MigL

I would think that the concept of a 'metric space' is more fundamental. "A set with the notion of distance between its elements." from Wiki. The corresponding metric, is then, the mathematical tool to measure those distances. And the example most people are familiar with is 3D Euclidian space with its usual distance between points. But another example would be on a sphere with angular distance. And not necessarily on a manifold, it could be the 'relation' between members of a set that aids transforms between them. My apologies if the terms I use aren't up to Studiot's standards, but this isn't my area of expertise.

Thanks guys.

V Putin doesn't care about ruining land, even his own. Look at what the former USSR did to Chernobyl, the draining of the Aral Sea, attempted nuclear extraction of their 'oil sands' equivalent, ground-based nuclear tests, and much more. He doesn't care about the people; he'll be in Moscow or St Petersburg. If the US does nothing about it, which is likely considering the President, I doubt Europe will do anything about it either. France possesses almost 300 strategic M51 submarine-launched nukes for deterrence but has as few as 20 TN81 tactical nukes. The UK gave up their air launched tactical nukes about 30 years ago and only have strategic Trident Submarine deterrence. I don't think either of them will launch hi-yield strategic ballistic missiles at Moscow/St Petersburg. Apparently the UK has ordered an additional 12 F-35As, with the capability of carrying tactical nuclear missiles or bombs and IIRC France was to build a model of the Rafale specifically for the carriage of tactical nukes ( although, not being 'stealth', I don't know how far it would get into heavily defended Russian airspace ) so maybe they will be both re-introducing tactical battlefield nukes.

Maybe if you'd looked up my reference on space-time diagrams Spacetime - Wikipedia Spacetime diagram - Wikipedia You would have realized that a light cone is a space-time diagram rotated about the time axis. And that any horizontal line, or plane as you describe, is space-like, and involves faster than light information transfer or motion: you know that's impossible, right ? ( actually, true horizontal implies instantaneous transfer or motion ) Until you exorcise these non-sensical preconceptions you have, by doing some basic research, or asking questions, you will be always pitching assertions that don't make sense. No matter how long you make your posts !

Do you realize that nothing Mars does can ever affect you before 22 minutes have passed ??? It could blow up, it could vanish, or even become a Black Hole; the fact is, it is not happening 'now', it is happening 22 minutes from 'now', as that is the only time it will have any effect on you that you can detect. Before the 22 minutes, you have no idea what, if any, changes have occurred and it might as well be in another space and time, for all you know. What were you told about 'common sense' and relativistic effects ? And what relativistic 'observations' have you made that are explained by your conclusions ? Anyone or any detection equipment in the same frame would not notice anything. If you and I are in space and a satellite passes by us, would you notice differently if I said I couldn't tell who is actually moving, us or the satellite ? No consideration involved; you are a peer. But we expect a little better from someone who's been active on this forum for over a dozen years, and has taken 'Physics at a University level". So why not learn about frames of reference, in what circumstances relativistic effects become noticeable ( and NOT common sense ), and the proper use of space-time diagrams ( light cones ), so you can make sensible contributions to the conversation ?

No wonder you're so confused about the science. You post one thing, and someone gives you advice on the proper reasoning and references, which you then ignore, and post another absurd claim regarding a different subject. Your scatter-brained postings are confusing to everyone else also. I think you need a hobby other than science.

Light cones are part of space-time diagrams and can't be moved as you please. There are certain rules they follow. Assuming the proper scale on your axis, the space-time diagram will have the speed of light at a diagonal 45o angle. Anything following that line is called 'time-like' and is moving at c . Anything moving more horizontally means that space is being traversed faster than c ( an impossibility ) and is termed 'space-like'. What we usually experience is motion more vertical, which means space is being traversed slower than c , and termed 'time-like'. How, exactly, are you moving your two light cones ? ( maybe understanding this will clear up your misconceptions Spacetime diagram - Wikipedia )

Interesting; did not know any of that. Thanks @toucana

Does that view extend to Quantum Mechanics ? Do you believe 'reality' is a probabilistic mathematical superposition of states until an observation/interaction collapses that 'reality' to a particle ? or do you believe in an underlying actual reality with non-local characteristics ? While I have no problem with no local reality for quantum effects, I cannot male the jump to considering space-time as 'something' that can be curved, stretched, compressed, or even cut like a fabric.

