# Curvature of space

The curvature of space is a physical fact of surface of a 2-sphere (our "ordinary" sphere, or a surface of a 3 dimensional ball) of surface area

r2

is enclosing a different volume than

(4 / 3)πr3,

where r is the radius of the 2-sphere.

In our physical universe containing energy, such 2-spheres may enclose larger volume of space than

(4 / 3)πr3,

and the 3-space with such properties is then called positively curved, or "convex". The 2-spheres in such 3-space contain more space than the same 2-spheres in Euclidean (flat) 3-space, which contains exactly

(4 / 3)πr3

volume of 3-space.

If a 2-sphere of surface area

r2

contains smaller 3-volume of 3-space than

(4 / 3)πr3

such space is nagatively curved and it is called "concave".

Curvatures of spacetime that is composed of space and time have to be described by more numbers than just the amount of space contained within 2-sphere of area

r2

and the volume of 3-space that it contains and then the "spheres" in spacetime are called 3-spheres and the curvature of spacetime is described by a set of 10 numbers composing a curvature tensor.

Luckilly, our physical spacetime is never curved since it has to be flat on Noether's Theorem to obey conservation laws. The tensors used in physics are specifically designed to obey these laws. Therefore the physical spacetime must be flat. Being flat it allows the space alone being curved and then the time compensates its curvature creating gravitation as described in Gravitation demystified.

According to Einstein, the space of our universe is such "curved space". One of its amazing feature is that it can be closed, in the same sense as the surface of the Earth is closed (no edges). But while the surface of the Earth is closed through the third dimension, the space of the universe is already 3 dimensional and there is no 4-th dimension in nature to close the space of our universe through. So we have to imagine how it works in 3 dimensions. What would happen when we travel through the universe and not being able to leave its space similarly as traveling on the two dimensional surface of the Earth we are going to make a big circle around the Earth and get back to the same point. How it can happen in the "curved space" of the universe. How the universe has to behave that such thing can happen while traveling through it along a straight line. Difficult to imagine, yet it is possible.

Let's start to describe the physical features of curved space.

As the first part of Einsteinian Gravitation is about what happens to time in the vicinity of masses (which explains all those things that Newtonian gravitation explains but without introducing any magical "gravitational attraction"), the second part is about what happens to space around masses.

Basically Einstein's theory states that there is more space around masses than there would be without those masses present. It means that if we make a spherical shell having volume of e.g. 1,000 gallons when empty then we pour into it water then the space inside the sphere becomes bigger than before with the outside shell ever changing its size. It is difficult to believe, yet it seems to be what actually happens. The diameter of the shell as measured from the outside will be the same as it was before, but the space inside will be bigger (measured with the same rulers that we used to measure the shell from the outside).

If we measured the diameter from inside, moving along a straight line through the center from one side of the shell to the other, we would find that the diameter of the shell inside is bigger than when measured from the outside. This is what is called "curved space". The space inside the sphere is bigger than the space taken by the sphere itself.

We may pour into a 1,000 gallon sphere (i.e. volume when empty) e.g. 1,001 gallons of some (very) heavy liquid which means that the space inside became bigger by one gallon.

To understand this strange thing with diameter of the shell being greater inside than outside (which happens to be a physical fact), we may think about it as rulers becoming shorter when put inside the shell containing mass inside. Just as clocks run slower in the vicinity of mass because time runs slower there (which is called time dilation), rulers become shorter in the vicinity of mass (which is called length contraction). So if our rulers are shorter inside the shell, the diameter of the shell measured from inside will be bigger.

But physically the clocks all run the same speed (maintaining local time). It is time itself which slows down in the vicinity of mass as seen from the outside. The rulers also remain the same, just there is more space in the areas in vicinity of mass than there would be in regular (flat, Euclidean) space, so compared to the diameter of the shell, the rulers seem shorter to us seen from the outside of the shell.

