Let's start by making some points clear:
1. We don't know what the Big Bang was.
Rather, we know that the Universe is expanding. If you extrapolate backwards, you'd expect the Universe to be denser and denser. More specifically, we talk about this as a change in the scale factor $a$, and this g...
I understand the expansion of the universe as actually an increase in the ratio of space to matter. Is this a correct understanding?
It isn't wrong. The ratio is increasing. But it isn't a "correct understanding". It's merely an observation of one of the results of the expansion of space.
...
If $v\in \bar C^1(\Omega)$, then for any $w\in\Gamma$, we let $\gamma_w(s)$ denote the geodesic emanating from $w$ with initial velocity $\nu(w)$, and define the "normal derivative" of $v$ at $w\in \Gamma$, $(\partial v/\partial\nu)(w)$, by $$\frac{\partial v}{\partial \nu}(w)=\lim_{s\uparrow 0}\langle (\nabla v)(\gamma_w(s)),\gamma'_w(s)\rangle.$$
@SirCumference : not quite. I've said we have no evidence that the universe is infinite. And that there's a non-sequitur wherein people claim a flat universe is an infinite universe. I've also said that I think that the expansion of the universe is evidence that it is not infinite.
I think the Universe is not flat, instead it has a small constant positive curvature (below measurement limit) and thus a spherical topology. But I can't prove it.
@SirCumference Still that does not stop something infinite from expanding. sure it will look like noting had happened, but that is far from not actually expanding at all given even in infinite volume, the spacing betwen galaxies are still increasing, and that is finite
@SirCumference Constant negative or zero curvature would close out a spherical topology, although I think it could be still cyclic, but not so naturally. I think the spherical Universe is coming from 3 things: 1) isotropy 2) homogenity 3) Aristotle's ultimate argument: everything should be spherical, because the sphere is the most perfect thing :-)
@SirCumference There is also a fourth, this time from Marx: qualitative changes always transform to quantitative changes after a point. It closes out the infinite universe being the same everywhere. If I add these 4 argument together, the result is a spherical Universe.
So recently, in chat, I was interested in about what if our universe contains only one CTC (thus making it unlike the Gödel metric, van Stockum dust where there can be more than one CTCs found) and that is the time dimension of the whole universe wrapped back onto itself (meanwhile the spatial di...
@peterh I'll let Baum & Frampton (2006) explain the problem with an oscillating universe:
> ...one principal obstacle is the second law of thermodynamics which dictates that the entropy increases from cycle to cycle. If the cycles thereby become longer, extrapolation into the past will lead back to an initial singularity again, thus removing the motivation to consider an oscillatory universe in the first place.
@SirCumference A recollapsing universe ends in a singularity (classically at least) and the equations of motion can't be integrated beyond that point. To claim cyclic behaviour seems to me at best a supposition.
@SirCumference I think gravity can somehow change the direction of the entropy. I suspect, if a quantum gravity will work, it will also find ways to reverse the direction of the entropy. This is my another thing what I suspect but I can't prove.
@Slereah My observation is that almost everyone in the lay public has room for only one notion of "particle physics" in their head at a time, and they focus on "most powerful" as the only figure of merit.
@SirCumference Look at the black holes, they give out energy and they will be hotter. Look a simple gas cloud, or any star, the denser components will concentrate around its center. If it is about gravity, there are always things in thermodynamics which somehow doesn't work as we accustomed to it, producing thermodynamical formulas based on hot gases on different cylinders.
It seems it could be reasonable that there could be a CTC along the comoving direction that is so large that the fields don't have time to accumulate enough to cause a divergence?
@SirCumference Yes. Smaller black hole has higher temperature. As the BH loses energy by HR, its mass shrinks, and its temperature grows. On the end stage, it explodes light a big, big nuclear bomb.
Guys, I’ve read Newton’s Third Law described in this way: "If two objects, sufficiently isolated from interactions with other objects, are observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant."
and the mass defined as such
However, I’m slightly confused by the definition of mass. What if we measure two electrons and how they repulse each other? We would then measure a huge mass. So should we only consider neutral objects then?
Suppose the universe has a CTC spanning billions of light years in diameter and for all observers in the comoving frame, they will agree the CTC is pointing along the time direction with no spatial components, then how would that differ from a cyclic universe?
@ShaVuklia: mass has dimensions so it's absolute value depends on the units. We could use kg or pounds or stones and the number we assign to the mass would be different with the different units.
