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 (Mitra [1] ). 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. [2] ). 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 7 categories:

· observational astrophysics with application of GC-formalism Kotake [3];

· review Naidu [4] Joshi [5] Lasky [6];

· gravitational collapse of fluid Lasky [6];

· gravitational collapse in star formation in Newtonian formulation Girichidis [7];

· extensions of OSD in COF (co-moving frame) with respect to heat-flux Herrera ( [8] [9] ) Goswami [10], radiation Sharma [11], neutrino-emission Nakazato [12], equation-of-state Sanwe [13] Joshi [14];

· singularity freedom in gravitational collapse Marshall [15], Mitra [1];

· gravitational collapse quantization from quantum-theoretical point of view Hajicek [16], Corda [17].

In this article we introduce 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 in chapters 2, 3, 4;

· the particular models and discussion of singularities in chapters 5, 6, 7;

· numerical methods and results in chapters 8, 9.

The most general spherically symmetric line element, in spherical coordinates (x^μ) = (t, r, θ, f), can be written as

d s 2 = − A ( t , r ) 2 c 2 d t 2 + B ( t , r ) 2 d r 2 + Y ( t , r ) 2 ( d θ 2 + sin 2 θ   d φ ) (1)

where A, B and Y are functions of the coordinates t and r.

The Einstein field equations with the above line element are:

R μ ν − 1 2 g μ ν R 0 + Λ g μ ν = κ T μ ν (2)

where R μ ν is the Ricci tensor, R₀ the Ricci curvature, κ = 8 π G c 4 , T μ ν is the energy-momentum tensor, Λ is the cosmological constant (in the following neglected, i.e. set 0), with the Christoffel symbols (second kind)

Γ μ ν λ = 1 2 g λ κ ( ∂ g κ μ ∂ x ν + ∂ g κ ν ∂ x μ − ∂ g μ ν ∂ x κ ) (3)

and the Ricci tensor

R μ ν = ∂ Γ μ ρ ρ ∂ x ν − ∂ Γ μ ν ρ ∂ x ρ + Γ μ ρ σ Γ σ ν ρ − Γ μ ν σ Γ σ ρ ρ (4)

The Einstein tensor is given by

G μ ν = R μ ν − 1 2 g μ ν R 0 (5)

we obtain from (3) and (4) the non-vanishing Einstein tensor components [4] [8],

G 00 = ( 2 B ° B Y ° Y + ( Y ° ) 2 Y 2 ) − A 2 B 2 ( − 2 B ' B Y ' Y + ( Y ' ) 2 Y 2 + 2 Y ' ' Y ) + A 2 Y 2 (6)

G 11 = B 2 Y 2 ( 2 A ° A Y ° Y − ( Y ° ) 2 Y 2 − 2 Y ° ° Y ) + ( 2 A ' A Y ' Y + ( Y ' ) 2 Y 2 ) − B 2 Y 2 (7)

G 01 = 2 ( B ° B Y ' Y + A ' A Y ° Y − Y ° ' Y ) (8)

G 22 = − Y 2 A 2 ( B ° ° B − A ° A B ° B + B ° B Y ° Y − A ° A Y ° Y + Y ° ° Y )                   − Y 2 B 2 ( A ' ' A − A ' A B ' B + A ' A Y ' Y − B ' B Y ' Y + Y ' ' Y ) (9)

G 33 = sin 2 θ   G 22 (10)

The dot ° represents ∂ t and the prime ' represents ∂ r .

The energy-momentum tensor T for a perfect fluid with 4-velocity u^μ, heat-flow q^μ, density ρ, and pressure P is

T μ ν = ( ρ + P c 2 ) u μ u ν − P c 2 g μ ν + q λ u λ (11)

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:

u μ = 1 A δ 0 μ

and the acceleration 4-vector is

u ° μ = A ' A δ 1 μ

In the external Schwarzschild space-time (ESM) with Schwarzschild-radius r_s

d s 2 = − ( 1 − r s r ) c 2 d t 2 + 1 1 − r s r d r 2 + r 2 ( d θ 2 + sin 2 θ   d φ ) (12)

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 [18], 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 r s = r s s M M s u n Schwarzschild-radius for r, M_sun for M, ρ s   = ρ s s / ( M M s u n ) 2 for ρ, P s = ρ s c 2 for P.