The space with such strange properties is called "curved" because the increase of volume inside our shell that contains mass is similar to the increase in surface area inside a ring when the ring is placed on a surface of a ball instead of on a flat table. The area of the surface inside a ring on a ball will be greater than the area inside the same ring placed on a flat table. We say that surface of the ball inside the ring is curved as opposed to the surface of the table inside the ring that is flat. Similarly we say that space inside our sphere is curved after mass, or energy, since every energy has mass according to the famous E = mc2 showed up inside it, as opposed to flat empty space.

Another interesting thing about the above is that the space in the vicinity of mass gets bigger (or "rulers shrink") by the same relative amount (by the same percentage) as the time slows down.

This relation of time to space near masses causes light rays to bend twice as much as predicted by Newtonian gravitation. As it was mentioned earlier, Newtonian gravitation only predicts accurately the gravitational effects caused by gravitational time dilation and none caused by curvature of space. That's why it predicts only half of deflection of ray of light and Einsteinian Gravitation predicts the whole angle as it is observed in the real world. And this is also how we know that the time dilation that causes half the deflection is the same as the increase in amount of space that causes the other half of deflection. This equality of spatial and temporal effects shows that the spacetime is more complex creature than space and time separately, and which are considered separately in Newtonian theory. That somehow time depends on the space, and space on time. It is similar to the married couple, where the woman and the man depend on each other and make a more complex structure than the two of them being considered together but independent form each other.

This interdependency of time and space explains why the universe appears to be expanding. There are masses in the universe as planets, stars, galaxies and all the other observed and not yet observed junk between them. Each of them curves the space a little bit (makes more of it in its vicinity) when we look through that increased space deeper and deeper into space we see time slowing down more and more. Such slowing of time simulates a Doppler effect, which makes it seem as if all sources of light in the universe are moving away from us with velocities proportional to their distance from us. Just as before Einstein the behavior of time in the universe simulated the existence of the universal gravitational attraction, post-Einstein it also simulates the universal expansion.

The astrophysicists prefer to insist that it is not possible to propose explanation of the observed phenomena other thanan Doppler effect, which according to them forces everybody to believe that the universe is expanding. And since it is not possible to propose explanation all papers on that subject are rejected by all scientific journals without even stating the reason for rejection other than the papers don't support the idea that the universe is expanding (small wonder). And it lasts already for over two decades. It will probably last well into this century until some Very Important Person, whose paper nobody will dare to reject, discovers that Einstein's theory explained all of it already. Then the big bang will also disappear from minds of astrophysicists.

Those of the readers who are interested in this subject enough to read to this point might have noticed that this illusion of the expansion of the universe is caused by behavior of time coupled with curved space, and that behavior of time is reflected in Newtonian gravitation. It might give them an idea that therefore it should be possible to demonstrate just with Newtonian formula that a non-expanding universe should appear to be expanding. That is indeed so, and this is what the author has done to convince astrophysicists (without much success though) that the expansion of the universe is an illusion. It has been shown with simple Newtonian math what should be the observed rate of apparent expansion if the universe didn't actually expand, and it turns out that the result is as it is really observed. It is also the same result as would be predicted by just following strictly Einstein's gravitation and the fact that time dilation is the same as curvature of space. Both methods derive Hubble's constant of apparent expansion as the speed of light divided by Einstein's radius of the universe. The details of the derivation of Hubble's constant for our universe are on this site in Essay:Hubble redshift in Einstein's universe. There is also an explanation for the general public, but more detailed than those two paragraphs above and with full mathematical support for those who want to see how it is derived in Gravitation demystified.

The above basically explains all the gravitational phenomena that are observed up to date. There also some predictions of what we might to encounter when we gather more data about the universe. All of them are quite interesting.

One such thing is that the more mass is placed inside the shell the more additional space there will be inside it, but the amount of space increases by a greater amount than the mass that creates that additional space. The math of this mechanism, described by Schwarzschild's solution of Einstein Field Equations (that describe Einsteinian Gravitation), indicates that for any sphere there exists a certain amount of mass that makes that additional space infinite. If there were such a mass inside our sphere we could pour infinite amount of water into it. Such an object, with an infinite amount of space inside is called a black hole. The name comes from a fact that if there were infinite space inside it, the light would need an infinite amount of time to travel though it to get out of it and come to us. So practically we would never see that light regardless how long we looked at that object. We would see something that does not emit light at all, and so it is perfectly black.

