Hyperbolic and Other Geometries (Part III)

Note: This is the conclusion of a three part post. For the first part, see Hyperbolic and Other Geometries (Part I) and for the second part, see Hyperbolic and Other Geometries (Part II).

It may seem that hyperbolic and elliptic geometry have no relevant meaning in the physical world, because everything here is purely, and simply, Euclidean. However, the surface of the Universe itself has a basis in the other types of geometry. In fact, there isn't any perfect Euclidean geometry in the physical world at all!

The geometry of the surface of space is not actually the shape of the Universe as a whole. Even if the geometry of the surface of the Universe was hyperbolic, the Universe could still be finite. This is because the surface of space isn't really a physical thing. In essence, it is what all the objects in the Universe "rest on". The gravitationally heavier objects, such as stars, make a dent in this surface and cause nearby objects to roll towards it. For example, the Sun creates a small depression in the center of the solar system, and all the planets roll toward it. However, the planets don't go hurtling into the Sun because each of them has a forward velocity. The closer you get to the Sun, the faster the forward velocity is needed to not roll into the Sun, because the depression is "steeper" in this area. Some objects are theorized to be so "heavy" on this surface, that they actually create a "hole" in space, with a steep well surrounding it. These objects are black holes. The hole in space could actually open a path to a parallel plane of space, an alternate universe!

With all these dents and holes, the Universe clearly isn't homogeneous or isotropic. Therefore, the crucial part is finding what the overall curvature of the Universe's surface is, i.e. what it would be without any of the inhomogeneities. As odd as it seems, the density of the Universe determines its geometry. It is known that there is a specified value in the Universe called the critical density of it. The critical density is the density of the Universe at which the curvature is zero and the surface of the Universe is Euclidean. If the the actual density is smaller than the critical density, the surface of the Universe is hyperbolic. If the density is higher than the critical density, the surface of space is elliptic.



In this image, the omega symbol represents the quotient of the actual density and the critical density of the Universe. For example, if the critical density equals the actual density, the omega would equal 1.

To calculate the density of the Universe, one must take into account not only density from the matter. One must also consider the mass contributed by energy. There is energy in empty space, called dark energy. The famous equation E=mc^2 shows that energy, E is equivalent to a certain amount of mass, m. Therefore, dark energy also adds density to the Universe. Regular matter and dark matter are attracted to each other gravitationally, but, oddly enough, dark energy seems to exert antigravity, or a force that pushes matter apart. The result of this is the creation of more space in the Universe, which in turn creates more dark energy, which accelerates the expansion even further. Despite the fact that the expansion of the Universe lowers the density of matter, the production of dark energy keeps the overall density constant throughout the lifetime of the Universe.

Using the Cosmic Background Radiation, one can easily infer how much matter existed at the beginning of the Universe. Using the speed at which different galaxies are receding from us, one can infer how much dark energy there is. Therefore, the densities of both dark energy and matter have been calculated. They do indeed add up to the critical density. To be exact, matter and dark matter makes up 27% of the actual density and dark energy makes up 73%. At the beginning of the Universe, the density of matter was minutely close to 100% of the critical density, and there was almost no dark energy. Since the amount of matter and dark matter hasn't changed for billions of years (since matter stopped spontaneously annihilating and turning into energy 380,000 years after the Big Bang) its density has steadily decreased. To varies with this change, the density of dark energy has increased. Therefore, the surface of the Universe is indeed flat. What does this tell us? It tells us that the Universe will never stop expanding. A value for the actual density that is the critical density or less dooms us to a fate of an infinitely expanding Universe. Only if the actual density is greater than the critical density will matter dominate. If this was the case, gravitational forces between matter would eventually cause the Universe to contract. Therefore, the Universe will continue to expand, at an ever increasing rate, until the Universe is infinitely large and infinitely old.

Overall, the fallacy of Euclid's parallel postulate has lead to to not only discover other geometries and theorems in mathematics, but has also determined the fate of our Universe.

This post and the others in this series were inspired by the following books: The Road to Reality by Roger Penrose Chapter 2, and Origins by Neil De Grasse Tyson and Donald Goldsmith Chapter 5.

Hyperbolic and Other Geometries (Part II)

Note: This is part II of a post. For the first part, see Hyperbolic and Other Geometries (Part I).

Now that it is true that the Parallel Postulate is false, we can return to the constant C, which is the proportionality constant between the difference between 180 and the angles of a hyperbolic triangle and the area of that triangle. The constant C can differ for different hyperbolic planes. This is because not all hyperbolic planes have the same curvature. To fully explain curvature, one must also look at the Euclidean plane.

The Euclidean plane is defined to have curvature 0. That means that the surface of Euclidean geometry is flat. Hyperbolic planes, however, have negative curvature. There are different hyperbolic planes because there are different possible values for the curvature of the hyperbolic plane. This is unlike Euclidean geometry, which only has one possible value for the curvature, i.e. 0. But what happens if there is positive curvature? If there is positive curvature another whole type of geometry is introduced, which is even more radical than hyperbolic geometry. This is elliptic geometry. This is the only one of the three types of geometry that has a finite plane, which is one of a sphere. In elliptic geometry, all structures are projected onto a sphere. The theorems resulting include the fact that the angles of a triangle add up to more than 180. A representation of a triangle projected on a sphere is shown below.



