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Why is the letter L for Georges LeMaîtres often, or even usually, left out?

Does he, or does he not, deserve some credit for this cosmological solution to Einstein's general relativity?

Well, to quote wikipedia, it's a matter of personal and historical perspective whom to credit (most):

Depending on geographical or historical preferences, the set of the four scientists - Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker - are customarily grouped as Friedmann or Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW) or Friedmann-Lemaître (FL).(… ) The FLRW model was developed independently by the named authors in the 1920s and 1930s

## Universe

The **universe** (Latin: *universus*) is all of space and time [a] and their contents, [10] including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. According to estimation of this theory, space and time emerged together 13.799 ± 0.021 billion years ago, [2] and the universe has been expanding ever since. While the spatial size of the entire universe is unknown, [3] the cosmic inflation equation indicates that it must have a minimum diameter of 23 trillion light years, [11] and it is possible to measure the size of the observable universe, which is approximately 93 billion light-years in diameter at the present day.

The earliest cosmological models of the universe were developed by ancient Greek and Indian philosophers and were geocentric, placing Earth at the center. [12] [13] Over the centuries, more precise astronomical observations led Nicolaus Copernicus to develop the heliocentric model with the Sun at the center of the Solar System. In developing the law of universal gravitation, Isaac Newton built upon Copernicus's work as well as Johannes Kepler's laws of planetary motion and observations by Tycho Brahe.

Further observational improvements led to the realization that the Sun is one of hundreds of billions of stars in the Milky Way, which is one of a few hundred billion galaxies in the universe. Many of the stars in galaxy have planets. At the largest scale, galaxies are distributed uniformly and the same in all directions, meaning that the universe has neither an edge nor a center. At smaller scales, galaxies are distributed in clusters and superclusters which form immense filaments and voids in space, creating a vast foam-like structure. [14] Discoveries in the early 20th century have suggested that the universe had a beginning and that space has been expanding since then [15] at an increasing rate. [16]

According to the Big Bang theory, the energy and matter initially present have become less dense as the universe expanded. After an initial accelerated expansion called the inflationary epoch at around 10 −32 seconds, and the separation of the four known fundamental forces, the universe gradually cooled and continued to expand, allowing the first subatomic particles and simple atoms to form. Dark matter gradually gathered, forming a foam-like structure of filaments and voids under the influence of gravity. Giant clouds of hydrogen and helium were gradually drawn to the places where dark matter was most dense, forming the first galaxies, stars, and everything else seen today.

From studying the movement of galaxies, it has been discovered that the universe contains much more matter than is accounted for by visible objects stars, galaxies, nebulas and interstellar gas. This unseen matter is known as dark matter [17] (*dark* means that there is a wide range of strong indirect evidence that it exists, but we have not yet detected it directly). The ΛCDM model is the most widely accepted model of the universe. It suggests that about 69.2% ± 1.2% [2015] of the mass and energy in the universe is a cosmological constant (or, in extensions to ΛCDM, other forms of dark energy, such as a scalar field) which is responsible for the current expansion of space, and about 25.8% ± 1.1% [2015] is dark matter. [18] Ordinary ('baryonic') matter is therefore only 4.84% ± 0.1% [2015] of the physical universe. [18] Stars, planets, and visible gas clouds only form about 6% of the ordinary matter. [19]

There are many competing hypotheses about the ultimate fate of the universe and about what, if anything, preceded the Big Bang, while other physicists and philosophers refuse to speculate, doubting that information about prior states will ever be accessible. Some physicists have suggested various multiverse hypotheses, in which our universe might be one among many universes that likewise exist. [3] [20] [21]

## Trip around universe

What is this "constant velocity" of which you speak? Constant relative to **what**? And that phrase "in a straight line" has the same problem.

OK, lemme try the question that (I think) you're really asking.

Q: If you travel in a locally straight straight line, free fall, no acceleration, no deliberate turning and twisting and course changes, through a curved universe, do the curvature effects mean that you aren't in a valid inertial frame?

A: You are always in a locally inertial frame. There is a region of space-time around you in which the curvature effects are too small to measure, and as long as you only do experiments within that region, you'll get the results predicted by special relativity, which works within inertial frames. If you do experiments on a scale large enough for the curvature to matter, you need general relativity. The stronger the curvature, the smaller the locally flat region - but (except at a singularity) you can always find a region small enough to be locally flat, and within that region you're always in a valid inertial frame.

Consider that when you're laying out the foundations of a house, you don't worry about the curvature of the earth's surface the earth is locally flat. If you're laying out a flight path between London and Tokyo, you do consider the curvature of the earth.

(Whether you would eventually return to earth or not depends on how and how much the universe is curved. Others have already commented on that).

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## What if the universe reaches a high enough density to become a black hole?

There seem to be a lot of questions here. If you don't understand the significance of the negative sign in the metric, you really need a textbook on special relativity and then general relativity (I'd recommend Spacetime Physics by Taylor and Wheeler). All relativistic metrics have one opposite sign - that's what makes it space*time*, not space.

Why would the universe start shrinking? This does happen in some variants of FLRW spacetime, but they don't end in a black hole. One way to see this is that the density of the universe is everywhere the same, so there cannot be any special points. A black hole is very different from the rest of the universe. But maybe you had some other reason in mind for why the universe should suddenly start shrinking.

The Schwarzschild metric does not describe a universe that is either expanding or contracting. So it is not relevant for the case you are describing.

