studiot

Hermann Bondi was a significant scientist, though his version of the steady state theory has fallen by the wayside. His 'Common Sense' relativity book contains nothing contrary to current explanations of Lorenz or Einsteins additions of relativistic velocites or why it is not possible to travel faster than light. Indeed he has chapters explaining mainstream theory on all of these. So please do not invoke this book as backup for your way out propositions.

I have posted my replies in a new thread since this one is in speculations and might easily be closed at any moment and the subject of a metric is really off topic here.

This thread is in response to questions in the Speculations forum about exactly what a metric is and this level of detailand discussion is neither speculation nor really on topic for the older thread. I am posting it in Physics as most instances of metric at SF are connected with coordinates. Also Maths is rather dry with its format of "Let us state an axiom or axioms and see what we can do with them." Physics can be more chatty. First a little backgorund. Yes indeed a set with a notion of distance between its elements. But we really need more than a notion, so let us examine the sort of things we want 'distance' to do for us or not do for us. As so often in maths we gather togther all the objects we want to find a distnce between and put them in a set. Voila we have a basic manifold or space. We will call the set X and the elements x1, x2 x3 etc Note these elements may be coordinate points in some coordinate system in which case our distance is coordinate distance. Or they may be binary strings or actual words in characters in which case our distance is known as the Hamming distance. So what do we want from our notion ? Well here are some suggestions. Our first notion of distance is that it is between two elements. So our set must contain at least two elements. We do not want any elements to be 'left out.' That is our distance determining function (D) must apply between each and every pair of elements (xn , xm) That is we we do not want the distances to be undefined or undefinable between any pairs of elements. We do not want any distance to be infinite (or do we?) Ideally we would like the distance to appear as a number that we can append a physical unit to. The function, D, must allow repeat values of distance from all different to all the same and everything in between. We want the distance from xn to xm to be the same as the distance from xm to xn. We want the distance to be zero if xn and xm are the same point or element. That is D(xn, xm) = 0 But we really do not want to cope with negative distances so we specify that D(xn , xm) is greater than or equal to zero. These can all be written very compactly as a couple of mathematical axioms, but I am going to add one further dersire that is very useful, but not essential. We want our D(xn , xm) to be the shortest or least value and that any distance via an intermediary point is greater than this. These conditions form the basis of Riemannian Geometry (and thus Euclidian Geometry) For Relativity we need to relax the restriction on negative values. I think this has rapidly swept through the questions in the anti-time thread so next time will be for examples and further answers along with clearer maths.

Why is that a problem ? We both sprang from a traceable common ancestor so have some common genetic material. But when push came to shove HS proved more adaptable than HN so outcompeted them.

Current thinking in anthropology actually proposes the opposes of this as the reason why H. Sapiens superceeded the more numerous H Neanderthalis. The last episode of the recent BBC series Human made this point most emphatically.

This is actually a very good question +1 and @TheVat has given a good shortform answer +1 If you like to explore this further the mathematics is extremely simple, yet reaches into most corners of the subject from classical mechanics to relativity to geometry to cryptography to error correction codes to statistics to topology to.... This all hinges on what is meant by 'distance' that is what we want from our 'tool'.

Thanks +1

Do you have a quotable source for this information ?

If you are going to quote the originator of the Butterfly Effect, then you should take note of what he actually said and / or wrote in his paper or his book. So read that instead of quoting rubbish garbled up by some experimental program. What was actually said is interesting because Lorenz said that if we are going to consider the scenario that s small sensitivity in the initial conditions such as the disturbance caused by a single flap of a butterfly's wing could cause a tornado, we should also consider that on another occasion a similarly small disturbance could prevent a tornado. As the implications of that are profound because unless you have reason to think that disturbances pushing in one direction are more numerous than disturbances pushing in another such disturbances should balance out and merely change the order of events. So can you control it ? Well the nature of the problem depends upon the deterministic equations involved. For instance autonomous first order differential equations cannot be controlled. But there are other equations that admit a form of control. (Not those to do with weather)

If you want to discuss your claim, then discuss it. But, when presented with a counterexample, please do not pretend you are the sole arbiter of nomenclature in such an offensive way.

Well the op certainly seems to have been frightened away, if that was your objective. As far as I know most students start with the definition a vector is something that requires both a magnitude and a direction. My treatment is linked to that and is designed to develop from there.

