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Space and time are features of the matter, not the matter itself. For instance, energy is another feature of the matter. If matter can be created or destroyed it's another question, the energy is said to be conserved (but, virtual particles?!). Though, matter comes with its features.

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The interpretation of gravity as curvature of spacetime is model-dependent. You already mentioned the teleparallel equivalent of general relativity, modelling gravity by torsion. Another possibility are bi-metric theories, where the metric is a more ordinary field on a fixed background (this should be more in line with how string theorists tend to think of ...

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In particle mechanics you integrate along a path, which is bounded by points, but in field theory you integrate over a spacetime volume, so your boundary is a hypersurface, not just points. For a typical quantum field theory process (at least the way it's formulated for calculations), there is some initial state consisting of noninteracting wavepackets, ...

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In field theory the values of the field at every point in space are independent degrees of freedom, just like the positions of different particles in a multi-particle system. So, AFAIK to specify the initial and final configurations for an action integral you have to give the values of the field at every point in space at the initial and final times. The ...

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If the Universe is infinite, then it had instantly grew from zero to infinity at the moment of Big Bang. Note, that this is a possible model from General Relativity (GR). But most probably GR is not applicable at the moment of Big Bang, because it contains singularity.

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Have a look at the Wikipedia article on Binary Black Holes. Essentially when two black holes come in close proximity they are believed to merge into a bigger black hole so you won't get one black hole sitting at the event horizon of another for infinity. Super massive black hole binaries are believed to form during galaxy mergers Regarding the answer ...

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It's not quite what you're looking for, but the article here from Physics World shows such a diagram for a charged Reissner-Nordström black hole. They note that Reissner-Nordström black holes were used to try to model the effects of incoming radiation being infinitely blueshifted at the inner (Cauchy) horizon, including the infinite blueshift of incoming ...

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There is no exact solution of Einstein's equation smoothly modeling the metric of a rotating star, so a diagram like this can only be a heuristic.

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Em, it will be no "image deformation" if your question is about it, just the biggest mass of the big planet will more curves the space time around it and the light will need more time to escape the big planet since the space time curvature will induce a longer distance for the light to travel throught.

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My guess would be that $\mathbb E^n$ denotes Euclidean space. In addition to having geometric structure (angles and distances) and motions (rotations, translations, reflections) - not all of it terribly useful in the 1-dimensional case - it is an affine space. Affine spaces have no notion of distinguished origin or zero point. We can use a vector space like ...

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First of all, if one talks about the mathematical problem, it is a mathematical problem and there is nothing such as "atoms" or "Planck length" or "Planck's constant" in mathematics. The sum is convergent and may be evaluated e.g. using Fourier series and the result is $$\sum_{n=1}^\infty \frac{1}{n^2} = \zeta(2) = \frac{\pi^2}{6} \approx 1.645$$ The Greek ...

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∆A+∆B+∆C = 180 in flat space. However, a triangle which was measured around the earth would have angles that sum up to an answer which is less than 180º. This is because space is bent downwards like a saddle by mass.

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Particles with zero rest-mass can only move at the speed of light, while massive particles can never reach it. This is a fact from special relativity and is independent of the source of said mass. The fact that the Higgs mechanism makes certain particles massive is not intrinsically related to this.

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According to Wald's GR [...] e.g. in a case where there are several massive bodies in relative motion--there exists no natural set of curves whose comparison with geodesics could be used to define gravitational force. Why not? Why not, indeed. What could be more "natural" than to make the required comparison ("with geodesics") for each participant ...

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First, note that there is no unified theory of QFT and gravity, so talking about geodesics and about the Higgs is really not possible within the framework of our current theories. Nevertheless, the confusion here seems to stem somehow from the idea that all particles are "initally" massless, and "then" the Higgs comes along and gives them mass. This idea of ...

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This is discussed in section 4.3 in my 1984 edition. The quote supplied can't really be understood in isolation - you need to consider the whole section. Wald's point is that in general relativity there are no inertial observers because in general spacetime is nowhere flat. In Newtonian physics or special relativity acceleration can be measured relative to ...