The relevant spacetime for a contracting universe is FRW spacetime in FRW spacetime, there is no black hole as the universe shrinks, no matter how high the density gets, because a black hole is a region of spacetime that cannot send light signals to infinity, and in FRW spacetime there is no infinity.

The current indications are that the expansion of the universe is accelerating, which means that it won't reach the critical density to become a black hole.

It's a bit surprising that the universal expansion is accelerating, but that's what the data is currently showing. See for instance <<https://en.wikipedia.org/w/index.php?title=Accelerating_expansion_of_the_universe&oldid=906250109>>. This is explained as being due to the cosmological constant - sometimes it's called other names.

If we imagine a universe obeying the laws of general relativity without a cosmological constant, the expansion would slow down rather than accelerate, and it's possible the expansion would stop and reverse, at which point such a universe would eventually re-collapse to form a black hole.

We'd expect that such a universe would become a black hole in that case. The details are unclear, however. Our cosmological models make assumptions to simply the problem, one key assumption which is called homogoneity, and it's likely that this assumption would break down sometime during the collapse process.

This difficulties also exists in our attempts to understand the realistic collapse on a lesser scale.

The evidence that there is a black hole at the center of our galaxy is pretty convicing, so we are pretty sure collapse is possible. Many of the details of a realistic collapse process are unclear.

The universe as a whole can't become a black hole. It has the wrong spacetime geometry. See my post #3.

Also, there is no critical density to become a black hole. (I'll expand on this in a response to the OP shortly.)

There is no critical density to become a black hole. The criterion for becoming a black hole is that an event horizon forms, which means a given amount of mass ##M## collapses in such a way that a 2-sphere with area ##16 pi G^2 M^2 / c^4## can enclose it. The larger the mass ##M## is, the smaller the density of the collapsing matter needs to be when this criterion is met.

In fact, a black hole doesn't even have a well-defined density since it doesn't have a well-defined interior volume. Some pop science sources will say it has the volume of a sphere with radius equal to the Schwarzschild radius, but that's not correct.

No, it's because the hole doesn't have a well-defined volume.

The Schwarzschild metric does not describe a universe that is either expanding or contracting. So it is not relevant for the case you are describing.

The relevant spacetime for a contracting universe is FRW spacetime in FRW spacetime, there is no black hole as the universe shrinks, no matter how high the density gets, because a black hole is a region of spacetime that cannot send light signals to infinity, and in FRW spacetime there is no infinity.

this kind of answers my question, what i was thinking is if all the matter in the universe suddenly started gravitating towards some point (i know this won't happen in our universe but lets assume magic happens), and a person outside this sphere of shrinking matter would see a black hole forming, and my question was what it would look like to people inside the shrinking sphere. but this shouldn't be possible? because the distribution of matter determines spacetime so there can't be large regions of empty flat space outside the shrinking sphere?

i should rephrase my question. suppose a very large nebular started shrinking (so it has a huge schwarzschild radius, many light weeks in radius), and there's an unlucky astronaut in it. when the nebular reaches the 2 sphere with critical area, what would the astronaut see? he will have plenty of time to observe the universe around him before all the nebular reach singularity due to extremely large schwarzschild radius. someone outside would see a black hole, but what would he see? would he suddenly see the universe change once the 2 sphere is reached?

i guess my question can be stated as what an interior observer would see once event horizon is formed, will he see a sudden change in everything around him etc. but for normal small black holes this is too far quick for him to have time to observe and think about everything, plus an astronaut cant see past the inside of a shrinking star (assuming he can somehow survive the temperature). that's why i thought about the universe in my OP question, but a very large nebular that's somewhat transparent would do.

## 6 Answers 6

The nature of singularities in GR is a delicate issue. A good review of the difficulties presented to define a singularity are in Geroch's paper What is a singularity in GR?

The problem of attaching a boundary in general to a spacetime is that there is not natural way to do it. for example, in the FRW metric the manifold at $t=0$ can be described by two different coordinate systems as: $

It might be tempting to define a singularity following other physical theories as the points where the metric tensor is undefined or below $C^<2>$. However, this is troublesome because in the gravitational case the field defines also the spacetime background. This represents a problem because the size, location and shape of singularities can't be straightforward characterize by any physical measurement.

The theorems of Hawking and Penrose, commonly used to show that singularities in GR are generic under certain circumstances have the conclusion that spacetime must be geodesically incomplete (Some light-paths or particle-paths cannot be extended beyond a certain proper-time or affine-parameter).

As mentioned above the peculiar characteristic of GR of identifying the field and the background makes the task of assigning a location, shape or size to the singularities very delicate. If one thinks in a singularity of the gravitational potential in classical terms the statement that the field diverges at a certain location is unambiguous. As an example, take the gravitational potential of a spherical mass $V(t,r, heta,phi)=frac

However, this prescription doesn't work in GR. Consider the spacetime with metric $ds^<2>=-frac<1>

Another approach is to define a singularity in terms of invariant quantities such as scalar polynomials of the curvature. This are scalars formed by the Riemann tensor. If this quantities diverge it matches our physical idea that and object approaching regions of higher and higher values must suffer stronger and stronger deformations. Also, in many relevant cosmological models like FRW and Black Holes metrics one can show that this indeed happen. But as mentioned the domain of the gravitational field defines the location of events so a point where the curvature blow up might not be even in the domain. Therefore, we must formalise the following: statement "The scalar diverges as we approach a point that has been cut out of the manifold.". If we were in a Riemann manifold then the metric define a distance function $d(x,y):(x,y)incal

So far, all the discussion has been for Riemann metrics, but as soon as we move to Lorentzian metrics the previous discussion can't be used as stated. The reason is that Lorentzian metrics doesn't define a distance function. They do not satisfy the triangle inequality. So we only have left the notion of geodesic completeness.

