- The choice of a time marker.
In the [t, r, q, j] coordinates, corresponding to the line element (25), the determinant of the metric is: (34)
which vanishes when r becomes zero. However, in 1924, Eddington [10] showed that the nullity of the metric depended on the coordinates. Let us first return to the initial form (35)
Emphasize the fact that the choice of coordinate system is purely arbitrary, because the metric, solution of the tensor equation (36)
S = 0
is fundamentally invariant under coordinate changes. We decide that particles follow geodesics. The arbitrarily chosen coordinates give a physical meaning to this geometric solution. We can choose x° = ct, c being a constant. But we can choose another frame of reference. It is up to us. The only requirement, for a chosen chronological marker x°, or t, x, is that the metric be asymptotically Euclidean: (37)
or: (38)
as expressed in a Cartesian coordinate system. Recall that a Riemannian metric is Euclidean if one can find a coordinate system where the quadratic form of the line element has constant coefficients. The set of signs is the signature. If this is (+ - - -), it is a Minkowski metric. (39)
being identified to an elementary distance, it seems reasonable to impose that the metric be asymptotically Euclidean "at large distance", regardless of the chosen definition for such a distance (r or r, as above).
The definition of the "cosmic time" or "space marker" remains a completely free choice. Conversely, we cannot modify the proper time s, or more precisely, the time interval Ds between two given points of the manifold, because it is fundamentally coordinate-independent. In addition, it is assumed that particles can move in both directions along a given geodesic.
**
Figure 12: The path of a particle along a given geodesic.
**
The path of a test-particle along a geodesic is a phenomenon. Another geodesic of the manifold is supposed to correspond to an "external observer at rest". But the state of rest depends on the coordinate choice (x°, x1, x2, x3), which is completely arbitrary.
This "external observer" is supposed to be located in a region of the manifold where the metric is Euclidean or quasi-Euclidean, i.e., of the form (37). Then the rest conditions mean that (40)
dx1 = 0
dx2 = 0
dx3 = 0
For such a rest-state observer, any proper time interval identifies to the arbitrarily chosen "cosmic time interval": (41)
Ds = Dx°
...The choice of cosmic time being purely arbitrary, the evolution of the test particle in time depends on this choice. Consider two points A and B on a given geodesic, which is supposed to refer to an external observer. These points are space-time events. On figure 13, the dotted lines are supposed to refer to constant cosmic time x°.
**
Figure 13: An "external observer at rest", "considering" the evolution of a test-particle on a geodesic. Cosmic time x°
**
Consider now another choice x for cosmic time. See figure 14.
**
Figure 14: An "external observer at rest", "considering" the evolution of a test-particle on a geodesic. Cosmic time x **
Let's clarify that the dotted lines are not the paths of photons. Photons travel along special, null geodesics, which are coordinate-invariant.
We still have Ds(O) = Dx° = Dx, but the intervals Ds'(TP) and Ds"(TP) can be very different, even though they refer to the same geodesic, because the pairs (A',B') and (A",B") can differ. Fundamentally, they depend on the chosen time coordinate, or the "time marker".
- Eddington's temporal coordinate transformation and its extended form.
The following coordinate transformation, introduced by Eddington in 1924, illustrating this point, is: (42)
The line element then becomes: (43)
As the term gxx vanishes on the sphere r = Rs, this becomes an infinite redshift surface (as in the classical Schwarzschild line element). The matrix becomes: (44)
whose determinant is: (45)
- r 4 sin2 q
and no longer vanishes, regardless of the value of r. For reasons to be explained later, let us extend this coordinate transformation to: (46)
Expressed in the coordinate system (x, r, q, j), the line element becomes: (47)
whose determinant has the same form (44). Note that Eddington's coordinate transformation corresponds to the value d = -1. We study the geodesics using the Lagrange equations, based on the function: (48)
with:
In addition, from the expression of the line element, we have classically for material particles (ds ≠ 0): (49)
A Lagrange equation gives: (50)
Consider the planar geodesic q = p/2, which gives: (51)
Along a geodesic, with respect to the proper time s, the evolution of j is monotonic. Another Lagrange equation gives: (52)
that is: (53)
Combining with (49), surprisingly, d disappears: (54)
Note that if dr = 0 (zero velocity) when r tends to infinity, this corresponds to l = 1. When r tends to infinity, according to (53): (55)
If l ≥ 1, when r tends to infinity, we obtain: (56)
with
we obtain (57)
In the frame [r, j], we recover, for non-null geodesics (ds ≠ 0), the classical differential expression: (58)
which provides the patterns of figures 7, 8, and 9. We can now define a new cosmic time by: (59)
x = ct
...The line element (43) remains asymptotically Euclidean. At "large distance", the proper time Ds of a stationary observer identifies to the interval Dt.
