Regge calculus from discontinuous metrics

2011

Gen Relativ Gravit

We consider a four-dimensional simplicial complex and the minisuperspace general relativity system on it. The metric is flat in most parts of the interior of every 4-simplex, with the exception of a thin layer of thickness ∝ ε along each three-dimensional face. In this layer the metric undergoes a jump between the two 4-simplices sharing this face. At ε → 0 this jump would become a discontinuity. However, a discontinuity of the metric induced on the face is not allowed in general relativity: terms arise in the Einstein action tending to infinity as ε → 0. In the path integral approach, these terms lead to the pre-exponent factor with δ-functions requiring that the metric induced on the faces be continuous. That is, the 4-simplices fit on their common faces. The other part of the path integral measure corresponds to the action, which is the sum of independent terms over the 4-simplices. Therefore this part of the path integral measure is the product of independent measures over the 4-simplices. The result obtained is in accordance with our previous one obtained from symmetry considerations.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gravity action on rapidly varying metrics

2011

Gen Relativ Gravit

“…Being defined in the extended configuration superspace with independent area tensors this measure then can be practically uniquely projected on the hypersurface in this superspace corresponding to the ordinary Regge calculus. The result is finite nonzero length expectation values [4,5].…”

mentioning

confidence: 99%

On the area expectation values in area tensor Regge calculus in the Lorentzian domain

2006

Physics Letters B

Self Cite

“…First, occurence of the Haar measure on the group SO(4) of connections on separate 3-faces DΩ σ 3 . This is connected with the specific form of the kinetic term π σ 2 • Ω † σ 2Ωσ 2 which appears when one passes to the continuous limit along any of the coordinate direction chosen as time. Here SO(4) rotation Ω σ 2 serves to parameterise limiting form of Ω σ 3 when σ 3 tends to σ 2 .…”

mentioning

confidence: 99%

“…The δ cont has been considered in our previous work [2], and now the question is about δ metric . Here situation is even simpler, because the problem reduces to that for a one 4-simplex and δ metric is the product over separate 4-simplices.…”

mentioning

confidence: 99%

Length expectation values in quantum Regge calculus

2004

Physics Letters B

Self Cite

Regge calculus configuration superspace can be embedded into a more general superspace where the length of any edge is defined ambiguously depending on the 4-tetrahedron containing the edge. Moreover, the latter superspace can be extended further so that even edge lengths in each the 4-tetrahedron are not defined, only area tensors of the 2-faces in it are. We make use of our previous result concerning quantisation of the area tensor Regge calculus which gives finite expectation values for areas. Also our result is used showing that quantum measure in the Regge calculus can be uniquely fixed once we know quantum measure on (the space of the functionals on) the superspace of the theory with ambiguously defined edge lengths. We find that in this framework quantisation of the usual Regge calculus is defined up to a parameter. The theory may possess nonzero (of the order of Plank scale) or zero length expectation values depending on whether this parameter is larger or smaller than a certain value. Vanishing length expectation values means that the theory is becoming continuous, here dynamically in the originally discrete framework.