The three kinds of vectors available in any Lorentzian metric define three nonequivalent notions of geodesic completeness depending on the character of the tangent vector of the curve: spacelike completeness, null completeness and timelike completeness. Unfortunately, they are not equivalent it is possible to construct spacetimes with the following characteristics:

- timelike complete, spacelike and null incomplete
- spacelike complete, timelike and null incomplete
- spacelike complete, timelike and null incomplete
- null complete, timelike and spacelike incomplete
- timelike and null complete, spacelike incomplete
- spacelike and null complete, timelike incomplete
- timelike and spacelike complete, null incomplete

Moreover, in the Riemannian case if $cal

Schmidt provided an elegant way to generalise the idea of affine length to all curves, geodesic and no geodesics. Moreover, the construction in case of incomplete curves allows to attach a topological boundary $partialcal

The procedure consist in building a Riemannian metric in the frame bundle $cal

In the case of the FRW the b-boundary $partialcal

Another types of boundaries can be attached. For example:

conformal boundaries used in Penrose diagrams and in the AdS/Cft correspondance. In this case the conformal boundary as seen here at $t=0$ is a three dimensional manifold.

Causal boundaries. This constructions depends only on the causal structure, so it doesn't distinguish between boundary points at a finite distance or at infinity. (See chapter 6, The large scale structure of spacetime)

I am unaware if in the two last cases explicit calculations have been done to the case of the FRW metric.

## Friedmann equations with $w <-1$

Under the assumptions that $a > 0$ and that the universe is expanding, we can derive some interesting results about the fate of such a universe.

From the Friedmann equations alone, we may derive

For $P = w ho$, as long as $w eq -1$, this yields

exactly as you stated in your question. So, yes, if the universe is expanding and $w < -1$, then the energy density does increase with time!

In the flat case with no cosmological constant, $Lambda = K = 0$, we may integrate the first Friedmann equation $3 H^2 = 8 pi ho$, with this expression for $ ho$, to yield

exactly as in the case $w > -1.$ The Hubble parameter is thus given by

In an expanding universe, we have $H > 0.$ Since $1 + w < 0$, we must have $ au < 0$ in order for the universe to be expanding. But since $a( au) propto au^

In a contracting universe, we have $H < 0,$ and so $ au > 0.$ Since this is past the $ au=0$ singularity, such a universe must have originated at a "big rip" at some finite time in the past.

So if we assume that the universe has existed for finite time, then it must be contracting (as you stated in your question), and it must have originated with a divergent scale factor. On the other hand, if we assume that the universe is expanding, then it will meet a singularity in finite time as the scale factor diverges.

A high enough energy density is a necessary condition but not a sufficient condition for black holes to form: one needs to have a center which will ultimately become the center of the black holes one needs the matter that collapses to the black hole to have a low enough velocity so that gravity may squeeze it before the matter manages to fly away and dilute the density.

The latter two conditions are usually almost trivially satisfied for ordinary chunks of matter peacefully sitting at some place of the Universe but they're almost maximally violated by the matter density right after the Big Bang. This matter has no center - it is almost uniform throughout space - and has high enough velocity (away from itself) that the density eventually gets diluted. And indeed, we know that it did get diluted.

In other words, a collapse of matter (e.g. a star) into a black hole is an idealized calculation that makes certain assumptions about the initial state of the matter. These assumptions are clearly not satisfied by matter after the Big Bang. Instead of a collapse of a star, you should use another simplified version of Einstein's equations of general relativity - namely the Friedmann equations for cosmology. You will get the FRW metric as a solution. When it is uniform to start with, it will pretty much stay uniform.

The visible Universe is, in some sense, analogous to a black hole. There exists a cosmic horizon and we can't see behind it. However, it is more correct to imagine that the interior of the visible space - that increasingly resembles de Sitter space because the cosmological constant increasingly dominates the energy density - should be viewed as an analogy to the exterior of a black hole. And it's the exterior of the visible de Sitter space that plays the role of the interior of a black hole.

The relationship between (namely the ratio of) the mass and the radius for the visible Universe is not too far from the relationship between (or ratio of) the black hole mass and radius of the same size. However, it's not accurate, and it is not supposed to be accurate. The mass/radius ratio is only universal for static (and neutral) black holes localized in an external flat space and our Universe is clearly not one of them.

I don't think that the question "what does the universe look like from the outside?" is very meaningful. Just because there is not outside for the universe. As for the black hole why should high density i.e. a lot of mass in little volume, cause the creation of a black hole? If you are thinking about the Schwarzschild solution (and radius), it describes a spherical object outside of which the space is empty, and as I said there is no outside for the universe.

The first thing to understand is that the Big Bang was not an explosion that happened at one place in a preexisting, empty space. The Big Bang happened everywhere at once, so there is no location that would be the place where we would expect a black hole's singularity to form. Cosmological models are either exactly or approximately homogeneous. In a homogeneous cosmology, symmetry guarantees that tidal forces vanish everywhere, and that any observer at rest relative to the average motion of matter will measure zero gravitational field. Based on these considerations, it's actually a little surprising that the universe ever developed any structure at all. The only kind of collapse that can occur in a purely homogeneous model is the recollapse of the entire universe in a "Big Crunch," and this happens only for matter densities and values of the cosmological constant that are different from what we actually observe.

