New solutions of gravitational collapse in General Relativity and in the Newtonian limit

We discuss the Oppenheimer-Snyder-Datt (OSD) solution from a new perspective, introduce a completely new formulation of the problem exclusively in external Schwarzschild space-time (ESM) and present a new treatment of the singularities in this new formulation. We also give a new Newtonian approximation of the problem. Furthermore, we present new numerical solutions of the modified OSD-model and of the ball-to-ball-collapse with 4 different numerical methods. 1.Introduction The gravitational collapse(GC) is, together with the Robertson-Walker-Friedmann-Lemaitre(RWFL) cosmological models, the most important dynamical model in General Relativity. In its OSD form it has, apart from RWFL, the only closed analytic solution in this area. In astrophysics and cosmology it is of utmost importance, because it is considered to be the valid model for the formation of stars from dust and gas clouds and for catastrophic events like star collapse to a neutron star or a Black Hole. However, the OSD formulation has severe drawbacks: its assumptions of homogeneous density and zero pressure are completely unrealistic and especially the latter can be even regarded as unphysical ([15]). Furthermore, the formation of the Black Hole asserted in OSD happens at infinite time for an external observer in the corresponding external Schwarzschild or Vaidya space-time (ESM) (see e.g. [1]). Finally, OSD uses co-moving coordinate frame as the formulation basis and introduces junction conditions (continuity of space-time function and derivative) to the external observer frame in ESM, which are difficult to calculate symbolically and to implement numerically. Recent publications on this subject can be divided into 4 categories: -observational astrophysics with application of GC-formalism Kotake[5] -review Naidu[12] Joshi[4] -extensions of OSD in COF with respect to heat-flux Herrera ([2],[3]), radiation Sharma[6] , neutrinoemission Nakazato[11], equation-of-state Sanwe[14] Joshi[7] -singularity freedom in gravitational collapse Marshall[8] Mitra[15] . In this article we introduce as the essential feature system equations for GC exclusively in ESM, which adds 1 differential equation to the 4 of OSD, and radial velocity as a new, fifth variable function. This makes the system more complicated, particularly a separation ansatz in r and t like in OSD is not possible anymore, but there are no junction conditions and all results are directly observable for an external observer. The article can be subdivided into three parts: fundamentals and the system equations for the different models chapter 2,3,4 the particular models and discussion of singularities chapter 5,6,7 -numerical methods and results chapter 8,9 2.GR fundamentals The most general spherically symmetric line element, in spherical coordinates (x) = (t, r,,), can be written as      d d r t Y dr r t B dt c r t A ds 2 2 2 2 2 2 2 2 2 sin ) , ( ) , ( ) , (      (1) where A, B and Y are functions of the coordinates t and r. The Einstein field equations with the above line element are:     T g R g R      0 2 1 (2) where  R is the Ricci tensor, R0 the Ricci curvature, 4 8 c G    ,  T is the energy-momentum tensor,  is the cosmological constant (in the following neglected, i.e. set 0), with the Christoffel symbols (second kind)                          x g x g x g g 2 1


1.Introduction
The gravitational collapse(GC) is, together with the Robertson-Walker-Friedmann-Lemaitre(RWFL) cosmological models, the most important dynamical model in General Relativity.In its OSD form it has, apart from RWFL, the only closed analytic solution in this area.In astrophysics and cosmology it is of utmost importance, because it is considered to be the valid model for the formation of stars from dust and gas clouds and for catastrophic events like star collapse to a neutron star or a Black Hole.However, the OSD formulation has severe drawbacks: its assumptions of homogeneous density and zero pressure are completely unrealistic and especially the latter can be even regarded as unphysical ( [15]).Furthermore, the formation of the Black Hole asserted in OSD happens at infinite time for an external observer in the corresponding external Schwarzschild or Vaidya space-time (ESM) (see e.g.[1]).Finally, OSD uses co-moving coordinate frame as the formulation basis and introduces junction conditions (continuity of space-time function and derivative) to the external observer frame in ESM, which are difficult to calculate symbolically and to implement numerically.Recent publications on this subject can be divided into 4 categories: -observational astrophysics with application of GC-formalism Kotake [5] -review Naidu [12] Joshi [4] -extensions of OSD in COF with respect to heat-flux Herrera ( [2], [3]), radiation Sharma [6] , neutrinoemission Nakazato [11], equation-of-state Sanwe [14] Joshi [7] -singularity freedom in gravitational collapse Marshall[8] Mitra [15] .In this article we introduce as the essential feature system equations for GC exclusively in ESM, which adds 1 differential equation to the 4 of OSD, and radial velocity as a new, fifth variable function.This makes the system more complicated, particularly a separation ansatz in r and t like in OSD is not possible anymore, but there are no junction conditions and all results are directly observable for an external observer.The article can be subdivided into three parts: -fundamentals and the system equations for the different models chapter 2,3,4 -the particular models and discussion of singularities chapter 5,6,7 -numerical methods and results chapter 8,9

