在研究虫洞几何结构的标准方法中，为了确保可穿越性，暗能量的存在是不可避免的。暗能量提供了负引力效应以保持虫洞喉部敞开。我们分析的问题是，是否能在没有暗能量的情况下实现同样的效果。结果表明，如果我们将能量-动量迹与标准的 Einstein-Hilbert 拉格朗日量耦合，并利用特定的状态方程，那么就可以避免使用暗能量。已知 Casimir 应力能量会导致能量-动量张量违反零能量条件（NEC）。这一现象使得此类状态方程（EoS）成为生成可穿越虫洞（WH）几何结构的理想候选者。这一经过实验验证的现象提供了一种机制，使得无需依赖暗能量即可维持虫洞喉部的敞开。我们在 f(R,T) 引力理论中由此生成了两类虫洞解，其中 R 代表 Ricci 标量，T 是应力-能量张量的迹。对于背景几何，我们选择静态球对称度规，并推导出了精确虫洞解的场方程。针对关联能量-动量张量分量的 Casimir EoS 的特定选择 [Kar 和 Sahdev: Phys. Rev D 52 2030 (1995)] 以及红移函数的不同选择，我们完全确定了虫洞的几何结构。所得的虫洞解违反了 NEC，并且满足物理上可实现的虫洞所要求的所有定性约束。这一点通过针对适当选取的 f(R,T) 耦合参数取值范围绘制的图形得到了证明。此外，我们的研究还涉及对引力排斥效应的调查，结果显示它的存在会导致沿零测地线传播的光子产生负偏转角。
英文摘要：
In the standard approach to studying wormhole geometry, the presence of dark energy is unavoidable to ensure traversability. The dark energy provides the negative gravity effect to keep the throat open. The question we analyse is whether the same can be achieved without dark energy. It turns out that if we couple the trace of energy-momentum with the standard Einstein-Hilbert Lagrangian and utilise a sppecific equation of state then dark energy may be obviated. The Casimir stress energy is known to result in the violation of the null energy condition (NEC) on the energy momentum tensor. This phenomenon makes such an equation of state (EoS) an ideal candidate to generate traversable wormhole (WH) geometries. The laboratory proven phenomenon provides a mechanism to sustain an open WH throat without having to appeal to dark energy. We generate two classes of WH solutions with this in $f(R,T)$ gravity theory, where $R$ represents the Ricci scalar and $T$ is the trace of the stress-energy tensor. For the background geometry we choose a static and spherically symmetric metric, and derive the field equations for exact WH solutions. For the specific choice of the Casimir EoS relating the energy-momentum tensor components [ Kar and Sahdev: Phys. Rev D {\bf 52} 2030 (1995)] and different choices for redshift functions, we determine the WH geometry completely. The obtained WH solutions violate the NECs and all qualitative constraints demanded of physically realisable WHs are satisfied. This is demonstrated via graphical plots for a suitably chosen range of values of the $f(R,T)$ coupling parameter. Furthermore, our study involved an investigation into the repulsive effect of gravity, which revealed that its presence leads to a negative deflection angle for photons traveling along null geodesics.
Dark matter halos are common in galaxies and globular clusters, making it possible to use dark matter density profiles to construct transverse wormholes. In reference [1], an approximate solution of the traversable wormhole satisfying the isotropic pressure condition is constructed by using the dark matter density profile. However, approximate solutions limit the discussion of traversable wormholes to a large extent. In this work, we construct the exact solution of the traversable wormhole in the case of rotation in the dark matter halo. We have written analytical expressions for the Kerr-like…
In this work we construct traversable wormholes geometries in the framework of the complexity factor. We provide the redshift function of a Casimir traversable wormhole which, in combination with a non--vanishing complexity factor, leads to a traversable wormhole with a minimum amount of exotic matter. The shape function and the embedding diagram are shown and discussed. The tidal accelerations and the time required to get through the wormholes are estimated.
In this paper, we study the perturbation problem of the scalar, electromagnetic, and gravitational waves under the traversable Lorentzian wormhole geometry. The unified form of the potential for the Schrödinger type one-dimensional wave equation is found.
We study the Lorentzian static traversable wormholes coupled to quadratic scalar fields. We also obtain the solutions of the scalar fields and matters in the wormhole background and find that the minimal size of the wormhole should be quantized under the appropriate boundary conditions for the positive non-minimal massive scalar field.
