[Mike Gale (OP)]

Johnson was certainly aware of relativity, but I'm pretty sure she didn't have to use it. The Newtonian equations are more than adequate to get you to the moon and back, but they are diabolically difficult to solve. You have to resort to numerical approximations like Euler's method. Relativity would not make that any easier and the increase in precision would be miniscule.

[Mike Gale (OP)]

I think I know why the new metric is invalid. The derivation of relativistic escape velocity assumes coordinate light speed is invariant and that is only true in the absence of gravity. If you apply the variable speed of light (VSL) hypothesis to the old metric, the equations of motion for the radial free fall case are:
dr/dt=-c*(1-r_s/r)*sqrt(r_s/r)
dT/dt=1-r_s/r
This solves the penetration paradox because everything (including light) stops at the horizon. It should also solve the dark matter dilemma because it predicts slower orbits near the horizon. Another interesting result is that PE=m_oc² at the horizon. Conservation of energy in the extreme!

[Mike Gale (OP)]

Here's an illustration of this result. Notice the v/c profile. It's a good match to a galaxy rotation curve.

[Mike Gale (OP)]

On 2nd thought, I'm not sure if this actually invalidates the new metric. However, the derivation of relativistic escape velocity on which it is based (http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm) uses relativistic mass at escape velocity to calculate the Newtonian potential. That's a dubious step because the mass is not moving in that context. In that case:
PE=GMm_o/r (rather than PE=GMm_o*gamma/r)
KE=m_oc²*(gamma-1)
The SC metric is consistent with these relations (https://en.wikipedia.org/wiki/Kinetic_energy). Jeffrey was right to suspect the validity of that derivation (https://www.thenakedscientists.com/forum/index.php?topic=69595.msg506434#msg506434).

[Mike Gale (OP)]

I'm still intrigued by the VSL interpretation though.

[Mike Gale (OP)]

I don't see any way to notify the author about his mistake on his website. He seems to be quite prolific (and consuming public funds in the process.) Hopefully the peer review process will catch it. In the mean time, be wary of publications by Joseph A. Rybczyk.

[Colin2B]

I don't see any way to notify the author about his mistake on his website. He seems to be quite prolific (and consuming public funds in the process.) Hopefully the peer review process will catch it. In the mean time, be wary of publications by Joseph A. Rybczyk.

There is little point in trying to contact him. Everyone in the trade is well aware of him and his ideas, but most have given up trying to argue with him. That's why you had a few shots across the bows in the original discussion then folks in the know stopped contributing when they saw where you were going. A lot of people have a limited amount of discussion time available so they cherry pick what they want to get involved in.


But you had fun getting there, yes?

[Mike Gale (OP)]

Absolutely. Thanks to all who contributed to my education in this matter.

[Mike Gale (OP)]

Rats. I thought I had it figured out, but if you correct for Rybczyk's error by removing gamma from the Newtonian potential then conservation of energy in the radial free fall case gives:
1-v²/c²=1/(1+r_s/2r)
That leads to this metric:
c²dT=c²dt²/(1+r_s/2r)²-(1+r_s/2r)²dr²-r²d(angle)²
Like my original formulation, this is not a vacuum solution in strong fields. (i.e. The Ricci scalar is zero, but the Ricci tensor is not.) However, this one is:
c²dT=c²dt²/(1+r_s/r)-(1+r_s/r)[dr²-r²d(angle)²
I don't understand why this metric is any less plausible than the Schwarzschild one.

[Mike Gale (OP)]

The scaling factor for a vacuum solution evidently requires a classical approximation of KE, but there are two ways to skin that cat depending on whether you're trying to approximate KE or E-KE. The distinction is moot for weak fields, but one way leads to an event horizon and the other does not. How do you choose?
I think it all boils down to the fact that PE is a classical concept. The ratio r_s/r is really PE/m_oc². That is, the ratio of a classical quantity (PE) to a relativistic one (rest mass energy.) The form of the SC metric depends on how you define PE and that's a sticky wicket because PE is a property of the system, not the test mass. I understand that gauge theory avoids the issue by representing PE in terms of 4-momentum, but I don't think it can distinguish between the two possibilities. Higgs treats it as a scalar field so he should be happy either way. I'm stumped.
Maybe we need to account for the fact that the center of gravity is affected by the presence of the smaller mass. It's a negligible effect if they are sufficiently separated, but not when they get up close and personal. The event horizon may be an artifact of that error.

[Mike Gale (OP)]

I should clarify my point about classical approximations because some GR theorists take exception to the idea that the scaling distance in the SC solution emerges from Newtonian dynamics rather than the other way around. I don't have access to authoritative texts on the subject, but all of the derivations I've seen invoke Newtonian dynamics to define the scaling distance. In either case though, the ratio r_s/r equates to v²/c² for radial free fall and the question is then how to determine v in strong fields because all GR can tell you is how it depends on time dilation.

[Mike Gale (OP)]

I found a derivation that seems to bypass Newton (link below.) The idea is that curvature imposes a pressure that cancels Newtonian gravity so I think it still involves Newton in a roundabout way, What puzzles me is why this metric doesn’t crop up as a possible solution for the integral of G_tt since it satisfies all of the same constraints. He doesn’t show his work for that part so it’s going to take some ciphering to diagnose. I suspect there's a +/- missing in one of the Christoffel symbols.

http://www.google.ca/url?url=http://gfm.cii.fc.ul.pt/events/lecture_series/general_relativity/gfm-general_relativity-lecture4.pdf&rct=j&q=&esrc=s&sa=U&ved=0ahUKEwioyZ6h5JjTAhUnI8AKHf_aDjMQFggUMAA&usg=AFQjCNFEW6NdUweWkTaqo6fe9_zEF-i84Q

[Mike Gale (OP)]

Never mind. I mixed up the time and radial coordinates so I was solving for the case of negative mass. It's interesting that there's no horizon in that case, but I can't imagine any practical applications for that solution. My local expert (Wilson) is aware of a researcher in Montreal who has pondered such a thing, but it evidently didn't produce any new insights.
So once again, the new metric is debunked unless there is any significance to solutions that only approximate the vacuum.

[Mike Gale (OP)]

The only useful take away from all of this is the VSL interpretation. It's not a new idea, but it may be worth a second look because it does solve the penetration paradox. More on that in the original thread: https://www.thenakedscientists.com/forum/index.php?topic=69595.0

[Mike Gale (OP)]

Kevin Brown talks about the alternate metric here: http://mathpages.com/rr/s5-05/5-05.htm

By his account, it's a viable solution except that it predicts the wrong value for the precession of Mercury's orbit.