A geometric construction of the Riemann scalar curvature in Regge calculus
aa r X i v : . [ g r- q c ] M a r A geometric construction of the Riemann scalarcurvature in Regge calculus
Jonathan R. McDonald and Warner A. Miller
Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USAE-mail: [email protected]
Abstract.
The Riemann scalar curvature plays a central role in Einstein’s geometrictheory of gravity. We describe a new geometric construction of this scalar curvatureinvariant at an event (vertex) in a discrete spacetime geometry. This allows one toconstructively measure the scalar curvature using only clocks and photons. Givenrecent interest in discrete pre-geometric models of quantum gravity, we believe is itever so important to reconstruct the curvature scalar with respect to a finite number ofcommunicating observers. This derivation makes use of a new fundamental lattice cellbuilt from elements inherited from both the original simplicial (Delaunay) spacetimeand its circumcentric dual (Voronoi) lattice. The orthogonality properties betweenthese two lattices yield an expression for the vertex-based scalar curvature which isstrikingly similar to the corresponding hinge-based expression in Regge calculus (deficitangle per unit Voronoi dual area). In particular, we show that the scalar curvature issimply a vertex-based weighted average of deficits per weighted average of dual areas.PACS numbers: 83C27, 83C45 and 70G45
Submitted to:
Class. Quantum Grav. he scalar curvature in Regge calculus he scalar curvature in Regge calculus h . Each of these hinges isthe meeting place of three or more 4-simplicies. In the traditional description of Reggecalculus, this hinge-based curvature is viewed as a conic singularity; however, it hasbeen shown that the areas h ∗ of the Voronoi lattice dual to the Delaunay simpliciallattice provides a natural area to distribute the curvature[21, 20]. The Voronoi latticeis constructed in the usual way by utilizing the circumcentric dual of the Delaunaylattice[10].The key to our derivation of the Riemann-scalar curvature is the identification I h ≡ I v of the usual hinge-based expression the Regge calculus version of the Hilbertaction principle [6, 21] with its corresponding vertex-based expression. We begin withthe Hilbert action in a d -dimensional continuum spacetime, which is expressible as anintegral of the Riemann scalar curvature over the proper d -volume of the spacetime. I = 116 π Z R dV proper (1)On our lattice spacetime, and following the standard techniques of Regge calculus, wecan approximate this action as a sum over the triangular hinges h . I ≈ I h = 116 π X hinges, h R h ∆ V h (2)Here, R h is the scalar curvature invariant associated to the hinge, and ∆ V h is the proper4-volume in the lattice spacetime associated to the hinge h . Following earlier work bythe authors[21], this curvature will be defined explicitely below. Though, non-standardin Regge calculus, we may also express the action in terms of a sum over the vertices ofthe simplicial d -dimensional Delaunay lattice spacetime. I ≈ I v = 116 π X vertices, v R v ∆ V v (3)It is the Riemann scalar curvature ( R v ) at the vertex v that appears in this expressionthat we seek in this manuscript, and it is the equivalence between (2) and (3) that willyield it. But first we must use the orthogonality inherent between the Voronoi andDelaunay lattices to determine the relevant 4-volumes (∆ V v and ∆ V h ). ‡ ‡ Although the primary concern of the authors is to apply these results to the 4-dimensional pseudo-Riemannian geometry of spacetime, our equations are valid for any Riemann geometry of dimension d . Therefore, in the text and equations to follow we will explicity use the symbol d to represent thedimensionality of the geometry, the reader interested in general relativity can simply set d = 4. he scalar curvature in Regge calculus v in the Delaunay lattice, and consider a triangle hinge h havingvertex v as one of its three corners. We define A hv to be the fraction of the area of hinge h closest to vertex v than to its other two vertices ( Figure 1). Dual to each trianglehinge, and in particular to triangle h , is a unique co-dimension 2 area, A ∗ h , in the Voronoilattice. This area necessarily lies in a ( d − h . The number of vertices of the dual ( d − h ∗ , is equal to the number of d -dimensional simplicies hinging on triangle h ,and is always greater than or equal to three. If we join each of three vertices of hinge h , with the all of vertices of h ∗ with new edges, then we naturally form a d -dimensionalproper volume associated with a vertex v and hinge h . This d -dimensional polytope isa hybridization of the Voronoi and Delaunay lattices, they completely tile the latticespacetime without gaps or overlaps, and they inherent their rigidity from the underlyingsimplicial lattice.∆ V hv ≡ d ( d − A hv A ∗ h (4)The simplicity of this expression (the factorization of the simplicial spacetime and itsdual) is a direct consequence of the inherent orthogonality between the Voronoi andDelaunay lattices, and its impact on this calculation, and in Regge calculus as a whole,cannot be overstated. These d-cells are the Regge-calculus hybrid versions of the reducedBrillouin cells commonly found in solid state physics, though they are hybrid becausethey are coupled to their dual structures in the underlying atomic lattice. We view theseas the fundamental building blocks of lattice gravity and at the Planck scale perhapsthe Regge calculus version of Leibniz’s Monads – Vinculum Substantiale . The scalarfactor in this expression, which depends on the dimension of the lattice, was derivedin the appendix of an earlier paper [21]. Furthermore we obtain the complete proper d -volume, ∆ V v , by linearly summing (4) over each of the triangles h in the Delaunaylattice sharing vertex v .∆ V v = X h | v ∆ V hv = 2 d ( d − X h | v A hv A ∗ h (5)We now can re-express the Regge-Hilbert action at a vertex in terms of these hybridblocks. I v = 116 π X v R v X h | v d ( d − A hv A ∗ h (6)We now return to the more familiar hinge-based Regge-Hilbert action ( I h ) of (2).The proper 4-volume associated to hinge h has been shown to be factorable in terms of he scalar curvature in Regge calculus Figure 1.
