Rolando

It is only the outside observer that is relevant in this context. This was new to me. I have to get informed about eddington-finkelstein coordinates. But do these really resolve the conflict? The problem arises already at the event horizon.

How can it be that intelligent people claim that there are black holes in the universe, e.g., at the center of galaxies, although the universe has only existed for a finite time in the cosmology they believe in? A black hole is an object whose mass is surrounded by an event horizon at which time is dilated infinitely. Such a state cannot be reached in a finite time. An event horizon is infinitely remote from an observer. This alone is sufficient to realize that black holes are merely mathematical constructs that do not exist in reality even if the universe is infinite in age. Isn’t it obvious enough that objects that need more than infinite time to form cannot exist in a universe whose age is finite? In science it happens that incompatibilities arise between predictions based on different foundations, such as general relativity and quantum theory. This is not mysterious, but in the case of black holes vs. a big bang universe, the foundation is the same: general relativity.

OK, then m^N means Σm. Could you also describe this "gravitational equilibrium with all other gravitating systems". What is it that is meant to be in equilibrium? Why is there no factor of 2 in the denominator of your equation (which I could imagine to be there)? Some parts of your text are hard to read, such as the last line the quotation. I don't get it, and so I am unable to follow your reasoning, although the equations that follow in your first post are intelligible to me. At one place, you multiply just one side of an equation by two, which is [insert adequate English word here]. But I find the question you raise, whether the universe could be in gravitational equilibrium, quite interesting.

Could you please explain what you mean by m^N to those who are not familiar with set algebra?

In response to Rolando saying "The volume that is gravitationally bound remains unchanged when the matter within this region collapses by gravitation", According to the simplest definition, gravitational binding extends to the distance r where the velocity due to expansion, v_exp=rH, begins to exceed the radial velocity of escape v_esc=(2GM/r)^1/2. This gives us r_gb=(2GM/H^2)^1/3. By ”the volume that is gravitationally bound”, I mean the volume that is enclosed within a sphere with radius r_gb. As you can see, this radius is given by M (and some constants). It remains the same when M (the total mass within this volume) is present in form of a low density gas or in form of a neutron star or a black hole or whatever. This is certainly so in the Schwarzschild metric, but the FLRW metric allows for expansion as well as for contraction. This remains a mystery to me.

The volume that is gravitationally bound remains unchanged when the matter within this region collapses by gravitation. The gravitationally bound volume can, however, increase by accidental mergers, which may happen occasionally. I see that his needs to be taken into account.

Collapse due to gravity is only possible within regions that are already gravitationally bound.

I am surprised about your comment, since

This is only true if the wavelenght (λ in Mordred's 1st equation) is compared with a standard (λ_0) that has remained unaffected by the expansion while the light was on its way. If the universe is completely homogeneous, which is presupposed in deriving the model from GR, there exists no such standard within the universe. Everything expands in proportion, so that any measured λ/ λ_0 remains =1. In this case, there is only a calculated λ/ λ_0 > 1, where λ_0 is the wavelength of a standard that has not been affected by the expansion. In reality, the redshift becomes measurable since coherent objects and standards of comparison do not expand. (In the models, this possibility is first excluded by pretending the expansion to be all-pervasive - and then nevertheless presupposed as a matter of course.) No idea why I think that the universe cannot ever have had a volume smaller than that of the regions that are gravitationally bound at present? I think this is clear enough as long as none of the gravitationally bound objects (remember the coins embedded in a rubber sheet) fuse. If they fuse, it requires calculating the volumes enclosed by their zero-velocity envelopes before and after fusion. Added: I have done this on paper and found that the non-expanding volume is proportional to the enclosed mass. In this case, it does not matter whether the objects fuse or disrupt or whatever. Huh? In order to convince you that you are wrong, it won't do with arguments that presuppose any of the things you reject (or put into question) to begin with. In my case, I was thinking of arguments based on the FLRW-models, whose adequacy I have put into question, but mainly of just teaching standard cosmology, as Mordred does. Mordred, your equations are correct, and at least the three first ones remain valid also if I am right in my criticism, but they tell nothing about the problem under discussion. I was unaware of the fourth one, although I had seen the handy summary by David W. Hogg before. It is mentioned there. elfmotat: I think my responses here may clarify even the points you comment. It's hard to respond to arguments that don't make sense. Why in the world would you think this? What led you to this thought? I will try to say it even more simply: The universe can never have been smaller than the present volume of those regions that have never expanded (because of their being gravitationally bound).

