See, you did not even understand my post....severe comprehension problem. I was referring to point (straightline issue) not about me.
Muddled? I don't believe so.......short hand version maybe, similar to John Wheeler's statement thus..... "Spacetime tells matter how to move; matter tells spacetime how to curve". Although I have heard and read that he actually did not refer to spacetime but just to space.....still, all reasonable people know what he meant. The "shortest distance" path is certainly a geodesic, but yes, it is possible to find examples in which a given geodesic between two points is not the shortest path connecting them. The real issue at hand though is that the twin paradox is not really a a paradox at all... Thank you though for your correction, although I don't see it as correcting the blunder that I was correcting and the crux of the matter.
As one who has been pointed out as "shifting the goal posts", my understanding of your remarks and general claims was spot on. Please Register or Log in to view the hidden image!
Mods, Can we withdraw these stupid emojis option from science thread ? Almost every second post of Paddoboy has few emojis, as if he is jumping from his chair...
That's because those were margin notes summarizing the content of part of the introductory section of a GR textbook. See page 5 of Misner, Thorne and Wheeler's Gravitation. As you suspect, the context makes it clear that we are talking about spacetime. The definition of a straight line is the definition of a geodesic. So "not all geodesics will be straight lines" communicates nothing and sounds like you are confusing something in coordinate language with the geometric concept of straightness. In the twin paradox we have two events with timelike separation, \(\mathcal{A}\) and \(\mathcal{B}\), in flat space-time with metric chosen so that \(ds^2\) is positive for time-like vectors and Cartesian coordinates so that straight lines have the property \(t'' = x'' = y'' = z'' = 0\). What destroys your statement that a geodesic is the shortest possible path is that the norm is not the Euclidean norm. Thus the non-inertial path from \(\mathcal{A}\) to \(\mathcal{B}\) always has a shorter proper time than the inertial path from \(\mathcal{A}\) to \(\mathcal{B}\). In manifolds which are locally Euclidean, we have geodesics being local solutions for the shortest path from \(\mathcal{A}\) and \(\mathcal{B}\) but they may not be global solutions, as we are not limited to just one path between \(\mathcal{A}\) and \(\mathcal{B}\). Thus while a path may a local extremum of small variations, it may not be the shortest path possible.
Agreed, and I'm sure my own extended explanations particularly in the "light propagation" thread, likewise explains a less then "muddled" explanation. Yes, I accept that, but again the crux of this matter imho stems from the frivolous and less than meaningful claim in the light propagation thread by the god thus..... "When almost nothing moves in straightline (through space), why photon ? The guy has got the momentum". In reply to origin's claim thus..... "Everything moves through space in a straight line unless acted upon by a force". Now while you correctly and rightly corrected origin [and myself later on] on the fact that in GR gravity is not a force, I'm pretty sure that most members understood the gist of what was trying to be said, as opposed to the fabricated suggested "anomaly" by the god.
In Minkowski spacetime and Cartesian coordinates, if \(\mathcal{C} - \mathcal{A} = t_1 \times ( 1 , \vec{u} ), \; \mathcal{B} - \mathcal{C} = t_2 \times ( 1 , \vec{v} ) \) then \( \mathcal{B} - \mathcal{A} = ( t_1 + t_2 , t_1 \vec{u} + t_2 \vec{v} )\). So \(c^2 (\tau_{ACB} - \tau_{AB} )(\tau_{ACB} + \tau_{AB} ) = (t_1 \sqrt{c^2 - \vec{u}^2} + t_2 \sqrt{c^2 - \vec{v}^2})^2 - \left( c^2 ( t_1^2 + t^2 + 2 t_1 t_2) - t_1^2 \vec{u}^2 - t_2^2 \vec{v}^2 - 2 t_1 t_2 \vec{u} \cdot \vec{v} \right) \\ \quad \quad \quad = 2 c^2 t_1 t_2 \left( \sqrt{1 - \frac{\vec{u}^2}{c^2}} \sqrt{1 - \frac{\vec{v}^2}{c^2}} - \left( 1 - \frac{\vec{u} \cdot \vec{v}}{c^2} \right) \right) \) Taking derivatives, we see that \(\vec{v} = \vec{u}\) is a local maximum. And the value of that maximum is zero. So \(c^2 (\tau_{ACB} - \tau_{AB} )(\tau_{ACB} + \tau_{AB} ) \leq 0\) and because everything else is positive, \( \tau_{ACB} - \tau_{AB} \leq 0\) or \( \tau_{ACB} \leq \tau_{AB} \) with equality only being reached when the two paths are congruent. So the straight time-like line in Minkowski space is the globally longest time-like path between events.
