Interior Schwarzschild Solution

The formation and evolution of a black hole.

Most people that have studied black holes to any extent are familiar with the exterior vacuum Schwarzschild solution. The interior Schwarzschild solution (for a body with uniform density) shown below is less well known. (See references here and here )

$\frac{d\tau}{dt} =\frac{1}{2}\left({3}\sqrt{1-\frac{2GM}{c^2\,R_o}}-\sqrt{1-\frac{2GM\,r^2}{c^2\,R_o^3}}\,\,\right)$

Here Ro is the surface radius of the solid body, d(tau) is the rate that proper time advances and dt is the rate that coordinate time advances (time measured by a clock at infinity). The equation is useful for studying the time and space relationships of a solid spatially extended body such as the Earth. It is also useful for studying what happens as a black hole forms.

By substituting the Schwarzschild radius the equation can be rewritten as:

$\frac{d\tau}{dt} = {3 \over 2}\left({1-{R_s \over R_o}\right)^{\frac{1}{2}}-{1\over 2}\left(1-{R_s r^2 \over R_o^3}\,\right)^{\frac{1}{2}}$

The radius where the proper time tau is measured is denoted by r.

The proper time of a clock at the centre of a solid body where r=0 is therefore:

$d\tau = dt\left({3 \over 2}\left({1-{R_s \over R_o}\right)^{\frac{1}{2}}-{1\over 2}\right)$

By using the above equation it is easy to see that when a massive body contracts to a radius of Ro = 9/8 Rs, the proper time of the central clock goes to zero. As the body continues to contract towards Ro = Rs the proper time of the central clock becomes negative. This is in effect the local time and it is clearly running in the opposite direction to coordinate time. The implication of this is that time reverses inside the collapsing body (that has not yet formed a classical black hole because at this point Ro is still greater than Rs.) and time reversal implies that gravity becomes repulsive rather than attractive. Particle fall up rather than down. Therefore it is impossible for the body to collapse further to a singularity of infinite density at the centre. It should also be noted that this negative proper time remains real despite the presence of the square root factor in the equation. This reversal of gravitational acceleration below the Schwarzschild radius as a black hole forms is in agreement with all the other arguments presented in this paper, even though none of the other arguments depended on assuming the reversal of proper time. Thus it is claimed that the arguments in this paper are self consistent and elsewhere it is shown that this new physical interpretation of General Relativity and more specifically the Schwarzschild solutions, removes all the remaining paradoxical contradictions from the traditional interpretation.

Additional observations.

Normally the Schwarzschild radius is synonymous with the event horizon but it should be noted that this is not always the case if we define event horizon as the boundary between positive and negative proper time (or in some cases the boundary between real and imaginary proper time). Essentially the event horizon is where proper time goes to zero in either case and perhaps zero time horizon might be a better name for event horizon to avoid confusion with the Schwarzschild radius. Any massive body has a Schwarzschild radius as does the Earth for example and without being rude so does the body of the reader. It is a fixed mathematical radius for any given fixed mass, and remains constant even for a gravitationally collapsing body if the total mass (or energy) of the body remains constant during the collapse. The zero time horizon on the other hand appears at the centre of the body when it has collapsed to a radius of Ro = 9/8 Rs. As the body continues to collapse the horizon expands outwards and asymptotically approaches the Schwarzschild radius. Particles below the horizon gravitate outwards towards the expanding horizon while particles outside the Schwarzschild radius continue to fall inwards creating two shells of matter asymptotically approaching each other at R=2m. To all outward appearances to an observer any distance away from the outer shell this body will appear like a classic black hole.

A possible objection is that real bodies do not usually have uniform density. To partially address this objection, it can be shown that for a body of monotonically increasing density the same basic principles apply. For this type of body the mass enclosed within a given radius is a function of r^2 rather the uniform density function of r^3. By making this assumption the interior Schwarzschild solution for a body of monotonically increasing density is:

$\frac{d\tau}{dt} = {3 \over 2}\left({1-{R_s \over R_o}\right)^{\frac{1}{2}}-{1\over 2}\left(1-{R_s r \over R_o^3}\,\right)^{\frac{1}{2}}$

For a clock at the centre where r=0 the equation still reduces to:

$d\tau = dt\left({3 \over 2}\left({1-{R_s \over R_o}\right)^{\frac{1}{2}}-{1\over 2}\right)$

as before. This is perhaps better demonstrated by looking at an alternative formulation of the alternative Interior Schwarzschild solution in terms of density which is:

$\frac{d\tau}{dt} = {3 \over 2}\left({1-{R_s \over R_o}\right)^{\frac{1}{2}}-{1\over 2}\left(1-{R_s \over r} \frac{p(4/3)\pi \,r^3 \over p_o(4/3)\pi \, R_o^3}\,\right)^{\frac{1}{2}}$

where p is the density inside radius r and po is the total density.

The equation can be restated in a more intuitive form as:

Since r cubed diminishes more rapidly than r it is reasonable to assume that as r tends towards zero the final term of the equation tends to = for any enclosed density other than infinite density.

Related links: See post 23 this physicsforums thread and post 8 of this thread for the full form of the interior solution including radial motion and orbital rotation.

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) 2008 KevPegrume