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You start from the (special) relativistic expression for energy: $$E=\sqrt{m^2c^4+p^2c^2}$$ Now, if the first term is much larger than the second ($mc^2\gg kT$ or $v\ll c$), we should take this first term out of the square root and taylor expand the (then in standard form) rest, discarding all but the leading term: ...

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I believe that this result only holds for $v\ll c$. Having assumed this, lets start: $$Total Energy = Rest Energy + Kinetic Energy$$ Keep in mind that since $v\ll c$, $c^2$ is the energy to mass conversion ratio and $K.E.=\frac{p^2}{2m}$ $$E=Mc^2+\frac{p^2}{2m}$$ There!

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Assuming we have the correct value for the cosmological constant the doubling time, that is the time it will take for the universe to double in size, is around 11.4 billion years. We have few hard theories about inflation, but suppose the universe expanded by $e^{60}$ as Danu suggests in his comment, then the number of doubling times is $60/ln(2) \approx ... 1 We model spacetime as a manifold and a metric. Broadly, the manifold gives us the dimensionality and connectivity while the metric provides a method of specifying distances. The equations of General Relativity allow us to calculate the metric from the stress-energy tensor (or vice versa if you're Miguel Alcubierre). The point that jinawee is making in his ... 0 I don't think we know for sure. For example see this article (the paper is on the Arxiv here) suggesting that the acceleration could be anisotropic. However if there is an anisotropy then it's small, and to a first approximation yes the acceleration is isotropic. 1 The (nearby) "separation between objects" you are referring to is the space-time metric. A metric in cosmology describes the expansion of space on large angular scales (low$\ell$on the angular power spectrum of the universe). Without going into the mathematics, the expansion of space is driven by cosmic inflation, and is affected by things the amount and ... 0 It is easier to make this calculation covariantly. Given a time-like vector field$n^\mu$representing the direction normal to the hyper-surfaces (in the Friedmann case the homogeneous and isotropic hyper-surfaces), you can always decompose the energy momentum tensor$T_{\mu\nu}$as $$T_{\mu\nu} = \rho n_\mu n_\nu + 2n_{(\mu}q_{\nu)} + p \gamma_{\mu\nu}+ ... 1 There is a three dimensional shell of galaxies (none currently observed) that have a gravitational redshift relative to us that is consistent with a relative motion at the speed of light. If you moved to any other galaxy in the universe, there is a very high probability that they would observe a different such bubble. If you look sufficiently far into ... 2 This going to be a rather approximate answer because it involves lots of estimated quantities like the current density of matter and the value of the cosmological constant. The second Friedmann equation tells us:$$ \frac{\ddot{a}}{a} = \frac{-4\pi G}{3}\left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3} $$It's conventional to take a = 1 at ... 1 Wolfram claims that the universe at its core is described by some Turing universal computations (it doesn't matter what specific form it takes, such as cellular automata, tag systems, Peano arithmetic etc)as all Turing universal computations are equivalent). He also claims that mathematical descriptions are a special case of general computations where you ... 1 It's just a rescaling of the time coordinate. Define t = f(\eta) and {\bar a}(\eta) = a(f(\eta)). Then,$$ds^{2} = -{\dot f}^{2}d\eta^{2} + {\bar a}^{2}d^{3}{x}$$Thus, if f(s) satisfies \frac{df}{d\eta} = a(f(\eta))\rightarrow \eta = \int \frac{dt}{a}, then you have transformed into conformal coordinates, and there is no special meaning for the ... 2 The CMB is relic light left over after photons decoupled from ions in the early universe, to a sufficient degree that photons could travel in a straight line unimpeded for 13 billion years. This happened everywhere in the universe, over a relatively short timescale (characterized by the thickness of the last-scattering surface). As you point out, each ... 0 Yes, you know - the many worlds theory is one of interpretations of quantum mechanics - so you can use quantum physics equation there (Like a probability by Born rule - the probability here can have a little different meaning). But the possible experiments what we would scientifically describe are (yet?) only speculations (Maybe because Copenhagen ... 1 We do have such structure in place: First, anything beyond the cosmological event horizon is effectively part of a different universe. An extension of that idea is the inflationary multiverse with bubble universes that can even have varying physical laws due to differently broken symmetries. String theory in turn adds its own flavour to that idea via the ... 1 My answer to your question is: no one can reliably answer your question. The model of the universe based solely on General Relativity says something about the beginning of the universe. If one follows the evolution of the universe backwards in time, one finds a singularity of infinite energy density "before" which the concept of time has no meaning. ... 1 Black Holes and Big Bang seed are both singularity (and, may look similar), but they are fully different. Black Hole is singularity in Time (meaning, at singularity, space component from Spacetime vanishes and if you fall inside a Black Hole, singularity would be in your direct future), but Big Bang seed is singularity in Space (in rough words; Big Bang seed ... 0 The horizon of the observable universe can be thought as a type of cosmic event horizon. Therefore the same paradox applies to it like in the case of black holes. 2 If you are sincerely asking about the fate of spaceship departing the solar system, then you are asking a question about the standard model of cosmology. Instead of worrying about relativistic effects we'll concern ourselves with a photon that leaves our galaxy in a direction so that it hits nothing, and just keeps traveling. Faster than any spacecraft that ... 3 Fundamentally, the misconception here is that something that is expanding must be finite. This is simply not true. When we say the universe is expanding, we mean the distance between two stationary observers, sitting still as best they can, grows over time. But it is entirely possible for an infinite thing to have this property. Imagine the real number ... -6 To start with, space was not created, space had always been there (yes even before the big bang). What was yet to be created was the incredible and ever expanding universe(Big bang ?). Space is infinite on it`s vastness and time because literally, space is the absence of stuff (sorry, couldnt think of a better word). Back to the answer: As you have heard ... 0 In cosmology, "flat" doesn't mean the opposite of "rough." It means the opposite of "curved in a global sense." The surface of a sphere is not flat, even if it is smooth, as the surface everywhere has positive curvature. A saddle has everywhere negative curvature. An uncurved plane is flat in the cosmological sense, even if it has some bumps and ripples on ... 3 Assuming that the FLRW metric is a valid description of the early stages in the expansion of the universe, then for any two simultaneous spacetime points the proper distance between them is given by:$$ D(t) = R(t) \chi$$for some constant$\chi$, where$R(t)$is the scale factor. The scale factor goes to zero at$t = 0$, but$R(t) > 0$for any$t > ...

