Tag Archives: Minkowski space

Causal cells: spacetime polytopes with null hyperfaces by Neiman

This week I have been reading a paper about polyhedra and 4-polytopes in Minkowski spacetime – in particular, null polyhedra
with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces.

A 3D projection of an tesseract performing an ...
A 3D projection of a tesseract performing an isoclinic rotation.

 

The paper presents the basic properties of several classes of null faced 4-polytopes: 4-simplices, tetrahedral diamonds and 4-parallelotopes. A most regular representative of each class is proposed.

The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four light rays in a tetrahedral pattern. This construct may have relevance for discretizations of curved spacetime and for quantum gravity.

In this paper, the author studies the properties of some special 4-polytopes in spacetime. The main qualitative difference between spacetime and Euclidean space is the existence of null i.e. lightlike directions. So, there exist line segments with vanishing length, plane elements with vanishing area, and hyperplane elements with vanishing volume. 3d null hyperplane elements are especially interesting. In relativistic physics, null hypersurfaces play the role of causal boundaries between spacetime regions. They also function as characteristic surfaces for the differential equations of relativistic field theory.

Important examples of null hypersurfaces include the lightcone of an event and the event horizon of a black hole.

SVG version of http://en.wikipedia.org/wiki/Im...

The prime example of a closed null hypersurface is a causal diamond – the intersection of two light cones originating from two timelike-separated points.

causal

Null Hyperplanes 

Null 3d polyhedra or polyhedra with vanishing volume reside in null hyperplanes, such as the hyperplane t = z. let’s look at the geometry of these hyperplanes. The normal ℓμ to the hyperplane, ℓμ (1, 0, 0, 1) is a null vector, i.e. ℓμμ = 0.  As a result, it is also tangent to the hyperplane. It’s integral lines  form null geodesics. The hyperplane
is  foliated into light rays. All intervals within the hyperplane are spacelike, except the null intervals along the rays.

Null Polyhedra

In 3d Euclidean space, each area element has a normal vector n. When discussing polyhedra, it is convenient to define the norm of n to equal the area of the corresponding face. The orientation of the normals is chosen to be outgoing. Not every set of area normals {ni}
describes the faces of some polyhedron. For this to be true, the normals must sum up to zero:

nullequ1

This can be understood as the requirement that the flux of any constant vector field through the polyhedron vanishes. In loop quantum gravity, this condition encodes the local SO(3) rotation symmetry.

 

Null tetrahedra

The simplest null polyhedron is a tetrahedron. Up to reflections along the null axis, null tetrahedra come in two distinct types: (1,3) and (2,2). The pairs of numbers denote how many of the tetrahedron’s four faces are past-pointing and future- pointing, respectively.

nullfig1

Null-faced 4-simplices

Null-faced 4-simplices have hyperfaces which have zero volume. A 4-simplex has five tetrahedral hyperfaces, which in this case will be null tetrahedra,

The scalar products  ημν(i)μ(j)νof the null volume normals are directly related to the spacetime volume of the 4-simplex and to the areas of the 2d faces. To express the spacetime volume, we must choose a set of four volume normals ℓ(i)μ. The time-orientation of the
normals should be correlated with the past/future status of their hyperfaces.  Next, we construct a symmetric 4 × 4 matrix  Lij =(3!)²ημν(i)μ(j)ν of their scalar products. The diagonal elements of Lij are zero. Elements corresponding to past- future pairs ij are positive, while those for past-past and future-future pairs are negative.
The spacetime volume can then be found as:

nullequ6

The area of the face at the intersection of the i’th and j’th hyperplanes can be found as:

nullequ7

Can also define the 4-volume directly in terms of triangle areas:

nullequ8

nullfig5

Null parallelepipeds

The six faces of a null parallelepiped are spacelike parallelograms. There are three pairs of opposing faces, such that each pair is parallel and congruent. In a given pair of opposing faces, one is past-pointing, and the other future-pointing.

nullfig2

 

Tetrahedral diamonds

Beginning with an arbitrary spacelike tetrahedron, situated at t = 0 hyperplane, this is the base tetrahedron. For each of the base tetrahedron’s four faces,  the lightcross of two null hyperplanes orthogonal to it are drawn. The tetrahedral diamond is then defined by the convex hull of the intersections of these null hyperplanes.

nullfig6

The 4-volume of a tetrahedral diamond can be found as twice the volume of a 4-simplex, with the spacelike tetrahedron as its base and the inscribed radius r as its height. The result is:

nullequ10

where V is the base tetrahedron’s volume.