This particular triangle is projected onto the Earth. The angles on the triangle add up to 230 degrees. Note that, in the close up image, the relative curvature of the area approximates zero, and the angles do add up to very close to 180 degrees. In elliptic geometry, Euclid's other postulates are violated. In this geometry, all but the fourth of Euclid's postulates are false (the fourth postulate is that all right angles are equal. The first postulate, that exactly one straight line can be drawn between any two points is false because, on a sphere, multiple lines can connect two points. Take longitude lines, for example. All longitude lines are drawn between the North and South poles, and in elliptical geometry, they all all straight lines. The second postulate, that a line can be continued indefinitely without ever intersecting itself is also false, because any latitude line on a globe eventually wraps its way around and intersects with itself. The third postulate (that a circle can be drawn with any center and radius) is false because if a circle is big enough, it will wrap around the sphere and intersect with itself, violating the very definition of a circle because the points on the edge of the circle would not all be the same distance from the center.

Just as the area of a triangle on a hyperbolic plane can be expressed with the equation 180-(x+y+z)=CΔ where C is a constant and Δ is the area of the triangle, the area of an elliptic triangle can also be expressed in terms of its angles, with the equation Δ=R^2(a+b+c-180). Again, Δ is the area of the triangle, while R is the radius of the sphere where the triangle is located. It is apparent that the quantity a+b+c-180 must be positive because the radius must be positive for the area of the triangle to be positive. If you combine these two equations and solve for C in terms of R, you obtain

C=-1/R^2

Going back to hyperbolic geometry, C, as the constant of variation for the area, must have a positive value. In hyperbolic geometry, therefore, for C to be positive, 1/R^2 must be negative (because then -1/R^2 would be positive). But this is impossible because all numbers square to a positive number or 0. Therefore, the solution to this equation for R must be imaginary, or the square root of a negative number. For information on imaginary numbers, see here. This seems impossible. How can a sphere have a radius that isn't positive or even a real number? It turns out an imaginary radius does have a geometrical meaning. The shape produced in this situation is called a pseudo-sphere. A representation of a pseudo-sphere is shown below.



In essence, this structure is kind of that of an inside-out sphere. The pseudo-sphere is wonderful, but what does it mean? It came about because the radius of a sphere that is a hyperbolic plane is a pseudo-sphere. It makes sense to say that hyperbolic geometry is actually projected onto this shape, just as an elliptic triangle is projected on a normal sphere. This may not seem like the case, but a pseudo-sphere is yet another Euclidean representation of hyperbolic geometry.

In conclusion, these are the only three types of geometry that fit the criteria. The criteria is that the geometries are homogenous and isotropic. Homogenous means that each point on the surface of the geometry is the same, and isotropic means that the perspective of the area around each point is the same as from any other point. There are other geometries that are combinations of the above three, and these usually include some regions with one geometry, and some regions with other types.

For the conclusion of this post, see Hyperbolic and Other Geometries (Part III)

Hyperbolic and Other Geometries (Part I)

Most people have some knowledge about regular plane, or Euclidean, geometry. The basis of many geometric theorems relies on five basic rules, or axioms, about Euclidean geometry that cannot be proved or disproved, but are simply guidelines that geometry must follow. People take these statements to be obviously true. The five are as follows (as stated by Euclid in Elements)

1. Exactly one straight line can be drawn between any two points
2. A finite line segment can be extended continuously without intersecting itself
3. A circle can be made from any center and radius
4. All right angles are equal to one another
5. Parallel Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if continued indefinitely, will eventually meet

In this post, the fifth of the five statements above is the one that will mainly be dealt with. This is the parallel postulate. In simple terms, as illustrated in the above image, if two lines are slightly inclined towards each other, in other words the angles between them and an intersecting line add up to less than 180 degrees, they will intersect eventually, whether it is a hundred units away, a million units away, or a googleplex units away, they will intersect. If alpha and beta in the picture above add up to more than 180 degrees, than the lines will incline away from each other and will meet only if you travel in the other direction. However, if the angles add up to exactly 180 degrees, the lines will be parallel. They will never meet. An equivalent property to this is the fact that all the angles of a triangle add up to 180 degrees. This is because if the lines are parallel, they never intersect. Yet another equivalent statement as the Parallel Postulate is the fact that for a line and a fixed point in a plane (that is a two-dimensional surface), only one line drawn through the point can be parallel to the given line (note that in three dimensions, two lines that never intersect aren't necessarily parallel. This is because they can be on two separate planes. Two lines with this relationship are called skew lines) All of the axioms above seem like common sense, right? However, most new branches of mathematics are opened up when one challenges the most simple axioms and properties of math.

