As I showed, the Einstein calculation should yield a radial velocity which can be written as a function of r.
Because of the similarity of my solution to the same problem, I can guarantee that my calculation will yield a radial velocity such that
where that function “F” is exactly the same function obtained from Einstein's theory. This means that I can write
Since that square root factor is less than one, it should be clear that the radial velocity obtained from my calculation (for the same radius: i.e., the same path) must be slightly larger than the radial velocity obtained from their calculation. (This can also be interpreted to imply they should have started with a larger radial velocity to begin with; if they wanted the thing to be where their plots show it.)
Right okay, here's where I had made a hidden assumption in my mind, and right now I'm backtracking a bit, trying to convince myself of your thoughts.
The assumption I had made was that, once a measurement of the speed of the satellite has been made, your expectation of higher radial velocity implied a larger path. So what I had in my mind when writing the previous post was that your solution would yield the expectations of the satellite moving along a path that goes farther away from the sun than the path of Schwarzschild's expectations.
So now I'm focused on that bit, I see the point is that the radial velocity is a function of "r", i.e. the radial velocity is higher at any given point of the orbit, and that is seen to essentially imply a shorter turnaround time for the same path?( (i.e. the satellite is just plain moving faster)
I can see that in that case, making a measurement of velocity and figuring out the radius from that, would give Schwarzschild the impression that the satellite is farther away than you'd figure it is. And yes in that case they would see unexpectedly large slow-down.
These brings up few questions to my mind, like whether the conservation of angular momentum is still valid in your solution and where did we get it when it was used in the derivation of this result. I'm not really sure because I'm not very good at reading the actual mechanisms of those equations :I
Well at any rate, it would be quite interesting to get in touch with someone who knows the measured data of Pioneer and could analyze the situation in terms of your solution. Should be fairly trivial since they don't even have to know where your equation came from...
Well yeah, the dark matter thing feels very much like a text-book example of how Kuhn describes things before a paradigm shift. I.e. adding complexity in terms of old terminology, until a new terminology washes it away as unnecessary.
Indeed. That would be interesting.
Btw, I don't know if you are aware of it, but there are attempts to modify the laws of gravity to explain the galaxy rotation problem, i.e. MOND and later TeVes;
I guess there has been few different flavours to it from day one (it's mostly based on guessing and error and trial), and it seems to be modifying somewhat different parameters than what your analysis shows (at least as far as I can understand that), but maybe you can tell whether they happen to be landing onto similar modifications as your analysis reveals, in any sense at all...
Well, it's interesting that they are saying that, because in my mind at least Hawking has been talking of things in terms of ontological correctedness a lot (like concerning himself with the possibility of time travel due to relativity etc), but at the same time I realize a lot of that is something he does with his "popular science" hat on, i.e. over simplification of things, and saying things that sell to general public. But, if they are now starting to say things like "it may not make sense to talk about what reality actually is.", my only problem is with the word "may" in there
Anyway, I do see physicists occasionally make comments about that issue, which is always nice. And perhaps the time would be ripe for Hawking to pay attention to your analysis now. If it was just made clear from the word go that your analysis is exactly concerning the issue non-sensibility of talking about reality itself, and the sensibility of talking about valid models.
Right now, judging by the recent discussion between me and Qfwfq, it seems physicists have somewhat different levels of understanding how deep that issue really goes. Qfwfq has been saying for a long time that he understands what I mean by reality being ontologically unknown, but he was still assuming all that time that the reality must bear some likeness to our everyday perception, which is essentially where all his comments about phenomological observations were coming from (and in my mind most of his misinterpretation of our commentary).
I think in his mind, allowing for the possibility that the everyday perception is just an interpretation of something entirely unknown (i.e. that there may not be any likeness there), is exactly the same as letting go of the only way to get any information about reality, so the first reaction to your analysis would be either "1. you are assuming idealism" or "2. you cannot be in connection with physics at all"
Also, when I say something like "no, there may not be any likeness", I know exactly how he reads that; he immediately starts to think about something like "but it's not possible to just interpret my kitchen table away".
Well, I'll be writing a response to the other thread, and as I said, I believe I know how he is thinking about this, and I'm just trying very hard to think about what would be a good way to communicate this issue in ways that would get around those few mental blocks that almost everyone seems to collide with head-on... The key thing here seems to be that there exists an interpretation to your analysis, that most physicists intuitively make, and it makes them convinced that they read you correctly, when they don't. Certainly a short version of it would be "they are not working exclusively with your definitions", but it would be nice to understand why exactly they find it so incredibly impossible to do that, or what are the things that spark them to make the wrong interpretation, so that could perhaps be avoided in the communication...
Easier said than done, I know!