# Bounded Observable Universe and General Relativity

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The book General Relativity for Mathematicians by Sachs and Wu defines a "spacetime" as a connected 4-dimensional, oriented and time oriented Lorentzian manifold. I emphasize the word "connected." There are those that say the observable universe is bounded, because the universe is expanding faster than the speed of light. A bounded observable universe that is isolated from the rest of the universe, strikes me as a disconnected manifold, and would not seem to comport with general relativity's notion of spacetime. Is this correct?

Following up on the above point, wouldn't there have to be some sort of discontinuity, at the moment when the universe splits off into isolated pieces? How many discontinuities are being hypothesized? Does general relativity contemplate discontinuities in spacetime and gravity?

The universe may be bounded (or it might not be) and parts of it are receding faster than the speed of light (in some sense) but these two notions are independent. It would be possible to have a universe that unbounded and expanding, or bounded and stationary etc.

Similarly it seems that "connected" means connected in the topological sense: the only simultaneously open and closed non-empty subspace of the universe is the whole universe. This is independent of any physical laws such as "information can't travel faster than the speed of light" which may make parts of the manifold inaccessible to us.

The universe could be said to be the disjoint union of the observable and non-observable universe, but these sets are not both open in the topological sense.

## Finite Yet Unbounded?

Albert Einstein, in his book Relativity: The Special and General Theory, dedicates a chapter to this idea, as its title suggests: The Possibility of a “Finite” and Yet “Unbounded” Universe. In the words of us laypeople, Einstein – among others – suggests a “spherical” universe, one in which we can venture out in a straight line, and circumnavigate back to our starting position. But how is such a cosmos possible, let alone fathomable?

## Bounded Observable Universe and General Relativity - Astronomy

By the discussion I was reading, being that there is 15 billion light years to either "side" of us, am I to assume that this means that earth is the center of the universe? Or was that just an example?

From our vantage point on the Earth, we infer that the observable Universe is 15 billion light-years in size in every direction that we look - in other words, we infer that we are at the center of a sphere 15 billion light-years in radius.

This does not mean, however, that we are at the centre of the Universe it just means that we are at the centre of our observable Universe. A fundamental principle in our understanding of the Universe itself, called the Cosmological Principle, states that the Universe is homogeneous and isotropic on the largest scales. That means that on the whole, the Universe as seen from any vantage point (even one that is 15 billion light-years away from us!) will measure a spherical observable Universe with a radius of 15 billion light-years.

How does this make sense? It turns out that there are a couple of possibilities. First, the Universe could be much, much bigger than the part which we actually observe. If the Universe has the geometry of a "flat sheet" that we assume everyday on Earth, then the Cosmological Principle implies that the Universe must be infinite, since every observer at every "Universe edge" must observe the same global parameters. On the other hand, it is possible that the Universe's geometry is not flat, but curved like a sphere or a saddle. In this case, the Universe would "wrap" around at the edges: just as on the surface of the Earth, you would come back to where you started if you walked in one direction for long enough. Recent observations indicate that the first scenario is most likely true - we see a piece of the infinite, flat Universe that is 15 billion light-years in radius.

#### Kristine Spekkens

Kristine studies the dynamics of galaxies and what they can teach us about dark matter in the universe. She got her Ph.D from Cornell in August 2005, was a Jansky post-doctoral fellow at Rutgers University from 2005-2008, and is now a faculty member at the Royal Military College of Canada and at Queen's University.

## Etymology

The word universe derives from the Old French word univers, which in turn derives from the Latin word universum. [33] The Latin word was used by Cicero and later Latin authors in many of the same senses as the modern English word is used. [34]

### Synonyms

A term for 'universe' among the ancient Greek philosophers from Pythagoras onwards was &tauὸ &piᾶ&nu , tò pân ("the all"), defined as all matter and all space, and &tauὸ ὅ&lambda&omicron&nu , tò hólon ("all things"), which did not necessarily include the void. [35] [36] Another synonym was ὁ &kappaό&sigma&mu&omicron&sigmaf , ho kósmos (meaning the world, the cosmos). [37] Synonyms are also found in Latin authors (totum, mundus, natura) [38] and survive in modern languages, e.g., the German words Das All, Weltall, and Natur for Universe. The same synonyms are found in English, such as everything (as in the theory of everything), the cosmos (as in cosmology), the world (as in the many-worlds interpretation), and nature (as in natural laws or natural philosophy). [39]

I will answer this as a question on the different possible ways you could answer what is the universe's usable energy.

Well, the energy of the universe is all positive energy, if you consider the energy in the stress energy tensor and the dark energy. The dark energy has negative pressure, but positive energy. I say below something about what it is if you include gravitational energy also.

Note: you can calculate numbers and get to answers below. Whether that makes sense as the source of energy for your theorem I won't speculate.

So if you want the total energy now (in the cosmological comoving time now) just calculate the critical energy density. We're pretty close to that, if you count the dark density and both normal and dark matter. And then multiply by the volume of the observable universe. Since you're probably going to be comoving that is your available total energy.

