Tolman Metrics Research Articles

Lie symmetries of Lemaitre–Tolman–Bondi metric

The aim of this paper is to investigate Lie symmetries including Killing, homothetic and conformal symmetries of Lemaitre–Tolman–Bondi (LTB) spacetime metric. To find all LTB metrics admitting these three types of symmetries, we have analyzed the set of symmetry equations by a Maple algorithm that provides some restrictions on the functions involved in LTB metric under which this metric admits the three mentioned symmetries. The solution of symmetry equations under these restrictions leads to the explicit form of symmetries. The stress–energy tensor is calculated for all the obtained metrics in order to discuss their physical significance. It is noticed that most of these metrics satisfy certain energy conditions and correspond to anisotropic fluids.

Constraints on the local cosmic void from the Pantheon supernovae data

In principle, the local cosmic void can be simply modeled by the spherically symmetric Lemaitre–Tolman–Bondi (LTB) metric. In practice, the real local cosmic void is probably not spherically symmetric. In this paper, to reconstruct a more realistic profile of the local cosmic void, we divide it into several segments. Each segment with certain solid angle is modeled by its own LTB metric. Meanwhile, we divide the 1048 type Ia supernovae (SNIa) of the Pantheon Survey into corresponding subsets according to their distribution in the galactic coordinate system. Obviously, each SNIa subset can only be used to reconstruct the profile of one segment. Finally, we can patch together an irregular profile for the local cosmic void with the whole Pantheon sample. Note that, the paucity of each data subset lead us to focus on the inner part of each void segment and assume that the half radii of the void segments are sufficient to constrain the whole segment. We find that, despite 2sigma signals of anisotropy limited to the depth of the void segments, the constraints on every void segment are consistent with Lambda CDM model at 95% CL. Moreover, our constraints are too weak to challenge the cosmic homogeneity and isotropy.

Adiabatic stability of stellar models in the unimodular framework

We investigate the behaviour of the Tolman metrics within the formalism of the trace-free (or unimodular) gravity. A solution generating algorithm is presented which influences the physical character of the models proposed. Specifically an additive constant of integration (K) is persistently in attendance and while it has little influence on the density, pressure, energy conditions and sound speed it exerts considerable influence on the equation of state, active gravitational mass and hence the compactification parameter (mass-to-radius ratio) as well as on the Chandrasekhar adiabatic stability index. A thorough study of the Tolman IV and V metrics is conducted and it is evident that the adiabatic stability criterion $\frac{\rho + p}{p} \frac{\mathrm{d}p}{\mathrm{d}\rho} > \frac{4}{3}$ is achievable in the presence of K but not without at least in the Tolman IV case. A range of parameter values for the Tolman V solution is considered and graphical plots corroborate that the Tolman choice $n = \frac{1}{2}$ only allows for the model’s compatibility with the elementary requirements for physical plausibility.

Impact of the Rastall parameter on perfect fluid spheres

We examine the effects of the Rastall parameter on the behaviour of spherically symmetric static distributions of perfect fluid matter. It was claimed by Visser [Physics Letters B, 782, 83, (2018)] that the Rastall proposition is completely equivalent to the Einstein theory. While many authors have raised contrary arguments, our intention is to analyse the properties of Rastall gravity through variation of the Rastall parameter in the context of perfect fluids spheres that may be used to model neutron stars or cold fluid planets. This analysis also serves to counter the claim that Rastall gravity is equivalent to the standard Einstein theory. It turns out that the condition of pressure isotropy is exactly the same as for Einstein gravity and hence that any known solution of the Einstein equations may be used to study the effects of the Rastall dynamical quantities. Moreover, by choosing the well studied Tolman metrics, we discover that in the majority of cases there is substantial deviation from the Einstein case when the Rastall parameter vanishes and in cases where the Einstein model displays defective behaviour, certain Rastall models obey the well known elementary requirements for physical plausibility. These empirical findings do not support the idea that Rastall theory is equivalent to Einstein theory as several deviations in physical behaviour are displayed as counter-examples.

Energy conditions and conservation laws in LTB metric via Noether symmetries

In this paper, we have investigated Noether symmetries in Lemaitre–Tolman–Bondi (LTB) metric. Using the Lagrangian associated with the LTB metric, the set of determining equations for Noether symmetries is obtained and then integrated in several cases. It is shown that the LTB metric can be classified in to eight distinct classes corresponding to Noether algebra of dimension 4, 5, 6, 7, 8, 9, 11 and 17. The obtained Noether symmetries are compared with Killing and homothetic vectors. The well known Noether’s theorem is used to find the expressions for conservation laws in each case. Moreover, it is shown that most of the obtained metrics are anisotropic or perfect fluid models which satisfy certain energy conditions and the equation of state.

Gravitational perfect fluid collapse in Gauss\u2013Bonnet gravity

The Einstein Gauss–Bonnet theory of gravity is the low-energy limit of heterotic super-symmetric string theory. This paper deals with gravitational collapse of a perfect fluid in Einstein–Gauss–Bonnet gravity by considering the Lemaitre–Tolman–Bondi metric. For this purpose, the closed form of the exact solution of the equations of motion has been determined by using the conservation of the stress-energy tensor and the condition of marginally bound shells. It has been investigated that the presence of a Gauss–Bonnet coupling term alpha >0 and the pressure of the fluid modifies the structure and time formation of singularity. In this analysis a singularity forms earlier than a horizon, so the end state of the collapse is a naked singularity depending on the initial data. But this singularity is weak and timelike, which goes against the investigation of general relativity.

