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The Friedmann equation describing the evolution of the universe can be written (in units where $\hbar=c=G=1$ as $$H^2+\frac{k}{a^2}=\frac{1}{3M_{\mathrm{pl}}^2}\rho,$$ where $H=\frac{\dot{a}}{a}$, $a$ is the scale factor, $k$ is 1,-1, or 0, depending on the curvature, and $M_{\mathrm{pl}}$ is the planck mass. $\rho$ is the energy density of a combination of ...


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When considering issues of this kind, you need to distinguish between a scientific model and the physical reality. When people say this or that is "what will happen" what they are really saying is "under the assumption of model X, this is what will happen" where model X is some sort of reasonable combination of what we know about gravity, ...


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Based on current accumulated astronomical evidence, the Big Freeze is closest to what I understand that most cosmologists accept as most likely. There is a slight detail you left out, which is the black holes will eventually convert all of their mass into particles as a result of Hawking radiation. Also, since gravity keeps the mass of a galaxy (and also ...


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The question is based on a natural misconception. In everyday life we encounter surfaces and objects embedded in normal space. We do not usually encounter things that have no embedding, so our intuitions tend to assume that there must be some kind of place or space to house everything, including the universe. However, in mathematics and physics there is no ...


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There are two possible solutions to this apparent “paradox”: The universe is infinite. The universe is finite, but wraps around on itself and so has no boundary. This is difficult to imagine, but it would a three-dimensional equivalent of the surface of a sphere or a torus, which are finite two-dimensional surfaces with no boundary. We have no evidence ...


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I think you are asking "Is it possible to have a static universe without incorporating a positive cosmological constant?". Please correct me if I'm wrong. If this is your question, FYI, a number of great minds has considered your idea before and, e.g., Schrödinger believed this can happen. I think this is a good question. A static universe implies $...


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When the universe is static (i.e., $\ddot a = \dot a = 0$), the Hubble parameter is always zero since $$H = \frac{{\dot a}}{a} = 0.$$ In this case, the dimensional densities, ${\Omega _i} = \frac{{8\pi G}}{{3{H^2}}}{\rho _i}$, are infinite, so, the right hand side of your equation is indefinite (actually, this is a common question about deriving Einstein's ...


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A common elementary coordinatization of the $2$-sphere of radius $R$ uses two angles $(\theta,\phi)$ as (co)latitude and longitude. If we embed the sphere into $\mathbb R^3$, then the points $(x,y,z)$ on the surface take the form $$\pmatrix{x\\y\\z} = \pmatrix{R \sin(\theta)\cos(\phi)\\R\sin(\theta)\sin(\phi)\\ R\cos(\theta)}$$ If we restrict $\theta \in (0,...


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The coordinates for a general sphere, with its centre at some arbitrary place, would be quite hard to capture in a single expression. But if you consider a sphere (I mean a 2-sphere located in the 3-sphere manifold) which is centred at the origin of coordinates, then the limits are easy: just pick one value for $r$ and let $\theta$ and $\phi$ vary over their ...


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It is impossible to be simultaneously spherical and flat, unless you refer to being locally flat. Spheres are a surface with positive curvature by necessity. As it is a curved surface. So, no, it is not possible to be a spherical Minkowski Space. Solvable by basic geometry. But, if we do live on a De-Sitter Space (Positive Curvature) it would appear ...


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An endless, fluctuation dominated universe, like ours will be according to theory, will produce anything and everything allowable by the laws of physics an infinite amount of times. That is just the logical conclusion based on the assumptions of heat death and expansion. That we are the type of observers we are (posses memories of the past, memories of a low ...


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The question is: Is the Boltzmann brain scientific? It might be sciency, but it is certainly not scientific. To believe that such a brain (or whatever structure, inclusive the entire universe that is visible now) can possibly come into existence is to deny the structures that can give rise to the non-Boltzmann brains. Why is that? Actual brains somehow must ...


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As the formation of a "Boltzmann Brain" seems vastly more likely than the formation of the universe (or even the local universe) that that brain could be imagining, would an eventually exact recurrence of it not seem likely to occur on a much shorter time scale than the full-scale recurrence of the universe or multiverse that it would be ...


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What it means is that Temark is a neo-Pythagorean. It was Pythagoras that first suggested that the nature can be explained with reference to mathematics. This was taken up by Plato who was influenced by Pythagoras in his $Timeaus$ where he used triangles as the basic building blocks of nature. Aristotle also referred to it as: The neccessary appears in ...


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It's just an opinion, but there seem to be a lot of useless speculative theories in physics at the moment. The theories should at least make testable predictions for us to take them seriously. The topic you described may well fit into the 'useless' category and mean nothing. Others include the multiverse theory, colliding braneworlds etc...


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I think the Boltzmann brain argument is similar to the cosmological constant problem: it's an apparent prediction of current models that is clearly unacceptable, but it's not clear how to evade it. There obviously is something wrong with the calculation, but that's no reason to dismiss it out of hand without knowing how it's wrong. The image of a lone brain ...


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I have heard about 'Boltzmann Brains'. I generally took them to be yet more wild speculations based upon the supposition that the wilder the suggestion the more likely it is true. This is just wrong. It captures the imagination of the public at large because of it's outlandishness. And it is of a piece with the current crisis over misinformation, ...


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If in my universe, there are $3$ electron charges and $4$ positrons, so that there is $1$ unit of positive charge. Then one day a photon reacts via $$\gamma \rightarrow e+e^+$$ so that now there are $4$ electrons and $5$ positrons. That makes your statement wrong! While still, the total charge remains $1$ unit. Edit: Please find the details of the pair ...


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I don't think that this question can be answered on a "fundamental" level. To judge the computational resources you must have an underlying problem that you are going to compute. But this underlying problem is always limited/based on our current understanding of physics. So, unless we manage to unify all branches of physics into a great grand super ...


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So, let's start with the fundamental vagueness of the question. The reason behind this vagueness is the lack of clear unpacking of what is meant by "computation" or "prediction" or "modelling". Let's suppose for a second that you actually do have an infinitely powerful computer that is able to simulate the dynamics of any number ...


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The question could easily introduce opinion-based answers. To avoid this danger, I'll try to stick to our present knowledge and understanding, avoiding any speculation on the future evolution of the field. I think that two related assumptions are underlying this question and should be elicited. The first is that computation always implies the possibility of ...


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First, no theorem can definitively demonstrate anything about the real world. They can only show that their conclusions follow from their assumptions. Second, even when the assumptions of this particular theorem are met, the conclusion seems quite modest: [A] cosmological model which is inflating – or just expanding sufficiently fast – must be incomplete in ...


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