scholarly journals Curvature Invariants for the Alcubierre and Natário Warp Drives

Universe ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 21 ◽  
Author(s):  
Brandon Mattingly ◽  
Abinash Kar ◽  
Matthew Gorban ◽  
William Julius ◽  
Cooper K. Watson ◽  
...  
Keyword(s):  
Constant Curvature ◽  
Constant Velocity ◽  
Shape Function ◽  
Skin Depth ◽  
Safe Harbor ◽  
Field Equations ◽  
Curvature Invariants ◽  
Unique Effect ◽  
Coordinate Mapping ◽  
Faster Than Light

A process for using curvature invariants is applied to evaluate the metrics for the Alcubierre and the Natário warp drives at a constant velocity. Curvature invariants are independent of coordinate bases, so plotting these invariants will be free of coordinate mapping distortions. As a consequence, they provide a novel perspective into complex spacetimes, such as warp drives. Warp drives are the theoretical solutions to Einstein’s field equations that allow for the possibility for faster-than-light (FTL) travel. While their mathematics is well established, the visualisation of such spacetimes is unexplored. This paper uses the methods of computing and plotting the warp drive curvature invariants to reveal these spacetimes. The warp drive parameters of velocity, skin depth and radius are varied individually and then plotted to see each parameter’s unique effect on the surrounding curvature. For each warp drive, this research shows a safe harbor and how the shape function forms the warp bubble. The curvature plots for the constant velocity Natário warp drive do not contain a wake or a constant curvature, indicating that these are unique features of the accelerating Natário warp drive.

scholarly journals Curvature Invariants for the Accelerating Natário Warp Drive

Particles ◽  
2020 ◽  
Vol 3 (3) ◽  
pp. 642-659
Author(s):  
Brandon Mattingly ◽  
Abinash Kar ◽  
Matthew Gorban ◽  
William Julius ◽  
Cooper K. Watson ◽  
...  
Keyword(s):  
Mathematical Description ◽  
Skin Depth ◽  
Ricci Scalar ◽  
Curvature Invariant ◽  
Curvature Invariants ◽  
Internal Structures ◽  
Coordinate Mapping ◽  
Key Features

A process for using curvature invariants is applied to evaluate the accelerating Natário warp drive. Curvature invariants are independent of coordinate bases and plotting the invariants is free of coordinate mapping distortions. While previous works focus mainly on the mathematical description of the warp bubble, plotting curvature invariants provides a novel pathway to investigate the Natário spacetime and its characteristics. For warp drive spacetimes, there are four independent curvature invariants the Ricci scalar, r1, r2, and w2. The invariant plots demonstrate how each curvature invariant evolves over the parameters of time, acceleration, skin depth and radius of the warp bubble. They show that the Ricci scalar has the greatest impact of the invariants on the surrounding spacetime. They also reveal key features of the Natário warp bubble such as a flat harbor in the center of it, a dynamic wake, and the internal structures of the warp bubble.

scholarly journals Kinematic formula and tube formula in space of constant curvature

1990 ◽  
Vol 33 (1) ◽  
pp. 79-88
Author(s):  
Sungyun Lee
Keyword(s):  
Euclidean Space ◽  
Euler Characteristic ◽  
Constant Curvature ◽  
Kinematic Formula ◽  
Curvature Invariants ◽  
Space Of Constant Curvature

The Euler characteristic of an even dimensional submanifold in a space of constant curvature is given in terms of Weyl's curvature invariants. A derivation of Chern's kinematic formula in non-Euclidean space is completed. As an application of above results Weyl's tube formula about an odd-dimensional submanifold in a space of constant curvature is obtained.

scholarly journals Lorentzian spacetimes with constant curvature invariants in three dimensions

2007 ◽  
Vol 25 (2) ◽  
pp. 025008 ◽  
Author(s):  
Alan Coley ◽  
Sigbjørn Hervik ◽  
Nicos Pelavas
Keyword(s):  
Constant Curvature ◽  
Three Dimensions ◽  
Curvature Invariants

