studiot

Perhaps you would like to consider the implications of this formula? I agree that it is best to place the origin at the centre, as you have done but consdier the implications for your integral. You have correctly identified that both density and the gravitational acceleration are functions of the radius. At the origin, gR = 0. Thus the integral must be zero at r = 0 At r = R, the integral must equal the surface pressure, by definition Thus your expression states P0 = PR + PR = 2 PR I think this difficulty arises because you have tried to include a constant of integration in a definite integral.

@lidal I looked for a (mathematical) response to this mathematical post from Joigus. But I could not find one. Joigus' proposition is mathematically sound and intriguing. I had not heard of it before so +1 to him for bringing it to my attention. This is unsupported dogma not rational conclusion drawing. The aether was a model. It was wrong and ultimately discarded, but it was not an unreasonable position given what was known about waves. Models have their disadvantages as well as advantages. A good knowledge (no one person can know everything) of the History of Science is useful to further the cause of understanding. In this case Maxwell did make a Model which fitted all the then known facts perfectly. But Maxwell himself also stated that it was false. It is known as Maxwell's Vortex Theory and is purely mechanical. Most of Science is a work in progress. You appear to want to cherry pick aspects (often popsci quotations) from partway through such development to support your notions, rather than doing the hard work of going through sift the available data - make hypotheses/proposal -- test those proposals -- refine those proposals in the light of the new data - return to step 1.

Do you not understand the difference between the North (and South) poles and The North (and South) Magnetic poles ? There is evidence that the core does not spin in the same way as the mantle and crust.

Any observer, Bob included, is stationary in their own frame. Why should I answer your question since you could not be bothered to answer mine. ? I was trying to help you simply.

Good job I read the article you linked to before I commented. Your article is not about the magnetic poles but the mechanical spin axis poles. The fact that spinning round object such as tops can suddenly flip their spin axis, thereby reversing the spin has long been known. The Victorians had curiosity devices based on this. In the 1980s Peter Warlow published a book https://www.amazon.co.uk/Reversing-Earth-Peter-Warlow/dp/0460044788 In which he attributes many past events to such flips. Yes the spin axis could, in theory, invert. But we have no hard evidence that it has ever done so. The inversion of the magnetic poles is a differnt matter. These are known to have inverted many times. Indeed we may well be approaching such a change at the moment as there is a growing inversion known as the South Atlantic Magentic Anomaly. https://en.wikipedia.org/wiki/South_Atlantic_Anomaly But I doubt that Ice Ages occur due to one single cause, although Milankovitch cycles do coincide with periodic precessional, spin, and orbital changes. https://en.wikipedia.org/wiki/Milankovitch_cycles

Please, The subject of this thread is mutations not masks.

@joigus ; @Markus Hanke I suggest there is a danger inherent in considering physical time as just one mathematical coordinate axis of several say x0, x1, x2, x3 in that physical time possesses at least one property not possessed by using the mathematical notion that time is a parameter that the other axes can be put in terms of. Time as a running parameter is excellent in many ways since it has a natural ordering that leads to causality (ie immutable sequence of events) But if you want invariants, independent of coordinate systems as in relativity, consider what happens when you change from instance from a cartesian to a spherical coordinate system.

Yes, but I didn't want to introduce the second part of the development before the first was digested. This comment also applies to swansont's more complete and comprehensive statements. Yes the next part of my offering would be for Bob to calculate his version of the journey and apply Lorenz to find his measurement of the distance travelled and thus find the same relative velocity as Alice observed. Then one could enquire about how the spaceship achieves constant velocity. Thank you for addition. Did you follow the first steps of my explanation, which give the correct sequence to consider things ? We have all said this is correct. No this is not correct and is the subject of the second part of my explanation that I have not yet provided. You need to fully understand why you have got the first part right before moving on to the second. Hopefully you will also take the opportunity to learn some correct terminology as you do this. 'Twist' is the wrong word. You should use 'Rotate'. If you swing a weight on a rope in a circle, the rope is straight not twisted. But it is rotating about an axis that is not along the axis of the rope. A twist would be about the axis of the rope.

Yes indeed, that is why I aksed for your definition of 'change'. Although that merely substitutes defining 'difference' for defining change. Your example suggests to me that you have not picked up my point. Sorry that point is very important but also very difficult to describe/define. Since I have not succeeded in providing enough clarity I will try again. The desert, savannah etc could have separate existance, each in their own right. As you move, they could also be changed (wink into and out of existence) like a Hollywood set. So they are different from a cup which is only a cup when all the parts are present together. A handle is not a cup, a base is not a cup, a bowl is not a cup etc. But thank you for the continued discussion.

This is not how I understand the original situation. I understand it to mean that The distance from Earth to planet 2 has already been measured (and will not change in the lifetime of Alice) by othe astronomical means. Similarly the spaceship has a 'cruise control so its velocity is predetermined at a fixed rate of 0.99c. No one measures this. Bob starts out from Earth when both his clock and Alice's clock read zero. So by Alice will know that when her clock reads 20.5 years Bob is just arriving at planet 2 with his clock. Bob's clock will be reading a value given by the Lorenz transform of this time.