The numbers are easy enough. Critical density is about $10^<-26>$ kg/$m^3$. This includes the mass equiv. The radius is 46 Gly, and volume you can compute to be about $10^<80>$ $m^3$. The total mass is then about $10^<54>$ kg. Multiple by $c^2$ and you get about $10^<70>$ joules. See some of the numbers from which you can calculate and verify at https://en.wikipedia.org/wiki/Observable_universe

However, the energy is not conserved. Over time the observable universe grows, and the density of matter energy decreases, except for the dark energy density which remains the same. That's 68% of the total density, or in order of magnitude the same as the total energy density. That density is about the same, and the universe becomes mostly a deSitter universe, expanding exponentially, which keeps growing to infinity. So total energy (which is all then dark energy) is infinite. In a comoving frame (remember the energy depends on the coordinate frame, if you change it it changes since the universe has no timelike symmetry). Also, although the observable universe at any one time is finite, the total universe is infinite, and so it is a not a bounded compact spacetime nor an asymptotically (I space or conformally) flat spacetime, and total energy is not conserved like it would be in an asymptotically flat spacetime. So in this view it just grows without limit. You can calculate it at $10^<100>$ years, it's about $10^<340>$ joules, order of magnitude assuming linear growth [I multiplied the $10^<70>$ joules now (roughly) by a factor of $10^<90>$ cubed, for a quick ROM]. This is a lower limit since the growth is exponential, so it may be closer to $10^<10^<340>>$ joules.

That would be one story. But dark energy is something we still don't understand. Another story would say most thinking leads to it probably being some kind of vacuum energy. Over time it is probable that that vacuum energy, could decay (say from false ground state), create matter energy, and then maybe form some other universe or bubbles of them. Whatever happens after our unknown dark energy starts decaying it really is impossible to predict now. This is highly speculative.

If you meant energy w/o the dark energy your number is above, as of now. Over time it goes to 0.

Either way, good luck with your Margolus-Levitin theorem. I know nothing of it. But the energy of the universe, depending on which energies you think you can use, are easily calculated as above. The less well defined in general relativity, the gravitational energy, you ignore unless you want to get to a total of 0, if you take the Einstein and the stress energy tensors. Or depending on how things fall out, you can get anywhere from 0 to about $10^<10^<340>>$ joules.

## How Big is the Universe, and How Do We Know?

On the 26th of April 1920, scientists and members of the public piled into a crowded lecture hall to hear astronomy’s most legendary debate. The night’s motion: was our galaxy the only one? Arguing for the motion was the young and ambitious astronomer Harlow Shapley, and against it was his more senior Herbert Curtis of Lick Observatory.

Many of Shapley and Curtis’ central points hinged upon observations of nebulae (the term used to apply to any diffuse astronomical object) and their distances from Earth. However, the data just weren’t there to come to a decisive conclusion.

Finding such data would take another three years, when Edwin Hubble observed a certain type of star called a Cepheid variable in one of the disputed nebulae. What made this so significant was that the brilliant — yet sadly overlooked — astronomer Henrietta Swan Leavitt had discovered a method of determining the intrinsic luminosity of a Cepheid by its period of dimming and brightening. Thus, by knowing a Cepheid’s period and the rate at which light from distant objects dims, you could calculate how far away it was. In this case, the Cepheid was measured to be 900,000 light-years away (later corrected to around 2.5 million light-years) — in any case far outside the known radius of the Milky Way. Hubble excitedly scribbled “VAR!” (as in “variable”) over his observations, gaining fame for his discovery while never crediting Leavitt.

Following in the long astronomical tradition of humbling our expectations of cosmic significance, Hubble’s discovery showed that Curtis was right. The universe was much bigger than we had imagined. But just how big?

Answering this would rely on another of Hubble’s results, this one made in 1929: that the universe is expanding. He realised this because the light emitted from distant celestial objects was redder than expected, due to a downward shift in their frequency as they receded from us. Since objects were getting further apart from each other, some scientists reasoned that there may have been some point in the past at which the whole universe was compressed into a single point. (It’s worth noting that other, more obscure, figures like Georges Lemaître and Alexander Friedmann made many of these contributions, and so my crediting Hubble is a gross act of simplification.)

After a long period of tweaking theory to match experimental data, this approach allowed astronomers to measure the age of the universe at about 13.8 billion years. You would be forgiven for assuming that the problem was solved there, and that a reasonable definition of the ‘observable universe’ — that part of the whole universe which we can theoretically see — is a sphere with a radius of 13.8 billion light-years centred on the Earth. After all, that’s only as long as light has had to reach us. This is a very common misconception: it would only be true in a universe that was static, i.e. not expanding or contracting.

You see, Hubble found from his data that the distance to a given galaxy was proportional to its recessional velocity, with a constant of proportionality now known as the Hubble constant H. Extrapolating this forward, there is a sphere past which everything is travelling away from us faster than the speed of light. This is known as the Hubble Sphere, and it has a radius of around 15 billion light-years.

You wouldn’t be the first to worry that this faster-than-light travel violates Einstein’s theory of relativity, which says that nothing can travel faster than light. However, what relativity actually says is that matter within space can’t travel faster than light, but nothing about how fast space itself can travel. Much ado is made about the fact that during inflation (the extremely rapid period of expansion in the early universe) pieces of matter were travelling apart from one another faster than the speed of light, but in fact, that’s still happening now!