Spin 3/2 Field Equation: Separation and Solution in a Class of Lemaître–Tolman–Bondi Cosmologies

The spin 3/2 field equation is studied in the general Lemaitre–Tolman–Bondi (LTB) space-time. The equation is separated by variable separation. The angular dependence factors out at the level of the general LTB metric. Due to spherical symmetry the separated angular equations coincide with those, previously integrated, relative to the Robertson–Walker and Schwarzschild metric. Separation of time and radial dependence is possible within a class of LTB cosmological models for which the physical radius is a product of a time and a radial function, the last one being further selected by the consistency condition of the radial equations. The separated time dependence, that can be integrated by series, results essentially unique. Instead the radial dependence can be reduced to two independent second order ordinary differential equations that still depend on an arbitrary radial function that is an integration function of the cosmological model. The generalization of the scheme to arbitrary spin field equation is suggested.

Analytical estimate of the effect of spherical inhomogeneities on luminosity distance and redshift

We provide an analytical estimate of the effect of a spherical inhomogeneity on light beams that travel through it. We model the interior of the inhomogeneity through the Lemaitre–Tolman–Bondi metric. We assume that the beam source is located outside the inhomogeneity. We study the relative deviations of travelling time, redshift, beam area and luminosity distance from their values in a homogeneous cosmology. They depend on the ratio H¯=Hr0 of the radius r0 of the inhomogeneity to the horizon distance 1/H. For an observer located at the center, the deviations are of order H¯2. For an observer outside the inhomogeneity, the deviations of crossing time and redshift are of order H¯3. The deviations of beam area and luminosity distance are of order H¯2. However, when averaged over all possible locations of the observer outside the inhomogeneity, they also become of order H¯3. We discuss the implications for the possibility of attributing the observed cosmological acceleration to the emergence of large-scale structure.

Formation of voids in the Universe within the Lemaître-Tolman model

We develop models of void formation starting from a small initial fluctuation at recombination and growing to a realistic present day density profile in agreement with observations of voids. The model construction is an extension of previously developed algorithms for finding a Lemaitre-Tolman metric that evolves between two profiles of either density or velocity specified at two times. Of the 4 profiles of concern -- those of density and velocity at recombination and at the present day -- two can be specified and the other two follow from the derived model. We find that, in order to reproduce the present-day void density profiles, the initial velocity profile is more important than the initial density profile. Extrapolation of current CMB observations to the scales relevant to proto-voids is very uncertain. Even so, we find that it is very difficult to make both the initial density and velocity fluctuation amplitudes small enough, and still obtain a realistic void by today.

Exact inhomogeneous models of cosmological inflation

Exact spherically symmetric solutions of the Einstein equations with quasi-vacuum source p = −e are found in the form of spatially inhomogeneous Tolman metrics describing the inflationary stage of the evolution of the universe.

A new class of inhomogeneous cosmological models with Yang-Mills fields

Exact solutions corresponding to spherically symmetric inhomogeneous nonstationary Tolman metrics are obtained for the self-consistent

On shell crossing in the Tolman metric

The initial conditions of the Tolman metric are studied. In the hyperbolic case an exact condition leading to shell crossing is found. it is conjectured that shell crossing is a good way to form glaxies and voids in a hyperbolic universe dominated by baryons. A comparison with the standard gauge-invariant theory of small perturbations is also presented

Models of voids in elliptic universes

Models for nonempty spherical voids in elliptic universes are constructed. The models consist of an underdense Friedman region (the void), an intervening Tolman region, and an exterior Friedmann universe. In order to achieve a significant underdensity, in fact as small as wanted but nonzero, it is necessary to allow for a prematurely born void. It is then possible to choose the arbitrary functions in the Tolman metric of the intervening zone in such a way that the models are exempt from shell-crossing during their whole history. In that case there is a first region in the intervening Tolman zone containing the oldest matter in the universe, and a second region with the youngest

SL(2,C) gravitational conserved current and Noether's theorem

Noether's theorem is applied to Hilbert's Lagrangian written as a functional of spinorial variables. The associatedSL(2,C) conserved current is obtained, and its expression for the Tolman metric is given explicitly.

Passage to a synchronous reference system and an estimate of the components of the metric tensor on the boundary of R and T regions

Conditions imposed on transformations leading to a synchronous reference system are obtained. The spherically symmetric case is considered in detail. It is shown that the known synchronous metrics (for example, the Lemaitre and Tolman metrics) can be obtained as a special case of those presented in the paper. The method developed in the work is applied to obtain estimates of the components egu and egl of the metric tensor in a rigid reference system on approach to the boundary of R and T regions from the side of an R region in the case where matter is present.

Relation between the generalized Schwarzschild metric and the Tolman metric

In the present paper the relation between the generalized Schwarzschild metric (the Schwarzschild metric including the four-dimensional curvature tensor) [1] and the Tolman metric is considered.

On the problem of the relationship between the schwarzschild and tolman metrics: PMM vol. 37, n≗4, 1973, pp. 739–745

The problem of the relationship between the Schwarzschild and Tolman metrics has occupied the attention of many workers. Although the solutions given in [1–3] satisfy the equations of the general relativity theory (OTO) ( ∗ ∗ Editors note. The abbreviations (OTO) and (CTO) are used in the relevant Soviet literature and stand for “general relativity theory” and “special relativity theory”, respectively. ),they contradict the correspondence principle. This means that for G → 0, the interval is not transformed into the interval of the special relativity theory (CTO) ( ∗ ), while for c → ∞, the solutions do not become Newtonian. This is apparently caused by the unfortunate choice of the coordinates in the Tolman frame of reference. Papers [4, 5] illustrate particular cases of a correct passage from one metric to the other. In the present paper a general method of obtaining solutions is proposed in which the passage from one frame of reference to the other satisfies the correspondence principle.