studiot

Thank you for coming back to tell us what happened. I'm so glad how well it turned out for you. +1 Please give us more notice next time.

Here endeth the lesson for today read by the Cookie Monster.

Ok to make a start on curvature. It's late so I am sorry that the sketch is rather rough. The point is that a sphere is a surface (mathematicians call the solid round object a ball) and is two dimensional. That is we need two numbers to specify any point on it. Conventionally we use latitude and longitude. Both of these are angles. Note that the radius is not needed to specify any point on the 2D surface. The radius takes us into 3D. But a different pair of numbers, representing x and y would give us a plane surface So what is the difference? Geodesics. These are the lines which have the shortest distance between any two points A and B on the surface. In a plane any straight line is a geodesic. Usually the gridlines are geodesics but this is not true of latitude, although it is true of longitude. Geodesics on a sphere are 'great circles' like the equator and lines of longitude. Geodesics also occur in higher dimensions than 2D surfaces, but they are still lines. Why so much emphasis on geodesics? Because they are the path taken by light rays through space and spacetime. Now look at the sketch, showing triangles on the surface of the earth. The 0o and the 90oW lines of longitude, AN and CN intersect at the pole, N, at 90o. The also intersect the equator at 90o, as do all lines of longitude. So the angles of triangle ACN on the surface of the Earth add up to a total of 90o +90o +90o = 270o, rather larger than the 180o for a plane triangle. This is called spherical excess (over 180o) and is dramatically large as the excess depends upon the area of the triangle and ACN is a large triangle. The excess for a smaller triangle, say ABN, can be seen to smaller at 60o. Furthermore if we follow AN to D and BN to E we have a yet smaller triangle, but the apex angle at N is still 60o. So the angles of intersection at D and E must be less than 90o to achieve a smaller spherical excess. So I have set up a real world example, sailing across the North Sea from Brora in Scotland to Kristiansand in Norway, which conveniently happen to be on the same 58th parallel of latitude. From the figures it can be seen that sailing along the parallel is about 1km longer than sailing a true great circle geodesic. Now translating this to the first experimental verification of general relativity (by Sir Arthur Eddington) which is conveniently shown here Light from a star hidden behind the Sun was deflected by the Sun's gravity a minute but measurable amount exactly as calculated by Einstein. There is more discussion of this and possible implications of GR https://www.scienceforums.net/topic/106314-what-warps-space/?tab=comments#comment-993433 https://www.scienceforums.net/topic/114528-gr-and-the-principle-of-reversibility-of-light/

At that time light was regarded as a wave phenomenon. For all waves, including light, the characteristic speed is independent of the speed of the emitter. The emitter is considered as a point source with its own 'speed'. That is once the wave has been lauched it is entirely controlled by the medium of transmission. An observer, travelling relative to the medium will observe a different velocity. So supersonic aircraft can 'catch up' as you put it with a sound wave. Einstein's innovation was to say that for light, the observed speed is independent of the observer's speed or the same for all observers. That is quite different form every other wave. This discussion is not about a transmission medium for light so don't get diverted into that. Now both of these are local in that the emitter and observer are points and they can only measure the speed near to themselves and in relation to themselves. They cannot measure a 'global' speed that is 'for the galaxy' or wherever. So they must rely on a principle such as Einstein's. No. You can see stationary em waves using lecher lines for instance. https://www.google.co.uk/search?source=hp&ei=Y22GXOLOF4GclwTz4oTQBQ&q=lecher+line+experiment&oq=lecher+lines&gs_l=psy-ab.1.1.0j0i10l2j0i22i30l2.826.2804..4822...0.0..0.214.1498.5j6j1......0....1..gws-wiz.....0..0i131j0i22i10i30.JYtj1z6og1w It was, as I said, formulated to recast Maxwell's equations in a form invariant way. I can write these out if you wish but this is a lot of Maths. Remember that our whole basis of Physics and Astrophysics/Astronomy is that the Laws of Physics are the same in Alpha Centauri as on Earth. That is a global statement that forms the basis of Cosmology, which is about the development and evolution of possible/credible universes. If the laws were not the same then we could not rely on spectroscopy to tell us what the stars are made of and so on. Thank you for your sketch. That's good I can do something today.