So, is the Hubble Sphere the solution to our problem? Is that the real observable universe? Not quite. The radius of the Hubble Sphere will be given by c/H, where c is the speed of light and H is the Hubble constant (take a second to think about why). Even though the discovery of dark energy in the 1990s showed us that the universe is not only expanding but accelerating in its expansion, the Hubble constant is actually decreasing — and thus not a constant at all! The reason for this is that the Hubble constant (I wish there were a better name) is the relative velocity of two objects divided by their distance apart, and the expansion of space means that the bottom of the fraction is increasing faster than the top.

The upshot of this is that our Hubble Sphere is expanding. If the Hubble Sphere expands fast enough, light leaving an extremely distant object can get ‘caught’ and drop from the superluminal (faster-than-light) region outside the sphere to the subluminal (slower-than-light) region inside it. Once this light is within the Hubble Sphere, it can make its way to our telescopes. If you work out their present distance from us, it turns out that all of the photons we receive from the first five billion years of the universe’s existence were all caught by the Hubble Sphere. These objects were, are, and always will be travelling away from us faster than the speed of light!

All of this is to say that the boundary of distant objects we can theoretically observe is much further out than the Hubble Sphere. The full extent of the observable universe is bounded by the Particle Horizon, which is based on the time that light has had to travel to us since the beginning of the universe, taking into account its expansion. The Particle Horizon has a radius of a staggering 46.5 billion light-years, implying that the observable universe is 93 billion light-years across.

The matter responsible for the famous Cosmic Microwave Background (CMB) radiation, sometimes called the universe’s baby picture, is at the Particle Horizon, but its light has taken “only” 13.8 billion light-years to reach us. In this case, we say that the comoving distance — the actual distance from point A to point B — is 46.5 billion light-years but the light-travel distance is 13.8 billion light-years. These two metrics would only be equal in a perfectly static universe.

Of course, we can only assume that the “whole” universe extends far beyond the Particle Horizon. Saying that the universe is infinite is as much an act of ignorance as an act of extrapolation. At the end of the day, we have no idea. After all, how could we?

The Particle Horizon limits how far distant we can see, and the Hubble Sphere limits how far back in time we can see. We could never see objects outside the Hubble Sphere as they are now, no matter how long we waited. Moreover, just as light can get caught in the Hubble Sphere, the expansion of space can push light out of it. It’s estimated that we lose visibility of 20,000 stars a second because of this! In a few billion years, the observable universe will consist of only our ‘local group’ of a few dozen galaxies. It’s melancholic to think that future astronomers would look to the skies and gaze upon a barren universe. Perhaps they would look back at our astronomy magazines and be left to wonder what all the fuss was about.

## Big Bang, Classic Confusions

One of the most confusing things about the Big Bang is that it involves an expanding universe. Any reasonable person, hearing about the Big Bang, will imagine something that he or she has seen expanding: a cloud of smoke exploding outward, or a balloon expanding as it is filled with air. This is very natural. And having imagined this, the reasonable person will ask, “But what is the universe expanding into?”

This reasonable though seemingly paradoxical question is simply the wrong question. It is a consequence of having imagined the wrong thing. My goal here is to set your thinking straight.

Let’s go back and look at Figure 3 from the Worlds of 1 Spatial Dimension article, the relevant part of which is shown at the top of Figure 1 below. Notice there are two very different representations of the aeolian line (the dimension of possible wind-directions, which include the directions north, south-east, west-north-west, etc.) One representation is as a line segment whose left end is the same as its right end. The other is as a loop drawn in a plane. Now wait a second, you might ask. These look different. The loop surrounds an area, so it has an inside and an outside. The line segment doesn’t seem to have this. So how can these represent the same thing?

Fig. 1: (Top) A circle can be represented as a line segment with its two ends the same point (marked in red) or by a loop drawn in a plane (where the open red dot is the same point as the red dots on the segment.) From the line segment representation it is easy to see that a circle has no inside and no outside. (Bottom) A torus can be represented as a rectangle with its upper and lower edges the same (marked in red) and its left and right edges the same (marked in blue.) The torus can also be represented as the surface of a tire, or doughnut. To see these two representations are the same, join the top and bottom to form a cylinder, then the two ends to form the tire. Though the tire appears to have an interior, the rectangle representation makes clear that a torus does not intrinsically have an inside or an outside. The joining seams are shown as dotted lines.

Ah! This is indeed a crucial question, and the answer is vital to understanding spaces. The two representations — the loop, and the line-segment with its two ends matched — do represent the same one-dimensional aeolian line. The area that the loop surrounds is a property of the representation that we have chosen, not a property of the aeolian line itself! We must never confuse properties of pictures that we use in visualizing a space with properties of the space itself. That is very easy to do, but it is crucial not to do it.