2.GR fundamentals
The most general spherically symmetric line element, in spherical coordinates (x  ) = (t, r,,), can be written as where A, B and Y are functions of the coordinates t and r.The Einstein field equations with the above line element are: where  R is the Ricci tensor, R 0 the Ricci curvature, T is the energy-momentum tensor,  is the cosmological constant (in the following neglected, i.e. set 0), with the Christoffel symbols (second kind) and the Ricci tensor The Einstein tensor is given by we obtain from ( 3) and ( 4) the non-vanishing Einstein tensor components [2], [12] The dot °represents t  and the prime ' represents r  .
The energy-momentum tensor T for a perfect fluid with 4-velocity u  , heat-flow q  , density , and pressure P is We will consider here only adiabatic systems without heat-flow to outside, so in the following q=0.For the gravitation collapse problem two coordinate systems are used.In the co-moving coordinate system (COF) the spatial part of the velocity is zero: and the acceleration 4-vector is In the external Schwarzschild space-time (ESM) with Schwarzschild-radius r s we take the coordinate system of the non-moving observer at infinity.In order to make the variables dimensionless, we introduce 'sun units' , like in [16], where r s means the Schwarzschild-radius.In the following, we make all equations and variables dimensionless, by using the "proper units" of the gravitating system

3.System equations for COF
We insert the above expression for u  into T and get the Einstein-equations as the system equations: eq00:
In the dimensionless representation and M=M sun : c 0 =3 .When calculating in COF, one uses in parallel an external space-time valid outside the gravitating mass system boundary , the usual choice is the Vaidya space-time for a radiating spherically symmetric mass system with total mass M(t) and initial mass M 0 in new coordinates r 2 , t 2 : When the system is adiabatic, i.e. heat-flow=0 and M(t)=M 0 , the Vaidya space-time becomes the Schwarzschild space-time in Eddington-Finkelstein coordinates: The line element functions of the inner and the external space-time must be continuous, and so must be their derivatives normal to the surface boundary, this generates junction conditions on the surface [2]: with the boundary surface space-time In the last equation, the dot °represents t  and the prime ' represents r  .

4.System equations for Schwarzschild space-time ESM
Here we give up the COF, and calculate exclusively in ESM.Now the radial velocity u(t,r) becomes a new variable function and we derive an additional system equation for it from the GR orbit equations, where  is the proper time: (24) we get the following differential equations for t=x 0 and r=x 1 and their derivative ' for  : And from that for the radial velocity u r =r' : We get for the energy-momentum tensor From this and the expressions for G (6…10) get the system equations for ESM, where u(t,r)=u r eq00: (25) where A(t,r)=exp(fA(t,r)) , B(t,r)=exp(fB(t,r)) , Y(t,r)=exp(fY(t,r)) , u(t,r)=exp(fu(t,r)) .

5.The gravitational dust cloud collapse model of Oppenheimer-Snyder-Datt
The Oppenheimer-Snyder-Datt (OSD) model of a gravitational collapse was the first exact GR-solution of a dynamic GR system.With the (unrealistic) assumption of a no-pressure dust cloud with homogeneous density and calculating in COF one can use a separation-ansatz in the system equations and gets an explicit analytic expression for the function variables.In the following we use the terminology in [1] and [4].In COF the solution space-time becomes (dimensionless) where the edge of the cloud is in COF and k=1/r 0 2 , and r 0 is the initial edge radius, v(0)=1 , density where the radius of the cloud R 1 (T col )=0 and there is a singularity with density blowing-up.In ESF, with coordinates t2 and r2 , the (dimensionless) equation for the edge is ([1] 45.10): so in the limit of infinite time t 2 the edge in ESM goes exponentially slowly (with the characteristic time t s =r s /c) to the Schwarzschild-radius r s : there is no singularity, as the Black Hole forms at infinity .This result can be found in [1], [8] and [15].In [15] the author gives a proof for non-existence of singularities in gravitational collapse in COF under certain conditions.