A black hole is defined as a region of space from which light rays can't escape to infinity. "To infinity" can be defined in a formal mathematical way,[HE] but this definition requires the assumption that spacetime is asymptotically flat. To see why this is required, imagine a black hole in a universe that is spatially closed. Such a cosmology is spatially finite, so there is no sensible way to define what is meant by escaping "to infinity." In cases of actual astrophysical interest, such as Cygnus X-1 and Sagittarius A*, the black hole is surrounded by a fairly large region of fairly empty interstellar space, so even though our universe isn't asymptotically flat, we can still use a portion of an infinite and asymptotically flat spacetime as an approximate description of that region. But if one wants to ask whether the entire universe is a black hole, or could have become a black hole, then there is no way to even approximately talk about asymptotic flatness, so the standard definition of a black hole doesn't even give a yes-no answer. It's like asking whether beauty is a U.S. citizen beauty isn't a person, and wasn't born, so we can't decide whether beauty was born in the U.S.

Black holes can be classified, and we know, based on something called a no-hair theorem, that all static black holes fall within a family of solutions to the Einstein field equations called Kerr-Newman black holes. (Non-static black holes settle down quickly to become static black holes.) Kerr-Newman black holes have a singularity at the center, are surrounded by a vacuum, and have nonzero tidal forces everywhere. The singularity is a point at which the world-lines only extend a finite amount of time into the future. In our universe, we observe that space is not a vacuum, and tidal forces are nearly zero on cosmological distance scales (because the universe is homogeneous on these scales). Although cosmological models do have a Big Bang singularity in them, it is not a singularity into which future world-lines terminate in finite time, it's a singularity from which world-lines emerged at a finite time in the past.

A more detailed and technical discussion is given in [Gibbs].

[HE] Hawking and Ellis, The large-scale structure of spacetime, p. 315.

This is a FAQ entry written by the following members of physicsforums.com: bcrowell George Jones jim mcnamara marcus PAllen tiny-tim vela

The standard ΛCDM model of the Big Bang fits obsersvations to the Friedmann-Robertson-Walker solutions of general relativity, which do not form black holes. Intuitively, the initial expansion is great enough to counteract the usual tendency of matter to gravitationally collapse. As far as we know, the universe looks about the same from every point on the large scale. It is a built-in assumption of the FRW family solutions, and sometimes called the "Copernican principle."

It doesn't absolutely have to be right, of course, though in a sense it is the simplest possible empirically adequate model, and so is favored by Ockham's razor. There have been attempts to fit the astronomical observations to an isotropic and *inhomogeneous* solution of GTR (meaning, we would be near "the center"), but to my knowledge they have been less than conclusive.

There is an oversimplified model of spherical stellar collapse assumes that the star has uniform density and no pressure, the interior of which comes out to be equivalent to the k = +1 (positive curvature, closed) contracting FRW universe. The interior is smoothly patched to a Schwarzschild exterior. The k = 0 (flat) and k = -1 (open) cases can be thought of as the interior of such a star in the limit of infinite radius, collapsing from rest and with some finite velocity, respectively. They too can be smoothly patched to a Schwarzschild exterior.

Our the observed universe is expanding, but we still can say that's it's possible for the isotropic and homogeneous region we observe to have an *edge*, or perhaps even be the the interior of a time-reversed black hole. But it should be emphasized that we have no empirical reason to believe that it's anything more exotic than a plain FRW universe. Though on a more serious alternatives, some models of cosmic inflation have our observed universe as one of many "bubbles" in an inflating background.

In many ways, the early universe was very similar in structure to a black hole, if one takes the singularity picture seriously. And even then, singularity free models still exist of black holes, so maybe the early universe does not require one either.

Anyway, this isn't important, what is important is that mathematics supports strongly an early universe with a structure similar to a black hole and in the later epoch where the universe has sufficiently cooled down and got large enough, seems to preserve the weak equivalence principle. (if you want more information on this I will elaborate).

It is possible, that these analogies to be taken seriously enough to speculate we live in a black-hole-like structure. Certainly there is a lot of arguments which attempt to support it. For instance, The radius of a black hole is found directly proportional to its mass $R propto m$ . The density of a black hole is given by its mass divided by its volume $
ho = frac

What does all this mean? It means that if a black hole has a large enough mass then it does not appear to be very dense, which is more or less the description of our own vacuum: it has a lot of matter, around $3 imes 10^<80>$ particles give or take a few power of tens of atoms in spacetime alone, the factor of $3$ to account how many spacetime dimensions there are - this is certainly not an infinite amount of matter, but it is arguably a lot yet, our universe does not appear very dense at all.

Early rotation properties resulting in centrifugal and torsion (the latter here to prevent singularities forming) as corrections to cosmology (if our universe is not a black hole analogy) could explain how a universe can break free from a dense Planck epoch (according to Arun and Sivaram). A lot of misconceptions concerning primordial rotation, exists even today.

Instead of going into great deal concerning equations I studied, I will give a summary of what I learned from it:

Hoyle and Narlikar showed that rotation exponentially decays with the linear expansion of a universe (this solves nicely why we cannot detect the background radiation ‘’axis of evil’’ expected to be like a finger print of rotation in the background temperatures).

Dark flow, an unusual flow that seems to show that galaxies are drifting in some particular direction at a very slow speed, could be the existence of a residual spin that has been left over.

Rotation explains cosmic expansion as a centrifigal force. Arun and Sivaram made a calculation from an expanding model.