We insert the above expression for u^μ into T and get the Einstein-equations as the system equations:

eq00: c 0 ρ = exp ( − 2 f A ) ( 2 f B ° f Y ° + ( f Y ° ) 2 )     − exp ( − 2 f B ) ( 2 f Y ' '   + 3 ( f Y ' ) 2 − 2 f B ' f Y ' ) + exp ( − 2 f Y ) (13)

eq11: c 0 P = exp ( − 2 f A ) ( 2 f A ° f Y ° − 3 ( f Y ° ) 2 − f Y ° ° )     − exp ( − 2 f B ) ( 2 f A ' f Y '   + ( f Y ' ) 2 ) − exp ( − 2 f Y ) (14)

eq01: − c 0 q = 0 = 2 exp ( − 2 f B − f A ) ( f B ° f Y '   + f A ' f Y ° − f Y ° '   + f Y ° f Y ' ) (15)

eq22: c 0 P = − exp ( − 2 f A ) ( f B ° ° + ( f B ° ) 2 − f A ° f B ° + f B ° f Y ° + f Y ° ° + ( f Y ° ) 2 )       + exp ( − 2 f B ) ( f A ' '   + ( f A ' ) 2 − f A ' f B '   + f A ' f Y ' − f B ' f Y '   + f Y ' '   + ( f Y ' ) 2 ) (16)

where A ( t , r ) = exp ( f A ( t , r ) ) , B ( t , r ) = exp ( f B ( t , r ) ) , Y ( t , r ) = exp ( f Y ( t , r ) ) .

The constant c 0 = 8 π G c 2 = 3 ( r s s ) 2 ρ s , explicitly and in sun-units.

In the dimensionless representation and M = M_sun:c₀= 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 massM₀ in new coordinates r₂, t₂:

d s 2 = − ( 1 − r s r 2 M ( t 2 ) M 0 ) c 2 d t 2 2 − 2 d t 2   d r 2 + r 2 2 ( d θ 2 + sin 2 θ   d φ ) (17)

When the system is adiabatic, i.e. heat-flow = 0 and M(t)= M₀, the Vaidya space-time becomes the Schwarzschild space-time in Eddington-Finkelstein coordinates:

d s 2 = − ( 1 − r s r 2 ) c 2 d t 2 2 − 2 d t 2   d r 2 + r 2 2 ( d θ 2 + sin 2 θ   d φ ) (18)

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 [8]: with the boundary surface space-time d s 2 = − c 2 d τ 2 + Y 2 ( d θ 2 + sin 2 θ   d φ )

A   d t = d t 2 ( 1 − r s r 2 M ( t 2 ) M 0 ) = d τ (19)

Y = r 2 ( t 2 ) (20)

A   d t 2 d τ = 1 − r s r 2 M ( t 2 ) M 0 + 2 d r 2 d t 2 (21)

− 2 ( B ° B Y ' Y + A ' A Y ° Y − Y ° ' Y ) = − B A ( 2 Y ° ° Y − 2 A ° A Y ° Y + 2 ( Y ° Y ) 2 ) + A B ( 2 A ' A Y ' Y + ( Y ' Y ) 2 − B Y 2 ) (22)

In the last equation, the dot ° represents ∂ t and the prime ' represents ∂ r .

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 τ is the proper time:

d 2 x κ d τ 2 = − Γ μ ν κ d x μ d τ d x ν d τ (23)

and the relativistic velocity normalization condition (dimensionless, c = 1)