6.Gravitational collapse in the Newtonian approximation
In the Newtonian approximation, we consider the total energy density of pressure, kinetic energy and gravitational energy, with the equation-of-state (eos) of the ideal gas which is conserved and therefore stationary under t  and r  , which gives 2 differential equations for the 2 function variables (t,r) and v(t,r) .
is the mass within the radius r .
In proper units, i.e. dimensionless the total energy density becomes The resulting (dimensionless) equations are of order 1 in (t,r) and v(t,r) .equt: with the abbreviation These equations can be simplified by using the integrability condition: With this the equations (33), (34) become Using (33a) and 1 2  v one can show that  cannot blow-up at r=0, so there is no singularity.The numerical results for the ball-to-ball model are given in 9.6 below.
, so with pressure non-zero, there is a physical limit for the density.Under these assumptions and taking only terms, which have the strongest divergence in t, the ESM equations ( 25

r)=exp(fAr(r)) , B(t,r)=exp(k*log(t)+fBr(r)) , Y(t,r)=exp(k*log(t)+frY(r)) , u(t,r)=exp(fut(t)+fur(r)
) , (t,r)=rhot(t)*rhor(r)) : with t-limit behavior a product ansatz in t and r is always possible.Now, as the density  must not blow-up (this would mean an unphysical solution before the Black Hole generation), the brackets with , and from (29s) also const fur  so that the space-time and the velocity is spatially constant, which is impossible.One can show, that other choices of divergent space-time functions lead to a blow-up of the density or to unphysical results.In essence, this confirms in ESM the results of [15], that (in COF) the gravitational collapse singularity can arise only with zero pressure, which is physically untenable.

8.The differential equations and their numerical form
The 5 differential equations for the gravitational collapse in SMF(Schwarzschild metric frame) have the form (25…29) .
The first 4 are the Einstein-equations for R 00 , R 11 , R 01 , R 22 , and the last is the equation for the radial velocity u from the GR equations-of-motion.The dot °represents Ak The chosen space-time range is with edge radius r 0 and final time T 1 : t=0...50(r s /c) , r=0…10(r s ) , r s =Schwarzschild-radius.The variable functions are Ak={fA(t,r),fB(t,r),fY(t,r),(t,r),fu(t,r)} , where

A(t,r)=exp(fA(t,r)) , B(t,r)=exp(fB(t,r)) , Y(t,r)=exp(fY(t,r)) , u(t,r)=exp(fu(t,r))
It is important to note the highest derivatives of the variable functions, they are: fA°, fA'',fB°°, fB',fY°°,fY'',, fu°,fu' For the variables with the differential degree 2 in t , the t-boundary-condition becomes t=0: , and correspondingly in r .The density  is the only algebraic variable function, so no boundary condition is imposed, its boundary value is calculated from the equations at the boundary, together with the other highest derivatives.All numerical solution methods used here operate on an equidistant 2-dimensional lattice {t i ,r j } with step size 1 h =r 0 /ndimx in t , and step size 2 h =T 1 /ndimy in r , ndim=8…32, depending on the required execution time .
For Ritz-Galerkin test functions and for fits a 2-dimensional trigonometric (Fourier) expansion , with basic angular frequencies  t  r and 1   I , of degree ngrad is used.Before the solution procedure starts, an appropriate "seed function" w varfit is chosen with the desired boundary conditions, and then corrected to solve the equations at the boundary , the boundary conditions are corrected appropriately .Its error is evaluated and a global trigonometric t-r-fit on the lattice {t i ,r j } is calculated.

