QG pheno w/o Lor violation: scattering on defects

1. Sep 4, 2013

marcus

Bee Hossenfelder has proposed a new approach to QG phenomenology, involving new ways to look for signs that space-time arises from a fundamentally non-geometric theory.

The basic hypothesis is that if what looks like a geometric continuum actually arose from myriad nongeometric entities then there will be DEFECTS (on which light can scatter.)

Traveling over cosmological distances a photon may SCATTER on such imperfections with a certain cross-section probability. Hossenfelder finds that it can be knocked this and that in such a way that effects cancel and on average there is no Lorentz violation---no dispersion on average.

Besides Lorentz violation, she identifies other observable effects depending parameters that are constrained by CMB data already collected. She plots the portions of parameter space which are still open, not having been ruled out so far.

The work is in two parts.

http://arxiv.org/abs/1309.0311
Phenomenology of Space-time Imperfection I: Nonlocal Defects
Sabine Hossenfelder
(Submitted on 2 Sep 2013)
If space-time is emergent from a fundamentally non-geometric theory it will generically be left with defects. Such defects need not respect the locality that emerges with the background. Here, we develop a phenomenological model that parameterizes the effects of nonlocal defects on the propagation of particles. In this model, Lorentz-invariance is preserved on the average. We derive constraints on the density of defects from various experiments.
25 pages, 7 figures

http://arxiv.org/abs/1309.0314
Phenomenology of Space-time Imperfection II: Local Defects
Sabine Hossenfelder
(Submitted on 2 Sep 2013)
We propose a phenomenological model for the scattering of particles on space-time defects in a treatment that maintains Lorentz-invariance on the average. The local defects considered here cause a stochastic violation of momentum conservation. The scattering probability is parameterized in the density of defects and the distribution of the momentum that a particle can obtain when scattering on the defect. We identify the most promising observable consequences and derive constraints from existing data.
18 pages, 5 figures

2. Sep 4, 2013

MTd2

http://arxiv.org/abs/1309.0652v1

Non-abelian Gauge Fields from Defects in Spin-Networks
Deepak Vaid
(Submitted on 3 Sep 2013)
\emph{Effective} gauge fields arise in the description of the dynamics of defects in lattices of graphene in condensed matter. The interactions between neighboring nodes of a lattice/spin-network are described by the Hubbard model whose effective field theory at long distances is given by the Dirac equation for an \emph{emergent} gauge field. The spin-networks in question can be used to describe the geometry experienced by a non-inertial observer in flat spacetime moving at a constant acceleration in a given direction. We expect such spin-networks to describe the structure of quantum horizons of black holes in loop quantum gravity. We argue that the abelian and non-abelian gauge fields of the Standard Model can be identified with the emergent degrees of freedom required to describe the dynamics of defects in symmetry reduced spin-networks.

3. Sep 5, 2013

MathematicalPhysicist

Physics without geometry seems like a good way to being cited.

I don't understand, space-time arises from a theory where you cannot measure distances?

4. Sep 5, 2013

marcus

No, that's not it.

The emergence of normal geometry (continuity, uniform dimensionality, smoothness...) from non-geometric or pre-geometric systems is an OLD subject of research and there have been many different proposals. Although various approaches and models have been proposed and investigated, I can't think of any corresponding to what you said. All those that come to mind have some obvious way of defining proximity in the pre-geometric phase.
The subject goes back to before 1992. Here are a couple of 1992 papers:

http://www.sciencedirect.com/science/article/pii/055032139290444G
"We consider a discrete model of euclidean quantum gravity in three dimensions based on a summation over random simplicial manifolds. The phase diagram as a function of the cosmological coupling constant and the gravitational coupling constant is studied in the search for a sensible continuum limit. An interesting cross-over behaviour is observed, where the universes change from being crumpled, of essentially zero radius, to being extended objects."

http://www.sciencedirect.com/science/article/pii/037026939290709D
"We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The phase diagram as a function of the bare gravitational coupling constant is studied in the search for a sensible continuum limit."

About what you imagined---the impossibility of defining distance in pre-geometric systems: as I say, I don't happen to know any cases of that. I'm not an expert, just moderately alert to what's going on in the field, so other people may know of cases. But all the pre-geometric schemes I can think of DO have an idea of distance or proximity that is prior to the emergence of a more regular geometry with some definite dimensionality.