As another example, it looks as though the doughnut (torus) drawn in Figure 1 has an inside and an outside. But it doesn’t. Just as a circle can be represented as a line segment with the left end and right end being the same point, a torus can be represented as a rectangle whose top and bottom edges are the same and whose left and right edges are the same. (To see that this is true, take a piece of paper. Attach the top edge to the bottom edge. You will now have a cylinder. Now you have to use your imagination to bend the left and right ends of the cylinder around so that they touch: but you will quickly see this will give you a torus.) There’s no inside or outside for our rectangle with matched edges, so there’s no inside or outside for a cylinder or for a torus. In other words, what is essential about a space is what you would learn about it if you traveled within it . To see the circle has an outside or an inside, you would have to travel across it: but if your circle is the aeolian line, that’s impossible. You cannot ask the wind to cut across the circle from north to south-east! It can only go around the circle, via east or via west and south. The only thing intrinsic to the aeolian line is the line itself!

And similarly, you cannot ask the tight-rope walker, from Figure 6 of the Worlds of 1 Spatial Dimension article, to cross from one side of the circular high-wire to the other. The only safe motion for the walker is to go around the circle one way or the other. So there is no way for the wind, or the walker, to find out whether there is or isn’t an interior or exterior to the circle.

This conceptual point is actually really important in understanding the expanding universe. If you are like most people, you’ve probably wondered (as I did when I was young), “What is it expanding into?” Well, in asking this question you are making the same mistake as for the circle: you are confusing something expanding with a representationof something expanding.

Fig. 2: Fig. 2: We are used to thinking about an inflating balloon expanding within three-dimensional space. It seems to have an interior and an exterior. But this just is a representation of an expanding sphere. An ant on the sphere could just as well represent it as a disk whose outer edge is a single point (as in a map of the globe with the north pole at center and the south pole at the edge.) It is then clear that it has no inside or outside, and does not expand "into" anything. It just expands, period, and this is evident to the ant from the fact that the distance between any two locations (such as the yellow and blue points shown) increases.

For example, you might imagine an expanding balloon. A balloon looks to us as though it is growing into the larger three-dimensional space in which it sits. But if you were an ant on the balloon, you would not know anything about some rumored interior or exterior all you would know is that the space on which you can walk has become larger. In fact you might (as an ant cartographer) represent the space as a disk whose edge is all one point (for instance, the point where the balloon is being inflated). You wouldn’t think about inside and outside all you’d know is that the distance between the yellow and blue points (and indeed between all pairs of points on the balloon) is growing.

Another two-dimensional space is the surface of the earth itself. Suppose you woke up one morning and the earth’s surface had doubled in size. You wouldn’t know whether the earth looked different to an observer out in space, or whether the diameter of the earth had grown. All you’d know is that when you walked to work or drove to get groceries it would take longer than it used to.

That the interior or exterior of the balloon is a property of the representation of an expanding space, and not of the space itself, is more obvious in the case of an infinite plane. It is possible for an infinite plane to expand, though it is not expanding into anything. It fills exactly the same space after it expands as before but the distances between objects on the space (for instance, the dots shown in the figure) has grown. The plane is intrinsically growing in size it isn’t inside a larger space, and so it obviously does not — cannot meaningfully — expand into that larger space. It just expands, period.

Fig. 3: An infinite plane can expand. It doesn't expand into anything but the distance between any two points, including between any pair of colored points drawn above, grows as the plane expands.

So it is with the universe. Like the plane just described, the space of the universe simply expands. There’s no way to see this from the outside there is no outside. But you can tell the universe is expanding from within the plane itself: the distance between all of the big objects in the universe (in particular, between the galaxies, the universe’s great cities of stars) grows and grows as the universe expands. Over time, it takes longer and longer to go from one big galaxy to the next. That’s what the Big Bang did: it took small regions of space and made them huge. It wasn’t an explosion it’s not like a bomb going off. It’s an expansion of space itself.

## THE UNIVERSE

According to the prevailing scientific model of the universe, known as the Big Bang, the universe expanded from an extremely hot, dense phase called the Planck epoch, in which all the matter and energy of the observable universe was concentrated. Since the Planck epoch, the universe has been expanding to its present form, possibly with a brief period (less than 10^-32 seconds) of cosmic inflation. Several independent experimental measurements support this theoretical expansion and, more generally, the Big Bang theory. Recent observations indicate that this expansion is accelerating because of dark energy, and that most of the matter in the universe may be in a form which cannot be detected by present instruments, called dark matter. The common use of the “dark matter” and “dark energy” placeholder names for the unknown entities purported to account for about 95% of the mass-energy density of the universe demonstrates the present observational and conceptual shortcomings and uncertainties concerning the nature and ultimate fate of the universe.

Current interpretations of astronomical observations indicate that the age of the universe is 13.75 ± 0.17 billion years, (whereas the decoupling of light and matter, see CMBR, happened already 380,000 years after the Big Bang), and that the diameter of the observable universe is at least 93 billion light years or 8.80×10 26 meters. According to general relativity, space can expand faster than the speed of light, although we can view only a small portion of the universe due to the limitation imposed by light speed. Since we cannot observe space beyond the limitations of light (or any electromagnetic radiation), it is uncertain whether the size of the universe is finite or infinite.

## If The Universe Is Expanding, Then Why Aren't We?

The fabric of expanding space means that the farther away a galaxy is, the faster it appears to . [+] recede from us.