@SirCumference HR is black body radiation. Calculating the temperature which gives the same black body radiation as a given black hole, you can order a temperature to the black hole. In this formula, the temperature of the black hole is reversely proportional to its mass. Thus, smaller BH is hotter.
@ShaVuklia: the point of the article is that the ratio of the accelerations gives the ratio of the masses and that's true regardless of what units we choose for mass.
It doesn't matter of what nature the force is - as long as it's not one of the situations for which Newton's third law doesn't work, that sentence holds true, regardless of whether there is charge or not.
@ACuriousMind I'm not entirely sure why I am thinking this, but I understand that if we are only dealing with the gravitational force, we can measure the mass. But what if we're only dealing with electrical force? This force is 24 digits larger, so wouldn't we get a huge mass?
@peterh Sure. The point of the quoted answer is that no matter how hard you jump, the ratio between your acceleration and the acceleration of the earth will always be the same.
@ShaVuklia No, and I don't understand why you would think so. The force between the two particles is much larger, but the ratio between the accelerations stays the same.
So the third law technically states that whatever interaction two objects have, if you place them together, their ratio of acceleration will always be the same?
@SirCumference The calculation of the Hawking Radiation is hard for commoners, but the results aren't. They are simple high school math formulas. Some constants, somewhere an $\frac{1}{M^3}$ or similar in them.
@SirCumference Playing with these formulas is very funny (for example, you can calculate that around Moon-sized black hole had been evaporated since the Big Bang).
But how do we know then for sure that we will always calculate the same mass when dealing with different forces? So say we have objects $X$ and $Y$, and $A$ and $B$, and an object $I$, that we consider to have unit mass. Say the force between $X$ and $Y$ is electrical of nature, and for $A$ and $B$ gravitational. And say our object $I$ has no charge. What happens between $X$ and $I$, and $Y$ and $I$ would then be gravitational I think, and between $X$ and $Y$ electrical.
So because it is always the same for $X$ and $Y$ by the law, we know that we should be able to get the ratio between the two, because it's also the same between $X$ and $I$, and $Y$ and $I$? Hmmm, I'm just really confused. I don't "see it" why we can always measure the mass this way.
Couldn't we have stated that the interaction between one object and all other objects will always be the same (i.e., constant acceleration)? And then deduce from that that the interactions between all those other objects are the same?
I'm just worried that if we fix all accelerations at once, that there could be contradictions, no?
Because in theory, there could be "random" accelerations
You have a ratio $c_{Ix}$ for all objects $x$ with the reference object $I$. Its mass is $m_x = -c_{Ix}$, if we define $m_I = 1$ to be our "unit" of mass. What is the problem?
Right. But the definition of $m_x = -c_{Ix}$ doesn't depend on a notion of force - it just needs the fact that there are constant accelerations, which is the content of the third law, and a choice of reference object
I suppose that you technically need to add that assumption to this variant of the third law. I.e. extend it as "the ratios of accelerations are constant, and for every triple of objects $A,B,C$ we have $c_{AB}c_{BC} = c_{AC}$".
In more sophisticated terms, I think it allows you to put utf8-encoded files instead of mere ASCII into pdfTeX, and therefore you can use unicode characters instead of having to jump through hoops
You don't need it with XeLaTeX or LuaLaTeX, iirc
But you should probably ask an actual TeXpert about that :P
If it's a porous surface like a fabric or carpet it's virtually impossible as blood contains cationic polymers that will bond strongly to the substrate. Use cold water and clean up before the blood starts clotting. But you'll never clean it well enough to escape detection.
If it's a non-porous surface like a ceramic use cold water then bleach.
Right, I'm off to drink beer and read about archaeology in the New World.
"We start from the assumption that at a fixed time, the fields $\Phi$ and their canonical conjugate momenta satisfy the same (canonical) commutation relations as the free fields $\Phi^0$ and that we may identify $\Phi$ and $\Phi^0$ as well as the respective canonically conjugate momenta at this time, say at $t = 0$"
Ah, that's the trick
We just ASSUME THEY'RE THE SAME
Dang it
that's not as clever a trick as I thought
Is there any motivation for this outside of "well at some point the particles will be far apart"
Let's start by making some points clear:
1. We don't know what the Big Bang was.
Rather, we know that the Universe is expanding. If you extrapolate backwards, you'd expect the Universe to be denser and denser. More specifically, we talk about this as a change in the scale factor $a$, and this g...
I understand the expansion of the universe as actually an increase in the ratio of space to matter. Is this a correct understanding?
It isn't wrong. The ratio is increasing. But it isn't a "correct understanding". It's merely an observation of one of the results of the expansion of space.
...
