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Well, without a metric we can’t really have a discussion about this. The thing here is that you don’t start with a metric - you begin with an energy-momentum tensor plus boundary conditions, then you use these to solve the Einstein equations. That gives you the metric. All solutions to the Einstein equations are metrics, but not all metrics are valid solutions to the Einstein equations.

@studiot How do you want to force criminals from using someone's address? Politicians have made enough s**t over the years with regulations like cookies, unsolicited mails, non-anonymous SIM cards, non-anonymous emails and online accounts, including private communication.. https://nypost.com/2022/08/09/facebook-gave-teens-private-messages-about-alleged-abortion-to-police/ "Facebook gave teen’s private messages about alleged abortion to Nebraska police" Did you see "Snowden" movie? https://www.imdb.com/title/tt3774114/ All these regulations are a "hacker's heaven".. If a hacker reached top politicians, he would ask for the kind of legislation they have implemented, because it helps hackers..

No, no, you don't need armies or weapons if you have feminine wiles and winsome smiles! Why do you think women keep pushing soy milk at men? Soy is a secret sterilizing agent that makes man-boobs appear and reduces any motivation to reproduce. Gradually, men are being pushed out (especially white men, who no longer control government, the Congress men are now just following marching orders from their wives!) and being replaced by females produced by parthenogenesis. Women are making up fake statistics to cover up this trend and conceal the secret war of extermination. Nolite te bastardettes carborundorum! I am sorry I said you are pontificating freely. That was wrong. Sometimes I am very stupid. From the soymilk and all the vegan entrees I am being fed. Did I mention the undescended testicles? Not important. Thanks again for your courageous struggle to get the word out on those deluded soymilk-addled kale-chewing elitist Liberals who are trying to exhume Ayn Rand's corpse and do unspeakable things to it! Onward, brother!

The argument still stands. Orthonormal is a particular case of orthogonal. Orthonormal=orthogonal and normalised. More specifically, a gravitational singularity is a region of spacetime in which any components of the Riemann curvature tensor become infinite. Read carefully @Markus Hanke's previous post. You do not invent the properties of the metric. You postulate the other (non-gravitational) fields. Then you obtain the energy-momentum tensor. Then you symmetrise it (with techniques like, eg, Belinfante's symmetrisation technique), because the canonical energy-momentum tensor is generally non-symmetric, and the source of the gravitational field must be symmetric in the space-time indices. Then you postulate boundary conditions, as Markus told you. Then you solve for your metric. Having done all that, you're still not home-free, because the particular coordinates that you use to solve for the metric can have false singularities, ie, singularities of your coordinate map that are not physical. So you must obtain the Riemann tensor and try to identify the singularities there. You have a lot of ground to cover still before you can meaningfully talk about your singularity. I hope the comments here you find helpful. The metric is not gauge-invariant. It's the Riemann tensor that's gauge invariant. This is in close analogy to electromagnetism. The vector potential in EM doesn't really give you the physics (except for the Aharonov-Bohm effect or the "holonomy" of the field). Infinitely many vector potentials give you the same physics. It's Faraday's tensor plus the holonomy which gives you the complete physics of electrodynamics. There is only one Faraday tensor (the E's and the B's) that define the physics. Gravity displays remarkable mathematical similarities to EM. It has a huge gauge arbitrariness. In modern GR we say space-time is not defined by a metric, but by an equivalence class of infinitely many metrics, all gauge-equivalent to each other. The matter is even more subtle. Sometimes you find a coordinate map that solves the equations. But the map has singularities of itself --fictitious. Then you introduce a change of coordinate maps that fixes the coordinate singularities. Example: Kruskal-Skezeres coordinates being a well-known example.

OK I have to come clean at this point and confess I do not know what "being more primitive" means in mathematics. I think it was Poincaré who tried to base everything mathematics in terms of group theory. Another attempt of basing maths on something "primitive" was Felix Klein's Erlangen program to unify geometry. Category theory seems to be another attempt at building a really primitive branch of maths. "Primitive" meaning something like "least number of assumptions." [?] Perhaps "primitive" means theory A can be based on concepts derived from theory B, but not the other way around, and therefore theory B being more "primitive" than theory A? I'm not sure of what mathematicians mean when they say they're trying to refer things to something more primitive.