 

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Polyhedra in spacetime from null vectors by Neiman

This week I have been studying a nice paper about Polyhedra in spacetime.

The paper considers convex spacelike polyhedra oriented in Minkowski space. These are classical analogues of spinfoam intertwiners. There is  a parametrization of these shapes using null face normals. This construction is dimension-independent and in 3+1d, it provides the spacetime picture behind the property of the loop quantum gravity intertwiner space in spinor form that the closure constraint is always satisfied after some  SL(2,C) rotation.These  variables can be incorporated in a 4-simplex action that reproduces the large-spin behaviour of the Barrett–Crane vertex amplitude.

In loop quantum gravity and in spinfoam models, convex polyhedra are fundamental objects. Specifically, the intertwiners between rotation-group representations that feature in these theories can be viewed as the quantum versions of convex polyhedra. This makes the parametrization of such shapes a subject of interest for  LQG.
In kinematical LQG, one deals with the SU(2) intertwiners, which correspond to 3d polyhedra in a local 3d Euclidean frame. These polyhedra are naturally parametrized in terms of area-normal vectors: each face i is associated with a vector xi, such that its norm
equals the face area Ai, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:

polyequ1

Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors xi that satisfy (1) and convex polyhedra with a spatial orientation. In
LQG, the vectors xi correspond to the SU(2) fluxes. The closure condition  then encodes the Gauss constraint, which also generates spatial rotations of the polyhedron.

In the EPRL/FK spinfoam, the SU(2) intertwiners get lifted into SL(2,C) and are acted on by SL(2,C) ,Lorentz, rotations. Geometrically, this endows the polyhedra with an orientation in the local 3+1d Minkowski frame of a spinfoam vertex. The polyhedron’s
orientation is now correlated with those of the other polyhedra surrounding the vertex, so that together they define a generalized 4-polytope. In analogy with the spatial case, a polyhedron with spacetime orientation can be parametrized by a set of area-normal
simple bivectors Bi. In addition to closure, these bivectors must also satisfy a cross-simplicity
constraint:

polyequ2

In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors Bi, they associate null vectors i to the polyhedron’s
faces. This parametrization does not require any constraints between the variables on different faces. It is unusual in that both the area and the full orientation of each face are functions of the data on all the faces. This construction, like the area-vector and
area-bivector constructions above, is dimension-independent. So we can parametrize d-dimensional convex spacelike polytopes with (d − 1)-dimensional faces, oriented in a (d + 1)-dimensional Minkowski spacetime.  These variables can be to construct an action principle for a Lorentzian 4-simplex. The action principle reproduces the large spin behaviour of the Barrett–Crane spinfoam vertex. In particular, it recovers the Regge action for the classical simplicial gravity, up to a possible sign and the existence of additional,degenerate solutions.

In d = 2, 3 spatial dimensions, the parametrization is  contained in the spinor-based description of the LQG intertwiners. There, the face normals are constructed as squares of spinors. It was observed that the closure constraint in these variables can always be satisfied by acting on the spinors with an SL(2,C) boost.  The simple spacetime picture presented in this paper is new. Hopefully, it will contribute to the geometric interpretation of the modern spinor and twistor variables in LQG.

The parametrization
Consider a set of N null vectors liμ in the (d + 1)-dimensional Minkowski space Rd,1, where i = 1, 2, . . . ,N and d ≥2.  Assume the following conditions on the null vectors liμ.

  • The liμ span the Minkowski space and  N ≥  d + 1.
  • The  liμ  are either all future-pointing or all past-pointing.

The central observation in this paper is that such sets of null vectors are in one-to-one correspondence with convex d-dimensional spacelike polytopes oriented in Rd,1.