9.Numerical solution 9.1.Numerical solution methods
Ritz-Galerkin global minimization with a trigonometric test function and the parameter vector qv2t operates on the differential-equation-set with the variable functions replaced by the test function.The goal function is the sum of the absolute error values on the lattice {t i ,r j }.First, an appropriate seed function w varfit is chosen, which fulfils the equations at the boundary.Its error is evaluated and a global t-r-fit on the lattice {t i ,r j } to the Ritz-Galerkin-test function is performed with a resulting parameter value vector qv20t .The boundary conditions t=0: are used as minimization precondition for the minimization procedure, the start vector for qv2t is qv20t , the initial fit parameter vector.The result of the procedure is the global parameter vector qv2t , parameter replacement in the trigonometric test functions gives the solution functions.The variable vector qv2t has the length 5*ngrad 2 .
Finite-difference-minimization is based on the finite-difference discretization of the differential equations.In the finite-difference scheme, the variables are the values Ak(t i ,r j ) on the lattice {t i ,r j }.The differences in the scheme are t-forward and r-backward, because it starts with the boundary condition: the start-values in {t i } and {r j } are replaced by the boundary values.The goal function is the sum of the absolute error values on the lattice {t i ,r j }.The start vector are the values {Ak(t i ,r j )(w varfit )} of the seed-function.The physical feasibility conditions are imposed via a penalty function added to the goal function.Finally, a global minimization with the variable-vector { Ak(t i ,r j )} and the start vector is performed, yielding the array {Ak(t i ,r j )} 0 as the solution.The variable vector { Ak(t i ,r j )} has the length 5*ndimx*ndimy , much more than global Ritz-Galerkin, but the minimization has no precondition and is therefore much faster.R-profile-wavefront for fixed t=t i , calculates r-profiles for a fixed t i , starting with the boundary t=t 0 =0.For each profile, the highest derivatives are calculated from the differential equations, the values of the lower derivatives are calculated from the predecessor profile via Euler step and inserted into the equations. .These functional algebraic equations in r can be solved point-wise consecutively starting with the r-boundary r=r ndimy =r0 and obeying continuity requirements (discrete algebraic profile calculation).Alternatively, they can be solved by 1-dimensional trigonometric Ritz-Galerkin minimization starting with a 1-dimensional trigonometric r-fit to the predecessor profile as start parameter-vector, and the r-boundary values as precondition (trigonometric Ritz-Galerkin r-profile calculation ).
Here, we used the trigonometric Ritz-Galerkin r-profile calculation, because it is faster and the continuity is guaranteed automatically.The initial step i=0 t=t 0 uses the boundary r-profile for t=t 0 as the predecessor.The result of the wavefront procedure is the array of the solution function values on the lattice vla (value local array).For each profile, the variable vector qvt has the length ngrad*5 , so the total number of variables is ndimx*ngrad*5 .
Finite-difference-wavefront for fixed t=t i , calculates discrete r-profiles for a fixed t i , starting with the boundary t=t 0 =0.Here, the differential equations are discretized via a finite-difference-scheme with variables Ak(t i ,r j ) on the lattice {t i ,r j }.For each profile t=t i , the discrete variables {Ak(t i ,r j )} j are calculated by minimization of the discretized equations, with start vector from the predecessor profile.The initial step i=0 t=t 0 uses the boundary r-profile for t=t 0 as the predecessor.The result of the wavefront procedure is the array of the solution function values on the lattice lattice vla (value local array).For each profile, the variable vector

9.2.Numerical result statistics
We present the results for the function variables as 3D-plots and for the rest in form of a table, which contains the essential numerical parameters: lattice size, execution time on a conventional 3GHz desktop and the error statistics.For the minimization methods Ritz-Galerkin and finite-difference the numerical parameters are: lattice size , execution time in sec, error=median error of the solution, error-testfunc=median error of the seed functions, total-error= total minimization error , for comparison testf-error= total error of the seed functions, and the mass= numerical mass of the solution over time (should be 1).The minimization is carried out globally (global-min) and with start-vector=seed functions (local-min).For the wavefront methods r-profile and finite-difference the numerical parameters are: lattice size , execution time in sec, error-fit=median error of the fit (spline, Fourier or polynomial interpolation) to the discrete solution values on the lattice, error-testfunc=median error of the seed functions, error-discrete= median error of the discrete solution array vla , and the mass= numerical mass of the solution over time (should be 1).In the 3D-plots the first coordinate is time t=0...50(r s /c) and the second coordinate is radius r=0…10(r s ) , r s =Schwarzschild-radius.

9.3.The equation-of-state
The equation-of-state, which we are using here for the GR-gravitational-collapse is for the non-interacting nucleon Fermi gas, with =5/3 near the critical nucleon density.In the limit of the Newtonian approximation, we use, as customary for sun-like stars, simply the dimensionless ideal-gas-equation (c=1) for r , M sun for M, and the relativistic velocity normalization condition (dimensionless, c=1) the nucleon mass.
in sun-units, and P is replaced by the equation-of-state mass M 0 =1(M sun ).
temperature, relative to the nucleon mass.