− 1 = g μ ν d x μ d τ d x ν d τ (24)

we get the following differential equations for t = x⁰ and r = x¹ and their derivative ' for τ:

t ' ' = − Γ 00 0   t ' 2 − 2 Γ 01 0   t ' r '   − Γ 11 0   r ' 2 = − A ( 1 , 0 ) A t ' 2 − 2 A ( 0 , 1 ) A t ' r '   − B B ( 1 , 0 ) A 2 t ' 2

r ' ' = − Γ 00 1   t ' 2 − 2 Γ 01 1   t ' r '   − Γ 11 1   r ' 2 = − A ​ A ( 0.1 ) B 2 t ' 2 − 2 B ( 1 , 0 ) B t ' r '   − B ( 0 , 1 ) B t ' 2

t ' = 1 + r ' 2 B 2 A

And from that for the radial velocity u_r= r':

u r ' = − ​ A ( 0 , 1 ) A ( u r 2 + 1 B 2 ) − 2 B ( 1 , 0 ) A u r u r 2 + 1 B 2 − B ( 0 , 1 ) B u r 2

We get for the energy-momentum tensor

u 0 = A u r 2 B 2 + 1

u 1 = u r B 2

T 00 = ( ρ + P ) ( u r 2 B 2 + 1 ) A 2

T 11 = − ( ρ + P ) B 4 u r 2 + P B 2

T 22 = P   Y 2

T 33 = P   Y 2 sin 2 θ

From this and the expressions for G_μν (6…10) get the system equations for ESM, where u(t,r)= u_r

eq00: c 0 ( ρ ( 1 + exp ( f u + f B ) ) + P ( exp ( f u + f B − f A ) ) ) = exp ( − 2 f A ) ( 2 f B ° f Y ° + ( f Y ° ) 2 )       − exp ( − 2 f B ) ( 2 f Y ' '   + 3 ( f Y ' ) 2 − 2 f B ' f Y ' ) + exp ( − 2 f Y ) (25)

eq11: c 0 ( P ( 1 − exp ( f u + f B ) ) − ρ exp ( f u + f B ) ) = exp ( − 2 f A ) ( 2 f A ° f Y ° − 3 ( f Y ° ) 2 − f Y ° ° )       − exp ( − 2 f B ) ( 2 f A ' f Y '   + ( f Y ' ) 2 ) − exp ( − 2 f Y ) (26)

eq01: − c 0 q = 0 = 2 exp ( − 2 f B − f A ) ( f B ° f Y '   + f A ' f Y ° − f Y ° '   + f Y ° f Y ' ) (27)

eq22: c 0 P = − exp ( − 2 f A ) ( f B ° ° + ( f B ° ) 2 − f A ° f B ° + f B ° f Y ° + f Y ° ° + ( f Y ° ) 2 )                 + exp ( − 2 f B ) ( f A ' '   + ( f A ' ) 2 − f A ' f B '   + f A ' f Y '   − f B ' f Y '   + f Y ' '   + ( f Y ' ) 2 ) (28)

eq44: f u ° exp ( f A ) + f u ' exp ( f B ) = − f A ' exp ( f u ) ( 1 + exp ( − 2 f B − 2 f u ) )       − 2 f B ° exp ( f B − f A + f u ) ( 1 + exp ( − 2 f B − 2 f u ) ) − f B ' exp ( f u ) (29)

where A ( t , r ) = exp ( f A ( t , r ) ) , B ( t , r ) = exp ( f B ( t , r ) ) , Y ( t , r ) = exp ( f Y ( t , r ) ) , u ( t , r ) = exp ( f u ( t , r ) ) .

The Oppenheimer-Snyder-Datt (OSD) model of a gravitational collapse was the first exact GR-solution of a dynamic GR system, Joshi [5]. 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 [2] and [5].

In COF the solution space-time becomes (dimensionless)

d s 2 = − d t 2 + v ( t ) 2 1 − k r 2 d r 2 + v ( t ) 2 r 2 ( d θ 2 + sin 2 θ   d φ ) (30)

where the edge of the cloud is in COF

R 1 ( t ) = r 0   v ( t )

and k = 1 / r 0 2 , and r₀ is the initial edge radius, v(0)= 1, density ρ ( t ) = ρ 0 / v ( t ) 3 .

The function v(t) = 0 at the dimensionless time T c o l = π 2 r 0 3 / 2 (see [2] ), where the radius of the cloud R₁(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 ( [2] 45.10):

d t 2 d r 2 = − 1 1 r 2 ( 1 − 1 r 2 )

which gives the relationship

t 2 = t 20 − ( 2 r 2 ( 1 + r 2 3 ) + log ( r 2 − 1 r 2 + 1 ) ) (31)

t 2 → ∞ ,   r 2 = 1 + c o n s t ∗ exp ( − t 2 )

so in the limit of infinite time t₂ 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 [2] [15] and [19].

