Junction conditions for composite matter in higher dimensions

2021 ◽  
Vol 38 (19) ◽  
pp. 195006 ◽  
Author(s):  
Sunil D Maharaj ◽  
Byron P Brassel
2003 ◽  
Vol 12 (06) ◽  
pp. 1035-1045
Author(s):  
A. BANERJEE ◽  
S. CHATTERJEE

We extend to higher dimensions a recent work of Bonnor, which generalizes the Einstein–Straus model utilizing the inhomogeneous Tolman–Bondi universe in place of the homogeneous Friedmann one. Following Israel's junction conditions, the criteria of matching between the higher dimensional Schwarzschild-like interior and the Tolman–Bondi-like exterior is obtained. We also give a new exact solution for a five-dimensional TB type of metric and use it to study the dynamical behavior of the vacuole boundary. Furthermore the transformation relations which transform the inhomogeneous TB metric to the homogeneous Friedmann model are explicitly given for any arbitrary dimensions. The frequency shift of radiation coming from the boundary surface is calculated and it is found that, depending on initial conditions both redshift and blue-shift are possible for an expanding vacuole. This is at variance with Bonnor's result where only redshift is possible under similar situation. It is also observed that higher dimensional models are less stable against perturbation than the usual 4D ones.


2021 ◽  
Vol 81 (4) ◽  
Author(s):  
Sunil D. Maharaj ◽  
Byron P. Brassel

AbstractIn this paper we study the junction conditions for a generalised matter distribution in a radiating star. The internal matter distribution is a composite distribution consisting of barotropic matter, null dust and a null string fluid in a shear-free spherical spacetime. The external matter distribution is a combination of a radiation field and a null string fluid. We find the boundary condition for the composite matter distribution at the stellar surface which reduces to the familiar Santos result with barotropic matter. Our result is extended to higher dimensions. We also find the boundary condition for the general spherical geometry in the presence of shear and anisotropy for a generalised matter distribution.


1993 ◽  
Vol 02 (04) ◽  
pp. 489-495 ◽  
Author(s):  
B. BHUI ◽  
S. CHATTERJEE ◽  
A. BANERJEE

Following O’brien-Synge’s junction conditions we find an exterior solution for a (n+2)-dimensional spherically symmetric distribution in comoving coordinates and match it with the zero-pressure dust interior. An expression for Schwarzschild-like mass is also obtained from the conditions of fit at the boundary. The relevant transformation relations which recast the comoving exterior into the static Schwarzschild-like form are also obtained. This generalizes to higher dimensions an earlier work of Raychaudhuri in 4D spacetime. Utilizing the transformation relations, an expression for frequency shift of radiation emitted from the surface of the sphere is also obtained.


Astérisque ◽  
2020 ◽  
Vol 416 ◽  
pp. 213-251
Author(s):  
Mikhail LYUBICH ◽  
Remus RADU ◽  
Raluca TANASE
Keyword(s):  

Astérisque ◽  
2020 ◽  
Vol 416 ◽  
pp. 213-251
Author(s):  
Mikhail LYUBICH ◽  
Remus RADU ◽  
Raluca TANASE
Keyword(s):  

2020 ◽  
Author(s):  
Deep Bhattacharjee

Gravity has been leaking in higher dimensions in the bulk. Gravity being a closed string is not attached or does not have any endpoints unlike photons to any Dirichlet (p)-Branes and therefore can travel inter-dimensional without any hindrance. In LHC, CERN, Gravitons are difficult to detect as they last for such a short span of time and in most of the cases invisible as because they can escape to higher spatial dimensions to the maximum of 10, as per 'M'-Theory. Gravity being one of the 4-Fundamental forces is weaker than all 3 (strong and weak nuclear force, electromagnetism) and therefore a famous problem has been made in particle physics called the 'hierarchy problem'. Through comprehensive analysis and research I have come to the conclusion that if dimension is 5 (or 4 if we neglect the temporal dimensions) then an old approach is there for the compactification of the dimensions as per Kaluza-Klein theory and the most important implications of this theory is that an unification of electromagnetism with gravitation occurs in the fifth dimensions, therefore we can conclude that both the charge (electric as well as magnetic and gravity) are dependent of each other in case of Dimensions greater than 4 (5 if time is added). Now, basic principles of electromagnetic theory states that the field-flux density through a closed surface like a T 2 Torus when integrated over the surface area leads to a zero flux. That means there is no flux outside this closed surface integral. However, if the surface is open then the field flux density is not zero and this preserves the concept of magnetic monopoles. However, in a paper in 1931,[1] Dirac approaches monopole theory of magnetism through a different perspectives that, if all the electrical charges of the universe is quantized[2] then there is a suitable (not yet proved though) existence of monopoles; however this are not well understood as of today's scenario. In condensed matter physics, plasma physics and magneto hydrodynamics, there are flux tubes and as the both ends of the flux tubes are independent of each other then the net flux through the cylinder is zero as the amount of field lines entering the tube on one side is equal to the amount of field lines exit from the other end. And in the sides of the cylinder or the flux tube there is no escape of field lines, hence, net flux is conserved. There also exists a type of 'Quasiparticles' that can act as a monopole.[3][4][5] Now, from the perspectives of the Guess law of electromagnetism, if there exists a magnetic monopole then the net charge or flux density over a surface is not zero rather the divergence of the flux density B is 4 [6]and an alternative approach of the 'monopole' can be achieved by increasing the spatial dimensions by a factor of 1 or more. The Gravity has no such poles and therefore can be considered as a unipolar flux density existing throughout the universe and is applicable to the inverse square law of decreasing magnitude via distance as 1/r 2. However, a magnet is always of bipolar with a north and South Pole. If a magnet can be broken then also the broken parts develop the other poles and become bipolar. However, there are tiny domains inside a magnet and if a magnet can be heated to approx. 700℃ then all the poles disappeared and if its cooled quickly, rather very quickly then the tiny domains inside the magnet would not get enough time to rearrange themselves and multipolar magnet is developed therefore to preserve the bipolar properties, the magnet should be cooled slowly allowing the time given to the tiny domains top rearrange themselves. Therefore, even multipole can be achieved quite easily but not the monopoles. So, the equation for a closed surface integral of a flux density without monopole is ∯(S) B dS = 0 or ∇ • B = 0 and that closed surface can be considered as 2 types namely (we will discuss about torus) as because in string theory compactification of higher spatial dimensions occurs in torus.


Author(s):  
S. G. Rajeev

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.


2021 ◽  
Vol 31 ◽  
pp. 100758
Author(s):  
Sanjar Shaymatov ◽  
Naresh Dadhich

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Konrad Simon ◽  
Jörn Behrens

AbstractWe introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the associated difficulties is a semi-Lagrangian based reconstruction of subgrid variability into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a comparison to standard methods to support our ideas and discuss possible extensions to other types of Galerkin methods, higher dimensions and nonlinear problems.


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