(Indeterminate, like me. Think outside the box, but when you step outside the box ... try to keep one foot in)
Is one analogy better than the other so we should forget the one and keep the other? Let's think it over now.The 2-dimensional analogy has the power of making people "feel" (from sensitive experience) how a ball thrown towards the mass will drift off of its initial trajectory because of the local curvature of that diminished space-time. In that sense, the 3-dimensional analogy does not do a better job.Best,
Well first, welcome back!2nd, yah I can see what you're saying, but my objection is in showing something 4-D as 2-D. I'd rather see it as 3-D, especially on a 2-D page, where 3-D is a minor intuitive leap.However, in Science Museums, the whole metal volute thing works fine, especially as extra income! I always enjoy rolling a penny down those things. You? :-)
Hi Steve,The problem of course is even in this picture space is only considered while time is ignored. That is in essence what we have is a four dimensional phenomena that serves to be inadequately served by either analogy. More importantly it has matter/energy and space as distinctly separate entities, rather than as Einstein would have had them taken being inseparable as enabling nature to demonstrate such a character. The defining situation of course being a black hole, where its often thought to be a place simply compacting matter and energy to extreme densities, while it is so often forgotten it having the same effect on space-time. So my personal way to look at it all is as to imagine the earth as a dense bit of foam, while the surrounding space-time having the density of foam almost gossamer like with its compression (density grandaunt) higher nearer the earth and lessening as one moves outward. This too admittedly is a feeble analogy, yet for me stands as being truer in relation to Einstein’s vision There is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field. Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space. This notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. It requires the idea of the field as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes’ idea, there exists no space "empty of field”."-Albert Einstein,” Relativity; The Special and the General Theory” (pages 176-177), Crown Trade Publishing Group Best,Phil
Thanks, Phil. Well, the black holes the thing, ain't it? It is a total warpage of space time. What the hey?!What I find frustrating about all these "experts" in Black Hole physics is how much they argue with each other, yet I find that mildly pleasant too as what's the point of studying cutting-edge stuff without discussion? The fact remains though that our knowledge is exceeded only by our ignorance, such that mathematical models, while pleasant, can't compete with actual observation and experiment.And the frustrating thing about black holes is thanks to the event horizon, everything that happens inside is akin to Gandalf's saying to the Balrog: "Thou shall not Pass!", meaning pfft it's impossible to observe anything inside one and we're left to Mathematical models and analogies.Sigh, I should really read Lee Smolin's first book.Back on topic, I can definitely "impart" in my head the time dimension to any static representation of space-time, for some reason I don't have a problem seeing one more dimension deep (although 5 dimensions is my limit). Back again to black holes. George Musser once explained (according to the maths) what happens inside a black hole (beyond the event horizon) which blew my mind and I will end with. You know I won't read Susskind's books because while brilliant the man's cockiness is a bit much even for me. Nevertheless and if you'd like to help, I sure would like the following to be explained more succinctly, no rather (since it is succinct), in layman's terms:SC: I spent a wonderful hour last night re-reading the grand chapters in your book on Black holes and Cosmology. Review mostly, but I have a question regarding Hawking radiation.You mention that the one of 2 virtual particles created near the event horizon goes into the hole, while the other flies off into the Universe (or goes in as well, but some fly off).Fine, I get and knew that.However, you then say this supports loss of mass via evaporation of the black hole? Huh? Why? Or rather, how?Incidentally, I found your overall explanations of BH's and Cosmology clearer and more lucid that Hawking's. Well done.GM: The particle that falls in acts to subtract mass from the hole. The reason has to do with what energy is: it is a quantity related to a symmetry of time. Momentum is similarly a quantity related to a symmetry of space. In the warped spacetime around the hole, the roles of space and time become switched, so the particle's momentum becomes its energy. Because the particle's momentum is the opposite of its partner's, the energy becomes the opposite - namely, negative. Negative energy corresponds to negative mass. SC: ?!?!
Hi Steven,I don’t know if I can do much better than Musser or as well for that matter. What I can say Hawking radiation has to do with is a consequence of QM as it allowing for the temporary existence of virtual particles that pop into existence in pairs, only to usually almost instantly annihilate each other as not to break the conservation law(s). This relates to the symmetry(s) that Musser mentions. However near a black hole’s horizon a particle can be taken in before it has a chance to annihilate with its partner. Since the one outside would permanently add to the total what’s drawn inside must balance this off by diminishing the mass of the black hole in it with becoming negative energy. In practice you could call this all smoke and mirrors, as the particle we see coming out (radiates) never went in or was part of the black hole to begin with. The thing being however for this to take place the region below the horizon must be time like rather than space like (which is consistent with GR) and yet how such a region would present itself (in terms of reality) runs counter to the imagination (well at least mine), where time becomes self evident dimensionally leaving space to be the more abstract notion.Best,PhilP.S. Please don’t ask me to give my explanation of the information problem as the reason for most of the infighting among the pros is they can’t agree on one themselves :-)
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