It’s tough to wrap your head around four dimensions. Scientists have known that the universe expands since the 1930s, but whether we expand along with it is still one of the questions I am asked most frequently. The less self-conscious simply inform me that the universe doesn’t expand but everything in it shrinks – because how could we tell the difference?

The best answer to these questions is, as usual, a lot of math. But it’s hard to find a decent answer online that is not a pile of equations, so here’s a conceptual take on it.

The spacetime in our local neighborhood, which is curved due to the gravitational influence of the . [+] Sun and other masses, is part of a much larger region that composes the observable Universe. Over that volume, the fabric of space expands.

The first clue you need to understand the expansion of the universe is that general relativity is a theory for space-time, not for space. As Herman Minkowski put it in 1908:

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

Speaking about the expansion of space, hence, requires us to undo this union.

It isn't the fabric of space itself that we can observe, but only the matter and radiation present . [+] within that fabric.

NASA, ESA, and A. Feild (STScI)

The second clue is that in science a question must be answerable by measurement, at least in principle. We cannot observe space and neither can we observe space-time. We merely observe how space-time affects matter and radiation, which we can measure in our detectors.

How matter (top), radiation (middle), and a cosmological constant (bottom) all evolve with time in . [+] an expanding Universe.

E. Siegel / Beyond the Galaxy

The third clue is that the word “relativity” in “general relativity” means that every observer can chose to describe space-time whatever way he or she wishes. While each observer’s calculation will then differ, they will come to the same conclusion.

Armed with these three knowledge bites, let us see what we can say about the universe’s expansion.

Cosmologists describe the universe with a model known as Friedmann-Robertson-Walker (named after its inventors). The underlying assumption is that space (yes, space) is filled with matter and radiation that has the same density everywhere and in every direction. It is, as the terminology has it, homogeneous and isotropic. This assumption is called the “Cosmological Principle.”

While the Cosmological Principle originally was merely a plausible ad-hoc assumption, it is meanwhile supported by evidence. On large scales – much larger than the typical intergalactic distances – matter is indeed distributed almost the same everywhere.

The various galaxies of the Virgo Supercluster, grouped and clustered together. On the largest . [+] scales, the Universe is uniform, but as you look to galaxy or cluster scales, overdense and underdense regions dominate, and the Universe appears very non-uniform.

Andrew Z. Colvin, via Wikimedia Commons

But clearly, that’s not the case on shorter distances, like inside our galaxy. The Milky Way is disk-shaped with most of the (visible) mass in the center bulge, and this matter isn’t distributed homogeneously at all. The cosmological Friedmann-Robertson-Walker model, therefore, just does not describe galaxies.

This is a key point, and missing it is the origin of much confusion about the expansion of the universe. The solution of general relativity that describes the expanding universe solves Einstein's equations on average it is good only on very large distances. But the solutions that describe galaxies are different – and just don’t expand. It’s not that galaxies expand unnoticeably, they don’t expand at all. The full solution, then, is both the cosmic and the local solutions stitched together: expanding space between non-expanding galaxies. (Though these solutions are usually only dealt with by computer simulations due to their mathematical complexity.)

You might then ask, at what distance does the expansion start to take over? That happens when you average over a volume so large that the density of matter inside the volume has a gravitational self-attraction weaker than the expansion’s pull. From atomic nuclei up, the larger the volume you average over, the smaller the average density. But it is only somewhere beyond the scales of galaxy clusters that expansion takes over. On very short distances, when the nuclear and electromagnetic forces aren’t neutralized, these also act against the pull of gravity. This safely prevents atoms and molecules from being torn apart by the universe’s expansion.

A massive galaxy cluster, like Abell 370 (shown here), can be made up of thousands of Milky . [+] Way-sized galaxies. The space inside this cluster is not expanding, but the space between this cluster and other, unbound, galaxies and clusters, is.

NASA, ESA/Hubble, HST Frontier Fields

But here’s the thing. All I just told you relies on a certain, “natural” way to divide up space in space-and time. It’s the cosmic microwave background (CMB) that helps us do it. There is only one way to split space and time so that the CMB looks, on average, the same in all directions. After that, you can still pick your time-labels, but the split is done.

Breaking up Minkowski’s union between space and time in this way is called a space-time “slicing.” Indeed, it’s much like slicing bread, where each slice is space at some moment of time. There are many ways to slice bread and there are also many ways to slice space-time. Which, as clue number 3 taught you, are all perfectly allowed.

The CMB sets the relationship between space and time in such a way that the Universe can be . [+] consistently sliced-up in a 3 + 1 (space + time) decomposition.

The reason that physicists chose one slicing over another is usually that calculations can be greatly simplified with a smart choice of slicing. But if you really insist, there are ways to slice the universe so that space does not expand. However, these slicing are awkward: they are hard to interpret and make calculations very difficult. In such a slicing, for example, going forward in time necessarily pushes you around in space – it’s anything but intuitive.

Indeed, you can do this also with space-time around planet Earth. You could slice space-time so that space around us remains flat. Again though, this slicing is awkward and physically meaningless.