- Time intervals for radial paths.
We can calculate the time interval Dt = Dx/c of a massive particle following a geodesic, from the differential equation: (60)
For the "radial geodesics" (h = 0): (61)
Near the Schwarzschild sphere, we obtain: (62)
l = 1 corresponds to a test-particle whose velocity tends to zero at infinity.
Consider this particular case: (63)
According to (54)
n = -1 corresponds to paths (dr < 0).
n = +1 corresponds to paths (dr > 0).
...Note that the particular Eddington coordinate transformation corresponds (for r ≥ Rs) to d = +1. When we calculate the radial travel time Dt of a test-particle, with respect to this new cosmic time, we find that this time depends on the direction of motion and the sign of d, that is, the product dn. When it is positive, the travel time of a test-particle along a radial geodesic (r ≥ Rs) is finite. When it is negative, this travel time becomes infinite.
...As a first consequence, if applied to the spherical symmetry black hole model, the Eddington coordinate transformation gives a finite free-fall time Dt. When r = Rs, the particle's velocity becomes: (64)
A test-particle, falling towards the Schwarzschild sphere, reaches it with the speed c.
- Speed of light.
Photons travel along null geodesics, corresponding to: (65)
Consider the speed: (66)
According to (65), we obtain: (67)
When r tends to infinity, vj tends to ±c.
When dr < 0, we have n < 1. Then, when r = Rs for the paths (dr < 0): (68)
When a test-particle falls towards the Schwarzschild sphere, along a radial path, it reaches it with the speed of light. In summary: (69)
(70)
The speed of light is different depending on whether we consider paths (dr > 0) or (dr < 0).
- Frame-dragging effect.
Consider the Kerr metric: (71)
where r is a spatial coordinate different from the one defined above. We simply reproduce equation 7.110 from reference [1]. Calculate the speed of the photon (ds = 0) for movements tangent to circles (q = p/2, r = constant). We find: (72)
that is, two distinct values. This corresponds to an azimuthal dragging and is a property of the Kerr metric. According to reference [1], 7.7, "The Kerr solution and rotation", we read:
An interesting physical effect arises from the rotational nature of the Kerr solution; a body moving along a geodesic experiences a force proportional to the parameter a, reminiscent of a Coriolis force. In vague terms, we can think that the rotating source "drags" the space around it. In a Machian sense, the sources "confront" the Lorentzian boundary conditions at infinity to establish a local inertial frame.
Rephrased in terms of Eddington coordinates, the black hole, considered as the source of the field, induces a radial frame-dragging.
- Black hole and white hole.
In section 4, we suggested a new interpretation of the Schwarzschild geometry where the Schwarzschild sphere, see figure 9, behaves like a throat connecting two "half-space-time folds". We can imagine a similar structure combining the following two Schwarzschild geometries: (73)
(74)
These two are derived from (43), the first expression (73) corresponding to d = -1 and the second (74) to d = +1. The connection poses no problem, as d does not appear in the calculation of the [r, j] representation of the geodesics. See equation (58). We obtain a pair "black hole - white hole", without "central singularity". Matter can enter the black hole, but cannot exit. On the other hand, matter can escape from the white hole, but cannot enter. The transit time is finite in one direction and infinite in the other. Calculated with the new cosmic time x, the finite transit time is similar to that calculated with the proper time s. For radial paths: (75)
This time is very short. As shown in this article, the black hole model relies on a particular choice of coordinates, notably the cosmic time. As indicated in section 6, the choice of the time marker is purely arbitrary. The classical choice gives a quasi-stationary system, where the matter falling into the black hole appears "frozen in time" from the perspective of an external observer. However, this article shows that another choice of time marker, derived from Eddington's idea, "unfreezes" the process. From this perspective, black holes, or the black hole-white hole pairs, cannot exist as permanent objects, since they could swallow dozens of solar masses per millisecond. There remains an open question:
- What happens when a neutron star exceeds its stability limit?
