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So I realise that 70+ questions have piled up since I used this site last year. It's not exactly the ideal format for asking anything of length, but I can try to get to these at some point. Apologies.
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What an absolutely fascinating question. I have no idea how one could give an honest answer to that though, because it would be pretty impossible not to have your thinking already biased by prior experience. I suppose I do have the advantage over the crows in that I am not limited to using up to three tools in sequence to obtain my tasty treat, but they certainly manage fairly well for themselves.
If we try to take it back to the minimum exposure that the human brain could have and still be functional, I've seen somewhat similar tests done with human babies, where they have to figure out how to use tools in combination to get a toy out of some contraption. Most babies can figure it out eventually, but not all (just like with the corvids). Since tool use is such a big part of our species' existence, it would be hard to imagine that an adult human couldn't out think a crow or other non-typical tool using species in a similar task, but all animals have their own unique intelligences.
Mo at Neurophilosophy has two great posts on avian intelligence: http://scienceblogs.com/neurophilosophy/2007/08/avian_intelligence.php, http://scienceblogs.com/neurophilosophy/2009/08/those_clever_corvids.php. -
Your thinking is correct, nothing should ever truly get sucked into a black hole in finite time, from the perspective of any observer outside of the event horizon. It would take an *infinite* amount of time for anyone to see the sun get pulled into a black hole.
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2 + 2 = 0 on the ring (ℤ₄, +, ⋅) ,
Where ℤ₄ = {0, 1, 2, 3} and, for any x, y in ℤ₄, x + y is defined to be their sum in ℤ mod 4; and, for any x, y in ℤ₄, x ⋅ y is defined to be their product in ℤ mod 4.
Or "2 + 2 = 5", if you're into Doublethink. -
Alight, manifolds are great fun! More often than not, when someone is talking about manifolds they mean a topological manifold.
⓿ Definition: Topological Manifold: a second countable Hausdorff space that is locally homeomorphic to Euclidean space. (Not the be-all and end-all definition of manifolds anymore – we can broaden the concept to include non-Hausdorff spaces and still call them manifolds).
This isn't necessarily the most helpful or “beginner” definition though (especially without topology), and it's probably better to start from some really basic real analysis and work up from there. (There is a scheme-theoretic definition which is actually quite nice, but probably the least friendly option).
So we'll go back to basics.
First, before we can actually define what a manifold is though, we need to talk about Euclidean spaces (in a way that might be a little more formal than fun, but I think it's important to set up our definitions properly before we can get to interpretations). I'm only going to assume really basic knowledge of what a set is (http://en.wikipedia.org/wiki/Set_%28mathematics%29), what n-tuples are (http://en.wikipedia.org/wiki/Tuple), what the real numbers are (http://en.wikipedia.org/wiki/Real_number), and what a limit is (http://en.wikipedia.org/wiki/Limit_%28mathematics%29) – so prepare for a first year (second year?) analysis course of definitions.
❶ Definition: Euclidean n-space (ℝⁿ): the set of all n-tuples (x¹,...,xⁿ), where xⁱ is a real number.
This is a pretty familiar one for most people because we're already familiar with ℝ¹ = ℝ = {x∈ℝ} (to write it really redundantly), the set of all real numbers, and if you remember plotting things on Cartesian grids, you remember ℝ² = {(x,y) | x, y ∈ ℝ} (the set of all ordered pairs), etc.
Now we need to talk about intervals:
❷ Definition: Open Rectangle: The collection of all n-tuples (x¹,...,xⁿ) = x, where xⁱ ∈ (aⁱ,bⁱ), A = (a¹,b¹) X ··· X (aⁿ,bⁿ) ⊂ ℝⁿ is an open rectangle.
❸ Definition: Open Set: The set U ⊂ ℝⁿ is "open" if for each x ∈ U, there is an open rectangle A such that x ∈ A ⊂ U.
Here's a classic picture from Spivak's “Calculus on Manifolds”, although again, this is all pretty regular stuff. http://individual.utoronto.ca/sck/Spivak_Manifolds_OpenSet.gif
Now we need to talk about functions:
❹ (Informal) Definition: Function or Map: A function f from ℝᵐ to ℝⁿ (f: ℝᵐ → ℝⁿ) is a rule that maps each point x in ℝᵐ to a point in ℝⁿ.
Simple and familiar enough.
❺ Definition: Differentiable at a Point: A function f: ℝᵐ → ℝⁿ is differentiable at a ∈ ℝᵐ if there is a linear transformation λ: ℝᵐ → ℝⁿ such that, for h ∈ ℝᵐ,
lim (h → 0) |f(a + h) – f(a) - λ(h)|/|h| = 0.
