One of the things I sometimes find myself writing about is the “bad” language used by physicists.  Sometimes we say Riemannian when we really should say psuedo-Riemannian, sometimes we call something a metric when it really is a line element – the kind of nitpicky pet-peeves that practically everyone has about literature in their field.  Today, I’m going to be talking about the bad language in physics in a totally different context however.

Teepee Lattices, Future-Pointing Wigwams and Polish Numbers

My secret love for discrete spacetimes comes from a beautiful little sub-field of general relativity (that is experiencing a little bit of a revival right now thanks to loop quantum gravity) called Regge calculus.  Regge calculus was a method suggested by John Wheeler and his student, Tullio Regge, in the early 1960s[1], to find approximate solutions to the Einstein Equations. Their basic idea was just to simplify spacetime and see what happened.  Instead of having one complicated, curved, four-dimensional Lorentzian manifold to work in, we would piece our spacetime together out of four-dimensional simplices (the higher order word for triangle), that would have, nice, simple, flat interiors, so the entire picture would show curvature, but each individual section would be easy to describe and work in.

Consider this 2D example

This two-dimensional “simplexification” might be able to give you a better idea of the process. Here, we have triangulated a small “curved” surface.  The interior of each triangle (a 2-simplex) is a flat, Minkowskian space, and the curvature is manifest at the vertices (0-simplices) where the triangles meet.  When we scale this concept up to four dimensions, we end up with flat, Minkowskian, 4-simplices, and then our curvature is contained at the 2-simplices (curvature is now manifest on triangles, not points) where the 4-simplices meet¹.

So that is the basic idea behind Regge calculus – we break up spacetime into simple, triangular, chunks.  The implementation of it is where the real difficulties arise.

Evolution

When we think of our favourite formulation of general relativity in the continuum limit, most of us probably think of the ADM formalism (No? I bet you’ve never even turned your alarm clock off in your sleep thinking that you still needed the lapse and shift to know what time it really was).

Creating a 4-geometry by "sandwiching" two 3-geometries (this is a "sandwich" of infinitesimal thickness): The 4-metric (ie. geometry) of the full 4D spacetime (which is what GR is all about) depends on the lapse and shift of the connectors ("sandwich filling") between the two 3-geometries as well as the 3-metric (the geometry of the "bread"). N is the lapse function (to get the proper time between the upper and lower surfaces you need Ndt) and Nⁱ is the shift function.

The ADM formalism says that we should foliate (slice) our continuous spacetime into three-dimensional spacelike surfaces that we can label by their time coordinates and then define our dynamics from there. It is an elegant, simple and well studied idea that is exceptionally powerful, so logically, it seems like a good set of concepts to work into Regge calculus.

Now, the original idea of Regge calculus was to model the four-dimensional Lorentzian manifold of spacetime by simplices (the Regge analogue of the tangent space), that have flat, Minkowskian interiors (although in his original paper [1] he actually used simplices with flat, Euclidean interiors, which effectively removed the ‘physics’ from the problem), but this didn’t leave much room for describing dynamics. To make Regge calculus more like our beloved ADM formalism, we approximate our differential three-manifolds² (our spacelike slices) by a collection of simplices, which allows us to preserve many important topological properties, while giving us room to describe dynamics (like the evolution of the universe, for example).

Simple Regge Evolution: (a) base triangulation (b) each vertex evolved (c) all connections to rightmost evolved vertex made.

So, the basic idea of evolution in Regge calculus is to take each vertex at a time t and “evolve” it up to some time t + dt (the lines connecting upper and lower vertices are in spirit with our connectors from the ADM foliation).  Since we also want to maintain some sense of reasonableness in our model (ie. that spacetime is locally path-connected, and is thus also connected), we also connect each initial vertex to each evolved vertex, so, for a single 2-simplex, we get:

2-Simplex Evolved: A lattice in 2 + 1 dimensions.

It’s a pretty simple idea, but, if you made it through all of the above, you probably found yourself asking, “But how do we actually know how to evolve a vertex? How do we know how long/at what angle to make all of those connections?”.  That’s a very good question, and one that doesn’t have a definite answer at this point (despite what certain individuals that have certain well known Regge evolution schemes named after them might think).  There is a sizable body of work dedicated to figuring out how to make those connections as physical as possible.

One such example came from Mark ﻿Galassi, in 1992, in his PhD dissertation on this very topic in which we were introduced to the concept of a teepee lattice [2].

Teepee (public domain image from Wikipedia)

From Collins English Dictionary:

teepee, noun: a cone-shaped tent of animal skins used by certain North American Indians [A]

It’s pretty obvious where this is going…

From Galassi's "Lattice Geometrodynamics"

The resemblance is uncanny(ish).

What’s nice about Galassi’s “teepees” is that they make the connection to ADM’s lapse and shift more obvious (if you’re familiar with both Regge calculus and the ADM formalism, that is) [3].  Other than that, it’s a rather funny name.  At least in Canada, gratuitous use of the word “teepee” makes a lot of people cringe.  In many circles, “teepee” is considered to be politically incorrect, because it has been so overused as part of stereotyping the Native “Indians”.  Political correctness be damned, says the physicist, it’s a fairly illustrative name for what’s going on in Regge evolution.  Is it offensive to some? Maybe.