To take a simple example, think of a complete graph with a large number of nodes. Every node is within unit length (say planck length) of every other. Then there is a dynamical rule according to which connections between most pairs of nodes eventually get broken and the system spreads out to form something like a regular lattice (but with a sprinkling of defects).

So you start out with say 10100 points where each point has approximately 10100 neighbors immediately adjacent to it (within unit distance) and you evolve towards a setup where each point has some more familiar number (like 4 or 6 or 8) immediate neighbors, depending on what dimensionality is favored by the graph dynamics.

Or we could be talking about simplicial complexes (collections of triangles, or tetrahedra, or 4-simplices). Dynamical triangulations instead of graphs.

You seem interested in getting citations, Ma-Phy. This paper (submitted in 1991) has accumulated 142 citations according to Inspire database:
https://inspirehep.net/record/322467?ln=en
Doubtless there are some papers/books on the subject that have gotten more citations than 142 and many that have gotten fewer. If citations are a special concern of yours for some reason, then I can try to get more info.

Last edited: Sep 5, 2013
5. Sep 5, 2013

marcus

Some people may not understand what the aim here is, and what is accomplished. The key word is phenomenology. In the past 20 years we've seen a variety of specific proposed models for how the macroscopic geometry we experience may have arisen from microscopic-level pre-geometry. What she presents is a general testing scheme not tied to any one specific theory. The point is that one typically expects DEFECTS to occur with some frequency, if spatial geometry is not an absolute given but has emerged by some process. One can test for defects, and that then covers a variety of diverse models.

Moreover a growing number of papers (following Ted Jacobson's 1995) treat GR geometry as a thermodynamic equation of state and/or geometry as having a thermodynamics---thus a statistical mechanics based on microscopic degrees of freedom---thus as something that emerges from a microscopic pre-geometry. BH temperature and entropy were earlier hints of this. Again, if geometry arises as a mass macroscopic thermodynamic phenomenon, it could have DEFECTS. And so Hossenfelder's phenomenological approach to testing could apply.

And apply very generally, not tied to any one theoretical picture. So I think this paper could be important, in particular, and might reward some of us taking a closer look:

http://arxiv.org/abs/1309.0314
Phenomenology of Space-time Imperfection II: Local Defects
Sabine Hossenfelder
(Submitted on 2 Sep 2013)
We propose a phenomenological model for the scattering of particles on space-time defects in a treatment that maintains Lorentz-invariance on the average. The local defects considered here cause a stochastic violation of momentum conservation. The scattering probability is parameterized in the density of defects and the distribution of the momentum that a particle can obtain when scattering on the defect. We identify the most promising observable consequences and derive constraints from existing data.
18 pages, 5 figures

6. Sep 6, 2013

MathematicalPhysicist

Perhaps I didn't understand what's a non-geometric theory?
Where in geometry you have distances, I mean even a discrete geometry is possible.

7. Sep 6, 2013

marcus

Hi Ma-Phy, it seems the issue here is semantics, and maybe there is no RIGHT (generally accepted) meaning of the words.
Did you have a look at Jacobson's 1995 paper about GR being the thermodynamic equation of state of some unknown unspecified bunch of microscopic degrees of freedom? The paper seems to have set off a train of thought that is running thru a lot of people's minds.

For me a geometry requires at least a *topological manifold* that is something with a uniform finite dimension because locally homeomorphic to Rd.
A topological manifold does not have to be smooth, it can be kinky wrinkly etc, and it does NOT need to have a unique distance function. But at least it has a well-defined uniform dimensionality.

The trouble with random graphs is they don't necessarily even have dimensionality. You can't say the dimension is 3, or 4,or 10. They might look like the dimension is 40 somewhere and have a long tail of dimension ONE trailing off somewhere else.