I see his school work has moved to Thermodynamics, after Chemical Bonding. Show us YOUR thinking/reasoning so far. ( hint; look at the effect of differing T, when the same amount of heat, Q, is added, in your equation for change in entropy, S )

In everyday experience, the effects of relativity are trivial to non-existent. I would have thought you'd know that after all these years. Go back, think about it, and come back with some good questions.

Except when you say 'implies' it could also imply that it acts like it is curved ( mathematically ) without necessarily being so, as space-time is not a thing, but a mathematical construct. It ( curvature ) cannot be measured directly; other effects are needed to 'imply' curvature. Remember that what we consider 'things' are three dimensional, and to join time to the three spatial dimensions into combined space-time, requires the time dimension to be orthogonal and therefore imaginary ( i ). But as I said, your view of 'reality' might be different from mine, and while I can't falsify yours, you cannot falsify mine ✌ .

Yes. At the Event Horizon the space-time curvature is such that no geodesics, or trajectories, lead outside the Event Horizon., not even for light. It is curved in a way that all paths lead to the future collision with the center of the Black Hole ( whatever that may be ). A neutron star isn't massive enough and its constituent neutrons can resist further gravitational collapse due to neutron degeneracy pressure.

A bit like wave/particle duality; is light a wave or a particle ? Well, it depends on the experiment used to detect it ( ie, measure it ). Could be both; could be neither. Because we're not actually measuring it; we measure how it acts on other things.

You can measure how light bends around massive objects, Shapiro delay, gravitational lensing or interference fringes due to gravitational waves ( LIGO ). You can measure how gravitational time dilation modifies clocks ( any type, even light frequency ). You can measure the trajectory along geodesics of massive objects in free fall. The bolded words are what is actually being measured. The left hand side of the EFEs represent the geometry of space-time, and g , the metric tensor is the tangent space connector ( affine ). This is not an actual measurement, but a mathematical construct to aid calculations. I would say we have different definitions of 'real'. ( I hope you don't also believe space-time has a 'fabric' 🙂 )

NO The particle over there doesn't know that you shook the particle over here, until the 'information' gets to it, and that information has a maximum speed, c . And you'd be wrong. Again.

You cannot put a measuring tape to space-time and measure the deviation from flat, but you can measure the effects of curved space-time on bodies, or phenomena, within it. Similarly, I can measure the effects of a Phonon in a crystal lattice. Should I take a Phonon as a 'real' particle ? Or should I consider it a mathematical construct that aids calculation ? We do not see 'reality', only its effects.

That was sarcasm, as I don't believe in imaginary beings that can observe the universe from 'outside'. Any observation you can make, I can give another which differs in time and location, because it is made from a different frame. And it, and all others, will be just as valid. If you want to use that 'excuse' for your failure to comprehend relativity, that's fine. The rest of us don't have that problem.

And the 'curvature at the singularity of a Black Hole is infinite. Which means that the equations which describe the singularity and the concept of curvature cannot be applied in that domain. Mostly because it is a 'made up' concept, and while you can measure 'effects' because reality acts ( for the most part ) like it is curved, the 'concept is not applicable in some extreme situations.

Yes, it may happen only once, but the relationship with the rest of the universe differs with the frames of reference of different observers in the universe. There is no absolute frame of reference, observer, or 'now'. If you won't believe me, ask the only absolute observer that I know of, God; let me know what He tells you.

As per our PMs Every different observer carries their own different perspective, shaped by different, and unique, trajectories through space-time; it is a tenet of Relativity.

Yes, the space-time of the model acts exactly like the reality ( in most situations ) and has been confirmed to a very high degree. In the model space-time can be curved and is malleable; whether reality is also, is a question for Philosophers to debate.

No it would not. Information, even mechanical, is limited by the speed of light. Is there something you wish to ask, before stating absurdities ?

The old forum software used to send Email notifications to my phone when someone on SFn had sent me a PM. The new software doesn't seem to do this anymore. Is that feature not supported, or can it somehow, be turned on in the settings ?

We are not big fans of the current state of AI on this forum. It is fine for people who want regurgitated, and often wrong, knowledge, albeit drawn from vast sources, but in terms of 'learning', AI isn't even at the level of a month old baby, or a dog for that matter. Quantum computing is in the same boat as Fusion reactors. For 40 years a breakthrough has been just around the corner. Except we have a working example of a Fusion reactor 93 000 000 miles away, so we know it works. Not so sure about Quantum computing. Changing games ? mRNA research changed ( and saved ) lives.