Because rotation is suggested to slow down, it would seem then at odds to why the universe is now accelerating. There may be two ways out of this problem. The light we detect from the further galaxies tend to tell us something about the past, not something about the present moment in that region of spacetime. What appears to be accelerating, maybe the light from an early universe when it was accelerating. This would explain nicely the Hubble recession in which the further the galaxy, the faster it appears to receed. A second option comes from recent studies, which it has been stated that cosmologists are pretty sure the universe is expanding, but they are no longer sure at what speed.

If particle production happened while the universe expanded due to centrifugal acceleration, then there is no need for inflation to explain why matter appears to be evenly distributed (as noticed by Hoyle). In fact, Inflation doesn’t answer for anything, according to Penrose because it requires a fine tuning. Though this bit is quite speculative, I have wondered whether the spin has ''taken'' the bulk energy of the vacuum in an attempt to explain the quantum discrepency, dubbed the ''worst prediction'' ever made.

The fact the universe could have a rotational property, would explain why there is an excess of matter over antimatter because the universe would possess a particular handednesss (chirality) - there is also a bulk excess of a particular rotation property observed in a large collection of galaxies with odds ranging between 1 to a million by chance.

But most importantly of all (and related to the previous statement), it suggests that there is in fact a preferred frame in the universe so long as it rotates. This will imply a Lorentz violating theory but one that satisfies the full Poincare group of symmetries. According to Sean Carrol, Lorentz violating theories will involve absolute acceleration.

Some people might say ''dark energy is responsible,'' and there would have been a time I would have disagreed with this, since dark energy only becomes significant when a universe gets sufficiently large enough - it's effects are apparent because we believe the universe is accelerating now.

But I noticed a while back, that this is not the case if the impetus of a universe was constant, but strong enough to break free from dense fields. The difference here, is that scientists tend to think of this situation as one where the cosmological constant is not really a constant - but I tend to think now that it is a constant and the real dynamic effect giving rise to acceleration, is a weakening of gravity as it gets larger.

The OP's question may have concerned the "direct collapse" of matter into a black hole, which has been astronomically verified on one recent occasion (in 2008), as discussed on the Astronomy Stack Exchange in 2018. However, for reasons I'll put forward here, it may also relate to stellar collapse, which is much more common.

On more than 90 occasions, clear evidence of the "stellar collapse" of matter into black holes has been found: Because all known stars rotate, and most stars are partners in binary pairs, most such evidence consists of a partner continuing to follow the elliptical orbit both had shared prior to the other partner's collapse, under its own weight, after the conversion of most of its nuclear fuel to radiation had left it without the internal radiation pressure adequate to prevent such a collapse.

The effects of each such collapse remain as permanent as anything we can detect: Theories of particulate decay that have been endorsed by Stephen Hawking's frequent collaborator, the mathematical physicist Sir Roger Penrose, predict that any particles within black holes will be the last to decay within any locality where particles might be observed. Although faint radiation outward from black holes has been hypothesized by Hawking, its observation is not, under such theories, expected to occur while any life that has originated on earth, or even any cybernetic descendants of it that might be composed of subatomic particles as we know them, remain viable. Except in some purely idealistic (and consequently imaginary) sense, the duration of BHs may, consequently, be considered infinite.

Virtual particles and their anti-particle partners are necessarily separated from each other (by at least the Compton wavelength, during at least the Compton time) by the event horizon of any black hole, during that horizon's extremely rapid propagation outward from the center of the volume which had been occupied by the collapsing star. The fermions among those separated particles on the inboard side of that horizon are materialized, in a reversal of the processes which can convert intense concentrations of heavy matter into nuclear energy.

Like all fermions, the newly-materialized ones spin, and the interaction of their spin with that of the star's own (vastly larger) fermions reverses and accelerates their trajectories outward, thereby forming a local universe (shaped approximately like the thick skin of a basketball), in a phenomenon that can be described as an expansion of the space within the region that has been causally-separated from the larger LU which had been its "parent".

After an inflationary (asymptotically exponential) expansion, the newer LU continues expanding quasi-inertially. As in the other inflationary cosmologies, the spatial expansion of the local universe continues into an infinite future.

(Readers finding this description too mechanistic for something as ethereal-sounding as spatial expansion might want to consider Rebhan's 2012 paper at https://arxiv.org/pdf/1211.1006.pdf, whose conclusions point out the fact that descriptions of such expansion are fundamentally identical to those describing an explosion, with their consideration as the latter being more appropriate for such outside observers as ourselves, given the fact that the causal separation between ourselves and the "parenting" LU would combine with the continuing expansion of that parent to render astronomical observations of it impossible.)

The process described above is described more formally in Nikodem J. Poplawski's "Cosmology with torsion", the first of many papers he's written about his past- and future-eternal cosmological model between 2010 and 2020: They're available free on the Arxiv site, and are also found in articles put out by Elsevier and other highly-reputed publishers of scientific material. His cosmology is based on Einstein-Cartan Theory, which was worked out through conversations between Einstein and the mathematician Cartan, 14 years after Einstein's publication of General Relativity. Although Poplawski's 2010 "Cosmology with torsion" sketches his model as an "alternative" to cosmic inflation, it's now more generally considered to be a version of inflation, and perhaps the only one that does not rely upon a hypothetical field of "inflaton" particles.