A complete graph is one where every vertex is connected to every other vertex. Maybe you have a complete graph with 100 vertices and you say the distance between any two pair of points is one centimeter. What is the dimension? Perhaps 99? But a general graph could have different regions of different degrees of connectivity. Distance can be defined on a set of points without it having geometry as we know it. The idea is that geometry as we know it (macroscopically gr) could emerge from something that is not at all like that---call it "non-geometric" if you like, or call it a "pre-geometry"

Something that does not have the usual features of a uniform small finite dimensionality, continuity, smooth structure---only has some, or none of what we associate with geometry.

In GR a *geometry* assumes more structure than just a topological manifold---you should have a differential (smooth) manifold that you can define a metric tensor on, and it is the metric tensor that specifies the geometry.

Just my personal take on it. Maybe there is no generally accepted meaning of "non-geometry".
Have to go out on some errands,maybe we can discuss this more later.

Last edited: Sep 6, 2013
8. Sep 6, 2013

Blackforest

Just to reenforce the concept of topological manifold (citation from the document called "Hamilton's Ricci flow", university of Melbourne).

A topological space Mn is an n-manifold if it looks like Euclidean space (Rn) near each point. The formal definition has some other technical conditions as well, to avoid certain pathologies that may arise:

Definition 1.1. A topological space Mn is a topological n-manifold if:
1. For each p ∈ Mn there is an open neighbourhood U of p and a function ϕ : U → Rn that is a homeomorphism onto an open subset of Rn. The pair (U, ϕ) is called a coordinate chart. We will frequently write ϕ(q) = (x1(q), x2(q), . . . , xn(q)). These xi(q) are referred to as local coordinates for Mn.
2. Mn is Hausdorff.
3. Mn is paracompact.
We will usually write M for a generic manifold, and Mn for an n-manifold if the dimension is of particular relevance. In this project we will ... (end of the citation).

The concept of distance comes later.

9. Sep 6, 2013

marcus

Thanks, BF, good to have a rigorous definition! I think you are right, working in the context of manifolds (although maybe in graph theory one could say that one of the first ideas is the primitive sort of distance where one simply counts the minimum number of links you need to get from one point to the other.)

In the context of manifolds, first one has a mere *topological space* a set of points with no shape at all and no dimensionality. Then it proceeds by successively adding structure:

1. topological space
2. topological manifold (locally homeomorphic to Rn for some n)
3. differential (aka "smooth") manifold (coordinate patches overlap smoothly, one can do calculus)
4. Riemannian manifold (give the differential manifold a metric)

And the metric is what finally defines what we think of as a geometry. In ordinary GR parlance I think a geometry is technically an equivalence class of metrics that are diffeomorphic to each other. IOW the idea of general covariance---changing coordinates shouldn't make a difference or even going to another manifold as long as you take everything along with you. So a geometry is basically specified by a metric defined on the tangent bundle (but technically you go up one level of abstraction.)
======================

But what about other contexts, like graph theory? Or "dynamical triangulations", or spin networks and spin foams? It seems like in some of these contexts you can have a *quantum state of geometry* without having all the elements we have discussed here.

I think of it this way: when we work with a labeled graph, such as a spin network, we are TRUNCATING the geometric degrees of freedom by imagining that we can only make a finite number of geometrical measurements, e.g. of areas, volumes, angles etc.

IOW the complete geometry in all its fully refined beauty is not represented by any mathematical object. The theory only allows truncation down to a finite number of geometrical particles, measurements, modes, atoms or whatchacallits. A finite number of geometric degrees of freedom, no limit on the number, it can be as large as you like, but finite.

Last edited: Sep 6, 2013
10. Sep 6, 2013

Blackforest

That's very interesting but at least for me very speculative. I am my self looking pictures to describe these pre-geometric states and that pre-geometry in general. IOW, what follows is (like some people around me say) coming from my "own kitchen".

What is embarassing is not only the problem with the dimensionality you mentioned. There is also a difficulty in imagining a universe which would only be a net made of small pieces of strings connected to each other and nothing else (absolute vacuum) in between (the problematic of a kind of discontinuity).

Concerning the discussion around the defects (I didn't read all details - its the nigth here) it's also embarassing because what are these defects: (a) the particles themselves or what we perceive as such? (b) the absolute vacuum I have mentioned previously?

If we just consider the relative proportion of what we, untill now, call massive particles, the concept of classical mass (going back to Newton) appears to be a rarity, an accident, a defect in an ocean of vaccuum. As you see I don't have my self a clear representation. But it is a real pleasure to have the opportunity of a discussion around that item.