(Note: f(a + h) – f(a) - λ(h) ∈ ℝⁿ).
(apologies for not being able to typeset that properly here)
This might seem like a slightly unusual definition of a differentiable function (perhaps the λ(h) part looks strange), but it is necessary to preserve the generality that I want. The discussion on what λ(h) really is would get a little lengthier than this post probably needs to be though.
⑤ Notation: The linear transformation λ is written as Df(a), “the derivative of f at a”.
❻ Definition: Differentiable On a Set: A function f: ℝᵐ → ℝⁿ is differentiable on A ⊂ ℝᵐ if f is differentiable at a for every a ∈ A.
❼ Definition: Continuous at a Point: A function f: A → ℝⁿ, for A ⊂ ℝᵐ, is continuous at a ∈ A if lim(x → a) f(x) = f(a).
❽ Definition: Continuously Differentiable at a: A function h : ℝⁿ → ℝᵐ is said to be continuously differentiable at a ∈ U, for open U ⊂ ℝⁿ , if Dᵢhʲ(x) exists and Dᵢhʲ is continuous at a.
We're getting really close now, I swear.
❾ Definition: One-to-One Function: A function f: A → ℝⁿ, for A ⊂ ℝᵐ, is said to be one-to-one if f(x) ≠ f(y) for x ≠ y, for x, y ∈ A.
❿ Definition: Inverse Function: For a one-to-one function f: A → ℝⁿ, for A ⊂ ℝᵐ, there is an inverse function fˉ¹: f(A) → ℝᵐ, such that fˉ¹(z) is the unique x ∈ A for f(x) = z.
We're very close now:
⓫ Definition: Diffeomorphism: For open sets U, V ⊂ ℝⁿ, a differentiable function h: U → V is said to be a diffeomorphism if it has a differentiable inverse hˉ¹: V → U.
⓬ Definition: Cᵖ-Diffeomorphism: If h and hˉ¹ are ᵖ times continuously differentiable, h is a Cᵖ-diffeomorphism.
⓭ Definition: A k-Dimensional Manifold: A set M is a k-dimensional manifold if for every x ∈ M, there is an open set U ⊂ ℝⁿ, such that x ∈ U, an open set V ⊂ ℝⁿ, and a Cᵖ-diffeomorphism, h: U → V, such that,
h(U ∩ M) = V ∩ (ℝⁿ x {0}) = {y ∈ V : yᵏ⁺¹ = ··· = yⁿ = 0}.
ie. U ∩ M is, up to a diffeomorphism h, ℝᵏ x {0}.
Or in picture form: http://individual.utoronto.ca/sck/Spivak_Manifolds_Manifolds.gif
So, we have *a* definition of a manifold now.
In other words (how you'll often see it phrased in analysis books), a set of points M is a manifold if each point of M has an open neighbourhood which has a continuous one-to-one map onto an open set of ℝⁿ for some n (or even more distilled, locally, M looks like ℝᵏ).
(apologies for the typos that are bound to exist in the above) -
Simple, we don't. We can only know things within some certainly; science can never make statements about any ultimate truths, but that doesn't mean that we shouldn't trust in science (we just have to remember that there is always some experiment that could come along and change what we believe).
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Ah, if wishes were horses. Sure, it would be nice if there was some magical being that could cheer me up when times get hard, but me wanting that doesn't make it true. I also would much rather live in a universe where I had an endless bathtub of money, but again, that isn't going to happen.
For me, I pick wanting logical consistency to wanting magic. I quite enjoy living in my godless world because that means I can rationally trust in the scientific method. As soon as you insert some physics-defying being, you lose the ability to actually trust in empirical methods and results. I can no longer imagine perfect experiments that I can preform n times and get the same outcome n times (science is about repeatability, after all). As long as there is some god-like being that can interact in our world, she has the ability to come down and stick her invisible hands in front of my detectors as many times as she would like without me knowing about it. Adding in magic results in absolute chaos! I'd have to pick a new career, because no laws of physics could rationally exist!
When I look across the street and see no cars, I can rationally trust in statistics and say that it is exceptionally unlikely that a car will appear (thanks quantum mechanics for making it a non-zero probability though) and kill me as I cross. I add in magic, my statistics are meaningless because they can't take into account the actions of some godlike being who exists outside of physics (and might want to throw a car at me one day). How could anyone rationally function in a world where they don't have any way to predict anything? Answer: They can't.