Given your background and the timescale involved, I would recommend concentrating on the Physics of Relativity, rather than the Maths. The Maths for Special Relativity is accessible to those with high school Maths, Gerneral Relativity is not. So in your essay go into the (Physics) principles of importance in Relativity. 1) The Principle of Relativity, both the pre Einstein version and the updates Einstein introduced. 2) The Principle of invariance of the speed of light to all observers. 3) The principle of equivalence. The first two form the basis of SR adding the third, ugrades to GR. (1) Expresses the desire in Physics to have isotropy and homogenity in space. (ask if you don't know what that means). Another way to put this is the desire to have the laws of Physics look the same to all observers. Well call this ' Form Invariance'. this can be made to work for the Laws of Mechanics (Newton's Laws) but not the Laws of electromagnetism (Maxwell's equations) (2) Is introduced to make Maxwell's equations compatible with Principle (1) (3) The first two are local principles. That is they apply to a small region ( a point) in space. (3) is used to create the Field equations which extends relativity over all space and time. I will post a couple of sketches showing the implications of (3) and the relationship to curvature. Please avoid the 'trampoline analogy'. It is just plain misleading. Is your maths up to knowing that a sphere can be represented by the equation radius = a constant so the position on the surface can be represented by the coordinates [math]\left( {r,\theta ,\varphi } \right)[/math] Where r is the radius, theta and phi are angles?

I know this is for an essay, But I think it is legitimate to ask your questions in the main relativity section. Homework help requires you to show your working here, which is not appropriate at your level. So ask (send them a private message) a moderator to move this for you. There are some really good folks here with great knowledge of relativity. I don't know how what your timescale is but this book could have been written for you. My local public library has a copy. Note that Special Relativity is founded on two principles General Relaticvity introduces a third principle and also the Einstein Field Equations. Note the plural - there are 10 of them, usually rolled into one big one that you will not yet have the maths to work. Go well with the writing, and tell us the timescale.

How does one collate a double limit with an infinitesimal?

That would be good if I could find it. I looked but could only find extracts in pdf.

I have ordered the two books you listed, thank you for the references. It will be interesting to see what they ahve to say. This book is hundreds of £ and out of my means, but I am pursuing a loan copy from our inter library loans system. Then would they also be infintessimals? I rather think it is the other way round. The limiting process can be usefully applied to Infinitesimals. However I think that the OP question can be answered as follows. Limits are the result of the limiting process. Infinitesimals are specially constructed abstract objects, ouside the normal number systems so no, they are not the same. The limiting process has wider applications than differentiation/integration but that is not the subject of the OP (for instance the relationship to asymptotes), so we should explore this in a new thread if you wish to take it further. There are huge and widespread applications in engineering and theoretical physics (relativity).

Asymptotes can be limits.

Only part of it.

What does that have to do with the OP question? Why can we not apply Science to look into any proposition we choose?

In what way? What are you thinking about?

I don't seem to have had an answer to this post I don't agree with this. Examples from statistics come to mind. Whilst I liked the style and presentation of your paper, I don't endorse everything in it. A further comment One thing you don't seem to have examined is the clear difference between infinitesimals and limits. Limits allows one to step over the process of accounting for an infinity of terms, which you correctly say begs the question of do the ignored terms add up to something significant, and go directly the an end result, or show that there isn't one. Convergence theory is all about this. I do not know of any equivalent process with infinitesimals, NSA notwithstanding.