With their expansion continuing inertially, the local universes of Poplawski's multiverse, evolving on sequentially smaller scales of spacetime, might eventually contain black holes of their own, although, due to local limitations on the time and energy available for use in magnification, their inhabitants might be able to observe any one of the black holes of any other sequence (even through the indirect process I described earlier) only if it would happen to be one that's on a scale approximately corresponding to the scale of their own astronomical surroundings: With expansion differing from relative motion and consequently not subject to the speed of light (which might itself vary between such causally-separated regions), the only exception might be their own LU, whose outermost spatial surface they would see simply as those parts of the night sky which are not occupied by stars.

Consequently, Poplawski's relativistic cosmology provides the simplest explanation for Olbers' Paradox, answering the question as to why the sky is not a lethal sheet of fire everywhere.

## Changing curvature as a universe evolves

@PeterDonis

Umm.. OK.

If the aim isn't to derive a GR solution but just to derive a set of formulae that does seem to describe the expansion of the universe, then you CAN derive that from Newtonian physics.

What is more surprising is that I don't believe anyone did derive those equations from Newtonian physics until AFTER the results were becoming apparent from G.R. and the work of Friedmann.

It's an assertion based on understanding the properties of FRW spacetime. Those properties are well known and understood and have been for decades.

Basically, you are saying you, personally, don't understand those properties well enough to see why @kimbyd's statement is true. So to you, it's "an assertion not a proof". But just saying that as a bald statement isn't going to make much progress in this discussion. Nor is continuing to just say you don't understand why certain things are the case, when you haven't taken the time to go build a better understanding of how FRW spacetime works on your own. If you are expecting to get *all* the information you need to improve your understanding from this thread, you are expecting too much. None of us get paid for this, and anyway just being told things won't make you understand them. You have to do the hard work of building your own understanding. The best we can do is try to point the way.

Perhaps that was the purpose of your OP in this thread, but it's not what the purpose of this thread has evolved into.

The simple fact is that, *if* we consider a single spacetime model from the class of FRW spacetimes, the normalized spatial curvature parameter (i.e., either ##+1##, ##0##, or ##-1##) *is* a constant for the model. If you want a "proof" of that simple statement, you will need to take some time to consult a textbook on cosmology that derives the FRW solutions from first principles, so you can understand how that derivation makes the statement true. Doing that explicitly here in this thread is well beyond the scope of what a PF discussion is supposed to be. I suggest Sean Carroll's online lecture notes on GR as a starting point IIRC he does the derivation in a fairly straightforward manner.

So the *only* way to construct a model in which the normalized curvature parameter can change is to take portions of two different FRW spacetime models that have two different parameters, and "glue" them together along some boundary. The obvious choice of boundary would be some particular spacelike hypersurface: roughly speaking, such a model would have curvature parameter ##k_1## up to some cosmological time ##t##, and then would have curvature parameter ##k_2
eq k_1## after that time. In post #16, for example, that was the kind of model I was thinking of when I talked about whether it's possible to match the two parameters continuously.

The Friedmann equations *are* a GR solution.

This common belief is wrong as the requirements for only the inside the sphere to govern the dynamics are not satisfied. The requirement is that the boundary conditions at infinity do not spoil the symmetry - which is true when you consider a potential that tends to zero at infinity. However, the Poisson equation with a constant density term is *incompatible* with any solution where the symmetry is maintained. This implies external fields - givenby the boundary conditions - also exist inside the sphere. Compare with adding an external electric field to the field of a soherical charge distribution. The external field will certainly affect the dynamics of a test charge inside the sphere.

I can only point to the lectures that are available online from Leonard Susskind. In those Cosmology lectures, the Friedmann equations were derived entirely using Newtonian physics. GR was only used at the very end and just to show that the same equations appear. I think @kimbyd has made this point.

The ability to derive the Friedmann equations from Newtonian physics may be a little surprising but it's not unlike the derivation of an object that was called a "dark star" and has many of the same properties for the object we now describe as a Black hole using GR.

Thanks again and I meant what I said. There's nothing wrong with what you've written. I'm just a stubborn student trying to understand why things have to be a certain way.

Why? This is an assertion not a proof.

This is very true. The full derivation is very complicated, and I haven't done it for a number of years. But basically when you try to describe a homogeneous and isotropic universe, you perform an integral at some point which results in a constant of integration. That constant turns out to be the spatial curvature.

If you tried to make it not constant, you wouldn't have a valid solution to Einstein's equations any longer.

I seem to remember this. My take-away is generally just that the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does: Newtonian gravity with this construction produces the exact same result as General Relativity. Thus the effects at infinity didn't spoil the solution when using a spherical construction.

My guess is that it's probably down to the fact that the spherical construction respects the homogeneity and isotropy conditions.

I seem to remember this. My take-away is generally just that the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does: Newtonian gravity with this construction produces the exact same result as General Relativity. Thus the effects at infinity didn't spoil the solution when using a spherical construction.

My guess is that it's probably down to the fact that the spherical construction respects the homogeneity and isotropy conditions.

The condition broken is homogeneity. By construction the spherical construction is isotropic. It is also not Gauss law that breaks down but the spherical shell theorem. That it by chance reproduces the correct result is not necessarily a sign of the derivation showing the same.

It *may* be possible to derive the same dynamic due to tidal forces regardless of the boundary condition at infinity (I have not had the time to look into this). This seems plausible to me as the volume change due to tidal forces should be proportional to a second derivative of the potential and the only such derivative is the Laplace operator. However, this is still a heuristic argument for obtaining the result, it does not show that the same equation is obtained in GR, and the solution is not necessarily isotropic and homogeneous (even if the density is).