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Ok I solved case 2. Let $t'=\tan^{-1}(\omega t)$ and $x'=x\sqrt{1+t'^2}$. Also observe that $x=A\cos \omega t + B\sin \omega t$ to represent simple harmonic motion. We get $x'=A+Bt'$. So only case 3 remains open. Also as u can see, I just did guesswork, without proper techniques for how to solve this in general.

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If you are allowing non-linear transformations of space-time (as you have done in your Case I), then I don't see why you can't write any motion of an object as x' = 0. i.e. Suppose the motion of the object in the (x,t) coordinates can be written as f(x,t) = 0 for some non-linear function f. Then, by setting x' = f(x,t), t' = t, the motion in (x',t') can be ...

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The Lorentz transformation simplifies to the classical transformation for all your cases since you took $c$ as infinity. There is no time dilation. The transform is just an identity multiplication for all 3 cases with: $$x' = x - vt$$ and $$t' = t$$ Now that you are in the classical world in one dimension, your first case is merely a tilted line with ...

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The singularity comes from the scale factor $a(t)$: $$ds^2 = -dt^2 + [a(t)]^2 ( dr^2 + r^2 d \Omega^2)$$ By solving the Friedmann equations for the scale factor we know that: $$a(t) = a_0 t^{\lambda}$$ where $\lambda$ is some positive number that depends on the matter-radiation ratio of the universe. At $t=0$ the scale factor becomes $a(0)=0$. So at ...

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This is an extended comment on Valter's answer, so please upvote his answer not this one. In Relativity (General and Special) there is no unique way to divide spacetime into space and time. Different observers, using different coordinate systems, will disagree about whether a four vector is just a displacement in time or just a displacement in space. So to ...

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In a certain sense (regime) acceleration is caused by the curvature of time more than the curvature of space. Actually, the curvature is of the spacetime so that, making rigid distinctions has no much sense. However, if you consider the motion of a particle free falling in a region of spacetime, the equation of its story is the geodesical one: ...

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Wald is a first rate relativist, and as such he is phrasing the concept of general covariance in terms of purely geometrical quantities, rather than resorting to the somewhat imprecise notion of coordinate transformations. In the discussion on pg. 57, he goes on to give an example of what it means to violate the principle of general covariance. In his ...

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I'm hardly a GR expert, so if you want a more technical analysis I'm sure others will be able to give you one. However, the answer to your apparent questions is fairly straight forward. It is not the curvature of space or the curvature of time that causes accelerations, it is the curvature of space-time. We live in a four dimensional universe (ignoring ...

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General covariance basically means you can change your coordinate system arbitrarily and express the laws of physics in the new coordinates. Because of this freedom, the relationship between coordinate distances, angles, etc. and physical distances, angles, etc. is variable and is expressed by the metric. So the quoted statement is basically saying that ...

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I'm not a GR expert (and there are some on this site), but I'll try to answer this. How does an object on the surface of earth stay fixed? Since there is no concept of "attraction" and only space time why cannot an object keep moving around on the surface of earth. The answer here is that there generally aren't any forces moving the object to the side. ...

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you should look for the theory of "Covariant Quantum Mechanics" originally introduced by M. Modugno and J. Jadcyk, in particular the "special algebra of quantizable functions". There is not so much literature since the theory is mathematically quite hard to enter, but afterwards it is worth. I have worked in that field for several years. The basic idea is to ...

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According to quantum mechanics the time evolution of the universe is described by a path integral that will sum over all histories. If we consider a robot whose processor runs at a clock cycle of $\tau$ to simplify things, then all the possible time evolutions during that period of $\tau$ will contribute to explain the robot's observations, including the ...

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Fortunately for experiments in physics we have better proxies than the accuracies of our five senses. We have detectors and computers and .... With these tools a theory of how the universe is made has been developed, from elementary particles with the theory of quantum mechanics building up the observables around us, to the astrophysical models that fit ...

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Quite a philosophical approach. There is still the reliance on our four other senses in order to make sense of our physical world, however the same approach can be imply to those senses also with the delay in neurological impulses. One must also take into account, as you would call it, the in between frames of other people's perceptions, as well as those ...

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