Drawing Kerr

The Kerr metric is analyzed using strictly geometrical elements. A 4-dimensional surface representing the geometry of the Kerr model is em-bedded into a 5-dimensional flat space. The field strengths of the model are explicitly worked out and understanding of the theory is supported by nu-merous figures. The structure of the field equations is analyzed.


Introduction
The metric of a rotating stellar object was found by Kerr [1] in 1963, albeit in a form that was hard to process. Two years later the metric was rewritten with the help of elliptical-hyperbolical coordinates by Boyer and Lindquist [2]. Enderlein [3] has given a good representation of the elliptical-hyperbolical coordinate system and Krasinski [4] has looked more deeply at the problem. We have analyzed the Kerr model in a series of papers, with a clear presentation in [5] and [6].
Nevertheless, we believe that the Kerr model is still too little understood. Thus, we have decided to present a very detailed presentation, supplemented by numerous drawings.

Basics of the Kerr Metric
The Kerr   we will realize that a geometric meaning can be assigned to the quantities of the metric (2.2). We will discuss this in the following.
First, we want to make clear that the line element x α α = is given by   [3] has depicted this coordinate net as shown in Figure 1.  Evidently r and A are the minor and major axes of the confocal ellipses. Although the coordinate r is not the radial direction of the system, the historical notation is maintained. The meaning of the quasi-polar angle ϑ can be found in Figure 3.
The two vectors are perpendicular and are in each case tangent to the other family of curves. In this way the term radial is defined in this geometry. "Radial" refers to the directions which are specified by the tangents of the hyperbolae.
Facing the 3 rd dimension, is the curvature radius of the circles, the parallels of the ellipsoid of revolution. Finally, one has the curvatures of the coordinate lines These quantities and their derivatives constitute the curvature equations of the system and satisfy the "field equations" 0 R αβ = , where R αβ is the 3-dimensional Ricci for the parabolic-hyperbolic system.

The Kerr Surface
So far, we have associated geometrical meaning to the quantities of the flat metric (2.4), written in elliptic-hyperbolic coordinates. Now we return to the metric (2.2), but we omit the timelike part. To explain the factor S A α δ = in the radial arc element, we extend (2.5) with the extra dimension 0' We will show that (3.1) provides an embedding of a surface into a 5-dimensional flat space, which we will discuss in more detail.
We define the angle ε by where ε has the orientation cw. Evidently tan ε tends to infinity for 0 δ = . The solution to r of this equation is The solution does not have a closed form; however, the integral can be evaluated numerically. If one suppresses the φ-dimension the surface appears as shown in Figure 5. If one extends to the φ-dimension and suppresses the ϑ-dimension, the surface builds up on concentric circles and is even more closely related to Flamm's paraboloid.
We must bear in mind that dr is the increase of the minor axes of the ellipses on the base plane. In contrast,   It is evident that the tangent related to the auxiliary angle χ which points to the next higher point of the surface depends on the angle ϑ . A short calculation would show that one does not arrive at the desired seed metric by using (3.6) either. For our model only the horizontal elliptical slices of the surface are of importance. If one follows the normal vector of the surface along an ellipse, one will discover that this vector also oscillates on its way, because the walls of the surface are round about differently scarped. In order to be able to use the surface, it has to be equipped with an additional structure. On the minor axes of the ellipses the elliptical factor is 1 R a = and the geometry is Schwarzschild-like.
That is where we start: we define a rigging vector in such a way that it coincides with the normal vector at this position, and that it always encloses the same angle with the base plane during its circulation. Then this rigging vector is no longer vertical to the surface and its vertical planes are no longer tangent to the surface. The family of all of these planes-and if one adds the φ-dimension, the family of the 3-dimensional hyperplanes-represents our graphic space, if we assume that we live in such a world. Those hyperplanes are anholonomic, as will be shown.
Now we replace the holonomic differential (3.6) by an anholonomic one that is no longer integrable. We call the family of hyperplanes that are orthogonal to these differentials and that are no longer V 3 -forming physical surface. It is the area of all possible physical observations.
However, the structure of the [r, φ]-part of the surface is simple. On this patch no elliptical properties and hence no anholonomies arise. It corresponds to Flamm's paraboloid of the Schwarzschild geometry. Sharp [7] has searched for such a surface for π 2 ϑ = . However, he did not start from the seed metric, but incorporated rotational parts of the Kerr metric into his computations. Hence, his result differs substantially from ours. From and the surface related to this metric we will call Kerr surface.
The anholonomic construction will be examined in more detail. Figure 6 shows the constellation of the holonomic and anholonomic vectors, namely the  Having explained the structure of the Kerr metric, we return to Equation (3.2) and recognize that the ascent of the integral curve is tan ε Thus, the integral lines are normal to the base plane at H r . For 0 ε = the ascent is tan 0 ε = and the geometry will be flat in the infinite.
We also read from (3.2) that for vanishing eccentricity a of the ellipses, i.e. We have good reasons to interpret as velocity of a freely falling observer in the exterior field of a rotating stellar object and S α as the correlated Lorentz factor of this motion. r A is the ratio of the axes of the ellipses. This means that the velocity of a freely falling observer depends on position in relation to the ellipses.
At the waist of the Kerr surface ( ) waist of the Kerr surface is not only a geometrical limit, but also a physical limit of the model. No object can cross the event horizon H r . In addition, we have to bear in mind that a radial motion in the fields of rotating objects in free fall is not possible. Since frame dragging acts on observers, the motion of an observer will get a circular component. Due to Einstein's law of composition of velocities the velocity of an observer will asymptotically reach the velocity of light before reaching the event horizon. The limit in this case is the ergosphere, as we have shown in [5] [6]. Our point of view is supported by similar circumstances in the Schwarzschild model. In contrast to the claims of Misner, Thorne, and Wheeler [8], we have shown in several papers and in [5] [6] that an observer starting from an arbitrary position can reach the Schwarzschild radius only asymptotically in infinite proper time. We have also shown [9] that the surface of a collapsing star, described by the Schwarzschild interior solution, collapses eternally, reaching the inner horizon of the model asymptotically in infinite proper time. Thus the final state of a collapsing non-rotating stellar object is an ECO (Eternally Collapsing Object) [10].
Thus, if we believe that this geometrical description of a rotating star is a good description of Nature, we have to dismiss the possibility of the formation of rotating black holes. We have supplemented the Kerr solution with an interior solution [11] [12], which has the property of developing into the interior Schwarzschild solution for 0 a = . However, we have made no attempt to implement a collapse for this model and we have not found any effort in the literature in this regard. If such an approach is possible, a RECO (Rotating Eternally Collapsing Object) would be expected.