It is shown in this paper that it is possible, at least in principle, to construct a traversable wormhole that is stable to linearized radial perturbations by specifying relatively simple conditions on the shape and redshift functions.
In this paper, we evaluate traversable wormhole solutions through Karmarkar condition in $f(R,T)$ theory, where $T$ is the trace of the energy-momentum tensor and $R$ represents the Ricci scalar. We develop a wormhole shape function for the static traversable wormhole geometry by using the embedding class-I technique. The resulting shape function is used to construct wormhole geometry that fulfills all the necessary conditions and joins the two asymptotically flat regions of the spacetime. We investigate the existence of viable traversable wormhole solutions for anisotropic matter configuratio…
We show explicitly that traversable wormholes requiring exotic matter in 4-dimensions nevertheless have acceptable stress-tensors obeying reasonable energy conditions in higher dimensions if the wormholes are regarded as being embedded in higher dimensional space-times satisfying Einstein's field equations. From the 4-dimensional perspective, the existence of higher dimensions may thus facilitate wormhole and time-machine constructions through access to "exotic matter".
The theoretical construction of a traversable wormhole from a Schwarzschild black hole is described, using analytic solutions in Einstein gravity. The matter model is pure phantom radiation (pure radiation with negative energy density) and the idealization of impulsive radiation is employed.
We consider the effects of Gravity's Rainbow on the self-sustained equation which is responsible to find new traversable wormholes configurations powered by their own gravitational quantum fluctuations. The form of the shape function considered is obtained by imposing the equation of state $p_{r}=ωρ$. We investigate the size of the wormhole as a function of the parameter $ω$ in the phantom region. We discover that a wormhole which is traversable in principle, but not in practice, can be produced.
A self--sustained traversable wormhole is obtained as a vacuum solution of a scale-dependent gravitational theory. Comparison with other approaches towards wormhole self--sustainability are presented, with emphasis on the running of the gravitational coupling and on a possible effective description of gravity near the Planck scale.
Analytic solutions are presented which describe the construction of a traversable wormhole from a Schwarzschild black hole, and the enlargement of such a wormhole, in Einstein gravity. The matter model is pure radiation which may have negative energy density (phantom or ghost radiation) and the idealization of impulsive radiation (infinitesimally thin null shells) is employed.
The purpose of this paper is to obtain exact solutions of the Einstein field equations describing traversable wormholes supported by phantom energy. Their relationship to exact solutions in the literature is also discussed, as well as the conditions required to determine such solutions.
This study aims to investigate the viability of a viable logarithmic f(R) gravity for inflation in giving wormhole solutions. We consider a static and spherically symmetric spacetime and two different shape functions. We found that the model provides traversable wormhole solutions shown by its violation to energy conditions. Violation to null energy condition indicates the existence of traversable wormhole and the existence of exotic matter. The equation of state parameter is less than -1 which shows the presence of phantom fluid. Furthermore, the stability of wormhole is analyzed using Tolman…
In this paper we study traversable wormholes in the context of $f(R)$ gravity. Exact solutions of traversable wormholes are found by imposing the nonconstant Ricci scalar. These solutions asymptotically match spherical, flat and hyperbolic FRW metric. By choosing some static $f(R)$ gravity models, we verify the standard energy conditions for the asymptotically spherical, flat and hyperbolic wormhole solutions. Unlike the Einstein gravity, we find that in the context of $f(R)$ modified gravity, the asymptotically spherical, flat and hyperbolic wormhole solutions can respect the null energy cond…
In this work, we construct a traversable wormhole by providing a suitable embedding function ensuring the fulfilling of the flaring--out condition. The solution contains free parameters that are reduced through the study of the acceptable conditions of a traversable wormhole. We compute both the quantifier of exotic matter and the quasi--normal modes through the $13^{th}$ order WKB as a function of the remaining free parameters. We obtain that the wormhole geometry can be sustained by a finite amount of exotic matter and seems to be stable under scalar perturbations.
Building on the work of Morris, Thorne, and Yurtsever, some particularly simple examples of traversable wormholes are exhibited. These examples are notable both because the analysis is not limited to spherically symmetric cases, and because it is possible to in some sense minimize the use of exotic matter. In particular, it is possible for a traveller to traverse such a wormhole without passing through a region of exotic matter. As in previous analyses, the weak energy condition is violated in these traversable wormholes.
