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The radius of the event horizon of a black hole of mass $m$ is given by: $$r_s = \frac{2GM}{c^2} \tag{1}$$ Let's consider your idea of taking $n$ black holes of mass $M$ and arranging them into a sphere. The total mass is $nM$, and the radius of the event horizon corresponding to this mass is: $$R_s = n\frac{2GM}{c^2} \tag{2}$$ Now let's see how ...

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Let me present a slightly different perspective to Luboš, though I'm saying basically the same thing. From our current location we can define an area of space called the future light cone. This is the region of spacetime that is connected to us by motion at less than or equal to the speed of light. If we draw a spacetime diagram then the lightcone looks ...

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As for the straight line, yes. All objects will continue moving along geodesics (a straight line in curved-space but sometimes a curved line in straight-space) if there are no external forces acting on them. Unless, by different velocity you mean the direction is not entirely radial to us. In that case, the expansion will cause the object's path to appear to ...

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It is in principle possible, at least for some time, to have a collection of black holes gravitating along the surface of a sphere such that one can still escape from the inside of that sphere. In other words, the inside of the sphere need not be hidden behind a gravitational horizon. However, as soon as the density of black holes exceeds a critical value, ...

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The main reason why physics isn't building on the assumption that the time is discrete is the fact that such an assumption is demonstrably incorrect. Physics is a natural science, a process of learning how Nature actually does work, not a movement to irrationally and indefensibly claim that there are some "cons" or "pros" about some arbitrary philosophical ...

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I am aware that my answer can sound surprising, too simple to be true, but please take a deep breath before downvoting.The answer has little to do with relativity. In SR it is the moving object that gets shorter , but space is stable. In such a universe, even if a body is receding at 2,3,30 c, its light will reach us sometime, and the time is short as it ...

