scholarly journals Geometric Flow Equations for the Number of Space‐Time Dimensions

2021 ◽  
pp. 2100171
Author(s):  
Davide De Biasio ◽  
Julian Freigang ◽  
Dieter Lüst
Keyword(s):  
Space Time ◽  
Geometric Flow ◽  
Flow Equations

scholarly journals BACH FLOWS OF PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250039 ◽  
Author(s):  
SANJIT DAS ◽  
SAYAN KAR
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Ricci Flow ◽  
Flow Characteristics ◽  
Geometric Flow ◽  
Flow Equations ◽  
First Order ◽  
Bach Tensor ◽  
Point Condition ◽  
Future Work

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.

scholarly journals ATTRACTOR FLOWS IN st2 BLACK HOLES

2011 ◽  
Vol 08 (05) ◽  
pp. 1031-1077
Author(s):  
RAJU ROYCHOWDHURY
Keyword(s):  
Black Holes ◽  
Central Charge ◽  
Space Time ◽  
Marginal Stability ◽  
Minimal Rank ◽  
General Solutions ◽  
Flow Equations ◽  
First Order ◽  
Extremal Black Holes

Following the same treatment of Bellucci et al. we obtain, the hitherto unknown general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing and vanishing central charge Z for the so-called st2 model, the minimal rank-two [Formula: see text] symmetric supergravity in d = 4 space-time dimensions. We also make useful comparisons with results that already exist in literature, and introduce the fake supergravity (first-order) formalism to be used in our analysis. An analysis of the BPS bound all along the non-BPS attractor flows and of the marginal stability of corresponding D-brane charge configurations has also been presented.

scholarly journals Signs and stability in higher-derivative gravity

2018 ◽  
Vol 33 (04) ◽  
pp. 1850031 ◽  
Author(s):  
Gaurav Narain
Keyword(s):  
Path Integral ◽  
Transition Point ◽  
Flat Space ◽  
Positive Definiteness ◽  
Space Time ◽  
Systematic Analysis ◽  
Flow Equations ◽  
Higher Derivative ◽  
Renormalizable Theory ◽  
The One

Perturbatively renormalizable higher-derivative gravity in four space–time dimensions with arbitrary signs of couplings has been considered. Systematic analysis of the action with arbitrary signs of couplings in Lorentzian flat space–time for no-tachyons, fixes the signs. Feynman [Formula: see text] prescription for these signs further grants necessary convergence in path-integral, suppressing the field modes with large action. This also leads to a sensible wick rotation where quantum computation can be performed. Running couplings for these sign of parameters make the massive tensor ghost innocuous leading to a stable and ghost-free renormalizable theory in four space–time dimensions. The theory has a transition point arising from renormalization group (RG) equations, where the coefficient of [Formula: see text] diverges without affecting the perturbative quantum field theory (QFT). Redefining this coefficient gives a better handle over the theory around the transition point. The flow equations push the flow of parameters across the transition point. The flow beyond the transition point is analyzed using the one-loop RG equations which shows that the regime beyond the transition point has unphysical properties: there are tachyons, the path-integral loses positive definiteness, Newton’s constant [Formula: see text] becomes negative and large, and perturbative parameters become large. These shortcomings indicate a lack of completeness beyond the transition point and need of a nonperturbative treatment of the theory beyond the transition point.

The Space-Time CE/SE Method for Solving Reduced Two-Fluid Flow Model

2012 ◽  
Vol 12 (4) ◽  
pp. 1070-1095 ◽  
Author(s):  
Shamsul Qamar ◽  
Munshoor Ahmed ◽  
Ishtiaq Ali
Keyword(s):  
Finite Volume ◽  
Numerical Test ◽  
Current Method ◽  
Space Time ◽  
Finite Volume Schemes ◽  
Reconstruction Procedure ◽  
Flow Equations ◽  
Flow Models ◽  
Two Fluids ◽  
Two Fluid

AbstractThe space-time conservation element and solution element (CE/SE) method is proposed for solving a conservative interface-capturing reduced model of compressible two-fluid flows. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term for accounting the energy exchange. The one and two-dimensional flow models are numerically investigated in this manuscript. The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building block of the suggested method. The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation. In order to reveal the efficiency and performance of the approach, several numerical test cases are presented. For validation, the results of the current method are compared with other finite volume schemes.

Einstein's hyperbolic geometric flow

2014 ◽  
Vol 11 (02) ◽  
pp. 249-267 ◽  
Author(s):  
De-Xing Kong ◽  
Jinhua Wang
Keyword(s):  
Global Existence ◽  
Wave Phenomenon ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Time ◽  
Einstein Metric ◽  
Geometric Flow ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold

We investigate the Einstein's hyperbolic geometric flow, which provides a natural tool to deform the shape of a manifold and to understand the wave character of metrics, the wave phenomenon of the curvature for evolutionary manifolds. For an initial manifold equipped with an Einstein metric and assumed to be a totally umbilical submanifold in the induced space-time, we prove that, along the Einstein's hyperbolic geometric flow, the metric is Einstein if and only if the corresponding manifold is a totally umbilical hypersurface in the induced space-time. For an initial manifold which is equipped with an Einstein metric, assumed to be a totally umbilical submanifold with constant mean curvature in the induced space-time, we prove that, along the Einstein's hyperbolic geometric flow, the metric remains an Einstein metric, and the corresponding manifold is a totally umbilical hypersurface in the induced space-time. Moreover, the global existence and blowup phenomenon of the constructed metric is also investigated here.

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