The triangle hinge h to the left is partitioned into three areas. The shadedregion ( A hv ) represents the portion of the triangle that is closer to the lower vertexthan its other two vertices. The darkened and pronounced vertex appearing in eachof the three line drawings of this figure is the circumcenter of the hinge, h . Eachhinge has its corresponding 2-dimensional dual Voronoi area ( A ∗ h ) shown in the upperright part of the figure as a pentagonal shaped polygon, and illustrate this dual areaas “encircling” the d -dimensional “kite” hinge. In bottom right portion of the figure,the “kite” hinge is connected to its dual Voronoi polygon by (4 × the area of the triangle hinge and its corresponding dual Voronoi area [21].∆ V h = 2 d ( d − A h A ∗ h . (7)Following the procedure discussed above, we can express the area of h a sum of itscircumcentrically-partitioned pieces (Figure 1). A h = X v | h A hv (8) he scalar curvature in Regge calculus I h = 116 π X h X v | h R h d ( d − A hv A ∗ h ! (9)A key step in this derivation is the ability to switch the order of summation, andfortunately action is unchanged if we reverse this order. I h = 116 π X v X h | v R h d ( d − A hv A ∗ h ! (10)The vertex-based action of (6) must be equal to this hinge-based action of (10). Weimmediately obtain the desired expression for the Riemann scalar curvature at a vertex. R v = P h | v R h A ∗ h A hv P h | v A ∗ h A hv = P h | v R h A ∗ h A hv / P h | v A hv P h | v A ∗ h A hv / P h | v A hv (11)Here we have divided the numerator and denominator by the same quantity leaving itunchanged. Both the numerator and denominator are in the form of a weighted averageover the ”Brillion kites” ( A hv ) at vertex v . We define, in a natural way, the “kiteweighted average” at vertex v of any hinge-based quantity Q h as follows: h Q i v ≡ P h | v Q h A hv P h | v A hv . (12)Given this definition, the scalar curvature invariant at vertex v can be expressed as a“kite-weighted average” of the integrated hinge-based scalar curvature of Regge calculus. R v = h R h A ∗ h i v h A ∗ h i v , (13)where it was shown in [21] that the Riemann scalar curvature at the hinge h is expressibleas the hinge’s curvature deficit ( ǫ h ) per unit Voronoi area ( A ∗ h ) dual to h . R h = d ( d − ǫ h A ∗ h . (14)Therefore the expression for the vertex-based scalar curvature invariant derived hereis strikingly similar to the usual Regge calculus expression for the hinge-based scalarcurvature invariant (14). The only difference is that the numerator and denominator of(14) is replaced by their kite-weighted averages. R v = d ( d − h ǫ h i v h A ∗ h i v (15)In a 4-dimensional spacetime the minimum number of events needed to measurethe scalar curvature at a vertex ( v ) is six. This occurs when the 4-dimensional Voronoicell dual to v is itself a 4-simplex. This corresponds to the minimum allowable number he scalar curvature in Regge calculus Figure 2. v (depected in the picture as the black circle) is shown in the central region ofthe figure. The dashed lines represent null edges while the solid lines are timelike. Wealso show each of its five 4-simplices exploded off into the perimeter of the diagram. of simplicies in a Regge calculus spacetime lattice that can meet at vertex v and isconsistent with earlier results on the minimum number of test particles needed tomeasure the twenty components of the Riemann tensor [3, 5]. Such a minimal 6-point scalar curvature detector can be constructed solely from null (laser) and timelike(clock) edges – the tools available to a spacetime surveyor [22]. (Figure 2). Suchchronometric constructions, we believe will be useful in their applications to discretemodels of quantum gravity. Acknowledgments
We thank Renate Loll and Seth Lloyd for stimulating discussions which provided us themotivation to continue this research, and we are especially grateful to John A. Wheelerfor providing the inspiration and initial guidance into this field of spacetime geodesy.We would like to thank Florida Atlantic University’s Office of Research and the CharlesE. Schmidt College of Science for the partial support of this research. he scalar curvature in Regge calculus References [1] Lloyd S 2006 “A theory of quantum gravity based on quantum computation” arXiv:quant-ph/0501135 [2] Ambjorn J, Jurkiewicz J and Loll R 2006 “Quantum gravity: the art of building spacetime” arXiv:hep-th/0604212v1 [3] Synge L 1960
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