Wormhole solutions in classical General Relativity are unstable and hence non traversable. Morris and Thorne discovered a traversable wormhole solution that required the energy momentum tensor of matter sources to violate various energy conditions and are out of the purview of the standard model of particle physics. The search for traversable wormhole solutions in Modified Theories of Gravity has been of significant interest in the decades after Morris and Thorne first published their results as such violations may be avoided in such theories. This work comprehensively reviews traversable worm…
In this paper, we examine the existence of traversable wormhole solutions within the Chern-Simons modified gravity. We find a non-trivial solution in the theory with dynamical Chern-Simons coefficient in the absence of matter sources. This result displays a situation opposite to GR where the matter sources violating the energy conditions are required.
The static spherically symmetric traversable wormholes are analysed in the Einstein- Cartan theory of gravitation. In particular, we computed the torsion tensor for matter fields with different spin S = 0; 1/2; 1; 3/2. Interestingly, only for certain values of the spin the torsion contribution to Einstein-Cartan field equation allows one to satisfy both faring-out condition and Null Energy Condition. In this scenario traversable wormholes can be produced by using usual (non-exotic) spinning matter.
This MSc thesis is divided in to two parts. The first, covers the foundations of theories of gravitation, and, the second incorporates original work on the subject of the existence of traversable wormholes in $f(R)$ modified theories of gravity. A short incursion in the field of scalar-tensor theories had to be made, owing to an apparent inconsistency in the result previously found.
Using thin shell formalism we construct two solutions of intra-universe wormholes. The first model is a cosmological analog of the Aichelburg-Schein timehole, while another one is an intra-universe form of the Bronnikov-Ellis solution.
Gauss-Bonnet formula is used to derive a new and simple theorem of nonexistence of vacuum static nonsingular lorentzian wormholes. We also derive simple proofs for the nonexistence of lorentzian wormhole solutions for some classes of static matter such as, for instance, real scalar fields with a generic potential obeying $ϕV'(ϕ) \ge 0$ and massless fermions fields.
This paper discusses a new variable equation of state parameter leading to exact solutions of the Einstein field equations describing traversable wormholes. In addition to generalizing the notion of phantom energy, the equation of state generates a mathematical model that combines the generalized phantom energy and the generalized Chaplygin gas models.
Wormholes have been always an interesting object in gravity theories. In this paper we make a little review of the principal properties of these objects and the exotic matter they need to exist. Then, we obtain two specific solutions in the formalism of (2+1)-dimensional gravity with negative cosmological constant. The obtained geometries correspond to traversable wormholes with an exterior geometry correspondient to the well known BTZ black hole solution. We also discuss the distribution of exotic matter that these wormholes need.
A one-parameter family of static and spherically symmetric solutions to Einstein equations with a traceless energy-momentum tensor is found. When the nonzero parameter $β$ lies in the open interval $(0,1)$ one obtains traversable Lorentzian wormholes. One also obtains naked singularities when either $β< 0$ or $β> 1$ and the Schwarzschild black hole for $β= 1$.
The general form of a stationary, axially symmetric traversable wormhole is discussed. This provides an explicit class of rotating wormholes that generalize the static, spherically symmetric ones first considered by Morris and Thorne. In agreement with general analyses, it is verified that such a wormhole generically violates the null energy condition at the throat. However, for suitable model wormholes, there can be classes of geodesics falling through it which do not encounter any energy-condition-violating matter. The possible presence of an ergoregion surrounding the throat is also noted.
We explore the possibility that traversable wormholes be supported by specific equations of state responsible for the present accelerated expansion of the Universe, namely, phantom energy, the generalized Chaplygin gas, and the van der Waals quintessence equation of state.
Wormhole solutions to the equations of general relativity have some spectacular local and global properties. As these unusual features are not explicitly forbidden by known physics, wormholes are considered in various astrophysical and cosmological scenarios. The paradigmatic traversable wormhole models are described by static spherically-symmetric Ellis--Morris--Thorne and Simpson--Visser metrics. We show that no dynamical solution of the semiclassical Einstein equations can have these metrics as their static limit. On the other hand, possible static limits of the dynamical solutions are not …
We construct a static axisymmetric wormhole from the gravitational field of two charged shells which are kept in equilibrium by their electromagnetic repulsion. For large separations the exterior tends to the Majumdar-Papapetrou spacetime of two charged particles. The interior of the wormhole is a Reissner-Nordström black hole matching to the two shells. The wormhole is traversable and connects to the same asymptotics without violation of energy conditions. However, every point in the Majumdar-Papapetrou region lies on a closed timelike curve.