That concept: that timelike geodesics represent the longest path "through time", I think is quite nonintuitive. But, it's the (special relativistic) reason one of a pair of twins ages more than the other in the twin paradox; it doesn't include gravity or say anything about "forces" (bare fact). I think, these days, the scientific understanding of force is that it's another way to explain physical laws, and there are ways to do that without invoking forces. There are four fundamental forces, presumably all measurable in Newtons (otherwise calling them forces would be inaccurate). I believe that the first problem might be addressed somewhat by getting one's head around an intuitive (physical) meaning of timelike, also spacelike, and lightlike. These terms all refer to distances ("separations") between two events, or alternately to ("causal") relations between events. Or at least that's what I think I know.
I'm not well versed in any of this, but remember from 5+ decades ago parts of my classical mechanic course, based on Goldstien's book by that name, that there are several different ways to describe motions without the force concept. One was via a Hamiltonian formation, (laying the foundations for later QM descriptions)* and another was via "action" - it had to be an extremium for any real motion. There were several others, which I forget. We solved the simple harmonic oscillator problem at least five different ways, only one used the concept of "force." As someone remarked: Using these powerful methods on such a simple problem was like opening a nut with a sledge hammer. *Force is not used in QM discriptions.
I am not able to figure out what is fabricated or anomaly in the OP, and I also don't understand what you mean by less than meaningful.....What is clear to me is that you have not learnt anything from this thread also....you are still babbling or copy/pasting without understanding an iota of underlying concept. How sad !!
Think harder, dig deeper, in the mean time brighten up! Please Register or Log in to view the hidden image!Please Register or Log in to view the hidden image!Please Register or Log in to view the hidden image!
And let me say you may well be correct, since that is the nature of science as our observations improve and technology enables us to experiment more thoroughly. I can though reasonably confidently say, that religious beliefs, all magical deities, or whatever magical spaghetti monster that you chose to put faith in and such, may be even further redundant then they are now, and possibly totally redundant...Now that would be one giant leap forward, you agree? Please Register or Log in to view the hidden image!
Great ! This is a problem area for you. I never mix my faith or religion etc with science. I am of the firm opinion that faith in God and Science both can co exist. You should be careful, by calling the God as monster or pixie etc, you are not making any point at all. There is absolutely no need to throw stones on somneone's faith and day to day beliefs. You know very well that even the great scientists or doctors or engineers may all have faith in God. The problem is cross interference.
I know....but what you probably do not know, or more correctly, do not want to know, is the fact that any validated theory extending the parameters of GR, will be doing just that....just as GR extended the parameters of Newtonian: If BH's are found to be invalid [a long shot] then something even more exotic [although not as exotic as any magical spaghetti monster] will have taken its place: But unlikely. WH's, not sure why you bring that up, as they are even now still only hypothetical, but like wormholes, allowed for by GR. Science/cosmology will though get as close to reality as possible. It will not be done by you or me on a science forum, I'm sure you'll agree, no matter how much you feel like twitching my nose. Please Register or Log in to view the hidden image! No, not at all...:shrug: Please Register or Log in to view the hidden image! [perhaps I'm twitching your nose? Please Register or Log in to view the hidden image! ] That would be easy to believe if you were not so fanatically anti science, anti mainstream and anti cosmology in particular. Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image! No point? So why comment on it? Please Register or Log in to view the hidden image! And secondly I am certainly not in the habit of throwing stones at anyone's faith, [I'm a live and let be type of bloke] only at the idiots, trolls and mugs that like to step up and deride anything and everything sciency on forums such as this, with no evidence, no credentials, and/or qualifications to do so. So again at this stage, if the cap fits, wear it. Please Register or Log in to view the hidden image!
7 emojis in a meaningless post in science section ?? Paddoboy, fallen in love with emojis at this age ? Vicarious love ??
The meaningless post is only meaningless to someone with an agenda, and does nothing more than correct the dishonest attempts to discredit science in general by someone who appears to be on some evangelistic crusade, and most certainly does not want to be reminded of that fact.
good grief, is this a science forum. you guys really need to take a long hard look at yourselves, really.