Surely Alice just reads it off her clock ?

BBC Science has an interesting article on this https://www.bbc.co.uk/news/science-environment-53486868 One thing they discuss is a second optological dating technique that corroborates the radiocarbon method. Back to the OP One of the commonalities is the 'Innuit' The Western author Louis Lamour wrote one of his epic novels using this fact about a US pilot that was shot down over Siberia, but made his way back to Alaska via the old route over several years. IMHO it is a really good read. (Wish I could remember the name of it)

Well yes, the UK government started well but managed to turn it into a shambles. But we also have some really first rate experts on the job (if only they were listened to). Apart from that, I am really glad for you that you only endured mild effects. I lost my Herfordshire cousin (who was youger than I) to the pandemic. Another ( long term) friend from Swindon has a lodger who worked as a night watchman in an othersie empty ( = shut down) hotel. Sadly nightwatchmen are the (apparantly from the expert figures) the most vulnerable group and he contracted the disease, which laid a formerly healthy young man low for several weeks. Here in Somerset we currently have no cases whatsoever in the local district hospital (and no deaths for the last two weeks) and an incidence curve more akin to that of Germany than the rest of the UK. I consider myself lucky. So should you.

1) Many mutations are appearing (covid is not a production factory or facility either). Most are unviable so fall by the wayside. But it only takes one bullet or mutation to kill you. 2) This thread is not an exercise in semantics. The experts have not yet fully agreed on the terminology, whcih itself has changed over the pandemic. Most members know what I am referring to

Nice point +1 On this subject it is more subtle than at first appears. I think someone has already attributed to Wheeler the quotation although that quotation is actually older than Wheeler. https://quoteinvestigator.com/2019/07/06/time/ (There is an interesting coresponding quote on space in the linked article) Let us return to Marcus' cup for a moment. () Now for there to be a cup, all the parts (handle, bowl, base etc) all have to happen all at once. It is the archetypal 'block universe' where all parts of the universe happen or exist, even when the 'focus' is not upon them. So the 'change' can be thought of as a comparison of the properties (eg curvature) from once place to another. So we can compare the curvature of the handle to that of the bowl and draw the conclusion that it must exceed the curvature of the bowl since it goes round the outside of the bowl. Time is not needed or involved in this comparison.

I thought it was one of those mythical cups of Old Ireland that are always full (though of wine).

Nice example +1 A little milk but no sugar in my tea please.

An exponential function is not suprising since my last line leads to three integrals two of which are standard logarithmic ones. The third one need to be handled by parts. So it is one of those situations where you have multiple competing functions and it depends (as you note) on aditional information (boundary conditions) which dominates. +1 to joigus for 'the physicist's method'

I think some people expect too much of mathematical 'true' and 'false'. They try to use them in situations where they are inappropriate. Take the integer equation a + b = 10 What is its solution? This has an infinity of solutions which are 'true' But it also has an infinity which are 'false'.

This could be a start. If I have read you correctly, then I have separated the variables for you. Edit, Whoopsy I have needed to edit last line as can be seen. Sorry.

Sorry I don't follow. The difference between what and what? Nice link however. Thank you +1. Personally as a pragmatic Mathematician I am not with the constructionists. A short quote from Joigus' link will serve to show what I mean Anyone who has studied more than the simplest differential equations will have met examples of We can prove the existence (there existence) of a solution, but that does not help us to find it. (construct). Worse, much of physics is controlled by, and therefore demonstrates the existence of solutions, to equations that we cannot solve.

Depends how you define change, but I would say yes. Transformation is not the only type of change.

No I don't think the first one is granite or even igneous. It looks like a fine grained sandstone or siltstone to me, perhaps a grewacke. https://en.wikipedia.org/wiki/Greywacke This would be consistent with the Faro peninsula, which is a small sedimentary piece on the edge of the alpine mountain building area in Iberia. (the Variscan Orogeny https://en.wikipedia.org/wiki/Variscan_orogeny The pebble does not look beach rounded but river worn. Perhaps it was brought down by the Chanca river from the mountains to the north and east. The second pebble definitiely looks like some form of volcanic glass, probably obsidian. It seems to be layed on duller form of lava, which is very common. The base layer appears to have bubbles or other structure, characteristic of other forms of lava. If that was basalt the pebble would be quite heavy. Again this would have formed during the Variscan. Yes the various discolourations of the veins in the samples will be dues to impurities in the quartz. Iron compounds give the brown colour. It should be noted that the quartz veining is quite different from the micro intrusions of liquid rock during vulcanism. The is no metamorphosis (cooking) of the parent rock along the boundaries. Quarz is slightly soluble in rainwater of geological timescales so precipitates out as the solution percolates small fissures and the water evaporates.

Dunno, what are you going to do with it? Sounds like an odd sort of greenhouse disinfectant, but you could easily drown in 1000L of water.

UV spectrophotometers used to use fused quartz prisms.