OK so you read a short paper which didn't tell you any of the back history leading up to this. Professor Millikan (famous for millikan's experiment with electrons) was a very careful and thorough worker and indeed justly famous for the work he did on the electron and its properties. But he also did a lot of work in Cosmic Rays. Not only was he a careful worker but he also documented his work in papers and in a book what was revised several times, starting in 1917 and called the electron. My 1947 revision is a model of how to report basic research, methods, results and conclusion and is now called electrons (+ and -) , protons, photons, neutrons, mesotrons and comic rays. It is the mesotrons that are the same particles we now call muons. The book also documents and correlates the work of other scientists in this field. The actualy work took place over a period 1917 to the early 1940s. It was published by Chicago University Press (and also by Cambridge University Press in England) On page 519 ff , you will find all the detail you want carefully explained with photographs, graphs, and more.

I liked James Blish and his spindizzy series, very imaginative. I assumed you know about Watership Down since it was a blockbuster film and hit for Art Garfunkel. But I asked because I was thinking about another such novel (actually a series of 3) tracing the semise of the red squirrel under pressure from the invading greys and based on fact, rahter than just being a nice personification story about animal colonies. For instance red squirrels bury nuts, but grey squirrels don't. The Silver Tide Michael Tod I also liked James Blish and his spindizzy series, very imaginative.

See Fleisch A students guide to Vectors and Tensors page 2 Cambridge University Press 2012 You should also note that this thread is only about directed line segments, or 'arrows', not any other type od vectro.

As a biologist you might then just appreciate the science behind the McCaffrey Pern books. Her other books are rather pedestrian and predictable though.

I fully understand virtual or imaginary forces as I prefer to call them (after D'Alembert who invented the method) But my point is that your example was not a good one. The walkers could just as easily stand still, involving no forces, pseudo or otherwise. https://en.wikipedia.org/wiki/Free_fall

I am not sure why you are introducing tensors, when you still have questions about vectors. So I suggest we defer the introduction of tensors till we have dealt with your questions about vectors. It would be helpful to tell us in what connection you wish to use vectors and tensors and if you understand matrices ? OK so your arrow is what is also called a 'directed line segment' I have shown this in Fig 1 as the arrow Op, where 9a,b) are the coordinates of p and V is the length of Op Op is a line segment of the extended line shown dashed. This line makes and angle theta with the x axis. As you say this vector is one of a family of vectors that all start at the origin and are totally determined by their length v = Op and the anticlockwise angle theta they make with the x axis. In other words they are 'bound' to one point (the origin) and indeed are called bound vectors. Yes indeed if these vectors begin somewhere else they are called unbound vectors, as vector W in Fig 2 Now I am going to leave x y coordinates behind for a moment since it is obvious that an unbound vector will, in general, point to some other coordinate than (a, b) - let us choose one and call this point q again as in Fig2. Now to specify a vector in a plane we need two numbers and if we do not want to be stuck with starting at zero we need to use different properties of the vector than its end coordinates. So I am going to introduce the vector V with length v = Op and making an angle ϴ with the x axis as V = (v, ϴ) Similarly we can consider the unbound vector W with length w = O'q and making an angle Φ with the x axis. So W = (w, Φ) So if v = w and simultaneously ϴ = Φ Then we have an unbound vector we can place anywhere on the plane parallel to our bound vector. Does this help with your first question ?

Of course its not irrelevant. A body cannot move itself without external interation. (newton's first law) So there must be The walking involves some real force (probably friction). The exzample is not a good one for gravity. A bodies motion through the curved space is free fall.

A thought experiment question. In a universe with no light at all, but everything else, would these phenomena still occur ?

One definition of force is rate of change of momentum, which is what happens when you bump into someone.

I'm not debating with you an explanation from some armchair of a measurement that has actually been made. Length contraction is a measured observable fact. This was first done in 1941. http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html

Sorry I don't understand your comment.

Not necessarily. Pratchett does nothing for me, but some fantasies can be fun. You generally have to suspend (dis)belief for some aspect of the story in SF, whether it's people magic or technology magic.

Unlike the first number I misquoted, where I made a silly copying mistake 156,423,343 is not prime. 156,423,343 = the product of two primes 22,343 x 7001. I should have constructed this example in the first place as it is beginning to show just how far you might have to go to find any factors. This of course is also beginning to demonstrate the interest in large or very large primes for cryptography. To recap Apologies for the earlier mistake 9,991,991 is not prime and = 2833×3527 9,999,991 is prime .

Gosh yes you are absolutely correct, I miscopied it I should have written 9.999,991 (which is the largest prime number less than 10 million) Thanks +1