What is a "pofinage"? And how can a basis remain orthogonal irrespective of the metric? The metric is what tells you whether a basis any given set of vectors is orthogonal or not. How can rest (which is an observer-dependent concept) be associated with a singularity? Does a frame change remove the singularity? Infinite value of what? Singularities in the metric are meaningless. It's only singularities in the Riemann tensor that are physically significant. Also, a 3pi angle points in the negative x direction, not in the negative y direction. I think you meant 3pi/2. Whatever x and y mean. There we go again.

Beatifully made, very motivational, educational videos on paleontology, geology, and the entire history of the Earth: The extra code just takes you to the list: https://www.youtube.com/watch?v=0_a_xU2KQdE&list=PLoOkB6QkDW_S7fdqmfW-_lwS_Kp_uwfeo

I understand but don't agree. We may know what to do to address the issues in theory-- but properly predicting where there will be an excess of water and a shortage of water is a necessary element-- and to a large degree we don't know until after it has happened. Did anyone predict the floods in Kentucky specifically-- or just somewhere in the south-east US? That's what is necessary in order to apply solutions-- unless we spend huge amounts of money everywhere. The inability to specifically predict is what I allude to in talking about only knowing after the fact. There is much more to this beyond politics and economics.

I don't like almonds. But I do like some monds.

It seems there have been studies on this: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4881732/ Fumes from hot oil are not combustion products, but they do contain a lot of species chemically altered by heat -by thermal cracking, in effect. There are some details in the link, though that study was mainly concerned with the epidemiology of lung cancers.

Inhaling the byproducts of any combustion (burning) is toxic and/or carcinogenic, so the question really is how much, how does it differ based on the substance being burned, and how do those things map to your personal level of risk tolerance. Keep in mind it’s not just the oil,in your pan, but the heating being applied below it that matter here.

I like it. +1 But correspondence is more primitive than counting - which requires numbers. Consider the following. The Australian Aborigenes had a number system one, two, many. 3 numbers only. So they could distinguish but not count. Marked tally bones (there may have been tally sticks as well but these will not have lasted as well as bones) for 30,000 - 50,000 years ago have been found. What were they for ? Well perhaps the head of the tribe sent scouts out to mark a tally stick by scoring marks onto the tally in one-to-one correspondence with either goats they were herding or hunting or the war aprty from another tribe or whatever. Perhaps the headman could count but the scouts ? A matter of conjectures. These marks predate known cave writing, though not pictures. So the scouts were distinguishing and tallying. Mathematics is about a whole lot more than machine calculation and numbers, geometry for instance. How would your machine tell the diffeence between a square and a rectangle or discuss the symmetries of a snowflake ?

Shortest recorded in the last ~50 years, but reconstructions show it was faster in the past, and we’ve gone to negative excess length of day in the atomic clock era. In the early 2000s ELOD was negative during part of the year. No leap seconds were inserted for 7 years. Then the days got longer again. https://www.ststworld.com/understanding-day-length-fluctuations-what-they-are-and-what-causes-them/ Even with the fluctuations, you can see the long-term trend is toward longer days, and that we’ve had negative ELOD in the 1930s and a much larger dip from ~1860-1900 https://geodesy.geology.ohio-state.edu/course/refpapers/dLOD_1800-2000.pdf Earth rotation rates have geologic- and climate-related contributions, so focusing on very short time scales is a tad misguided

If f is in the direction of r'(t) then it is in the direction of increasing r(t). So, the directions of f(t) and r'(t) will also be same. Definition of dot product: And ds =|| r'(t)|| ⇒ [math]\sqrt{x'(t)^2 + y'(t)^2}dt [/math] so, work = force × distance = [math] \displaystyle\int_C f\cdot dr= \displaystyle\int_C \left\vert\vert{f} \right\vert\vert ds[/math]

If my understandings are wrong then I will forever be replying to posts opposing my arguments and consequently will NEVER be able to present any evidence supporting what I say. This is simple logic. Chronological censorship in action, in this instance driven by the X chromosome's influence, (traitor males mostly) and intellectual arrogance.