Denver, Colorado, USA, exhibiting the street grid typical of large cities in the southwestern USA. . [+] If we so demanded, we could define space so that this city shrank, grew, or remained stationary, but it isn't particularly meaningful.

This brings us to the relevance of clue #2. We really shouldn’t be talking about space to begin with. Just as you could insist on defining space so that the universe doesn’t expand, by willpower you could also define space so that a city, like Brooklyn, does expand. Let’s say a block down is a mile. You could simply insist on using units of length in which tomorrow a block down is two miles, and next week it’s ten miles, and so on. That’s pretty idiotic – and yet nobody could stop you from doing this.

But now, consider that you make a measurement. Say, you bounce a laser-beam back between the ends of the block, at fixed altitude, and use atomic clocks to measure the time that passes between two bounces. You would find that the time-intervals are always the same.

The atomic transition from the 6S orbital, Delta_f1, is the transition that defines the meter, . [+] second and the speed of light.

Atomic clocks rely on the constancy of atomic transition frequencies. The gravitational force inside an atom is entirely negligible relative to the gravitational force – it's about 40 orders of magnitude smaller – and fixing the altitude prevents gravitational redshift caused by the Earth’s gravitational pull. It doesn’t matter which coordinates you used, you’d always find the same and unambiguous measurement result: the elapsed time between bounces of the laser remains the same.

In cosmology, too, it helps to first clarify what it is we measure. We don’t measure the size of space between galaxies – how would we do that? We measure the light that comes from distant galaxies. And it turns out to be systematically red-shifted regardless of where we look. A simple way to describe this – a space-time slicing that makes calculations and interpretations easy – is that space between the galaxies expands.

The 'raisin bread' model of the expanding Universe, where relative distances increase as the space . [+] (dough) expands. The galaxies (raisins) themselves, though, don't change. It's only that the light coming from them get redshifted (or stretched) in an expanding Universe.

So, the brief answer is: no, any bound object in the Universe doesn’t expand. But the more accurate answer is that you should ask only for the outcome of clearly stated measurement procedures. Light from distant galaxies is shifted towards the red, meaning those galaxies are retreating from us. Light collected from the edges of a city like Brooklyn isn’t redshifted. If we use a space-time slicing in which matter is at rest on the average, then the matter density of the universe is decreasing and was much higher in the past. To the extent that the density of Brooklyn has changed in the past, this can be explained without invoking general relativity.

It may be tough to wrap your head around four dimensions, but it’s always worth the effort.

## Teeny Tiny Milky Way Galaxy

How small is the Milky Way Galaxy compared to the Universe? This might give you an idea. I found it in my new Astronomy calendar. It is put out by Firefly Books and edited by Terrence Dickinson and was on the June 2018 page.

If a grain of sand represented a star, a thimble would be filled with all the stars you can see on a moonless, clear, dark night in the country. It would take an entire dump truck to represent all the stars in the Milky Way Galaxy. To represent all the stars in the Universe you would need a freight train filled with hoppers of sand. That train would be so long it would circle the Earth 25 times. If you wanted to watch the train pass by at the rate of one hopper per second, it would take 3 years! Wow. and I get impatient if I have to wait 3 minutes at a crossing. Happy New Year!

### #2 GamesForOne

Perhaps it should read, "To represent all the stars in the observable Universe. "?

### #3 xiando

The milky way galaxy is but a grain of sand on the beach in comparison to number of galaxies in the universe. But having gobbled up so many smaller ones over the eons, at least it's a whole grain.

### #4 NiteGuy

That's a great perspective! It's a bit hard to take that all in after watching and being in the center of the college football universe for 12 consecutive hours yesterday.

### #5 Stacyjo1962

What a great comparison! Thank you for sharing this!

That's a great perspective! It's a bit hard to take that all in after watching and being in the center of the college football universe for 12 consecutive hours yesterday.

Ah yeppa, that's what we did too! LOL.

### #6 CygnuS

Perhaps it should read, "To represent all the stars in the observable Universe. "?

---

Michael Mc

Okay, so now you got me thinking. How many more stars does the Universe have compared to the observable Universe?

### #7 GamesForOne

Perhaps it should read, "To represent all the stars in the observable Universe. "?

---

Michael Mc

Okay, so now you got me thinking. How many more stars does the Universe have compared to the observable Universe?

"Both popular and professional research articles in cosmology often use the term "universe" to mean "observable universe". This can be justified on the grounds that we can never know anything by direct experimentation about any part of the Universe that is causally disconnected from the Earth, although many credible theories require a total universe much larger than the observable universe. No evidence exists to suggest that the boundary of the observable universe constitutes a boundary on the Universe as a whole, nor do any of the mainstream cosmological models propose that the Universe has any physical boundary in the first place, though some models propose it could be finite but unbounded, like a higher-dimensional analogue of the 2D surface of a sphere that is finite in area but has no edge. It is plausible that the galaxies within our observable universe represent only a minuscule fraction of the galaxies in the Universe."

So. we may never really know the answer.

### #8 xiando

The only way we' ll know is to get out farther so we can expand our viewing radius. Gotta find that stargate, wormhole, or hyperdrive!

Edited by xiando, 02 January 2018 - 09:39 PM.