Curvatures of the Elliptic-Hyperbolic Geometry
So far we have shown that the Kerr model is based on an elliptic-hyperbolic system, endowed with an integral surface with elliptical horizontals, which could be envisaged as an elliptically deformed Flamm's paraboloid. The Boyer-Lindquist coordinate system, with its curved coordinate lines, contributes to Einstein's field equations, which have little to do with the physical content of the model but are incorporated in the connexion coefficients of the physical quantities and must be treated for this reason.
Still suppressing the timelike part of the seed metric, we have to deal with the curvature vectors All the quantities in (4.2) obey the structure 2 d 1 1 0 dr r r + = .
The "field equations" for B, N and C refer to the elliptic-hyperbolic system and drop out from Einstein's field equations. But we cannot omit these quantities, because we need them for the covariant derivative of the physical quantities which we will discuss later on.
We start with the detailed discussion of these quantities. The quantity B is related to the curvature radii E ρ of the ellipses. We know that the curvature B is normal to the ellipses. Thus, we introduce an auxiliary reference system " 0",1", 2" a = and suppress the φ-dimension for the sake of simplifying the problem. In this system B has only one component   The components of From the figure it can be seen that E θ is the angle of ascent of the curvature ra- as can be seen in Figure 9.  Next, we discuss the quantity C, which is related to the curvature radii of the circular parallels of the ellipsoids of revolution, i.e.  This is depicted in Figure 11. Recall that the angle ε is cw and thus 0 C is pointing into the opposite direction of the local extradimension 0 x .
The quantity M still needs to be discussed. It does not belong to the elliptic-hyperbolic system, but to the integral surface. M is related to the curvature radii of the integral lines and has the components M is situated in the direction of the local extradimension as shown in Figure 12.
The quantity is only important for the use of 5-dimensional covariant derivatives. Since we do not use this formalism in this paper, we will not discuss this quantity in detail.
With the help of these one can calculate all the expressions of this section. Now we are prepared to discuss the full seed metric for the Kerr  The metric is written in the original Hilbert notation with index 4, i.e. (++++) and the timelike element is interpreted as an arc on a pseudo circle, shown as a pseudoreal representation in Figure 13. Sometimes another pseudoreal representation is used, a hyperbola instead of a circle. This has the advantage that the infinite can be better visualized, but the hyperbola shows a position-dependent curvature, while the pseudo circle has a constant curvature, including the infinite. Therefore a pseudo circle is also called hyperbola of constant curvature. Unfortunately the pseudoreal representation with hyperbolae misleads some authors to take the hyperbolae literally, but this hinders a geometric explanation of the Kerr