In [19] the author gives a proof for non-existence of singularities in gravitational collapse in COF under certain conditions.

In the Newtonian approximation, we consider the total energy density of pressure, kinetic energy and gravitational energy,

E d = P s ( ρ ) + ρ v 2 2 − G M ( r ) ρ r (32)

with the equation-of-state (eos) of the ideal gas P s ( ρ ) = ρ   T r which is conserved and therefore stationary under ∂ t and ∂ r , which gives 2 differential equations for the 2 function variables ρ(t,r) and v(t,r).

M ( r ) = 4 π ∫ 0 r ρ ( r x ) r x 2 d r x is the mass within the radius r.

In proper units, i.e. dimensionless the total energy density becomes

E d = ρ   T r + ρ v 2 2 − 3 ρ 2 r ∫ 0 r ρ ( r x ) r x 2 d r x

Into the eos enters the relative temperature T r = k T m n c 2 , relative to the nucleon mass.

The resulting (dimensionless) equations are of order 1 in ρ(t,r) and v(t,r).

equt: ∂ t E d = 0

ρ ° ρ = ( v °   v − 3 ρ 2 r I [ r 2 ρ ° ] ) 3 ρ 2 r I [ r 2 ρ ] − ( T r + v 2 2 ) (33)

equr: ∂ r E d = 0

ρ ' ρ = ( v ' v + 3 ρ 2 r 2 I [ r 2 ρ ] − 3 ρ 2 r 2 ) 3 ρ 2 r I [ r 2 ρ ] − ( T r + v 2 2 ) (34)

with the abbreviation

I [ r 2 ρ ] = ∫ 0 r ρ ( r x ) r x 2 d r x and I [ r 2 ρ ° ] = ∫ 0 r ρ ° ( r x ) r x 2 d r x

and mean density ρ m = I [ r 2 ρ ] r 2 .

These equations can be simplified by using the integrability condition:

∂ t equr = ∂ r equt

3 ρ 2 r I [ r 2 ρ ] − ( T r + v 2 2 ) = 3 ρ 2 2 r ρ ° ( I [ r 2 ρ ° ] r 2 − 1 ) (35)

With this the Equations (33), (34) become

ρ ° m ( 1 + r 2 ) = 1 + 1 3 r ρ ( v 2 ) ° (33a)

ρ ' ( 1 − ρ ° m ) = ρ ° ( ρ   r 2 − r ρ m − 1 2 ρ ( v 2 ) ' ) (34a)

Using (33a) and v 2 < 1 one can show that ρ 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.

In this section we examine the singularity behavior in Schwarzschild space-time gravitational collapse.

We assume that there is a singularity in one or several of the function variable at the time t = t₀, i.e. A(t₀) or B(t₀) or Y(t₀)= 0. For simplicity, by a coordinate shift we can make t0= 0.

We make the ansatz B ( t , r ) = t k exp ( f B r ( r ) ) , t → 0 , Y ( t , r ) = t k exp ( f Y r ( r ) ) , t → 0 ,

A ( t , r ) = exp ( f Y r ( r ) ) , t → 0

that is, B and Y vanish at t = 0 like t^k(k ≥2) and A remains non-zero, this behavior is modeled at the OSD-COF singularity (where k = 2), but any other choice of vanishing space-time function with k = 2 will bring the same result.

For the logarithmic function variables it means f B ( t , r ) = k log ( t ) + f B r ( r ) , t → 0 , f Y ( t , r ) = k log ( t ) + f Y r ( r ) , t → 0 both B and Y have a genuine logarithmic singularity at 0, where they become negative infinite.