Gravitational collapse and dark universe with LTB geometry

2016 ◽  
Vol 26 (06) ◽  
pp. 1750045 ◽  
Author(s):  
M. Zaeem-ul-Haq Bhatti ◽  
Z. Yousaf
Keyword(s):  
Black Hole ◽  
Electromagnetic Field ◽  
Time Interval ◽  
Curvature Scalar ◽  
Field Equations ◽  
Dark Sector ◽  
Curvature Invariants ◽  
Text Model ◽  
Polynomial Formula ◽  
Electrically Charged

The objective of this paper is to examine the influence of polynomial [Formula: see text] dark sector cosmic terms on the collapse of electrically charged Lemaître–Tolman–Bondi geometry. We explored a class of solutions for [Formula: see text] field equations in the existence of electromagnetic field and under the constraint of constant curvature scalar. The influence of [Formula: see text] model on the dynamics of collapsing object have been discussed by studying its black hole and cosmological horizons. Also, the effects of these dark sources on the time interval between the corresponding singularities and horizons have been studied. We investigated that the process of collapse slows down due to the higher order curvature invariants of polynomial [Formula: see text] model and electromagnetic field.

scholarly journals Null hypersurfaces in de Sitter and anti-de Sitter cosmologies

2018 ◽  
Vol 27 (04) ◽  
pp. 1850045
Author(s):  
P. A. Hogan
Keyword(s):  
Cosmological Constant ◽  
Gravitational Waves ◽  
Constant Curvature ◽  
Focus Attention ◽  
De Sitter ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Null Hypersurfaces ◽  
Einstein's Field Equations ◽  
Line Elements

The study of gravitational waves in the presence of a cosmological constant has led to interesting forms of the de Sitter and anti-de Sitter line elements based on families of null hypersurfaces. The forms are interesting because they focus attention on the geometry of null hypersurfaces in spacetimes of constant curvature. Two examples are worked out in some detail. The first originated in the study of collisions of impulsive gravitational waves in which the post-collision spacetime is a solution of Einstein’s field equations with a cosmological constant, and the second originated in the generalization of plane fronted gravitational waves with parallel rays to include a cosmological constant.

scholarly journals On the existence and stability of traversable wormhole solutions in modified theories of gravity

2021 ◽  
Vol 81 (8) ◽  
Author(s):  
Oleksii Sokoliuk ◽  
Alexander Baransky
Keyword(s):  
Shape Function ◽  
Energy Condition ◽  
Null Energy Condition ◽  
Modified Theories Of Gravity ◽  
Energy Tensor ◽  
Field Equations ◽  
Traversable Wormhole ◽  
Existence And Stability ◽  
Theories Of Gravity ◽  
The Stability

AbstractWe study Morris–Thorne static traversable wormhole solutions in different modified theories of gravity. We focus our study on the quadratic gravity $$f({\mathscr {R}}) = {\mathscr {R}}+a{\mathscr {R}}^2$$ f ( R ) = R + a R 2 , power-law $$f({\mathscr {R}}) = f_0{\mathscr {R}}^n$$ f ( R ) = f 0 R n , log-corrected $$f({\mathscr {R}})={\mathscr {R}}+\alpha {\mathscr {R}}^2+\beta {\mathscr {R}}^2\ln \beta {\mathscr {R}}$$ f ( R ) = R + α R 2 + β R 2 ln β R theories, and finally on the exponential hybrid metric-Palatini gravity $$f(\mathscr {\hat{R}})=\zeta \bigg (1+e^{-\frac{\hat{{\mathscr {R}}}}{\varPhi }}\bigg )$$ f ( R ^ ) = ζ ( 1 + e - R ^ Φ ) . Wormhole fluid near the throat is adopted to be anisotropic, and redshift factor to have a constant value. We solve numerically the Einstein field equations and we derive the suitable shape function for each MOG of our consideration by applying the equation of state $$p_t=\omega \rho $$ p t = ω ρ . Furthermore, we investigate the null energy condition, the weak energy condition, and the strong energy condition with the suitable shape function b(r). The stability of Morris–Thorne traversable wormholes in different modified gravity theories is also analyzed in our paper with a modified Tolman–Oppenheimer–Voklov equation. Besides, we have derived general formulas for the extra force that is present in MTOV due to the non-conserved stress-energy tensor.