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Time is relative. When it comes to Time Dilation, you actually see dilated time of another observer. So, your own time flow won't get frozen in any case. Hypothetically, you can see another one's time frozen if she is traveling at speed of light (time dilation by speed) or she is at event horizon of Black Holes (gravitational time dilation). Unfortunately, ...

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The answer to (2) is simply that no-one knows, and further that it's unlikely we will ever know. It's impossible to prove that the universe is infinite, but it's just possible we might prove it closed and therefore finite if the length scale is around the size of the currently observable universe. The paper Topology of the Universe: Theory and Observations ...

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Rotation and Translation The distinction is crucial for understanding Newton’s mechanics. In broad terms, translation indicates motion in a straight line; and rotation indicates motion around an axis. When Newton muses in Principia that maybe there is no such thing as a body truly at rest, he is also implying that maybe there is no such thing as motion in ...

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The molecules do not expand, because they are kept together by the electromagnetic interaction. The same applies to hadrons (strong force). These interactions have their own coupling constants, independent of gravity. So space expanding in an atom is like trying to pull the atom apart very slowly by a force which does not hold the atom together. The ...

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You can see the expansion with the Doppler Effect. Nothing else is expanding in the Universe except from the Universe. The Planck length is the same because it's a constant length. The stars and the Galaxies are getting pushed away from each other due to the red shifts. Here's a video on redshifts: http://www.youtube.com/watch?v=TLwiOToY79I The objects in ...

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The density of a black hole is defined simply as the mass within the event horizon divided by the volume within the event horizon. This gives an average density, but doesn't imply that the density is uniform within the event horizon. So when you hear statements like the density of a supermassive black hole is the same as water don't take this too literally. ...

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Photons or cosmic rays don't (normally) emit gravity waves. Consider the comparison with radio waves. A moving electron doesn't emit radio waves. It has to be accelerating to emit EM radiation. Specifically radio waves are only emitted when there is a changing dipole moment. So you wouldn't expect a particle moving at constant velocity (photon or ...

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In light of this, why do photons traveling from the most distant reaches of the observable universe not lose energy due to the gravitational radiation they must emit? There is a misconception here in "gravitational radiation they must emit" . There does not yet exist a unified theory of elementary particles and the three interactions well described by ...

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The light from the universe outside the event horizon of the black hole should be visible as a small disc or point light source in the direction facing away from the singularity. This will happen even before you reached the event horizon. You do not need to cross the horizon to see this effect. At ergosphere distance the BH will cover half of the sky ...

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Why questions ultimately in physics have the answer "because". What physics does is to model mathematically from existing observations the observed universe, in an ideal case a theory of everything towards which physics aims. The models are validated by predictions for new unrecorded at the time the model appeared, phenomena. The mathematics of the ...

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How to explain the fine tuning of the Universe (This is not so much an answer as an extended comment before this question is closed.) The unspoken premise is that there is an 'explanation' that stands apart from, is independent of, is not a part of, the Universe. But, the Universe is all there is, all there was, and all there will be. There is no ...

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Particles in physics in current terminology is "elemenntary particles", which are the building blocks that form atoms molecules and radiation that we observe macroscopically in bulk. These can be created and destroyed during the processes of stellar evolution, and particularly photons, which are bosons, have no limit on their number at all. There is no limit ...

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I think that Wolfram is arguing that the study of cellular automata and perhaps similar computational systems could serve as an organizational principle, providing a coherent framework to look at different problem (just like the more familiar frameworks provided by physics and chemistry). This explains the title of his new book, A new kind of Science (i.e. ...

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That depends on whether the Universe is finite or not, which we don't know and probably will never find out.

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Gravity has played a dominant role in the development of the large-scale structure of our universe. The largest structures of matter in our universe (most of it dark matter) grew out of over-densities in the primordial matter distribution that emerged shortly after the big bang. Dense areas began collapsing under their own gravity, and over time as their ...

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This is really a comment, but it got a bit long for the comment box. It's a comment because I kept meaning to go off and research this properly but have failed to find the time (and probably never will). So I'll post my initial thoughts, but treat this as suggestions for things to look at rather than a definitive answer. When you say I also understand that ...

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