### #9 B l a k S t a r

There was a race that "folded" space in Frank Herbert's Dune series. Large beings that lived in water if I remember correctly - long since I read that.

### #10 xiando

There was a race that "folded" space in Frank Herbert's Dune series. Large beings that lived in water if I remember correctly - long since I read that.

Afaik, they were once humans but were genetically altered by their massive ingestion of spice over thousands of years..

### #11 CygnuS

Perhaps it should read, "To represent all the stars in the observable Universe. "?

---

Michael Mc

Okay, so now you got me thinking. How many more stars does the Universe have compared to the observable Universe?

"Both popular and professional research articles in cosmology often use the term "universe" to mean "observable universe". This can be justified on the grounds that we can never know anything by direct experimentation about any part of the Universe that is causally disconnected from the Earth, although many credible theories require a total universe much larger than the observable universe. No evidence exists to suggest that the boundary of the observable universe constitutes a boundary on the Universe as a whole, nor do any of the mainstream cosmological models propose that the Universe has any physical boundary in the first place, though some models propose it could be finite but unbounded, like a higher-dimensional analogue of the 2D surface of a sphere that is finite in area but has no edge. It is plausible that the galaxies within our observable universe represent only a minuscule fraction of the galaxies in the Universe."

So. we may never really know the answer.

It sounds like I'm going to need to buy one of those fancy $300 eyepieces to solve this mystery. ### #12 xiando Perhaps it should read, "To represent all the stars in the observable Universe. "? --- Michael Mc Okay, so now you got me thinking. How many more stars does the Universe have compared to the observable Universe? "Both popular and professional research articles in cosmology often use the term "universe" to mean "observable universe". This can be justified on the grounds that we can never know anything by direct experimentation about any part of the Universe that is causally disconnected from the Earth, although many credible theories require a total universe much larger than the observable universe. No evidence exists to suggest that the boundary of the observable universe constitutes a boundary on the Universe as a whole, nor do any of the mainstream cosmological models propose that the Universe has any physical boundary in the first place, though some models propose it could be finite but unbounded, like a higher-dimensional analogue of the 2D surface of a sphere that is finite in area but has no edge. It is plausible that the galaxies within our observable universe represent only a minuscule fraction of the galaxies in the Universe." So. we may never really know the answer. It sounds like I'm going to need to buy one of those fancy$300 eyepieces to solve this mystery.

Heh. you missed a few zeroes. Maybe more than a few

### #13 Maxtrixbass

Perhaps it should read, "To represent all the stars in the observable Universe. "?

---

Michael Mc

Okay, so now you got me thinking. How many more stars does the Universe have compared to the observable Universe?

I took a physics cosmology class in college and I recall the answer presented being "there are probably more galaxies in the entire universe than there are particles in the universe we can see".

If true, that represents a scale that I simply cannot wrap my brain around. Not even close..

### #14 CygnuS

Perhaps it should read, "To represent all the stars in the observable Universe. "?

---

Michael Mc

Okay, so now you got me thinking. How many more stars does the Universe have compared to the observable Universe?

I took a physics cosmology class in college and I recall the answer presented being "there are probably more galaxies in the entire universe than there are particles in the universe we can see".

If true, that represents a scale that I simply cannot wrap my brain around. Not even close..

It looks like we're going to need a bigger train.

### #15 CygnuS

Perhaps it should read, "To represent all the stars in the observable Universe. "?

---

Michael Mc

I took a physics cosmology class in college and I recall the answer presented being "there are probably more galaxies in the entire universe than there are particles in the universe we can see".

Clearly then the author was writing about the observable Universe since he stated that the sand in the train would amount to all the sand on all the beaches on Earth. It makes me wonder how often people make that mistake. If Terrence Dickinson can make it I suspect it's common. It sounds like there are an infinite number of stars. Yet, that too would not be a correct statement since all numbers, no matter how large are tiny by comparison.

### #16 Ultimatetelescope

Wow, really makes you realize how small we are.

### #17 xiando

Wow, really makes you realize how small we are.

Well, fortunately, we can always look "the other way" and be proud of the staggering number of atoms that we're made of.

### #18 SeaBee1

Interesting discussion. The simple thing is, we don't have a clue.

I suspect the "Universe" has no bounds. it is infinite in scope. I don't believe that this means the infinite universe is filled with an infinite number of galaxies. I believe there is a finite amount of mass. there is just A LOT OF IT.

Love the Dune references, BTW. big fan here.

### #19 BillP

I suspect the "Universe" has no bounds. it is infinite in scope. I don't believe that this means the infinite universe is filled with an infinite number of galaxies. I believe there is a finite amount of mass. there is just A LOT OF IT.

Infinity is an interesting "concept" like many other conceptual constructs though not one yet found with any empirical certainty in reality. Lots of conceptual things exist only in that space and do not have a reality counterpart. Part of the beauty of math is its boundlessness. I'm sure the oceans felt infinite to the initial explorers. it still feels infinite, and scary, when you fly out into it in a small private plane. Would be nice if the universe were infinite, but being a finite creature certainly makes that imagining difficult to comprehend.