Sure, you can imagine that only wonderful and benevolent beings can posses this outside-of-physics-magic, so they won't be throwing any cars or adding in or removing any particles in my interferometers, but how can you be sure? If you pick up any version of the Christian bible that contains the Old Testament, there are numerous stories of that god causing incredible mayhem without much provocation. I certainly don't blame Christians for rewriting her personality for the sequel, because without the New Testament's god being reasonably kind are more predictive, I don't know how believers could sleep at night.
When times are hard, I don't need faith in a god to feel better, I use my scepticism and appreciation for probabilities: "I wonder what the odds are that things could actually get worse?" -
I do agree that "whys" get meaningless at some point ("Why does the universe exist?", for example), but it would be very premature to stop asking questions about the dimensionality of our spacetime and on the differences between space and time in general, because they aren't just idle questions. If we want to have an effective theory of quantum gravity, there is a good chance that we actually require those answers. It appears that the number of dimensions we see at the distance/energy scale we live in, may be different than the number of dimensions that exist at a smaller, perhaps more fundamental, distance/energy scale. Interestingly, while most people who are familiar with popular science these days have heard about compactified dimensions ("small", extra dimensions rolled up, so we shouldn't be able to perceive them at our distance/energy scale) and the 11-dimensional M-theory, there is an even more interesting (in my opinion) apparent 2D-ness about spacetime at very small scales that seems to be true although not understood. Questions, like "why that is?", are rather important in understanding the fundamental nature of our universe, and we're not just spinning our wheels by asking them. The fact that we can, fairly rationally, talk about compactification of dimensions, emergent dimension, or the apparent dimension of spacetime these days, suggests our understanding of the universe is really progressing along those lines.
I do think that we'll eventually hit meaningless "why" questions, but how can we actually know until we ask them? -
The second part of your question: "gravity to separate space from time to give independent analysis of space or time, I won't go any further with this line of inquiry. Thank you for your time."
Nope I'm afraid not; gravity is part of spacetime as a whole, not space or time separately, so we can't talk about them totally disconnected from each other.
We do separate space and time quite often though when we're actually doing general relativity to make things conceptually (and mathematically) easier on us. In the ADM (and other) formalisms of GR, we foliate spacetime into a family of spacelike surfaces, each one at a different time so we can deal with "space" and "time" separately. However, there isn't a unique foliation for a spacetime manifold, so we could do it again for a different time coordinate/"observer" and get another family of spacelike surfaces, both describing the same spacetime.
So, we can analyze (and often have to) space and time separately in GR, but that analysis is dependent on the definition of time we are choosing.
There are many people who are working on a more fundamental sense of space-time separation though (see anything by Julian Barbour on the static universe/"end of time"), but that work is still fairly disconnected from reality. In much of quantum gravity, we come upon some interesting issues when dealing with things on exceptionally small scales (spacetime loses a lot of its meaning when we are forced to add/remove dimensions when scaling), but again, there is no clear and agreed upon phenomenological meaning behind most of that at this point.
Space and time are quite different from each other (Why does time have a direction but space doesn't? Why, for our scale, are there three spatial dimensions and only one temporal dimension?), and figuring out why that is is still an exciting puzzle for relativists, mathematicians, and philosophers alike. -
1) You're basically right. Quantum electrodynamics (QED) tells us that the vacuum is actually full of energy in the form of virtual particles. Interestingly, and a little problematic for theorists, QED doesn't allow for a maximum number of virtual particles to be created, it's in fact infinite (countably infinite), so the vacuum is actually full of an infinite number of virtual photons with all different wavelengths. Now the Casimir Effect: We place two mirrors/plates in the vacuum some (small) distance apart from each other. Some of the wavelengths will be short enough to fit between the plates and some will be too large. Despite there being an infinite number of waves in the vacuum (the logic becomes awkward and still contested by some here - renormalization doesn't actually make people overwhelmingly happy when they have to think about it), since the plates are close together, we are limiting the waves that can exist between them to just be the "very small" ones (so the longer waves are only outside of the plates), thus, there will be less vacuum energy (fewer waves) between the plates than there will be outside of the plates. This energy differential results in the plates getting pushed together (or "attracting" one another). One could also imagine a scenario where the plates would repel each other if the distance between them was large and the universe was relatively small and compact (see http://en.wikipedia.org/wiki/Casimir_effect for more on the Casimir Effect).