In this work, the study of traversable wormholes in $f(R)$ gravity with the function $f(R)=R+αR^n$, where $α$ and $n$ are arbitrary constants, is taken into account. The shape function $b(r)=\frac{r}{\exp(r-r_0)}$, proposed by Samanta et al. \cite{godani1}, is considered. The energy conditions with respect to both constant and variable redshift functions are discussed and the existence of wormhole solutions without presence of exotic matter is investigated.
It is well-known that traversable wormhole solutions to the Einstein equations require the existence of an exotic matter source violating the null energy condition. An apparent exception is the overcharged Kerr-Newman-NUT solution of the Einstein-Maxwell equations, which has all the properties of a geodesically complete, traversable wormhole spacetime. We show that the exotic matter sourcing this consists in two counter-rotating tensionless straight cosmic strings -- the Misner-Dirac strings -- which, as expected, violate the null energy condition. The wormhole possesses an ergoregion, but no …
In this paper, we first use the optical metrics of black-bounce traversable wormholes to calculate the Gaussian curvature. Then we use the Gauss-Bonnet theorem to obtain the weak deflection angle of light from the black-bounce traversable wormholes. Then we investigate the effect of dark matter medium on weak deflection angle using the Gauss-Bonnet theorem. We show how weak deflection angle of wormhole is affected by the bounce parameter $a$. Using the Gauss-bonnet theorem for calculating weak deflection angle shows us that light bending can be thought as a global and topological effect.
Traversable wormholes necessarily require violations of the averaged null energy condition; this being the definition of ``exotic matter''. However, the theorems which guarantee the energy condition violation are remarkably silent when it comes to making quantitative statements regarding the ``total amount'' of energy condition violating matter in the spacetime. We develop a suitable measure for quantifying this notion, and demonstrate the existence of spacetime geometries containing traversable wormholes that are supported by arbitrarily small quantities of ``exotic matter''.
In this work, we calculate the deflection angle of light in a spacetime that interpolates between regular black holes and traversable wormholes, depending on the free parameter of the metric. Afterwards, this angular deflection is substituted into the lens equations which allows to obtain physically measurable results, such as the position of the relativistic images and the magnifications.
In this talk I discuss pertinence of the wormholes to the problem of circumventing the light speed barrier and present a specific class of wormholes. The wormholes of this class are static and have arbitrarily wide throats, which makes them traversable. The matter necessary for these spacetimes to be solutions of the Einstein equations is shown to consist of two components, one of which satisfies the Weak energy condition and the other is produced by vacuum fluctuations of neutrino, electromagnetic (in dimensional regularization), and/or massless scalar (conformally coupled) fields.
A process for using curvature invariants is applied as a new means to evaluate the traversability of Lorentzian wormholes and to display the wormhole spacetime manifold. This approach was formulated by Henry, Overduin and Wilcomb for Black Holes in Reference [1]. Curvature invariants are independent of coordinate basis, so the process is free of coordinate mapping distortions and the same regardless of your chosen coordinates. The four independent Carminati and McLenaghan (CM) invariants are calculated and the non-zero curvature invariant functions are plotted. Three example traversable wormho…
This paper addresses the following issues: (1) the possible existence of macroscopic traversable wormholes, given a noncommutative-geometry background, and (2) the possibility of allowing zero tidal forces, given a known density. It is shown that whenever the energy density describes a classical wormhole, the resulting solution is incompatible with quantum field theory. If the energy density originates from noncommutative geometry, then zero tidal forces are allowed. Also attributable to the noncommutative geometry is the violation of the null energy condition. The wormhole geometry satisfies …
We present a static, spherically symmetric, traversable wormhole solution to multimetric gravity which is sustained by only non-exotic matter, i.e., matter which satisfies all energy conditions. The possibility of this solution arises from the fact that under certain conditions the multimetric gravitational field equations reduce to the Einstein equations, but with a negative effective gravitational constant. We show that the Arnowitt-Deser-Misner mass of this wormhole vanishes, so that it appears massless to observers in the asymptotically flat spacetime. We finally speculate on the feasibili…
In this paper I present a new class of traversable wormholes. This is done by surgically grafting two Schwarzschild spacetimes together in such a way that no event horizon is permitted to form. This surgery concentrates a non--zero stress--energy on the boundary layer between the two asymptotically flat regions. I shall investigate this stress--energy in detail using the "junction condition" formalism. A feature of the present analysis is that this class of traversable wormholes is sufficiently simple for a (partial) dynamical stability analysis to be carried out. The stability analysis places…
In this paper we show that, analogously to as it occurs for black holes, there exist three well-defined laws for Lorentzian wormhole thermodynamics and that these laws are related with a thermal phantom-like radiation process coming from the wormhole throat. It is argued that the existence of wormholes could be manifested by means such a radiation. These results are obtained by analyzing the Hayward formalism of spherically symmetric solutions containing trapping horizons, the phenomenon of phantom accretion onto wormholes and the development of phantom thermodynamics.