The Field Equations
All the quantities , , , B N C E discussed in the previous section are members of the Ricci-rotation coefficients and satisfy the subequations of Einstein's field equations. For a detailed discussion we again refer to [5] [6]. Here we restrict ourselves to noting some relations. For the elliptic-hyperbolic basic system we ob- The occurrence of the quantity Ω is the second surprise of the elliptic-hyperbolic system. It is a second-rank tensor describing rotational effects which we will meet with the genuine Kerr metric. Evidently the sum of the above equations vanishes. In a covariant form the curvature equations of the elliptic-hyperbolic system can be written as i.e. the field of gravity is coupled to itself, due to the non-linearity of Einstein's field equations. The seed metric does not provide a vacuum solution, it is an auxiliary metric, a forerunner to better explain the Kerr metric. Thus, we will not continue with this model but will turn to study the rotational effects in the next section.

Rotation
Starting with the seed metric (4.15), we define an anholonomic transformation ω is the angular velocity and ωσ the orbital velocity of an observer subjected to frame dragging by the field of a rotating source. The 4-bein system obtained in this way is named system C after Carter and is one of the preferred reference systems attributed to the Kerr model. The coordinate system is oblique-angled, the Carter tetrads are mutually perpendicular by definition.
In contrast, if we perform a Lorentz transformation operating on the tetrad indices of the seed metric, we obtain instead of which differs from the genuine Kerr metric (6.2) only by the position of the gravitational factor S a concerning the last brackets. Although the two metrics are very similar, they have a quite different physical interpretation. While in (6.2) the rotation is inherent, the metric (6.5) is still static, but observers are rotating around the source producing the exterior field. We use the similarities of the metrics to make clearer the structure of the rotational effects of the Kerr metric.
It will be shown that the metrics (6.2) and (6.5) exhibit the same fundamental rotational structures. Since it turns out that (6.5) is much easier to treat, we make use of this property, and we will compare the results for both metrics at the end of the section.
Evidently, it is sufficient to consider the [3,4]-piece of the metric. We make a  The centrifugal force is coupled to the field energy, which is composed of qu-   Thus, one can write in vector notation Evidently, the field equations have a similar structure to the Maxwell equations of electrodynamics. The genuine Kerr Equations (6.14)-(6.20) also have almost the same structure. The similarity of gravitation and electrodynamics was first discovered by Lense and Thirring [13] and Thirring [14] [15] [16] in weak field approximation and treated in general form by Hund [17]. In the last decades this problem was investigated by many authors and called gravito-electromagnetism (GEM).   From this surface a band has to be cut off and the remaining surface has to be matched horizontally to the auxiliary surface of the Kerr metric. R is the radius of the circular arc at the minor axes of the ellipses. All individual 'radial'

Kerr Interior Solution
curves have hyperbolic contributions in their properties. For 0, r A a = = the horizontal ellipses reduce to a distance which is clamped by the common foci of the ellipses. If one now adds the third dimension, these points rotate through ϕ . For π 2 ϑ = a circle emerges. The radius of curvature of the ellipses is zero on this circle and the assigned field strengths are infinitely large. This is the Kerr ring singularity.
Since the junction condition is satisfied, both solutions, the interior and the exterior, match, as can be seen in Figure 17. The Kerr interior shows centrifugal, Coriolis, and gravitational forces, which can be geometrically explained as we have done with the forces of the exterior solution. The interior has a complicated stress-energy-momentum tensor, consisting of gravitational energy, current, and stresses. All that is treated in [5] [6] in detail.

Summary
We have revisited the Kerr model with the methods of tetrads. These are orthogonal local reference systems. The components of the field quantities represented in these systems are measurable quantities and have a clear geometrical or physical meaning. We have visualized the curvature of space with surfaces and have demonstrated how these quantities emanate from geometrical structures with several drawings.
In addition, we have separated the quantities B, C, and C describing the curvature of the elliptic-hyperbolic system from the physical quantities , , E F Ω , and D describing the physical content of the model. We have shown that the field equations of these physical quantities satisfy Maxwell-like equations. Thus, there is hope for a better understanding of the Kerr model.