Constructing the polytrope

Consider a set {liμ} ,take the sum of the liμ normalized
to unit length:

polyequ3

The unit vector nμ is timelike, with the same time orientation as the liμ. Now take nμ to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to nμ define the projections of the null vectors liμinto this hyperplane:

polyequ4

The spacelike vectors siμ  automatically sum up to zero. Also, since the liμ span the spacetime, the siμ must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the siμ are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors li define a d-polytope oriented in spacetime.

Basic features of the parametrization.

The vectors  are liμ  associated to the polytope’s (d −1)-dimensional faces and are null normals to these faces. The orientation of a spacelike (d − 1)-plane in Rd,1 is in one-to-one correspondence with the directions of its two null normals. So each liμ carries partial information about the orientation of the ith face. The second null normal to the face is a function of all the liμ. It can be expressed as:

polyequ6

where  nμ is given by

polyequ3

Similarly, the area Ai of each face is a function of the
null normals liμ to all the faces:

polyequ7

The total area of the faces has the simple expression:

polyequ8

A (d+1)-simplex action

To construct a (d + 1)-simplex action that reproduces in the d = 3 case the large-spin behaviour of the Barrett–Crane spinfoam vertex.

At the level of degree-of-freedom counting, the shape of a (d +1)-simplex is determined by the (d + 1)(d + 2)/2 areas Aab of its (d − 1)-faces. These areas are directly analogous to the spins that appear in the Barrett–Crane spinfoam. Let us fix a set of values for Aab and consider the action:

polyequ9

Then restrict to the variations where:

polyequ10

The stationary points of the action  have the following properties. For each a, the vectors  labμ define a d-simplex with unit normal naμ
and (d − 1)-face areas Aab.

polyfig1

A (d − 1)-face in a (d + 1)-simplex, shared by two d-simplices a and b. The diagram depicts the 1+1d plane orthogonal to the face. The dashed lines are the two null rays in this normal plane.

 

The d-simplices automatically agree on the areas of their shared (d −1)-faces. The two d-simplices agree not only on the area of their shared (d − 1)-face, but also on the orientation of its (d − 1)-plane in spacetime. In other words, they agree on the face’s area-normal bivector:

polyequ15a

The area bivectors defined  automatically satisfy closure and cross-simplicity:

polyequ17

We conclude that the stationary points are in one-to-one correspondence with the bivector geometries of the Barrett-Crane model with an action of the form:

polyequ25

 

Numerical work with Vpython and sagemath12: The Quantum Tetrahedron

In a number of recent posts including:

I have been studying the ‘time-fixed’ quantum tetrahedron in which a quantum tetrahedron is used to model the evolution of a state a into state b as shown below:

physical boundary fig 1

In this toy model the quantum tetrahedron evolves from a flat shape:

Background independence fig3

via an equilateral quantum tetrahedron

Background independence fig1

towards a long stretched out quantum tetrahedron in the limit of large T.

Background independence fig2

This can be displayed on a phase space diagram,

sagemath12fig6

Where “Minkowski vacuum states” are the states which minimize the energy.

Using sagemath I am able to model the quantum tetrahedron during this evolution by varying the lenght of c – which is used to measure the time T variable;

sagemath12fig1

By using Vpython I am able to model this same system as shown in these animated gifs, the label in the diagram indicated the value of T.

An animated gif showing the edge a

sagemath12vpython.gif

 

 

An animated gif showing a different view of the quantum tetrahedron:

sagemath12vpython2.gif.gif

Analyzing the {6j} kernel

An important object in the modelling of the quantum tetrahedron is the Wigner {6j} symbol. As we saw in the post: Physical boundary state for the quantum tetrahedron by Livine and Speziale The quantum dynamics can be studied as by Ponzano-Regge, by  associating with the tetrahedron the amplitude

physical boundary equ 4

Using sagemath I was able to evaluate the value of this in some extremal cases: when b=0 and b=2j.

Case b=0

sagemath12fig2

sagemath12fig3

Case b=2j

sagemath12fig4

sagemath12fig5

In the next post I will be looking at the mathematics behind the Wigner{6j} symbol in more detail.