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Static, spherically symmetric bodies are studied by the use of flat space-time theory of gravitation. In empty space a singularity at a Euclidean distance from the centre can exist. But the radius of this singular sphere is smaller than the radius of the body. Hence, there is no event horizon, i.e. black holes do not exist. Escape of energy and information is possible. Flat space-time theory of gravitation and quantum mechanics do not contradict to one another.

In this paper the theory of gravitation in flat space-time [

In a recent article of Hawking [

We shortly summarize the theory of gravitation in flat space-time. Let

A special case is the pseudo-Euclidean metric where

Put

Define

The Lagrangian for the gravitational potentials

where the bar / denotes the covariant derivative relative to the metric (2.1). Let k be the gravitational constant. Put

and define the differential operator of order two

Then, the field equations for the potentials from the Lagrangian (2.3) are

where

It is worth to mention that the energy-momentum of the gravitational field

The equations of motion for matter are

The derivative of these results can be found in the articles [

Here, we follow along the lines of paper [

We set

The potential are

A body at rest has the four-velocity

The matter tensor can be written in the form

where

We use the abbreviation

and define

Then the energy-momentum tensor of the gravitational field is

The differential equations for the gravitational field have the form

In addition, there are boundary conditions for _{0} denote the boundary of the spherically symmetric body which gives

Put

then we get the gravitational mass

Elementary calculations give by a suitable linear combination of the Equations (3.6) and the boundary conditions

It is worth to mention that static, spherically, symmetric bodies with pressure equal to zero do not exist. It holds (see [

The derivation of all these results can be found in the article [

The gravitational field in the exterior of the body, i.e.,

with

It follows from (4.1) by the substitution

Elementary calculations give the asymptotic solutions, i.e. for

where the parameter A must be fixed by the use of the interior solution. It is worth to mention that no additional parameter arises by the use of general relativity.

Numerical methods are used to get the solutions in the exterior of the body for different parameters A. For small values

There are different types of solutions:

1) regular solutions, i.e. for all

2) singular solutions, i.e. there exist a critical value

mention that this singularity arises at the Euclidean distance r_{c} from the centre of the body. It is real and not a property of the coordinate system as by the use of the general theory of relativity which implies an event horizon.

All these results can be found in the article [

We will now study the solutions in case (2) in the neighbourhood of

With suitable constants

with positive constant A_{0}, B_{0}, C_{0}. We get by the substitution of (5.1) and (5.2) into the Equation (4.4c)

implying

The differential Equation (4.4b) gives by (5.1) and (5.2)

without fixing

The relations (5.3a) and (5.3c) imply

Hence we have

It follows from (5.4) that the inequality (5.3b) is fulfilled for

This inequality is in agreement with (5.4) and

Summarizing, we have

Hence, β and γ are always positive whereas

for

We get by the use of (5.2) and (5.5) that the solution cannot be continued to

symmetric bodies with radius

but in analogy to Rosen’s biometric theory of gravitation [

We will now study a spherically symmetric body with radius

We get from the relations (5.1) and (5.2) by the use of (4.3) for

Therefore, it follows from (5.7) for

Hence, relation (3.8) yields for

It follows from (5.8)

We get from this inequality

The condition P = 0 implies that the mass M = 0 by virtue of

implying

The equation of state

gives

in contradiction to (5.10).

Hence, there exists no static, spherically symmetric body with Euclidean radius

Summarizing, we can state that static, spherically symmetric bodies have no singular solutions. There exists no event horizon, i.e. black holes do not exist.

This result is in agreement with quantum mechanics in contrast to black holes of the general theory of relativity.

The differential equations for a collapsing, spherically symmetric body are given in Chapter III of the book [