quad \text{or} \quad \frac{qQ}{r_{\text{H}}} < \frac{qQ}{M}\leq E <\frac{qM}{Q} .

 


Summary and the Third law of Black-Hole-Dynamics

As we have shown in the above calculations: For M > Q one can always find values E, m and q for a particle such it will enter through the event horizon and will increase its charge-to-mass parameter \lambda.
But for M = Q no particle can enter and therefore we can not reach \lambda > 1 by this method.
That means, we proved in the case of a non-rotating black hole, that the cosmic censorship holds and no naked singularity can occur by 'overcharging' a black hole.

As a last side remark I want to mention that this is in perfect agreement with the Third law of Black-Hole-Dynamics. If \kappa denotes the surface gravity at the horizon, the law states that \kappa = 0 can not be reached within a finite steps. Therefore we also can not 'jump' over or reach  \kappa = 0. This is here analogue to \lambda =1.


 Literature

To prepare this talk is used the following sources:

Mathematics - curse or blessing?!

With the begining of this semester I also started my bachelor in the topic of mathematics.
By studying  a lot of very dry definitions of algebraic sets and topological features of closed set the question emerges:

Do we need all of this as theoretical physicists?!

I.e. does "nature" really care about e.g. closed or open sets?
This was and is an absolute non-trival question. Most physicists would argue that these mathematical concepts only emerge from our logical understanding as a human. But on the other side: Humanity has build up logical concepts from watching at nature itself. Consider for example the integer numbers \mathbb{N}. These arise naturally from counting stuff like apples or so.
But the question is then, what is couting? It is the ability to see similarities between two things and to say, that these two or more objects are equal. This then give us naturally the concept of \mathbb{N} in order to specialize how many "equal" things there are.

This was just ment to give a little example of how mathematical concepts arise from nature and this is also the reason why I think that mathematics is much more a blessing as a cure for any theoretical physicists. It gives the ability to "see" structures, similarities and relations between objects and theories, that somehow might look completly different.
Due to this face I would encourage everyone, who would love to work in theoretical physics, to put his effort a little bit more towards mathematics. Of course one should not lose his mind in the big sea of defintions, lemma and propositions but being a little bit more rigorious about math will benifit a lot. At least that is was I think and I hope to get from my studies in these fields.

Holographic Gravitation

At the last Wednesday, the 24.09, I have a talk at my old high school with the title:

Holographic Gravity
Black Holes and Information.

I tried to explain to people of normal high school level to explain by a lot of analogies like fishes in a pool with a drain hole are "analog" to a black hole and several others, how the idea of holography came up during the last decades and how we can understand this.

This talk was very close to the one Leonard Susskind gave some years are and I have chosen his way, expect of the String Theory part, because it summarizes the whole concept of holographic gravity in a very basic way that is even for pupils easy to understand.

This kind of talk is the second on I gave (the first one was exactly one year ago in the same school and had something to do with Quantum Information Theory) and I will definitely continue to give more talks like this on a public level because I think it is very import that we should not sit in our universities and wait for students to come to us. Rather we should try to go to them and fascinate them we the things we accomplished.

Here you can find the slides to the talk (German):

http://prezi.com/qg5chgbpd-g2/holographische-gravitation/

And here is also a report of my talk from the high school:

http://elze.cjd.de/elze/pages/index/p/6968/9231