My own kitchen is bringing me the repetitive image of small tetrahedrons because they have some remarkable properties and because they allow a concept of ternary distance. But once more time (and I beg my pardon) this is speculation.

Nevertheless I have the intuition that our binary way of thinking is limiting our understanding. We have the greatest difficulty to think global. A net can evidently be characterized by a set of classical distances, but that kind of description certainly forgets the angles and the same net can also be seen globally, exactly in the same manner that we proced when we analyze a form, a person coming to us (trying to guess what he/she thinks just in observing his/her global behavior).

11. Sep 7, 2013

MathematicalPhysicist

I guess it's a problem of terminology, even if smooth geometry isn't the basic fundamental building block we still some sort of geometry.

Can you give me a link to Jacobson's article you referring to?

12. Sep 7, 2013

marcus

http://arxiv.org/abs/gr-qc/9504004
Thermodynamics of Spacetime: The Einstein Equation of State
Ted Jacobson
(Submitted on 4 Apr 1995)
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation $\delta Q=TdS$ connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with $\delta Q$ and $T$ interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
8 pages, 1 figure.
Phys.Rev.Lett.75:1260-1263,1995

I see from the Inspire entry that this paper has 629 citations:
http://inspirehep.net/record/394001?ln=en
As I recall, Jacobson has (co-)authored one or more follow-up articles. I vaguely remember the date of one of them as around 2004. Maybe, if you want to read more on this, someone can supply a link or two to interesting further development by Jacobson himself.

Last edited: Sep 7, 2013
13. Sep 7, 2013

marcus

Ma-Phy, I succumbed to the temptation to offer Smolin's 2012 article as a sample of the followup to Jacobson 1995.

http://arxiv.org/abs/1205.5529
General relativity as the equation of state of spin foam
Lee Smolin
(Submitted on 24 May 2012)
Building on recent significant results of Frodden, Ghosh and Perez (FGP) and Bianchi, I present a quantum version of Jacobson's argument that the Einstein equations emerge as the equation of state of a quantum gravitational system. I give three criteria a quantum theory of gravity must satisfy if it is to allow Jacobson's argument to be run. I then show that the results of FGP and Bianchi provide evidence that loop quantum gravity satisfies two of these criteria and argue that the third should also be satisfied in loop quantum gravity. I also show that the energy defined by FGP is the canonical energy associated with the boundary term of the Holst action.
9 pages

It shows where this idea can lead. The idea is that Einstein GR equation of about 100 years ago is analogous to "PV = nkT", that is it is telling us the overall statistical behavior of some tiny microscopic (pre-)geometrical degrees of freedom that we can't see!

And so we can try to imagine what these microscopic entities could be that underlie macro geometry analogous to how gas molecules underlie our experience of P, V, and T in that other equations of state. And Smolin presents an argument that the underlying (pre-)geometric d.o.f. could be something like the discrete 2-cell complex that spinfoam QG is built on.

In any case, he argues that whatever the microscopic d.o.f. are (that GR is the EoS of) this microscopic picture has to satisfy three conditions. And then he argues that the spin foam QG picture is plausible because it seems to satisfy those conditions.

And Smolin's conjecture (or case for plausibility) is not the only such! I can only offer it as a sample of the various idea people have. If Einstein GR field equation is an EoS then it becomes an interesting question to try to guess what the underlying micro d.o.f. could be.

And if, for example, it is something analogous to a CONDENSED MATTER picture, say remotely analogous to a crystalline structure, then it could have DEFECTS. So you get a phenomenologist like Hossenfelder struggling with the problem of how could you look for those defects, and possibly *rule out* certain pictures because we do or do not see evidence of defects.

Last edited: Sep 7, 2013
14. Sep 7, 2013

DennisN

Marcus, just to make sure I am picking up this correctly, by "EoS" you mean Equation of State, right?

15. Sep 8, 2013

marcus

Exactly. I neglected to mention that. Thanks for clarifying--it helps other people who might be reading the thread. BTW nice picture of Zoidberg and Bender!