Interestingly, this Native American naming concept didn’t come from Galassi, but came from Kheyfets et al. a few years earlier during the “Null Strut calculus” craze, with the introduction of spacetime wigwams [4].  Null Strut calculus is a variant of Regge calculus that insists that the connection between a “t” vertex and its “t + dt” counterpart is to be null-like.

Ojibwa wigwam (public domain image from Wikipedia)

From Collins English Dictionary:

wigwam, noun: any dwelling of the North American Indians, esp one made of bark, rushes, or skins spread over or enclosed by a set of arched poles lashed together [B]

The relation to Regge calculus doesn’t seem as obvious here, as a wigwam is basically just a rounded house, but that’s only if you actually know that.  A sizable number of people, including some physicists, don’t know that wigwam doesn’t equal teepee.

From Kheyfets's "Null Strut Calculus II. Dynamics". The main visual difference in Kheyfets's sketch and Galassi's sketch above is that Galassi's is representing a 1+1 spacetime while Kheyfet's is representing a 3+1 spacetime, ie. the general shape is the same, it's just the dimensionality that is different.

Null strut calculus allows one to “erect past-pointing wigwams” or “future-pointing wigwams” in “thin null strut sandwiches”, depending on how the vertices are aligned with time.  For silly names alone, null strut calculus is a blast to read.

Apparently, so the story goes, one of the authors on the Kheyfets paper, who choose the name wigwam to describe their lattice, did so because he thought it meant the same thing as teepee (ie. that a wigwam was a triangular looking construction).  In any case, neither wigwam nor teepee are especially politically correct words to name your structures these days, but, that wasn’t(/isn’t) always part of the thought process.  Regardless of correctness, wigwam really isn’t the most illustrative name for a simplex lattice structure (as again, it has nothing to do with triangles).

This whole idea does lead nicely into one of my favourite papers, from title alone, “Pitching Tents in Space-Time” by Üngör and Sheffer [5].

“Tent pitching”, and other methods for tessellating spaces, is where Regge calculus and computer science, a field that has its own share of semi-politically incorrect naming practices, come together.  Polish notation, a form of notation in arithmetic, algebra and logic, is one such example.  Since it is parenthesis-free, to the untrained eye, it can look pretty unruly and confusing to begin with.  Like many other students who first meet it, I was taught that “Polish” notation was named as such because it was confusing and backwards like the Polish language (seriously, I know many other people who learned it was named because of this).  In actuality, Polish notation is called “Polish” because of its creator, Jan Łukasiewicz, who was Polish, but that doesn’t make for nearly as memorable of a story.

Every now and then, you come across a little sub-pocket of a field that has naming practices that just have to make you laugh.  Sometimes, they’re full of silly, vaguely politically incorrect terms, like all the teepees and wigwams of Regge calculus (and I know a read a paper with spacetime “lean tos” in it too a few years back, but I couldn’t find it), sometimes they’re just odd attempts at being clever, like the “graphity” (rhythms with “gravity”) in Quantum Graphity, and other times they’re the over the top acronyms, like the “MiSaTaQuWa equations” found with the Gravitational Self-Force.  Every now and then, it’s worth it to just pause and smile at the whole “research” process.

References

[1] Regge, T. (1961). General relativity without coordinates Il Nuovo Cimento, 19 (3), 558-571 DOI: 10.1007/BF02733251

[2] Galassi, M (1992). Lattice Geometrodynamics State University of New York at Stony Brook, Ph.D. Dissertation [pdf].

[3] Galassi, M. (1993). Lapse and shift in Regge calculus Physical Review D, 47 (8), 3254-3264 DOI: 10.1103/PhysRevD.47.3254

[4] Kheyfets A, LaFave NJ, & Miller WA (1990). Null-strut calculus. II. Dynamics. Physical review D: Particles and fields, 41 (12), 3637-3651 PMID: 10012308

[5] ALPER ÜNGÖR, & ALLA SHEFFER (2002). PITCHING TENTS IN SPACE-TIME: MESH GENERATION FOR DISCONTINUOUS GALERKIN METHOD International Journal of Foundations of Computer Science , 13 (2) : 10.1142/S0129054102001059

[A], [B] “teepee.”, “wigwam.” Collins English Dictionary – Complete & Unabridged 10th Edition. HarperCollins Publishers. 30 Aug. 2010. <Dictionary.com http://dictionary.reference.com/browse/>.

Slight Technical Remarks

¹ The curvature of the manifold is manifest in the way that the 2-dimensional faces (bones) of the 4-simplices fit together. In flat space, the sum of the dihedral angle about a particular leg would be 2π, divergence (or deficit) from this value indicates curvature (local non-Euclideanism).

² In the continuum limit, we will be considering a 3-dimensional, differential manifold, to be a given spacelike hypersurface, called σ, that will be given by x₀ = t = constant.  The distinction here between space and time is given to correspond to the observations made by a set of observers distributed throughout the 3-volume, each with their time coordinate normal to the hypersurface, σ. It is this three geometry that is to be fixed on each hypersurface.