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Replacing $GL(4, \mathbb R)$ with $M(4, \mathbb R)$ we have $${\cal V}:= \{M \in M(4, \mathbb R)\:|\: \mbox{if x\in {\cal T}_+, then Mx \in {\cal T}_+}\}\tag{1}\:,$$ where $${\cal T}_+ := \{x \in \mathbb R^4 \:|\: x^T\eta x \geq 0\:, x^0 \geq 0\}$$ so that we can equivalently restate the given definition as $${\cal V}:=$$ \{M \in M(4, \mathbb R)\:|\: ... 3 The relative speed between two objects is only restricted within the special theory of relativity. These restrictions are only guaranteed to apply in general relativity – the theory of curved space that you need for the Big Bang theory – if the space surrounding the objects is the flat Minkowski spacetime, or at least can be approximated by the flat ... 3 Are you thinking of something like neutronium? This is the (hypothetical) matter formed when you compress the electrons into the protons to make neutrons, then pack the neutrons tightly together. If so, then the density is 4 \times 10^{17} kg/m^3. However you should note that even neutronium isn't pure matter, because neutrons are made up from quarks ... 3 Suppose classical "pure matter" as you describe it existed and suppose a spherical volume V of 1\,\text{m}^3 of this stuff has mass M. Since it exists of pure matter only, one expects a uniform mass density \rho and M is just \rho V. So you'd have to define the mass density of "pure matter" to answer your question. Say you make it 1 Planck mass ... 3 They are just saying that in our universe of 3 spatial dimensions the event horizon is a 2-sphere. Ignoring time, our universe is a 3 dimensional manifold because it takes 3 numbers to specify a point within it. Likewise, an event horison is a 2 dimensional manifold because it takes 2 numbers to specify a point within it. Judging by the comments there is ... 2 If you are asking about the mechanism that causes gravitation, it's mass. And energy. Why? General relativity doesn't say why. It says what happens. If you are asking why Einstein chose to use the equivalence principle as a guiding concept in his development of general relativity, in a very real sense he had no other choice. There's a general concept that ... 2 There is a known example where it looks like a black hole on the outside but is flat spacetime on the inside. Imagine the funnel shaped exterior of a black hole and take the entire part outside of the event horizon, which is a spherical shell of surface area 4\pi r^2. Then take a spherical ball of Minkowski space of radius r and sew the two together ... 2 You're confusing the definition of a vector. A vector always has magnitude and orientation regardless of the dimensionality and they are not independent. In typical physics applications, the magnitude is the Euclidean norm of the vector. So in 3D, you have 3 components defined by scalars multiplied by the unit, or basis, vectors. In 2D, you have 2 ... 2 Of course there are forward and backward. Now reduce the number of dimensions to just one, leaving just a magnitude Note that it is a magnitude, not an absolute magnitude. a direction paramaterized by two discrete symbols +,- has been added. No. There is only one value there which is a member of \mathbb{R}. The sign is part of the value. ... 2 I would guess that Kaku is referring to the brane world scenario, though he has had to simplify it for popular consumption to the point where it is barely recognisable. String theory is most naturally formulated in ten spacetime dimensions, one time dimension and nine space dimensions, so the question is why don't we see all these dimensions. The brane ... 2 In some extra-dimensional models, such as brane cosmology, the fields (except gravity) are indeed confined to a lower-dimensional surface, which is sort of like "sharing almost the same coordinates in the extra dimensions". In Kaluza-Klein theory with compact extra dimensions, the fields are basically spread evenly across the entire size of the extra ... 2 Yes, if all the dimensions are compact, well, we really mean that all spatial dimensions are compactified on a torus T^9, then (multiple) T-duality may map any simple Dp-brane aligned with some dimensions to a D0-brane. Under T-duality, Dp-brane is mapped either to a D(p+1)-brane or a D(p-1)-brane, so its dimension either increases or decreases. ... 2 Relativity treats spacetime as a four dimensional manifold equipped with a metric. We can choose any system of coordinates we want to measure out the spacetime. It's natural for us humans to choose something like (t, x, y, z), but this is not the only choice. Even in special relativity the Lorentz transformations mix up the time and spatial coordinates, so ... 2 Your diagram looks like an illustration of the Ekpyrotic universe. In this model the extra dimensions are not compactified (i.e. curled up) so there is no uncurling of them. The reason we don't see the extra dimensions is because our universe is confined to a 3D brane, not because the extra dimensions are curled up. One well known theory for what determines ... 1 OP is asking about 3+1 dimensions, but let us here work out the corresponding construction in 1+1 dimensions. The 1+1 dimensional result may be used as a toy model to gain some intuition of what might (or might not) hold in higher dimensions. We use light-cone coordinates x^{\pm}. I) The future light-cone is\tag{1} {\cal T}_+~=~\{(x^+,x^-)\in ...

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It's tempting to think of spacetime as something like the rubber sheet that is so popular in analogies for spacetime curvature. In that case it's quite reasonable to ask why can't matter slide over the rubber sheet as it expands. However this is a misleading idea of what spacetime is. Spacetime isn't a physical object, it's a mathematical structure$^1$ that ...

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I am sorry to say that I can not agree with previous answers. We believe, but do not know for sure, that light from some galaxies will never reach us. This has nothing to do with the fact that they are moving away from us at more than the speed of light. Rather, it is assumed that these galaxies, like us, are not moving relative to the special frame in ...

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It's certainly possible, though on current evidence it looks unlikely. The past bound isn't really a bound in the usual sense of the word, but instead it's a singularity. If we solve Einstein's equations for the universe with a few apparently plausible assumptions we find that the universe is described by a scale factor, normally written as $a(t)$, and as ...

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Preliminary remarks: if a book say that invariance of $c$ is a direct consequence of M-M experiment, stop reading it. Answer to your question: as to kinematics the answer to your question is yes (but I don't know how to carry on with dynamics): we can derive relativistic kinematics from different postulates than the one of invariance of $c$. Consider these ...

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Well, i would say no. Why? Because an absolute center of mass would require a uniform covering (coordinate system) over the whole manifold, which, even if it exists, will probably not be on the manifold itself. An analogy would be the center of mass of a spherical surface/manifold. It would be exactly on the center of the sphere (i.e not on the sphere ...

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General Relativity allows black holes of any size (though making a small one might be hard or worse than hard). So as a thought experiment that means that you can consider a small black hole that curves spacetime exactly as much as the sun does. This black hole would be much smaller than the sun, but to us out here spacetime would look the same (except we ...

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