**Concerning post #28 in general.**

I seem to have caused a problem or offence and I can only apologise for that. Perhaps I could explain the motivation for my actions so that you might see that no harm was intended and how easily a new foum member can get things wrong.

I was looking for a place to discuss some Physics and a Google search came up with Physics Forum. I skim read some of the terms and background for this website and found statements like this -

*Our mission is to provide a place for people (whether students, professional scientists, or others interested in science) to learn and discuss science as it is currently generally understood and practiced by the professional scientific community*

So I started to use the site. When this thread was created it was my intention to discuss a topic and learn as I went and I tried to make that clear in the OP and early posts. I thought a forum like this one might work as something comparable to a "study group" or just a coffee shop inside a place of learning, where one person can discuss both their ideas and their difficulties in understanding with others. That is what I have tried to do.

I have listened to all of the posts from others, engaged with every person that took the time to discuss anything and openly admitted where they were right and I had been wrong. See post #6 and #7 as a short example of an exchange between @Orodruin and myself. If I hadn't done this it would have been wrong and that's not the sort of person you would want in a study group or any discussion. However, you wouldn't want a person in a discussion or study group if they don't pull their weight either. This seemed to be especially true in a forum where you are the original poster. For example, in post #5 I got a message from a moderator that seemed to be saying something along those lines.

So I increased the amount of writing in my own comments and replies and set about demonstrating why a solution should exist. In later posts, I continued writing more and learnt how LaTeX works in this forum. In all likelihood I went too far, my posts were too long and may have seemed like some direct challenge backed up with half-correct references and nonsense mathematics. There were then comments from others along those lines. There is a phrase we use locally, "you're damned if you do and you're damned if you don't". Making a discussion without any references or avoiding Mathematics wouldn't have been good either. The situation is understood, I hope, when you see that I was just trying to discuss things as if in a study group.

I have made a similar mistake while trying to keep a forum thread on topic and updated. I thought it was generally good to stay on the main topic and drifting off topic is often considered as "hijacking". In my limited experience of forums I've also noticed that very old and very long threads with many replies are almost impossible to join in any constructive way. It is not possible to read and appreciate all the replies that have gone before and to see what the current state of the discussion has become. It is useful if someone, often the OP, can occassionally update the thread somehow. This is the sort of thing I was trying to do in post #23. I seem to have got this completely wrong and attracted comments along those lines.

It is a narrow line to walk, learning to use a forum and new members will get it wrong. My sincerest apologies.

I did not know but suspected that may be the case. I have always been and remain grateful for the time spent. You have never been under an obligation to reply. One of the ways in which this thread has failed is that it has failed to attract attention from people in a similar position to myself and instead consumed time from you. I am very sorry, it is not working as I hoped and I will cease. I cannot offer to pay you for your time since that seems to conflict with the terms and conditions of use, however there are many banners indicating ways in which I can support PF as a whole and I will be looking into those.

## Why is the FLRW universe (general relativity solution(s)) sometimes called the 'FRW universe'? - Astronomy

We are in an expanding Universe as per the big-bang cosmology.

Does it really mean that the outward thermal pressure now beats the inward gravitational pressure in the Universe?

What is the exact mechanism behind such an expansion (or, contraction later)?

The expansion of the universe can be explained by General Relativity using the Friedmann-Robertson-Walker metric, which contains a scale factor a(t) that determines the size of cosmos. Upon solving Einstein's field equations for ordinary matter (eg. dust), the scale factor is found to increase as t to the two-thirds power. Thus, GR predicts the universe expands from a point, called the big bang, and its expansion slows down with time. This solution does not involve pressure, as dust is assumed to be pressureless.

However, in the 1990's it was discovered through Supernova Type Ia observations that the expansion of the universe is not slowing down, but is accelerating. Cosmic acceleration at first appeared to contradict GR. A number of modified gravity theories have been proposed to explain it, but none have gained general acceptance. The most favored theory today is the Lamda-Cold-Dark-Matter (LambdaCDM) model, whereby cosmic acceleration is described by adding a cosmological constant Lambda to Eintein's field equations. This constant is often interpreted as "Dark Energy", a phantom substance with pressure equal to minus the mass-energy density. The model has a number of outstanding problems, and is the subject of ongoing research.

Nobody knows the answer to your question, but, I have given a possible detailed answer in the Daon Theory ( see my profile).

In my opinion, if the universe has hyperspheric form, its expansion is due to simple centrifugal force

Preprint ALL SPECIAL RELATIVITY EQUATIONS OBTAINED USING GALILEAN TRA.

Besides, it fits perfectly with the difficulty of measuring this expansion (it does not expand equally) and with the latest trends of grouping zones according to their density to calculate the Hubble constant.

The expansion of the universe can be explained by General Relativity using the Friedmann-Robertson-Walker metric, which contains a scale factor a(t) that determines the size of cosmos. Upon solving Einstein's field equations for ordinary matter (eg. dust), the scale factor is found to increase as t to the two-thirds power. Thus, GR predicts the universe expands from a point, called the big bang, and its expansion slows down with time. This solution does not involve pressure, as dust is assumed to be pressureless.

However, in the 1990's it was discovered through Supernova Type Ia observations that the expansion of the universe is not slowing down, but is accelerating. Cosmic acceleration at first appeared to contradict GR. A number of modified gravity theories have been proposed to explain it, but none have gained general acceptance. The most favored theory today is the Lamda-Cold-Dark-Matter (LambdaCDM) model, whereby cosmic acceleration is described by adding a cosmological constant Lambda to Eintein's field equations. This constant is often interpreted as "Dark Energy", a phantom substance with pressure equal to minus the mass-energy density. The model has a number of outstanding problems, and is the subject of ongoing research.