In this case, we are challenging Euclid's axioms. The fifth of the statements, the Parallel Postulate, seemed the least obvious, and for centuries, various mathematicians attempted to prove this. Many of these mathematicians used a method known as proof by contradiction. Proof by contradiction begins by assuming what you're trying to prove is false and then trying to find a contradiction to a known axiom directly resulting from this fact. If a contradiction is reached, then the statement cannot be false and is therefore true. This method has be repeatedly used by mathematicians, and one notable example includes the proof that the square root of two is irrational (see here). The point is that many mathematicians assumed that the parallel postulate is false. If this is assumed to be false, then more than one line can be drawn through a point parallel to a given line. The many theorems that were derived from this assumption were very odd, indeed. However, no contradiction could be found. We now know that the Parallel postulate is neither true or false; it is true only in some cases. While searching for a contradiction, mathematicians found something much more interesting: a whole new type of geometry! In this geometry, a theorem that the angles of a triangle add up to less than 180 degrees can be derived. This new geometry is called hyperbolic geometry. It is very difficult to represent this geometry with images in Euclidean geometry, but one visual representation is shown below.



This is a representation of hyperbolic geometry in a Euclidean circle. In hyperbolic space, each angel or devil are precisely the same size. However, this representation has squeezed the entire plane of hyperbolic geometry into one circle, called the bounding circle. To the inhabitants of this Universe, you can go as far as you want, but you will never reach the bounding circle because the plane goes on forever. Shapes, lines, and other Euclidean structures can all be represented in hyperbolic geometry, although they may look different than in Euclidean space. Here is a triangle as it looks like on the hyperbolic plane.



The values shown are of each angle along with the total sum off to the right. Note the the sum of the angles, 107.1 degrees, is much smaller than what it "should" be, 180. In fact, it was discovered that the value of 180 minus the sum of the three angles is proportional to the area of the triangle, a relation somewhat unexpected by mathematicians. The following equation represents this relation.

180-(x+y+z)=CΔ

x, y, and z represent the three angles of a triangle and the symbol Δ represents the area of the triangle. In this equation, the constant C is not a set value for all hyperbolic planes. In fact, unlike Euclidean geometry, there are different types of hyperbolic planes, which will be mentioned later. Note that if the line segments on the above triangle were continued into lines, they would continue curving and also "reach" the bounding circle (obviously the finite curve that we see intersecting the bounding circle is indeed infinite from a hyperbolic standpoint).

Returning to the origin of this geometry, is the parallel postulate indeed false on this plane? Is is possible to draw more than one parallel line through a given point? Yes. The picture below is a different visual representation of hyperbolic geometry, but the structures are similar.



This representation may look different, but it is viewing the same plane. The lines in the lower right are parallel, but they curve in opposite directions, rather than remaining a fixed distance fro m each other. From this image, it is clear that infinitely many lines could be drawn through the same point as the given line, but would still never intersect it. Euclid's parallel postulate is indeed false in this geometry.

For the next part of this post, see Hyperbolic and Other Geometries (Part II).

Heat Death of the Universe

This fate of the Universe (note 5) begins 100 trillion years from now when stellar formation has come to a halt.  Over the next 10^30 years, planets, stars, and eventually galaxies will decay gravitationally and start to flow freely in space.  From 10^30 to 10^40 years, all protons will decay and all matter will be converted into photons and gamma rays.  All that remained after this era was black holes.  As the age of the Universe soars to a googol (10^100) years Black holes will evaporate due to Hawking Radiation and all the will remain are gamma rays and the occasional photon.  

The Big Rip

The Big Rip is yet another theory describing the possible fate of the Universe.  In this theory, the expansion of the Universe reaches infinite speed (due to to balance between Dark Energy and Dark Matter), and everything in it, even atoms, are ripped apart.  According to the theory, 60 million years before the end of the Universe, every galaxy but the Milky Way  would be out of visible range from the Earth.  Right before the end, the galaxy and the solar system (if the solar system still exists) would be gravitationally unbound.  A split second before the end, everything down to atoms would be ripped apart as the expansion speed of the Universe reaches infinity.  If this theory is correct, the Big Rip will happen in approximately 50 billion years.

The Big Crunch

If the amount of Dark Energy in the Universe was reduced to lower than the amount of Dark Matter a phenomena called the Big Crunch will occur.  The Universe would slow down its accelerated expansion and reverse.  Black holes would engulf stars and eventually entire galaxies as everything contracted.  As the shrinking increased in speed, black holes would swallow each other and the universe would end as an almost indefinitely tiny singularity, all matter squeezed into a small speck at the center of a black hole.  A variation of this theory is called the Big Bounce.  It includes the possibility that a Big Crunch would lead to another Big Bang, therefore creating a  second universe.  This theory opens up another important question.  Are we the first, five hundredth, or the one trillionth universe?  This question is explored in another article.

The Big Freeze

Of all the Ultimate fates of the Universe the Big Freeze is the most supported. This theory proposes that the Universe reaches a temperature too cold to sustain life. This would happen because as the Universe expands, the Cosmic Background Radiation (formed immediately after the Big Bang) would fade away completely causing the temperature of the Universe to drop. Once a star dies its gases used to create another star wouldn't group together and begin fusion because extreme cold means no moving particles. The Universe's temperature would reach absolute zero (-459.67 Fahrenheit) and stay that way forever.