scholarly journals CHARGED PERFECT FLUID CONFIGURATIONS WITH A DILATON FIELD

2005 ◽  
Vol 20 (11) ◽  
pp. 821-831 ◽  
Author(s):  
STOYTCHO S. YAZADJIEV
Keyword(s):  
Equation Of State ◽  
Nonlinear Equation ◽  
Helmholtz Equation ◽  
Linear Equation ◽  
Perfect Fluid ◽  
Constant Curvature ◽  
Interior Solution ◽  
Dilaton Field ◽  
Field Equations ◽  
Interior Solutions

We examine static charged perfect fluid configurations in the presence of a dilaton field. A method for construction of interior solutions is given. An explicit example of an interior solution which matches continuously the external Gibbons–Maeda–Garfinkle–Horowitz–Strominger solution is presented. Extremely charged perfect fluid configurations with a dilaton are also examined. We show that there are two types of extreme configurations. For each type the field equations are reduced to a single nonlinear equation on a space of a constant curvature. In the particular case of a perfect fluid with a linear equation of state, the field equations of the first type configurations are reduced to a Helmholtz equation on a space with a constant curvature. An explicit example of an extreme configuration is given and discussed.

Lorentzian spacetimes with constant curvature invariants in four dimensions

2009 ◽  
Vol 26 (12) ◽  
pp. 125011 ◽  
Author(s):  
Alan Coley ◽  
Sigbjørn Hervik ◽  
Nicos Pelavas
Keyword(s):  
Constant Curvature ◽  
Four Dimensions ◽  
Curvature Invariants

scholarly journals Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 690 ◽  
Author(s):  
Ali Alkhaldi ◽  
Mohd. Aquib ◽  
Aliya Siddiqui ◽  
Mohammad Shahid
Keyword(s):  
Constant Curvature ◽  
Necessary And Sufficient Condition ◽  
Einstein Field Equations ◽  
Statistical Manifold ◽  
Field Equations ◽  
Space Forms ◽  
Equality Case ◽  
Statistical Manifolds ◽  
Necessary And Sufficient ◽  
Statistical Space

In this paper, we obtain the upper bounds for the normalized δ -Casorati curvatures and generalized normalized δ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be η -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.

scholarly journals A new shape function and some specific wormhole solutions in braneworld scenario

2021 ◽  
pp. 2150167
Author(s):  
Bikram Ghosh ◽  
Saugata Mitra
Keyword(s):  
Energy Density ◽  
Shape Function ◽  
Energy Condition ◽  
Energy Conditions ◽  
Shape Functions ◽  
Field Equations ◽  
Strong Energy Condition ◽  
Strong Energy ◽  
Arbitrary Positive Constant ◽  
Anisotropic Case

Considering an energy density of the form [Formula: see text] (where [Formula: see text] is an arbitrary positive constant with dimension of energy density and [Formula: see text]), a shape function is obtained by using field equations of braneworld gravity theory in this paper. Under isotropic scenario wormhole solutions are obtained considering six different redshift functions along with the obtained new shape function. For anisotropic case, wormhole solutions are obtained under the consideration of five different shape functions along with the redshift function [Formula: see text], where [Formula: see text] is an arbitrary constant. In each case all energy conditions are examined and it is found that for some cases all energy conditions are satisfied in the vicinity of the wormhole throat and for the rest of the cases all energy conditions are satisfied except strong energy condition.

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