Last edited: Sep 8, 2013
16. Sep 8, 2013

Paulibus

This thread reminds me of a ship setting out across an unknown sea — in this case, the sea of innovation. The ship may encounter the tips of unexpected icebergs which, hopefully, it will safely navigate around. I wish it bon voyage. But in the past physics has often foundered on such icebergs. Long ago, what we now call the vacuum was as modelled as an elastic continuum. It was then postulated by Burton, in 1892, that matter was simply modifications in the structure of such a continuum, and in 1897 Larmor considered local defects that he called ‘strain-figures’ as entities in such a medium. And then there was the vortex atom theory of the closing years of the 19th century, supported by Kelvin; vortices can be regarded as stable dynamic defects that alter local spatial symmetry. So much for past discussions of defects in spacetime.

For me a ‘defect’ is synonymous with ‘a local change of symmetry’. Defects in crystals are various local changes in the symmetries that characterise these solids. Point, line, areal and volume defects locally change the global symmetries of crystals (translation, rotation, reflection and inversion symmetries). These defects are then labelled, respectively, vacancies /interstitials , dislocations/ disclinations, stacking faults and, in polycrystals, grains. The long twentieth-century history of the defect solid state suggests to me that space-time defects may turn out to be more complicated than those envisaged by Bee Hossenfelder.

Points that for me need clarification are:

What symmetries, if any, are to be changed locally by the sort of defects Hossenfelder proposes? She writes
that “Unlike defects one normally deals with (e.g. in crystals), the defects under consideration here occur on space-time points and do not have world-lines.” Does this preclude symmetry changes?

What about localised points of mass/energy; are they defects or not? They induce local spacetime curvature, thus changing the local symmetry of flat Minkowskian spacetime, and in my view qualify as defects in vacuum homogeneity.

Perhaps it’ll turn out that elementary particles are just point defects that change local symmetries of the
vacuum in still-unknown but distinctive ways!

Last edited: Sep 8, 2013
17. Sep 9, 2013

marcus

Yes, a risky venture. But some people think there are strong hints that there exist microscopic d.o.f. from which the experience of macroscopic geometry arises, and which allow geometry and matter to interact. Something you suggested, in a way, at the end of your post. geometry and matter fields have in some way a common root and so they naturally interact (they reciprocally guide and bend each other). So, if one believes there are strong hints of microscopic "pre-geometry" d.o.f., then one thing to do is venture to model them, and (as Hossenfelder does) prepare to TEST models by the predictions derived from them.

Paulibus, you might be interested to learn of a project and an upcoming conference about the conceptual basis of approaches to quantum gravity. There are several links (I list here in no particular order)
http://philosophyfaculty.ucsd.edu/faculty/wuthrich/ [Broken]
Sample quote:
"I am an associate professor of philosophy and of science studies in the Department of Philosophy at the University of California at San Diego (UCSD).
My philosophical interests most prominently include foundational issues in physics, particularly in classical general relativity and quantum gravity. Of course, I also get excited about the implications of philosophy of physics for general philosophy of science and metaphysics. More specifically, I also enjoy thinking about issues such as space and time, persistence, identity, laws of nature, determinism, and causation. I have also historical interests, particularly in the history of physics. My Erdös Number is 3 (Erdös → Tarski → Andreka and Nemeti → me).
Emergent Spacetime in Quantum Theories of Gravity
Nick Huggett (University of Illinois, Chicago) and I have started a large new research project funded (inter alia) by the American Council of Learned Societies on the emergence of spacetime from fundamental, non-spatio-temporal structures as suggested by many approaches in quantum gravity..."

Here's Nick Huggett's CV, notice his two BOOKS IN PROGRESS, one coauthored with Wüthrich:
http://tigger.uic.edu/~huggett/Nick/Home_files/CVweb.pdf

Here's their project/conference webpage, notice the abstracts of some of the talks to be discussed!:
http://beyondspacetime.net/discussion/ [Broken]

Here is the 3-day schedule of talks:
http://beyondspacetime.net/seminar-program/
More stuff:
http://beyondspacetime.net/people/

The book that Huggett and Wüthrich are working on is titled "Out of Nowhere – on the emergence of space and time in quantum gravity"

Last edited by a moderator: May 6, 2017
18. Sep 10, 2013

Paulibus

Marcus: you seem to have total recall of an astonishingly eclectic range of references. Thanks for taking the trouble to steer me towards to this philosophy-centred perspective on the mysteries of spacetime. But: long live the Baconian tradition!