It has: if everything expands in proportion, which will be the case if the universe is completely homogeneous, there will be no observable redshift caused by the expansion. If there are expanding voids and coherent bodies that do not participate in the expansion, there will be such a redshift. Only in this case, the standard of comparison does not expand in proportion to expanding light waves. Also in response to Strange: My point is is very simple: In my understanding, the universe cannot ever have had a volume smaller than that of the regions that are gravitationally bound at present. If this is true, then the FLRW-approach, which is derived on the basis of homogeneity at all scales is dramatically inadequate. There cannot have been a big bang scenario. Speculations about "inflaltion" are, then, also irrelevant. In order to convince me that I am wrong, it won't do with arguments that presuppose any of the rejected things to begin with.

I agree, but I think we also agree that it is only the voids that expand, since the rest is gravitationally bound, and that this inhomogeneity is crucial. If the universe was completely homogeneous, there would be no redshift, because the standards of comparison would expand in proportion to the light waves. The universe can never have been smaller than the total volume of the gravitationally bound regions within it, and, to my understanding, this volume cannot have been smaller in the past than it is now. Sure, but it makes no difference to the size of the gravitationally bound regions whether the mass in them is present in form of gas, stars or planets. One can also say that the universe was more homogeneous in the past, since a smaller fraction of its volume was void (gravitationally unbound).

Even now, the mass of galaxy clusters consists mainly of gas clouds (and, supposedly, of even more dark matter), and the size of the clusters (”coins”) is given by the mass enclosed within the envelope at which the gravitational attraction balances the expansion of the universe. It makes no difference to the size of the ”coins” whether the mass enclosed by the envelope is condensed into stars or present in form of gas and dark matter. This approximation may be acceptable at present, when most of the volume of the universe consists of voids, but it becomes grossly inadequate when we go back in time, e.g., to z=10 (of the most distant galaxy that has been observed). The scale factor a(t) was then roughly 0.05 and the volume of the universe about a factor of 10^-4 smaller than it is now. There was not much room for expanding voids then.

OK. The stretching rubber sheet represents the expanding universe as described by Friedmann, Robertson-Walker, the ΛCDM concordance model, etc. These models have been derived on the basis of GR, assuming that the mass of the universe is distributed uniformly. They assume that there are no coherent objects (neither small ones nor larger ones, such as galaxy clusters etc, which probably house nearly all the matter in the real Universe). Further, they describe the possible evolution of the universe in a system of reference that is not itself affected by this evolution, like that in which a coin on the rubber sheet is static. However, coins on the rubber sheet are not within the universe. If anything, they represent separate, metrically static worlds outside the universe. In order to model the actual Universe, to which there is no outside, it would be necessary to have the coins embedded within the rubber sheet. But this would make it clear that the Universe could never have been smaller than the volume of the coins and that the mentioned models and the big bang are grossly unrealistic.

Elfomat & Strange, thanks for your replies, in which you explain how an expansion of the universe can be understood and why galaxies do not expand. This may be valuable for newcomers, but I am quite familiar with it already. My question concerned, instead, the claim that metric space expands. How can one reasonably claim this when any standards of comparison for length that are shorter than the diameter of a galaxy cluster have to be exempted from the expansion? (Otherwise, there would not either arise an observable redshift.) I am not either quite sure about why this idea has come to be diffused.

Sure, in the real world, the ruler (= the meter) does not expand, but to my understanding this blatantly contradicts the claim that the metric space expands. Then, what is the reason for ascribing the expansion of the universe to an expansion of metric space? Sure, but in order to stay the same size in their non-expanding metric space, meter rules need to contract in the assumed expanding metric space (which stays the same size in comoving coordinates).

This is what I meant. While the light is on its way, our ”meter rule” and the light waves expand in proportion. Hence no redsdhift. This is how most astronomers think about it. In an ordinary, static metic space, galaxies etc. are held together by gravity, while the universe as a whole is said to expand. There is no problem with the logics of this reasoning, but on a very large scale, it leads to velocities > c, which are considered as unphysical. I think that this is the reason why the expansion has come to be attributed to space itself. The logical problem arises if the metric space is assumed to expand. In this case, it needs to be explained why meter rules and galaxies etc. contract within this space. i.e., why their comoving linear size shrinks by the inverse of the scale factor, i.e., just as much as necessary to keep their sizes constant in a non-expanding metric space.

