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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 ...


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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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@benrg I got it. Follow your first half suggestion. Pick $s\in I^+(r)$, so $r\in I^-(s)$. $I^-(s)\cap I^+(p)\ne\emptyset$. Choose any point $q\in I^-(s)\cap I^+(p)$. There is a trip from $q$ to $s$, and at the same time, there is trip from $p$ to $q$, so glue the 2 trips to get a 3rd one from $p$ to $s$. Therefore, $s\in I^+(p)$. $s$ is an arbitrary point in ...


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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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1st Law of Motion Every object in a state of consistent motion tends to remain in that state of motion unless an external force applied to it. 2nd Law of Motion It is pertaining to the relationship between an object’s mass, its acceleration, and the applied force. In this law, the direction of the force vector is the same as the direction of the ...


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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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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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As the incredible difficulty to finding solutions to the three dimensional Ising model proves beyond doubt, discrete problems are fundamentally just as hard as continuum problems, in many cases probably even more so. More interestingly, physically relevant solutions of the higher dimensional Ising model correspond to modes of certain continuous equations, ...


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 D$p$-brane aligned with some dimensions to a D0-brane. Under T-duality, D$p$-brane is mapped either to a D$(p+1)$-brane or a D$(p-1)$-brane, so its dimension either increases or decreases. ...


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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 ...


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Not a cosmologist but there some interstellar objects known as neutron stars that are extremely dense. Some neutron stars can have masses of 500,000 times that of earth with a diameter of about 25km. In the core of these stars, it is hypothesized that "quark liquid" exist which is when quarks get pushed into each other with no spacing between them. In such a ...


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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 ...


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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 ...


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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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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 ...


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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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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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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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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 ...


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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 ...


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If u think of time as we percieve it as a fourth dimension time is actually all space and vice versa. We can only percieve the present at any time so every moment I guess is its own dimension or universe making time a record of all the space every moment in a way. The past can never be changed and the future can't be affected until it's the present. That's ...


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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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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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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 ...


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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

We can't say much about closed timelike curves with any certainty; they are an artefact of the existence of solutions to the general relativity equations which allow them. It is possible (and quite a few physicists believe this) that a theory of quantum gravity may preclude CTC from occurring, or that CTC may occur but the information might be censored by ...


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If somehow, it is only possible to create one alcubierre drive, and it can never turn around, then you won't get closed timelike curves. Otherwise, any construction is going to have them. The reason is that you can "zoom out" far enough that the distortions to spacetime caused by the drive are no longer present, and the person flying the drive then just ...


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Ok, this is going to be a math-free-ish answer. Let's get something straight right off the bat. Inflation is over. Inflation refers to the first ~$10^{-34}s$ of the universe after the Big Bang. What we have now is accelerated expansion. Second, asking if this accelerated expansion "costs energy" is not particularly meaningful. What you'll find is that most ...


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One of the Friedmann equations is a "conservation" equation: $$\dot \rho_i= -3\frac{\dot a}{a} (\rho_i + p_i) \tag{1}$$ where $\rho_i, p_i$ describes energy density and pression for a particular "fluid" (dust, relativist particle, dark energy/cosmological constant). For each fluid, there is a relation between $p_i$ and $\rho_i$ (respectively $p_i=0, p_i ...


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The statement that space is Euclidean is a broad statement, not meant to hold near very massive bodies or in arbitary volumes. One sense in which it can be meant is to hold "on average" for the whole spacetime - the universe - as such. The best candidate for the overall metric of spacetime is the FRLW metric, which is the exact solution for a universe that ...


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It takes time for light to bend, i.e. the bending of light is a consequence of the curvature of geodesics in four dimensional space-time. On the other hand, a three dimensional snapshot of space appears to have geodesics that are straight lines (some specifics shown below). This is exactly why we thought space was Euclidean for so long and why all ...


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We can't peek deeper than a certain distance away from our current cosmic position, but we know for sure that the universe extends far beyond that cosmic horizon. In fact, for all we know, although our observable universe is finite, the full universe is infinite in size. If this is indeed true (and there is no single piece of evidence against an infinite ...


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There is no edge if you come along with the infinite concept. Update: Harvard complements my answer: http://www.cfa.harvard.edu/seuforum/faq.htm#s1 Galaxies extend as far as we can detect... with no sign of diminishing.There is no evidence that the universe has an edge. The part of the universe we can observe from Earth is filled more or less uniformly ...


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Mathematically, that's due to superposition. Both masses produce some gravitational field, which add together to give the "net field". (The same goes for electromagnetism, where one may add electric/magnetic field strengths,electric potentials etcetera for every point in space.) Ever so slightly changing the field strength at the well. As gravity gets weaker ...


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If two events both have a spacetime interval of zero, can they both be said to be happening “now”? There is an interval associated with any two events but there is not an interval associated with an event. From the Wikipedia article "Spacetime": In spacetime, the separation between two events is measured by the invariant interval between the ...


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I'm not sure this is a complete answer to your question, but thinking about special relativity that way will get you into trouble. Essentially, that way of interpreting special relativity attributes all of its weirdness to signal delay. Here's how I think you're interpreting the barn door experiment: The ladder is put stationary in the barn and is found to ...


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OK, just one more try to end this stupid question. There IS a way to formulate physics using light rays as your basis: a double null coordinate system${}^{1}$. If you have a ray moving in the $+x$ direction, define the two coordinates $$2\xi = t + x\;\;\quad\quad\quad2\eta = t - x$$ Then, the metric becomes $$ds^{2} = -4d\xi \,d\eta + dy^{2} + dz^{2}$$ ...



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