The eco (furniture) factory described in the book is still going strong. https://grimshaw.global/projects/herman-miller-factory/

I agree with this, unfortunately those promoting these shifts tend to be less well resourced and less articulate than the establishment. For example the small book 'Cradle to Cradle' by Braungart and McDonough Is very well reasearched, and contains some excellent examples of actual (successful) cases. However it is very hard to read as its style is not coherent or progressive. When our back have been to the wall (eg WW II) coherent (joined up) Government did indeed lead and direct a successful collective response to an emergency. But real and imminent emergencies do appear to be the only drver for such action. In the Netherlands (Holland) last year the banned the use of gas in new homes. The UK government is considering a similar ban. But another part of the UK government is still offering grant suport and promoting gas boilers. Worse the UK government has fragmented its policy to provide a relaible and stable electricity alternative.

Thank you for the reply Wasn't there a question in the title?

Many think that there are multiple environmental crises assailing the planet. Here is an article on a fresh report from the https://www.ippr.org/, by the BBC https://www.ippr.org/ The degradagation of arable land is cited in the New Scientist book https://www.amazon.co.uk/dp/1473629772/ref=asc_df_147362977258469057/?tag=googshopuk-21&creative=22110&creativeASIN=1473629772&linkCode=df0&hvadid=310865071345&hvpos=1o1&hvnetw=g&hvrand=18039469306525565984&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=1007149&hvtargid=pla-563590850360&th=1&psc=1

I don't agree with this. Examples from statistics come to mind. Whilst I liked the style and presentation of your paper, I don't endorse everything in it.

Takazu Seki, pioneer Japanese mathematician, accountant and chief of the National Bureau of Supply b.early 1640s Edo or Huzioka, d. 1708 We had a long discussion about this subject in a recent thread. https://www.scienceforums.net/topic/116421-definition-of-derivative/?page=3 Many of the references I referred to in my first post here appeared there. You do not seem to have heard of at least some of these.

You have references to this? In discussing the OP paper (the purpose of this thread) I note a psarsity of references to Newton and his input compared to that of Leibnitz and the continentals. This bias towards one or the other european originator is common in articles. It is also common to entirely fail to mention Seki.

Compared to the usual rambling rants against the establisment this paper was clean and tight, with good references. All in all a worthwhile addition to the knowledgebase on this subject and a reminder that no subject is static and that fashions come and go, even in Mathematics. A pleasure to read, I did not know there was/is a movement to re-examine the history of this subject in the light of modern thinking and I can personally vouchsafe the value and veracity of your opening sentence in most disciplines. Nor it seems did wtf. My library goes back to the earlier period you speak of , so I will be comparing some of the texts from that time So welcome and +1 for introducing an interesting subject.

Energy is a property not a thing. The 'thing' theory of energy (caloric) was disproved centuries ago. Exactly so. Space on its own is a general term for the stage where stuff happens. We need to tie it down with qualifiers to properly identify which stage we are talking about. This is why, for instance, my comment used the term free space. But 'Space' could be limited to area or even a linear measurement, rather than volume. Which brings in measure. Many useful spaces (including all geometric spaces) possess a measure or distance property as mathematically defined and called a metric. Unfortunately Physics has (once again) a different definition of the word metric, but it is equally important, especially when considering Relativity questions. Back to the mathematical definiton leads us to consider those spaces without a metric. These are topological spaces and non metric topological spaces lead us directly to wormholes with the 'gluing' rules of topology. Computer programmers use another such space with packman type games on screen. To understand Space and its qualifiers we need to look into set theory, functions, mappings and containers. A good simple example of this would be to explore this view of 'vectors'; this readily shows how you need a 'container' filled with several different sets to develop useful a useful theory - that of vector spaces.

I think that simply saying space is volume merely replaces one word with another. You could equally ask "What is volume?" In fact space is not just any old volume, it is a particular sort of volume. So to say something more useful than "space is volume you" have to detail its particulars. Furthermore that particular volume does indeed interact with energy (in the form of EM waves since there is no such thing as 'pure energy') as evidenced by the easily measurable complex impedance of free space, usually denoted by the symbol Zo.

That about says all that needs to be said. It is clear that there is only one poster expounding inflexible views, regardless of anything anyone else says. I thought it all sounded familiar. Thank you for preventing any further waste of my time. +1