- Representation space.
Before trying to present an alternative model project, a few words about what we could call "representation spaces". At the beginning of the article, we studied a 2D surface defined by its line element. It turned out possible to immerse this surface in R3, which gave us an isometric representation of this geometric object. By the way, we mentioned a representation [r, j].
It is not possible to give an obvious representation of a four-dimensional hypersurface, because we cannot draw it or show figures of it. However, the hypersurface can be represented in many representation spaces, corresponding to various choices of coordinates, because the object is fundamentally invariant under coordinate changes. For example, we can introduce the change (6). The line element then becomes: (76)
for r > 0
and: (77)
for r < 0.
The "radial" geodesics (for example q = p/2, dj = 0) converge towards the geometric center O of the system (in this particular representation). This point is comparable to a "hypercone point". A symmetry with respect to a point in a 3D space is a P-symmetry.
In this frame [t, r, q, j], the Schwarzschild line element is P-symmetric. It is also time-independent (invariant under temporal translation, i.e., corresponding to a stationary state) and T-symmetric, invariant under:
t → -t
Rephrased in terms of Eddington's coordinates, the black hole, considered as the source of the field, induces a radial frame-dragging.
- Black hole and white fountain.
In section 4, we have suggested a new interpretation of the Schwarzschild geometry where the Schwarzschild sphere, see figure 9, behaves like a throat sphere, linking two "half space-time folds". We can imagine a similar structure, combining the two following Schwarzschild geometries: (73)
(74)
These two are derived from (43), the first line element (73) corresponding to d = -1 and the second one (74) to d = +1. The linking causes no problem, since d does not appear in the computation of the [r, j] representation of geodesics. See equation (58). We obtain a pair "black hole - white fountain", without "central singularity". Matter can enter the black hole, but cannot exit. On the other hand, matter can escape from the white fountain, but cannot enter. The transit time is finite in one direction, and infinite in the other. Calculated with the new cosmic time x, the finite transit time is similar to that calculated with proper time s. For radial paths: (75)
This time is very short. As shown in this paper, the black hole model is based on a particular choice of coordinates, especially of the cosmic time. As pointed out in section 6, the choice of the time marker is purely arbitrary. The classical one gives a quasi-steady-state system, where the fall of matter, poured into the black hole, is "frozen in time", with respect to an external observer. But this paper shows that another choice of the time marker, derived from Eddington's idea, "defrosts" the process. From this point of view, black holes, or black hole-white fountain pairs, cannot exist as permanent objects, for they could swallow dozens of solar masses per millisecond. There remains therefore an open question:
- What happens when a neutron star exceeds its stability limit?
- Representation space.
Before trying to present an alternative model, a few words about what we could call "representation spaces". At the beginning of the paper, we studied a 2D surface, defined by its line element. It turned out to be possible to embed this surface in R3, which gave us an isometric representation of this geometric object. By the way, we mentioned a [r, j] representation.
It is not possible to give an evident representation of a four-dimensional hypersurface, since we cannot draw it or show figures. However, the hypersurface can be represented in many representation spaces, corresponding to various coordinate choices, because the object is fundamentally coordinate-invariant. For example, we can introduce the change (6). Then the line element becomes: (76)
for r > 0
and: (77)
for r < 0.
"Radial" geodesics (for example q = p/2, dj = 0) converge towards the geometrical center O of the system (in this particular representation). This point is comparable to an "hyperconic point". A symmetry with respect to a point in a 3D space is a P-symmetry.
In this [t, r, q, j] frame, the Schwarzschild line element is P-symmetric. It is also time-independent (invariant under time translation, i.e. corresponds to a steady state) and T-symmetric, invariant under:
t → -t