The Brouwer fixed-point theorem in topology states that for any continuous mapping $f$ on a compact convex set into itself admits a fixed point, i.e., a point $x_0$ such that $f(x_0)=x_0$. Under certain conditions, this fixed point corresponds to the throat of a traversable wormhole, i.e., $b(r_0)=r_0$ for the shape function $b=b(r)$. The possible existence of wormholes can therefore be deduced from purely mathematical considerations without going beyond the existing physical requirements.
The `theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic world is plagued by the violation of the well--known energy conditions of General Relativity. In this brief article we show : (i) how the extent of violation can be quantified using certain volume integrals (ii) whether this `amount of violation' can be minimised for some specific cut--and--paste geometric constructions. Examples and possibilities are also outlined.
In this paper, a traversable wormhole in the Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) model with one extra spacelike compact dimension is studied. We have chosen dynamical compactification as the evolution of the fifth dimension. In this respect, we study how the existence of the extra dimension, affect the behavior of the energy density, the shape function and the scale factor. It is shown that the total matter can be non-exotic and the violation of the weak energy condition can be avoided.
We present new, asymptotically flat, static, spherically symmetric and traversable wormhole solutions in General Relativity which are supported by a family of ghost scalar fields with quartic potential. This family consists of a particular composition of the scalar field modes, in which each mode is characterized by the same value of the angular momentum number $\ell$, yet the composition yields a spherically symmetric stress-energy-momentum and metric tensor. For $\ell = 0$ our solutions reduce to wormhole configurations which had been reported previously in the literature. We discuss the eff…
Noncommutative geometry, as conceptualized by Nicolini, Smailagic, and Spallucci, may be viewed as a slight modification of Einstein's theory. The same can be said for $f(R)$ modified gravity for an appropriate choice of the function $f(R)$. Since such an $f(R)$ could be determined from the noncommutative-geometry background, these gravitational theories make very similar predictions in the discussion of (a) dark matter and (b) traversable wormholes; they can therefore be taken as equally viable models.
Recent studies have shown that (a) quantum effects may be sufficient to support a wormhole throat and (b) the total amount of "exotic matter" can be made arbitrarily small. Unfortunately, using only small amounts of exotic matter may result in a wormhole that flares out too slowly to be traversable in a reasonable length of time. Combined with the Ford-Roman constraints, the wormhole may also come close to having an event horizon at the throat. This paper examines a model that overcomes these difficulties, while satisfying the usual traversability conditions. This model also confirms that the …
The analytic solution of the general relativity equations for spherically symmetric wormholes are given. We investigate the special case of a "traversable" wormhole i.e., one allowing the signal to pass through it. The energy-momentum tensor of wormhole matter is represented as a superposition of a spherically symmetric magnetic field and dust matter with negative matter density. The dynamics of the model are investigated. We discuss both the solution of the equation with a Lambda-term and without it. Superposing enough dust matter, a magnetic field, and a Lambda-term can produce a static solu…
In this paper thermodynamics of static Morris-Thorne wormholes has been discussed in the context of $f(R)$ gravity. The generalized surface gravity, unified first law of thermodynamics and wormhole dynamics have been studied at trapping horizons. We have investigated thermodynamics in non-minimal coupling of curvature and matter which produces very complex equations. Our results generalize the results that have already been derived in Einstein's gravity in the absence of curvature-matter coupling.
We analytically explore the effect of falling matter on a spherically symmetric wormhole supported by a spherical shell composed of exotic matter located at its throat. The falling matter is assumed to be also a thin spherical shell concentric with the shell supporting the wormhole, and its self-gravity is completely taken into account. We treat these spherical thin shells by Israel's formalism of metric junction. When the falling spherical shell goes through the wormhole, it necessarily collides with the shell supporting the wormhole. To treat this collision, we assume the interaction between…
