Aq Id, my dear fellow, you seem to be becoming quite paranoid and seeing creationists under the bed!
If you think Fednis is a crank, or an agent of "SPAZ" (love the concept, by the way!), please do me a big favour and read the exchange he or she and I have had on the virtual particles thread I started yesterday.
Personally, I think this is evidence that he or she is quite clearly a practising physical scientist, who can communicate clearly and authoritatively, at any rate in their own field.
Thanks for that feedback, exchem. Lately I've been willing to forego overreacting to posters I don't know. If Fednis really has training, then it means I should be able to arrive at some common ground, even if only by definition of terms. My rationale went like this: first, I've seen RJ's posts quite a bit, and it never dawned on me that he was a crank until now. Then when I saw RJ summoning Fednis to the rescue, my SPAZometer started pegging. That's one of the benefits of having "sleeper" sock puppets that can be summoned to shore up the crank argument. Having seen how hard it has been for the mods to root out and put an end to the SPAZzes, and how manic they are, it seems a worthy cause for me to risk being abrasive to people I would ordinarily have only wanted to reach out to.
Again, thanks. I value your opinions a lot, first of all because you knew who Josquin des Prez was--which may seem a little silly to you, but these are the bits of common ground that serve to "identify friend or foe". Plus, it speaks to another world we don't get to talk about much here, one that's just freaking awesome.
Ok, I'm glad you responded; I know you're an intellectually honest fellow, so I assume you'll be able to give me a no-nonsense answer here. And even if you think I'm a crank, please don't waste time saying as much or posing Socratic questions. Just point out the error in the following reasoning:
1. Your source says any accelerometer in an inertial frame will read zero.
2. Your source says any two inertial frames are in a constant state of rectilinear motion with respect to one another.
3. Since the Earth is a single object, all parts of it are in a constant state of rectilinear motion with respect to all other parts of it (at least in Newtonian mechanics, which is the context we're dealing with right now).
4. From 3, any point on the surface of the Earth is in a constant state of rectilinear motion with respect to the center of the Earth.
5. From 2 and 4, a frame centered on the center of the Earth is inertial iff a frame centered on any point on the surface of the Earth is inertial.
6. You claim the a frame centered on the center of the Earth is inertial.
7. By 5 and 6, you claim that a frame centered on any point on the surface of the Earth is inertial.
8. By 7 and 1, you claim any accelerometer at any point on the surface of the Earth will read zero. This is false.
As far as I can tell, this proves that either your claim is wrong or you're misinterpreting your sources. Where is my mistake?
Your mistake is trying too hard to nail me with "any" accelerometer, since I did not clarify that we calibrate our accelerometers against g. But first this:
Based on the positive character reference from exchem I'm going to zeroize my SPAZometer reading of you. Let me restate a few things that set it off and we can put this behind us. First, I saw what I thought was an attempt to arrive at relative acceleration, by unnecessary and superficial application of the Hamiltonian. One of the markers of SPAZdom is to wave jargon around like this for no purpose but to try to hack the speech of science itself in order to get traction in a thread, so they can set their bait traps out. My point was that you only needed to state the allegations directly, something like this:
$$F\quad =\quad ma\quad =\quad \frac { G{ m }_{ \oplus }{ m }_{ 0 } }{ { r }^{ 2 } } \\ \\ { \therefore \quad a }_{ rel }\quad \equiv \quad { a }_{ 0 }\quad -\quad (-{ a }_{ \oplus })\quad =\quad \frac { G({ m }_{ \oplus }+{ m }_{ 0 }) }{ { r }^{ 2 } } \\ \\ r\quad =\quad \frac { 1 }{ 2 } \quad { a }_{ rel }\quad { t }_{ 0 }^{ 2 }\\ \\ \therefore \quad { t }_{ 0 }\quad =\quad \sqrt { \frac { 2r }{ { a }_{ rel } } } =\quad \sqrt { \frac { 2h{ r }^{ 3 } }{ G({ m }_{ \oplus }+{ m }_{ 0 }) } } $$
Now we just need to decide whether this is valid. I say no, that the notion of relative acceleration is an artifact of a non-inertial reference frame. To arrive at the truth of the matter, we should be able to dispense with the confusion over what constitutes an inertial or non inertial frame, and we should be able to test the truth of which opinion is correct by using the correct coordinate transformation that ports the vectors between Galileo's frame and the thing I'm labelling non-inertial, which I call RJ-space. I will try to avoid the unnecessary calculations, although I might come up with a less tedious way to do it. I just prefer the transform approach, since it seems to be the main deficit of cranks and SPAZzes, and a little education might go a long way to disarming them the minute they pop up.
So for now I'll simply rely on the more scholarly method, which rejects vector math done outside of inertial reference frames as something "warped" by the nature of the coordinate transformations involved. That way, at least the educated people will follow my logic. And it gives RJ an incentive to feast on the mathematical analysis he's laid on his plate, should he be a glutton for punishment.
First, let's restate the operative assumptions. RJ is assuming the Earth is a perfect sphere of uniform density, free from any rotational or orbital acceleration.
Now let's pick a reference frame. The acceleration at the center of the Earth is ~0.05%g with an additional ~0.3%g at the equator. RJ is neglecting these. We begin by placing a Cartesian coordinate system at the center of an Earth-sized sphere. We pick the point where the z-axis intersects the sphere and we designate this the position of impact.
A frame at the center of the sphere is inertial because, in RJ's model, it is not accelerating. An accelerometer placed there reads zero. An accelerometer placed on the surface reads -1g in the z-direction. We calibrate the accelerometer to remove the -1g bias, and thereafter it serves as a benchmark, against which we can validate all other reference frames. We can now place the accelerometer anywhere on the surface of our spherical Earth and it will continue to read zero. Further, we can place the accelerometer in constant rectilinear motion and after a brief spike, the accelerometer will read zero. Thus we find Galileo is in our inertial reference frame, at a distance h from our origin directly below him at the surface of the sphere, which is the vertical height from which he drops the object. We run the experiment and conclude that the time it takes from release until impact is
$${ t }_{ 0 }\quad =\quad \sqrt { \frac { 2h }{ g } } $$
But RJ is not in an inertial reference frame. He has chosen as his origin the center of mass of the Earth-object system. When we place our accelerometer at the origin of RJ-space, it at first reads zero, but then, as the experiment proceeds, his origin begins to accelerate. The accelerometer reads nonzero. Alternatively, we can say RJ-space is non-inertial since it is accelerating against a known inertial reference, the center of our idealized spherical Earth.
Conclusion: the relative acceleration method of calculating the time of impact is invalid since it relies on a non-inertial reference frame. Galileo was right and RJ was wrong.
Also, I really like the idea of "creationists under the bed." Sounds likes a Goosebumps story, or maybe a Doctor Who episode.
