James
You say; "I do not see math as being an abstraction".
I'm suggesting that our long evolved assumption that maths is naturally reality is itself what has kept us from the truth. Points and lines are not real but abstractions. Descartres xyz co-ordinates were conceived as 'describing' a body, and Einstein correctly specified them as 'rigidly attached to a body'. But we forgot. Moving points invalidate geometry and time invalidated 4D space. But we abstracted all to numbers, and again forgot the need to reverse the process. I'm just approaching it the way Einstein did, by thought, but with 100 years better information.
In logic, just one tiniest wrong initial assumption can invalidate a whole theory and 100 years of physics. Maths is one way to describe reality, but easy to trust too much or abuse. I suggest Charles Dodgeson was right, as soon as we forget maths is abstraction and forget to 'remormalise' it we end up in Alice's wonderland wondering why physics is in two giant halves that won't fit together, twins are older than each other and space is full of anomalies.
Einstein said "we won't solve our problems with the same kind of thinking that created them." Bragg thought similarly. I agreed and learned a different way, with conceptual visualisation, logic, and Reality, the subject of this essay. It means there's quite a gauntlet to run, but my family motto is "I have the strength of ten men as I am pure in heart."
A bit sickly, but hey ho!..you can't change them. You also asked;
"However, my question about "Can you explain why you expect a similar curve? Are you assuming that this must be true or have you worked it out mathematically?" was meant to apply directly to your explanation about accumulating mass as velocity is increased. If that extra mass was accumulated linearly then the relativistic effects would not hold. So, I assume that your point is that mass is added on by the accumulated "shock wave" and that this accumulation is very minor for most of the increase in velocity, but, as the particle approaches light speed, there is a great increase in "shock wave" material that greatly increases the mass. What I was wondering was: What support, theory would be welcome here, is there to show that the accumulation of "virtual particles" follows a relativistic curve? I was looking for an explanation that did not rely upon referring to observed results. Those results may be caused by something else. Each explanation has to find its own logic so that the result may be predicted. If you think my logic here is incorrect please point that out.
Yes I agree with your logic, and that is what has of course been assumed. And No. In fact I was wrong, though non critically. Eckard correctly pointed out that the curve is not of course exponential. Essentially - for the propagation I've assumed we can roughly use a similar curve and orientate it differently for each of the 3 functions. However, there are a number of other factors which will make accurate calculation impossible, so logic and experiment can only be backed up by guesstimates!
1. The size of the mass. A bunch of protons the size of Earth propagates more than a single electron or smaller bunch.
2. The background (accelerator magnetic/solar wind/CMBR field). Which varies.
3. The size of the pipe, and relationship to other mass. I don't think Earth's in a pipe, but at the LHC as soon as the moving bunch starts hitting the pipe wall the propagation rate is boosted by secondary Pe's.
And probably more, and Yes, there should be the opportunity for much more work to be done here, at the LHC and with maths. (Could it even save the Tevatron?)
Did you disagree, or have any other thought on how virtual particles might pop up from a void at up to 10^13/cm^-3 due to something 'moving' through nothing? (Current theory, when considered, seem to assume they 'come from' the em field or pipe walls).
Best wishes
Peter