I did not think of space having a substance. It is rather my belief that substance (objects) need a space that causes the trouble. How can one claim that space expands if the volume that is occupied by objects, even by whole galaxy clusters, is to be exempted?

I meant that the idea that space itself could expand or contract is not very common outside the field of cosmology. Within cosmology, it is often claimed that space is expanding and that this is the cause of the cosmic redshift. In this context it is also often said that it is the "metric space" that is meant to expand. When waves expand in an ordinary, static metric space, their wavelengths increase. Light will, thus, be redshifted. The same observations would be made if waves remained unaffected in a contracting metric space. If waves expanded or contracted in a metric space that evolved in proportion, there would be no observable shift in wavelengths. A unit of length such as the meter, which by definition is equal to a certain number of waves of a certain type of radiation, would expand in proportion with the observed light waves. If waves remained unaffected in an expanding metric space, there would be a blueshift. None of these alternatives describes the situation. Coherent objects up to the size of galaxy clusters are actually not assumed to evolve in proportion with the scale factor, which leaves just the voids between them to do that. My question is about the logic of those who claim that the redshift results from an expansion of metric space, when this applies to no coherent object at all (neither atoms nor galaxy clusters or anything in between). They must consider objects not to occupy space. Can you make this palatable to me?

The spectral lines in the light we receive from distant galaxies are shifted towards the red. There is a “cosmic redshift”. The waves arrive stretched. We observe expanded waves in a static space. The idea that space itself could expand or contract is not very common, but it is diffused by some cosmologists and their followers. These claim that space is expanding and that this is the cause of the cosmic redshift. But this is unintelligible to me. What do they mean? An expanded wave in an unexpanded space is equivalent to an unexpanded wave in a contracted space – not in an expanded one.

Thank you, hoola and petrushka.googol, for your responses. I have thought about this myself a little more and I realize that it is difficult to model exactly. This is how one can think about it within the frame of orthodox cosmology: There are gravitationally bound objects up to the size of galaxy clusters. These can be thought of as enveloped by a ”zero-velocity surface” inside which gravitation is strong enough to prevent expansion (see e.g. Karachentsev, 2005, doi:10.1086/426368). However, this is not without exceptions. For a galaxy to escape beyond the zero-velocity surface, it requires in some cases only very little extra acceleration, which can arise by gravitational interaction with another galaxy. There is only a strong tendency for galaxies to remain gravitationally bound to their naighbors. If the universe expands, this tendency may explain why there are so large voids in it. There appears to be no force that counteracts the expansion of voids.

According to concordance cosmology, the universe expands, and its increase in linear size is described by a function a(t). However, any objects that are gravitationally bound (up to the size of galaxy clusters) do not participate in this expansion. They retain their size. Minor question: How large is the fraction x of the volume of the universe that is at present, i.e., in our cosmic vicinity (up to z = 0.1 or so), occupied by gravitationally bound objects? If a(t) increases with time and gravitationally bound objects retain their size, it is obvious that x must have been larger in the past, so that at an earlier time (not extremely long ago) most or all of the matter in the universe was gravitationally bound and so would never expand. Major question: How is this to be handled properly in concordance cosmology? What is it that is assumed to have made gravitationally bound objects to expand in the past? Can you point me to a model?

The slowing down is only observed from a distance.

Gravity causes objects to accelerate, but it is also the cause of gravitational time dilation, which affects all processes to the same extent, but this becomes substantial only in extreme gravitational fields - in the perspective of a distant observer.

The movement we see slows down to the same extent to which the ticks slow down. And this is due to gravity.

Let's say the object is a clock. As it approaches the event horizon, we see it tick slower and slower. When it reaches the event horizon, it stops. Time is frozen, but we can no longer see the clock, because the light waves are also frozen. Light waves become longer and longer when the clock approaches the event horizon and ultimately they would become infinite in length. However, this cannot happen within a non-eternal Universe, in which nothing can pass an event horizon or even reach it.