19. Sep 16, 2013

Paulibus

In my post # 16 I suggested that space-time points (which are the entities Hossenfelder labels
as ‘defects’) can be so labeled if they are point-like elements of mass/energy that distort
otherwise flat spacetime by causing it curve, as described by non-Euclidean geometry from a
strictly continuum perspective.

I’d like to add (for anybody interested in Hossenfeld’s ‘struggle’) that continua may
be variously imperfect, both in theory and in practice. For example a mathematical function, or
its derivatives, can have 'imperfections' or 'defects' like singularities or discontinuous steps. And, at an above-atomic resolution, stuff like plastic kitchen-foil, liquids, and amorphous solids are all conveniently treated as continua. Yet foil can be crumpled, or flawed by cuts and holes. Similarly, liquids may contain bubbles, while voids, cracks, strains and Volterra dislocations occur in solid continua. These defects are all discontinuities of a sort, simple to label but not quite so simple to describe quantitatively and predictively, i.e with mathematics and physics. Indeed it has proved impractical to design strong materials by applying only such tools.

For me, a question that may have an interesting answer is: if familiar continua can be
imperfect in several ways, why should spacetime be imperfect only in one, continuous, sort of
way, namely just by being curved?

For instance, from a defect perspective the always-attractive nature of the mass/energy
interaction, which contrasts with the attractive-repulsive nature of the electrostatic interaction
could signal that (+ and -) charge is associated with some spacetime defect with a similar dichotomy of character, akin to various crystalline defects.

Some of the looked-for ‘evidence’ for defects that Marcus mentioned might yet prove to have been staring us in the face for most of the last century!

Last edited: Sep 16, 2013
20. Sep 16, 2013

marcus

If it's OK with you, since we've turned a page, I'll bring your #16 forward. I'd like to have it in front of us since it raises some important issues.
I think HOSSENFELDER in her papers is trying to establish a general phenomenology of defects that is not tied to any specific model or specific idea of what defects ARE. Rather the game is about a general class characterized by what they DO. If you are a photon, a "defect" is something that throws you off course, sets you back, or bumps you forward.

She wants her calculus to apply to whatever model of space-time. As an intuitive example, it might be a nearly perfect regular lattice with occasional mistakes---pairs of points which should not be connected but which by some sort of random error which occurred in the growth of the lattice are nevertheless connected. Her idea is that any type of DISCRETE model, whatever it is "made of" mathematically (points, lattice links, tetrahedra, ...), it can have a small approximately uniform distribution of probability that a photon's progress through it will be disrupted.

She proposes to model the effect of these random disruptions (however they are depicted mathematically!) by a continuously distributed probability of of encounter. A kind of "cross-section" attached to the population of defects.

Further, I think she assumes that ON AVERAGE you have Lorentz INVARIANCE. Correct me if I am mistaken: I don't fully understand this but I gather these little jolts have zero net effect on the speed of light. Or perhaps they are extremely rare and have no measurable Lorentz violating effect. You may be able to read more closely and get a better idea of what's being said. What she discovers, it seems, is that that even though there is no gross overall Lorentz violation there still could be something (some more subtle fuzziness) which the phenomenologist can look for!

As I say, what I like about her approach in these papers is that she is not tying analysis to any particular ontology, or any specific quantum geometry of spacetime. She wants to be about to use the same analytical tool to compare/constrain the different models belonging to a broad class.

BTW I like this way of putting it: "For me a ‘defect’ is synonymous with ‘a local change of symmetry’." Because it is local, it necessarily has a LOCATION. I don't think that Hossenfelder identifies defects as points especially. Mathematically they could consist of various things depending on the qg model. But like you, she assumes that, whatever it is, a defect has a location, and it has a probability of encounter. A cross-section-type chance that a photon passing through the neighborhood will be effected by it.

So what she uses, mathematically, is a kind of continuously distributed scattering cross-section, with no assumption about ontology. I think

Last edited: Sep 16, 2013