Another interesting feature of such an object would be that it would be at infinite distance from us. It would be so because when there is more space around a big mass and time also slows down in this region, light needs more time to get to that mass and back. If we use radar to measure the distance to that heavy object the photons we send to it come after longer time than they would if the mass of the object were small. In case of a hypothetical black hole, that time, and therefore the distance too, would be infinite.

It is not known whether such objects as black holes exist in nature. Some scientists maintain that they can't exist (Einstein was one of them) because of those infinities they produce like the effect of time slowing to halt at the surface of the sphere, and so no more objects can fall into such a sphere in a reasonable time. So a real black hole couldn't be formed during the lifetime of the universe (regardless how long it were, unless it were infinite as well). The surface of the sphere which contains enough mass to stop the flow of time is called event horizon since at that surface the time stays still and so nothing can ever happen. No events are possible on and beyond that surface. Some scientists believe that black holes exist but no reasonable hypothesis telling how the objects may fall into them through the event horizon to form them has been proposed yet (however unreasonable hypotheses were proposed and many scientists, not understanding Einseinian gravitation, believe that those who proposed those hypotheses did check that they make sense). Consequently the black holes are more popular in SF texts than in science. And often this SF is presented to the public as science by naive astrophysicists who don't understand Einsteinian gravitation but believe what they are told by experts many of whom don't understand it either. Since we want to keep this text as close to science as possible let's forget about the black holes. There are even stranger and observable things in the nature so we don't need to get into SF to be baffled.

It should be mentioned that the curvature of space has very little influence on gravitational phenomena in our solar system and next to none in vicinity of the earth. It applies mostly to deep space of the universe. To understand what is really going on in the universe it would be good to learn what happens to its space when each mass curves it only a little bit.

The main thing that happens to space is that if the space behaves this way it has to be closed. It means that from whatever point in the universe one starts to travel in a straight line in whichever direction, if one travels long enough along that straight line, one is bound to return to the starting point.

It seems unbelievable, perhaps much more than to the people who believe that the earth is flat, that going due West one may return one day to the starting point from the East, never even getting to the edge of the earth. Of course we know that the two dimensional surface of the earth is closed via the third dimension. The three dimensional space of the universe can't be closed via the fourth dimension since it is easy to notice that the fourth spatial dimension does not exist (it is impossible to make four mutually perpendicular directions). So let's try to explain how our space can still be closed despite that the lack of fourth spatial dimension.

While reading popular science books about the universe one may find a quasi explanation of the closed three-dimensional curved space. That "it is the same" as the surface of the earth (or of a balloon), which is also closed, and that space has "just one dimension more" being three dimensional, while surface of the earth is two-dimensional. This is a good example of magical thinking which might be explained in this example, so the reader may be aware of it while reading other popular texts about science which may contain other instances of magical thinking. The magical thinking is very natural to humans, and showed up in human civilizations most likely just after invention of language. It is thinking rather about the names of things than about the things themselves. Analogies get created based only on similarity of words or ideas. And those analogies are very often false (another name for magical thinking is false analogy), and therefore they don't really explain anything.

Since there are only three spatial dimensions, the curvature of space can't be explained with analogy to a two-dimensional sphere that is curved into third dimension. To explain the curvature of three-dimensional space one has to do it within three dimensions. This might seem to be possible if there is a way of explaining spherical geometry in two dimensions, by using only two dimensions, just on the flat surface. It turne out there is, as we'll see below. Besides, for many curved geometries (as e.g. for Lobachevskian geometry) a three-dimensional model can't be even constructed: there is no three-dimensional surface in Euclidean space that would have the Lobachevskian geometry and yet there are phenomena in nature that this geometry describes. So we have to find a way of understanding curved geometries in different ways than geometries of some curved surfaces in Euclidean three-dimensional space. E.g. a simple geometry of a surface of a ball or even simpler one of a surface of a cone that may be unrolled into a flat Euclidean surface. The mathematicians say that the geometry of the surface of a cone or a cylinder is flat despite that to a lay person it looks curved. It is flat because all distances between points on such a surface are the same as on a flat surface of a table. Bending of that surface into a cylinder does not change those distances while bending it into a sphere would.