Self-gravity doesn't exist at this scale of cosmos.

Here is a non-technical but an idea based answer that is supported by Hubble's law.

The accelerating expansion of the universe is due to another fundamental force called Dark Gravity. This means variation of space-time field points in such a way that the points are farthest at the center of the universe and getting closer as one moves towards the edges. Consequently, gravitational constant tends to change and a point comes where spacial point and temporal point collapses. Hence, favors the big bang at the center but denies the big crunch near the edges.

Thus, galaxies speed up to faster than the speed of light near the edges and the total expansion theory comes into play causes nothingness of everything near the edges.

This also implies the universe to be more than what we observe.

Dear Khalid Ansari Universe does not act mechanically, the Universe is a Quantum Mechanics phenomenon. Universe like anything in it, it is growing by its nature, as it is growing new galaxies are born. Think of an apple tree, as it is growing new apple produce. Apple tree is getting its nitration from outside of its body, the universe is acting similar to this apple tree.

Space-time of GTR is not working with any solar system neither with ours, and spacetime is just a mechanical theory for one galaxy universe, while our universe has billions of galaxies and it is expanding rapidly.

And Gravity is not mechanical either, a mass can not create any gravity, i.e. if the earth had gravity, do you think the earth would let the mass of water goes up?

We have to think out the box, GTR is not working.

Article Gravity is an Internal Force

Per the General Theory of Relativity, mass causes space-time to curve in a positive direction, causing what we perceive as gravity (namely, things get closer together in the future than we would expect if there were no force acting on them), while the absence of mass causes space-time to curve in a negative direction, causing what would be called anti-gravity if some thing were causing the effect (namely, things get further apart in the future than we would expect if there were no force acting on them.

In this scenario, some place in empty space will either move toward objects or away from them, depending on the overall ratio of mass to empty space within the region near it. If the mass in a given region is big enough (as it is in planetary systems, stellar systems, galaxies and clusters of galaxies), things will move relative to each other according to what we would consider the force of gravity between them. But if the mass in a given region is too small in comparison with the amount of empty space, then things will move further apart, at a more or less constant rate per unit of space. At the time that Einstein predicted this, the Universe as we know it was not know to exist -- only a part of the Milky Way galaxy was thought to be the whole Universe -- and it is neither collapsing under its own gravitational force, nor expanding, so he presumed that somehow, the ratio of mass to empty space must automatically balance, using the constant capital lambda to account for that result. But once Slipher showed that most galaxies were moving away from us, and Hubble and Humason showed that the further a galaxy was from us the faster it was moving away from us, everyone (especially Einstein) realized that the mass of the Universe must not be large enough to balance the amount of empty space, and so the Universe is expanding at a more or less constant rate (constant in terms of being about the same in every part of space) at a given period of time. As it turns out that rate is NOT constant over time, because early on there wasn't as much empty space as now, so the ratio of mass to space was larger, and mass was actually slowing the rate of expansion for the first 8 billion years or so of the Universe's existence. But about 6 billion years ago the amount of empty space had become so great that the amount of mass was no longer adequate to slow the expansion and as the Universe continued to grow and the ratio of mass to empty space continued to decrease (it is currently only about 27% of the "balance" or "critical" value), the expansion slowly increased. My expectation is that it will asymptotically approach a value not much bigger than now over the next umpteen billion years (not much bigger than now because 27% of the "critical" mass is already far closer to zero than to the "balance" point).

The way this works is that on the average, for every Megaparsec (3.26 million light years), more or less empty space expands at about 70 km/sec (the actual value is somewhere between 67 and 73 km/sec/Mpc, but 70 is the average of that range, and I've been using it ever since I was working on the distribution of galaxies and clusters of galaxies with George Abell back in the 1960's, and have never seen any reason to change the value). Objects that are about 10 megaparsecs (32.6 million light years) away from us have a tendency to move away from us at about 700 km/sec, give or take a few tens or hundreds or km/sec due to their "peculiar velocities" (their motions relative to their neighbors, independent of the overall expansion of space). Within that distance, the peculiar velocities are large enough that the recessional velocity relative to the Cosmic Background Radiation (which is the most fundamental way of determining the rate of expansion) does not necessarily correspond directly to their distance. But once you get to a distance of 100 megaparsecs (about 326 million light years), the Hubble Flow, as it is sometimes called, is about 7000 km/sec, which is far larger than the peculiar velocities of any galaxy, even galaxies inside rich clusters of galaxies, where peculiar velocities of a thousand or so km/sec can be caued by the enormous mass and gravity of the cluster. As a result, at such distances and all distances beyond that, the recessional velocity relative to the CMB is a very good indicator of the distance of a galaxy or cluster of galaxies, and by the time you get to about 4000 Megaparsecs, the recessional velocity due to the cumulative expansion of 70 km/sec/Mpc for every one of those 4000 Mpc is 280 thousand km/sec, or nearly the speed of light and not much beyond that the cumulative recessional velocity becomes equal to or even larger than the speed of light, so that at some point, no light from such distant regions can reach us.

Note that in this discussion there is absolutely no mention of any kind of force, thermal pressure, or any other method of causing space to expand. It is just "natural" for large amounts of sufficiently empty space to expand, and that is all that the Universe is doing, or will ever do, for the rest of eternity (not that any of us or any of our descendants will be around long enough to see anything significantly different than what is going on at this exact moment in the history of the Universe).