Before the singularity is reached, all physical feasibility conditions must be met, i.e. in particular 0 ≤ u [ t , r ] < 1 , v s o u n d = ∂ P ∂ ρ = k 1 γ   ρ γ − 1 < 1 , mass condition M 0 = ∫ 0 r 0 ρ [ t , r ] 4 π r 2   d r , 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)…(29) take the form

eq00: 0 = − c 0 ρ − exp ( − 2 f B ) ( 2 f Y r ' '   + 3 ( f Y r ' ) 2 − 2 f B r ' f Y r '   − exp ( 2 f B r − 2 f Y r ) ) (25s)

eq11: 0 = − c 0 k 1 ρ γ − exp ( − 2 f B ) ( 2 f A r ' f Y r '   + ( f Y r ' ) 2 − exp ( 2 f B r − 2 f Y r ) ) (26s)

eq01: 0 = f B t ° ( f Y r '   + f A r '   − f Y r ' ) = k t k − 1 ( f Y r '   + f A r '   − f Y r ' ) (27s)

eq22: 0 = − c 0 k 1 ρ γ + exp ( − 2 f B ) ( f A r ' '   + ( f A r ' ) 2 − f A r ' f B r '   + f A r ' f Y r '           − f B r ' f Y r '   + f Y r ' '   + ( f Y r ' ) 2 ) (28s)

eq44: 0 = − f u ° exp ( f A ) − 2 f B ° exp ( − f A ) (29s)

where A ( t , r ) = exp ( f A r ( r ) ) , B ( t , r ) = exp ( k ∗ log ( t ) + f B r ( r ) ) , Y ( t , r ) = exp ( k ∗ log ( t ) + f r Y ( r ) ) , u ( t , r ) = exp ( f u t ( t ) + f u r ( r ) ) , ρ ( t , r ) = r h o t ( t ) ∗ r h o r ( r ) : with t-limit behavior a product ansatz in t and r is always possible, where ρ is the density and u is the velocity.

Now, as the density ρ must not blow-up (this would mean an unphysical solution before the Black Hole generation), the brackets with exp ( − 2 f B ) must vanish identically.

This gives f A r ' = 0 , then f B r ' f Y r ' = 0 , and finally f B r ' = 0 = f Y r ' , and from (29s) also f u r = c o n s t 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 [19], that (in COF) the gravitational collapse singularity can arise only with zero pressure, which is physically untenable.

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 R⁰⁰, R¹¹, R⁰¹, R²², and the last is the equation for the radial velocity u from the GR equations-of-motion. The dot represents ∂ t A k and the prime’ represents ∂ r A k . The constant

c 0 = 8 π G c 2 = 3 ( r s s ) 2 ρ s , explicitly and in sun-units, and P is replaced by the equation-of-state P ( ρ ) = f e o s ( ρ ) , the total mass M₀= 1(M_sun).

The chosen space-time range is with edge radius r₀ and final time T₁: t = 0...50(r_s/c),r = 0…10(r_s), r_s = Schwarzschild-radius.

The variable functions are A k = { f A ( t , r ) , f B ( t , r ) , f Y ( t , r ) , ρ ( t , r ) , f u ( t , r ) } , where A ( t , r ) = exp ( f A ( t , r ) ) , B ( t , r ) = exp ( f B ( t , r ) ) , Y ( t , r ) = exp ( f Y ( t , r ) ) , u ( t , r ) = exp ( f u ( t , r ) ) .

It is important to note the highest derivatives of the variable functions, they are:

f A ° , f A ' ' , f B ° ° , f B ' , f Y ° ° , f Y ' ' , ρ , f u ° , f u '

For the variables with the differential degree 2 in t, the t-boundary-condition becomes

t = 0 : A k = f b x [ r ] , ∂ t A k = f D b x [ r ] ,

for degree 1 it becomes

t = 0 : A k = f b x [ r ] ,

and correspondingly in r.

The density ρ 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 h₁= r₀/ndimx in t, and step size h₂= T₁/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 ∑ k , l = 0 n g r a d a k , l exp ( I   k ω t t ) exp ( I   l ω r r ) , with basic angular frequencies ω_t ω_r and I = − 1 , 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.

The author declares no conflicts of interest regarding the publication of this paper.

Helm, J. (2022) New Solutions of Gravitational Collapse in General Relativity and in the Newtonian Limit. Journal of High Energy Physics, Gravitation and Cosmology, 8, 457-485. https://doi.org/10.4236/jhepgc.2022.82034