Hopefully the infinity of the vacuum energy gave you pause (why should there be "fewer" waves between the plates? Nothing stops us from having infinitely many arbitrarily small waves); one would usually expect that the cardinality of the set of virtual photons within the plates should be the same as the cardinality of the set of virtual photons outside of the plates. That's where renormalization jumps in. While we know that there are an infinite number of photons (and should expect an infinite amount of energy), we kind of pretend that it's not really infinite and in fact, generally say that it's "zero". Once we make this step out of convenience, we're able to calculate the vacuum energy inside and outside of the plates by just looking at a finite number of virtual photons and measuring the difference from the two sets. Then, we say that this difference is preserved from the finite to the infinite case (a statement that makes most mathematicians choke). Weirdly, this procedure that lacks in rigour seems to completely work and gives overwhelmingly precise predictions (QED holds up phenomenally well in experimental tests). Why? That is an excellent and baffling question (new, improved, and rigorous renormalization schemes are constantly being sought).
So yes, the Casimir Effect does result from the fact that the vacuum is full of an infinite number of virtual particles. However, it's a very strange kind of (non-mathematical?) infinity.
2) No, not in the slightest. -
I don't know any futurists on twitter, but that doesn't bother me. Tipler's Omega Point is total non-science and not something I've ever felt the need to lose any sleep over. As for multiverse people, none of the active theorists I know who really work with multiverse ideas are on twitter, but again, as it's not science, I'm not that broken up about it (mind you, I do have some friends who work with multiverse concepts in cosmology, but they don't do it in a pseudo-science/"Physics of Christianity"/hand-waving way).
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Is that really a thing people have to overcome? I'm thrilled to have peers that know more than I do about subjects, mathematics or other (if I wasn't, being in academia would be pretty unpleasant, as practically everyone is the number one expert in at least some subfield). Having mathematicians around is great because they are wonderful to discuss things with and learn from. There will always be people who know more than you in some area, that's life. You should make the most of it and try to learn from them.
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So, Quantum Electrodynamics tells us to picture the electromagnetic force as being mediated by the exchange of virtual photons between charged particles, which we draw very nicely in Feynman diagrams. (http://en.wikipedia.org/wiki/Quantum_electrodynamics)
Sometimes I think that this Feynman diagram image becomes a little too ingrained in our minds and we start to get confused into thinking that there are these particles being tossed back and forth between charges. Virtual particles get the special privilege of being allowed to be off the mass shell (http://en.wikipedia.org/wiki/Mass_shell), ie. virtual photons can carry mass, thanks to energy-time uncertainty. It's the energy-time uncertainty, that gives virtual photons their existence in the first place, that allows EM to be both a repulsive and attractive force.
You can imagine it by a sloppy example: If you hit someone with a real, massive, ball, the momentum they absorb from the moving ball pushes them backwards a little bit (repels them, if you will). Now imagine it were possible to have this special kind of virtual ball that could could have momentum that could knock them backwards or "knock" them forwards. That's the rough idea of what is happening with virtual photons and EM being able to be both attractive and repulsive.
This classical ball example makes it confusing when it comes to wondering how the virtual photon going off knows which "momentum" to carry, but that's when we have to remember that we are dealing with quantum mechanical particles, and the notion of a classical path really doesn't apply. -
For starters, we have very little reason to believe that the graviton is actually a real particle, so I won't bother addressing it.
The mass of the photon, on the other hand, is actually a really interesting question. Much of our modern physics is built around the assumption that the photon has no mass (most of our gauge theories in standard particle theory demand the photon to be massless via symmetry breaking, in the Standard Model, the Higgs can't couple to it, so it can't obtain mass from the Higgs mechanism, also, if it wasn't massless, we'd have to rename "the speed of light", because if the photon had mass, it couldn't actually travel at c).
Now, the basic Standard Model has the Higgs multiplet breaking SU(2)xU(1) to U(1)*, which only has single generator (this lazy group theory speak translates to saying that there must be one massless gauge boson (we match up with the photon) and three massive gauge bosons (the Ws and Z). See http://en.wikipedia.org/wiki/Standard_Model#Field_content for details.
However, just because that version of the Standard Model has the photon being massless as a trivial consequence, it doesn't mean that it's a closed question (who says that's the right model for particle physics anyway?). Experimental tests have been able to support the photon having a very, very small mass (<1×10−18 eV), but that is not zero (after all, we used to think neutrinos had no mass and the tiny mass that they do have allows for exciting new physics, ie. neutrino oscillations).
There are theories, modifications of the Standard Model, that allow for massive photons (see Stueckelberg Theory http://en.wikipedia.org/wiki/Stueckelberg_action and Proca Theory http://en.wikipedia.org/wiki/Proca_action), which could, in fact, turn out to be true.