Edited by BillP, 03 January 2018 - 10:49 AM.

### #20 SeaBee1

Hi Bill! I guess I have always wondered about the construct of the "Universe", even as a young pup. the thing I always wondered about when hearing or reading of a "bounded universe" was what is beyond the boundary. if there is a boundary, there has to be something on the other side of said boundary. even if it is "nothingness". and isn't that "nothingness" still part of the "Universe"?

I suspect we will never know for sure.

### #21 CygnuS

Hi Bill! I guess I have always wondered about the construct of the "Universe", even as a young pup. the thing I always wondered about when hearing or reading of a "bounded universe" was what is beyond the boundary. if there is a boundary, there has to be something on the other side of said boundary. even if it is "nothingness". and isn't that "nothingness" still part of the "Universe"?

I suspect we will never know for sure.

But if it were true nothingness it couldn't be "part" of anything. Yet the nothingness that the Universe sprang from couldn't be true nothingness. The least it could have been was potential in order to have the potential to create the Universe.

### #22 CygnuS

Interesting discussion. The simple thing is, we don't have a clue.

I suspect the "Universe" has no bounds. it is infinite in scope. I don't believe that this means the infinite universe is filled with an infinite number of galaxies. I believe there is a finite amount of mass. there is just A LOT OF IT.

Love the Dune references, BTW. big fan here.

CB

If the Universe were infinite wouldn't it be filled with an infinite amount of everything?

### #23 BillP

Hi Bill! I guess I have always wondered about the construct of the "Universe", even as a young pup. the thing I always wondered about when hearing or reading of a "bounded universe" was what is beyond the boundary. if there is a boundary, there has to be something on the other side of said boundary. even if it is "nothingness". and isn't that "nothingness" still part of the "Universe"?

I suspect we will never know for sure.

So imagine for a second that the Universe is infinite and maybe that space-time is just a finite bubble in the larger infinite universe. Who knows what is on the other side of that bubble? No way to really tell unless you go there. If you were born inside a closed opaque box you can forever imagine and theorize what might be outside the box. But no matter what you imagined or scientifically proposed, the reality will be much larger of what is outside because what you can glean from the inside is based on very little data. In an infinite universe, no real way of telling what is all out there because if all you have is observation and deduction, then all will be able to come up with is an imaginative conclusion based on a myriad of assumptions and probabilistic deductions -- not something to take to the bank. The imagining and scientific theorizing is still fun regardless. But "knowing" I think is something we will have to let go of and be satisfied with just our imaginings, scientific or otherwise.

Edited by BillP, 03 January 2018 - 11:35 AM.

### #24 SeaBee1

Since the law of conservation of mass and energy states that it can neither be created nor destroyed, it seems to me this would imply a finite amount, regardless of how big the universe is. And, of course, there may be other things at play here that I am not up on. High school physics was a loooong time ago for me

Hi Bill! I guess I have always wondered about the construct of the "Universe", even as a young pup. the thing I always wondered about when hearing or reading of a "bounded universe" was what is beyond the boundary. if there is a boundary, there has to be something on the other side of said boundary. even if it is "nothingness". and isn't that "nothingness" still part of the "Universe"?

I suspect we will never know for sure.

But if it were true nothingness it couldn't be "part" of anything. Yet the nothingness that the Universe sprang from couldn't be true nothingness. The least it could have been was potential in order to have the potential to create the Universe.

I used the term "Nothingness" not as a meaning of "void" really, but as an indicator of "we don't know what is there". and even if it were a "void", why couldn't it be a part of "something"? That something being the "Universe".

Interesting discussion. The simple thing is, we don't have a clue.

I suspect the "Universe" has no bounds. it is infinite in scope. I don't believe that this means the infinite universe is filled with an infinite number of galaxies. I believe there is a finite amount of mass. there is just A LOT OF IT.

Love the Dune references, BTW. big fan here.

CB

If the Universe were infinite wouldn't it be filled with an infinite amount of everything?

Not necessarily. not according to the law of conservation of mass and energy. we might not know exactly how much mass/energy is present in an infinite universe, but if this law is correct, it is measurable.

Hi Bill! I guess I have always wondered about the construct of the "Universe", even as a young pup. the thing I always wondered about when hearing or reading of a "bounded universe" was what is beyond the boundary. if there is a boundary, there has to be something on the other side of said boundary. even if it is "nothingness". and isn't that "nothingness" still part of the "Universe"?

I suspect we will never know for sure.

So imagine for a second that the Universe is infinite and maybe that space-time is just a finite bubble in the larger infinite universe. Who knows what is on the other side of that bubble? No way to really tell unless you go there. If you were born inside a closed opaque box you can forever imagine and theorize what might be outside the box. But no matter what you imagined or scientifically proposed, the reality will be much larger of what is outside because what you can glean from the inside is based on very little data. In an infinite universe, no real way of telling what is all out there because if all you have is observation and deduction, then all will be able to come up with is an imaginative conclusion based on a myriad of assumptions and probabilistic deductions -- not something to take to the bank. The imagining and scientific theorizing is still fun regardless. But "knowing" I think is something we will have to let go of and be satisfied with just our imaginings, scientific or otherwise.