So to understand how three dimensional space can be curved despite that there is no fourth dimension it might be good to understand first how two dimensional surface may be curved without being curved into third dimension as if the third dimension didn't exist at all.

The model of such a curved surface is quite simple. Let's imagine a big flat disk e.g. of 20,000 km radius that has such a property that whatever moves from its center towards its edge keeps its length in the radial direction (towards the center of the disk) but gets a little longer in the perpendicular direction. It gets back to its original size when it returns to the center. If it is a man walking on that disk, if he makes circles around the center of the disk, the circle that is twice as far from the center wouldn't have twice as long circumference but a little less than that.

Lets assume that the relation between getting longer (perpendicular to the radial direction) and distance from the center is such that circumferences of those circles are exactly the same as those of parallels on the Earth at the distances from the Earth's pole the same as the radii of those circles. In such a case the man drawing circles and measuring them may conclude that he is not on a flat surface of a disk, but on a curved surface of a sphere of the size of the Earth. For him, for all practical purposes the surface would be a curved surface with the same geometry as the surface of the Earth. So he wouldn't be surprised when at the distance of 10,000 km from the center his circle would have length of only 40,000 km instead of about 62,832 km as it would on a flat surface. Or, that inside this circle there is more area than within the circle of the same circumference 40,000 km on a flat (Euclidean) surface. And even that all circles with radii greater then 10,000 km become smaller instead of getting bigger. And yet he would be on a "flat" two-dimensional surface of a disk. A surface that is not "curved" into any third dimension.

The trick with changing size of a ruler in direction perpendicular to the line from the ruler to the center of the disk would make the geometry appear as if the surface were a surface of a sphere. The last circle that he would make, about 20,000 km from the center, would be so small that he could just slide around it and walk beyond it for another 20,000 km, getting back to the center of the disk from the opposite direction. So we see here a model of a curved two-dimensional surface without the necessity of introducing the third dimension (this model is known as hot plate model in literature of curved surfaces).

It is good to add that since the geometry is the same as the geometry of a surface of a sphere there is really no center in it despite that the disk has a center. The trick with changing sizes makes impossible to tell which point of the disk is the center. All the points look the same, as they look the same on the surface of the Earth. The trick with sizes turns a flat two-dimensional space into a curved two-dimensional closed surface of a two-dimensional "sphere".

Now this two dimensional model can be changed to three dimensions in the way that instead of walking and making circles the man can fly in any direction and build spheres around some center. If sizes change in the same way as before, the surfaces of his spheres will be growing a little slower than their radii, and the largest of them all will have a circumference of 40,000 km. Then circumferences of the farther spheres (with largest radii) will be smaller and smaller until the last one, at 20,000 km from the center will be so small that that he could just pass by it. He could fly beyond it for another 20,000 km, getting back to the center of the disk from the opposite direction. For all practical purposes he will be in a closed three-dimensional space with its weird properties, going East along a straight line and coming back from the West along the same straight line. The trick with sizes turns a flat three-dimensional space into a curved three-dimensional closed space of a three-dimensional "sphere". And all of it happens without introducing any fourth dimension.

The above shows that it is possible by playing with sizes to change a flat space into a curved space. This is what seems to be going on in our universe, except that it is not the size of the objects or rulers that the movement is changing. The distances in that space change so that those rulers appear longer in some places and directions than in others, like they appeared shorter in the vicinity of some particular mass. Also the radius of that three-dimensional sphere into which the space is curved is much larger than the radius of the Earth. This radius is called Einstein's radius of the universe and it's size is about 20 billions light years (give or take a few billions). It happens to be the same radius called Einstein's radius of curvature of space, also known as Hubble's constant of the apparent expansion of the universe, which as mentioned above can be derived via a Newtonian formula. This shows how many additional questions about the universe, except why Mercury moves differently than predicted, and why light rays bend more than predicted, the Einsteinian Gravitation explains.