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## Main Question or Discussion Point

In another thread, Nerid mentions

The Magnetospheric Eternally Collapsing Object (MECO) Model of Galactic Black Hole Candidates and Active Galactic Nuclei.

also at http://arxiv.org/abs/astro-ph/0602453

This apparently has been published, though I'm not positive if where it was published was a peer-reviewed journal.

I thought I would comment on it from a GR perspective since there appears to be some interest in the topic.

[add]

True statement:

The velocity of an object falling into a black hole approaches 'c' relative to an observer hovering, with rockets, a constant Schwarzschild 'r' coordinate. As the hovering observer getrs closer and closer to the event horizon, the velocity of the falling object get's closer and closer to 'c'.

False statement:

The geodisic of an object falling into a black hole is null. One simply cannot infer that the geodesic is null from the above true statements, even though the limiting velocity is 'c'.

[end add]

The nature of the Lorentz interval between two points on the geodesic (time-like, null, or space-like) can be calculated in any coordinate system where the metric is not singular at the horizon. This excludes standard Schwarzschild coordinates if the calculation is done rigorously, as g_tt = 0 and g_rr is infinite. One can show however, that geodesics have the expected timelike behavior in the limit as one approaches or leaves the event horizon even in Schwarzschild coordinates.

In addition, one can use any one of several well-behaved coordinate systems such as Eddington-Finklestein or Krusak-Szerkes to eliminate the coordinate singularity at the event horizon entirely, and show that the trajectory is purely timelike, as expected.

See for instance MTW, "Gravitation", pg 824 for a solution for r parameterized in terms of the arbitrary parameter n for an object falling at rest from a starting height r_max into a black hole.

r = (1/2) r_max (1 + cos(n))

tau = sqrt(r_max^3/8M)*(n+sin(n))

The region of interest is 0<n<Pi, during which it can be seen that proper time tau is finite, and increasing, and r is decreasing from r_max to zero.

The fact that the proper time exists shows that the geodesic is timelike, though this can be confirmed by detailed caluclations.

I've omitted the solution for the Schwarzschild 't' coordinate for the geodesic as it is rather difficult to type in - only if there is serioius interest will I go into more detail.

[add]

Note that as E_0 is constant for an object following a geodesic in the Schwarzschild metric, g_00 dt/dtau will be constant. This should allow anyone sufficiently interested to solve for the expression for t, though it would certainly be easier just to look it up in the reference.

The Magnetospheric Eternally Collapsing Object (MECO) Model of Galactic Black Hole Candidates and Active Galactic Nuclei.

also at http://arxiv.org/abs/astro-ph/0602453

This apparently has been published, though I'm not positive if where it was published was a peer-reviewed journal.

I thought I would comment on it from a GR perspective since there appears to be some interest in the topic.

This quote strikes me as being seriously confused. Horizon crossing geodiscs for infalling matter of positive rest mass are definitely not null geodesics!For black holes to exist, gravity must be able to do what no other force of nature can do; namely, to accelerate the physical three-speed of a finite mass to exactly the speed of light. But this means that horizon crossing geodesics, in realistic coordinates, would become null rather than timelike.

[add]

True statement:

The velocity of an object falling into a black hole approaches 'c' relative to an observer hovering, with rockets, a constant Schwarzschild 'r' coordinate. As the hovering observer getrs closer and closer to the event horizon, the velocity of the falling object get's closer and closer to 'c'.

False statement:

The geodisic of an object falling into a black hole is null. One simply cannot infer that the geodesic is null from the above true statements, even though the limiting velocity is 'c'.

[end add]

The nature of the Lorentz interval between two points on the geodesic (time-like, null, or space-like) can be calculated in any coordinate system where the metric is not singular at the horizon. This excludes standard Schwarzschild coordinates if the calculation is done rigorously, as g_tt = 0 and g_rr is infinite. One can show however, that geodesics have the expected timelike behavior in the limit as one approaches or leaves the event horizon even in Schwarzschild coordinates.

In addition, one can use any one of several well-behaved coordinate systems such as Eddington-Finklestein or Krusak-Szerkes to eliminate the coordinate singularity at the event horizon entirely, and show that the trajectory is purely timelike, as expected.

See for instance MTW, "Gravitation", pg 824 for a solution for r parameterized in terms of the arbitrary parameter n for an object falling at rest from a starting height r_max into a black hole.

r = (1/2) r_max (1 + cos(n))

tau = sqrt(r_max^3/8M)*(n+sin(n))

The region of interest is 0<n<Pi, during which it can be seen that proper time tau is finite, and increasing, and r is decreasing from r_max to zero.

The fact that the proper time exists shows that the geodesic is timelike, though this can be confirmed by detailed caluclations.

I've omitted the solution for the Schwarzschild 't' coordinate for the geodesic as it is rather difficult to type in - only if there is serioius interest will I go into more detail.

[add]

Note that as E_0 is constant for an object following a geodesic in the Schwarzschild metric, g_00 dt/dtau will be constant. This should allow anyone sufficiently interested to solve for the expression for t, though it would certainly be easier just to look it up in the reference.

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