At this point, unless you accept the Minimal Standard Model with the normal Higgs (doublet) Coupling Mechanism as religious truth, there is no deep reason to assume the photon can't have a very small mass - and experiments may actually find that it does someday (although I would be pretty shocked - relativity would be completely unscathed, but high energy physics would get a little turned on it's head). -
For starters, it's not just electrons and protons that make up all regular matter - what about the neutron? Today though, even the protons-neutrons-electrons definition of matter is a little antiquated because we're familiar with other forms of matter, although very uncommon in day to day life, made up of a whole menagerie of different quark and lepton combinations (see: http://en.wikipedia.org/wiki/Matter#Quarks_and_leptons_definition).
Since you're asking a relativist what spacetime is, you might not get the answer you're looking for, but spacetime is just spacetime, it's not made up of anything, unless you want to consider the discrete points (events) its composition. Spacetime is just a manifold, so it's like asking what's a manifold made up of (a set?). The only thing spacetime has in common with matter is that they both exist (neither is "nothing", so to speak). So the classical relativist in me answers: "nothing" or "events", depending on your interpretation.
Even from the quantum gravity perspective, when we are modelling discrete spacetimes (ignoring the continuous kind that string theorists use, because those are really no different than the simplest classical ones), whether we're using lattices or some form of causal sets, spacetime is made up of events - nothing tangible. -
Here seem to be the continuation of your "question":
"The creation of space is by virtual particles popping in and out of the void, and only become real when they act one upon the other through the temporary space that is also created momentarily, thus the metamorphosis transpires."
"To me space is not priori, the void is priori."
"Virtual particles create the environment in which real particles can inhabit and real particles create the space between them. Void transformed to space."
"Three assumptions have to be made: An initial singularity (not necessarily a point) must exist. A void must exist. The singularity must be contained in a void. "
"Time is created by the interplay between the void and the singularity. Time is contained within the Universe, the Universe itself (as a whole) does not know time. I know this sounds strange and I do not expect you to accept this… "
"OK, when I used the term void I am not refering to http://en.wikipedia.org/wiki/Void_(astronomy) but to an initial TRUE void."
You can believe that all you want but it doesn't contain a speck of physics. It is completely meaningless to discuss this "true void" that you speak of and outside the realm of science - science only deals with things within spacetime and spacetime itself (and as a relativist, you're not going to get sympathy for some non-spacetime centric model of physics from me). Virtual particles can not exist outside of spacetime, it's as simple as that. You can't have matter, virtual or otherwise, without it being contained somewhere (and this "outside of the universe" void doesn't count as "somewhere").
This kind of stuff has a place in the Dr. Who universe, maybe, but not in physics. -
For whatever reason, when we talk about particles of that scale, we switch from curvature language to the language of self-force and thus we want to use the, beautifully named, MiSaTaQuWa equations.
For a great primer on the gravitational self-force (and intro to the MiSaTaQuWa equations) see http://www.physics.uoguelph.ca/poisson/research/self.pdf by E. Poisson or Wald's introduction to the topic: http://arxiv.org/abs/0907.0412 -
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Depending on what dark matter candidate you like, you can probably touch dark matter.
Axions are one dark matter candidate (although an especially contrived one) that are supposed to be a fundamental particle with a definite mass, just like any other real particle, so there is no reason why you couldn't touch them as well as you could touch an electron (although what that would mean is anyone's guess - "touching" is really a matter of interpretation here).
MACHOs (Massive compact halo object), another candidate, are supposed to be massive astronomical objects composed of normal baryonic matter (pretty much what everything else is made up of), so those would be fair game to handle (although seeing as the list of popular MACHOs include black holes - no one will ever know if you did actually touch it -, neutron stars, and brown dwarfs, the "gravity" of handling them may be undesirable).
RAMBOs (Robust associations of massive baryonic objects) are in the same boat as MACHOs, you could logically touch them.
WIMPs (Weakly interacting massive particles), the most popular dark matter candidate, only interact weakly and through gravity so even though they have mass, and you might want to be able to think that you can hold anything that has mass, "touching" is really an EM interaction, so you could "hold" but not "touch" a WIMP.
Anything that interacts via electromagnetism AND gravity you can touch, unless it is relatively very small, and then quantum uncertainty comes in and it becomes a matter of interpretation (but I'd still say it's fair game). -
That's because there are forces holding things together - Electromagnetism is a very strong force and it holds the majority of material objects together on the macroscale, regardless of what the rest of the universe is doing (the Strong force is also certainly helping, although it's main work is happening at the quark-quark interaction scale). When you scale up from EM, gravity is what keeps our solar system and galaxy together. The universe is expanding, but that doesn't change the laws of physics.
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S.C. Kavassalis’s Bio
Permanent student of mathematics, physics, and sometimes, the philosophy of their intersection.
