Beyond Euclidean Geometry

Euclidean Geometry ruled mathematics for 2 millennia. Even today a beginner in mathematics would believe that Euclidean geometry is perfect and it is impossible to prove it wrong. Why is it so? The answer is Euclidean Geometry is based on our common sense and our day-to-day experiences. However modern physics and mathematics is beyond common sense.

What does our common sense say?  Famous philosopher Aristotle in his book On heaven wrote “The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all.” Further astronomer Ptolmey came up with proof for this. His proof was simple: there can be only three lines mutually perpendicular to each other.

Bernhard Riemann, a student of the famous mathematician Gauss, came up with an alternative to Euclidean Geometry. He proposed Hyperspace. Before understanding hyperspace let us imagine a race of two-dimensional creature living on a sheet of paper. Their day-to-day experience will make them conclude that they live in a two dimensional world and as they don’t experience anything three dimensional, mathematicians in this flat world would conclude that only 2 dimensions exists and more than two dimensions is nonsense.

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Riemann imagined the same world of flatlanders but on a crumpled sheet of paper. What would these flatlanders feel now? Riemann concluded that flatlanders would still find their world flat as they themselves are crumpled. However when they try to walk in this crumpled world they would experience unseen mysterious force acting on them making them move right and left like a drunkard. However we know that this force experienced by the flatlanders is nothing but effect of the uneven terrain in which they live.

He then applied the same to our world. He predicted force to be a result of unevenness of the terrain of the three dimensional world we live in viewed from forth spatial dimension.

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Another thing proposed by Euclid is that sum of all the angles of a triangle is always 180^o. Riemann realized that this holds only in a flat surface. He realized that a surface can have positive curvature (like outer surface of a sphere) or negative curvature (a saddle) as well. Euclidean geometry didn’t hold in such surfaces.

Table 1

Type of plane	Parallel Lines	Sum of angles triangle
Flat Plane	Never meets	is exactly 180o
Positively Curved Plane	Never meets	is greater than 180o
Negatively Curved Plane	Always meet	is less than 180o

Riemann’s main aim was to mathematically explain how crumpled the sheet is. He was successful in this by the introduction of set of number defining how crumpled the space is at a given point. Let us take the example of four dimensional space. Riemann realized that to define a point at a point he needs 10 numerical values. This was represented in a 4x4 board as follow (note that g­₁₂=g₂₁ and so on. Therefore six values are redundant)

Table 2

g11	g21	g31	g41
g12	g22	g32	g42
g13	g23	g33	g43
g14	g24	g34	g44

These sets of numbers are called metric tensor. It is used to get all mathematical information needed to describe how curved or crumpled a point in N-dimensions.

A tesseract (4D of a cube) undergoing double rotation in 4D space.

What is "Time"?

            In the modern globalized world every person keeps thinking about time. Time is very important for all of us now. A famous saying in English "time and tide waits for no one" proves its importance. So is it for physics lovers. Time is a very interesting concept in physics and is also a very beautiful concept. Let’s start with the simplest thing about time, its unit.

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            SI unit of time is second. What is one second? One second can be defined as the time taken by light to cover 299,792,458 meters. As speed of light is 299,792,458 m/s, the time taken to cover that distance will be obviously, 1 second. Will one second be same for everyone? A common person will say yes, one second is one second for all how can it vary from person to person. If you think the same way sadly it’s wrong. Time varies with your velocity. As you approach the speed of light, you will feel the time to have slowed down. Why?

            Before understanding it let us understand few other concepts.

• 1.     Speed of light in a vacuum is same for all (3 x 10^8 m/s), irrespective of the velocity the observer is travelling. In the sense if a person is travelling at velocity of let’s say 1.5 x 10^8 m/s (for example), he would still measure the speed of light to be 3 x 10^8 m/s and a person at rest will measure the velocity of light to be 3 x 10^8 m/s.
• 2.     Law of physics remains the same in all inertial frame of reference.
• 3.     Speed of light is a cosmic speed limit; it is not possible to go faster than speed of light.

          
           Now let us look at a situation from two different frames of reference. Let’s say a person is travelling at velocity of 1.5 x 10^8 m/s, with a photon clock. In a photon clock a ray of light keeps getting reflected perpendicular between to reflecting surfaces, let's say mirrors. 


            Case (i)
            Let's say that the person who is travelling sees the clock. He will find the clock to be stationary and the ray of light will be getting reflected perpendicular to the mirrors, as per him.

            Case (ii)

            Now let's say another person is standing and watches this person with the photon clock moving. How will he see the ray of light in the clock? Is the clock stationary as per him? The answer is no. The clock is moving therefore by the time the ray reaches from one mirror the other mirror would have moved so the diagram will be like this:

            It is obvious that the distance covered by ray in Case (ii) is greater than that it Case (i). We know that distance covered is directly proportional to time taken. Therefore the observer without the clock will feel that the moving person is aging slowly. This phenomena is called time dilation. This phenomena is negligible at small scale however the when the velocity is huge this phenomenon is observed clearly. This gave rise to an interesting paradox called twin paradox.

            Therefore we can say that one second is not the same for all. So our speed makes a huge difference at large scale. What are the other affects it has? We will discuss about it later.

Gravitation

 Let’s start with a famous story in science…

                Newton was sitting under an apple tree investigating the nature of force which makes moon go around the earth. All at a sudden an apple fell from the tree and this raised a question in Newton's mind ‘what made the apple fall down and not go up?’ This gave rise to a very important concept in physics called Gravitation.

                Now let’s discuss what Newton said about Gravity. Two bodies in this universe attract each other with a force which is directly proportional to the product of mass of both the objects and inversely proportional to the square of distance between them. Let’s put it in an equation:

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What’s that ‘G’? G is gravitational constant which is 6.67 x 10-11 Nm² kg^-2.

                Now let’s think about a situation. All at a sudden sun disappears! Sounds stupid isn’t it? As per Newton the planets will immediately start moving tangential to its orbit. This is because gravitational field will disappear and planets, which rotates around the sun due force of gravity, will be free to move and hence will move tangential.

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                However Einstein disagreed with this. As per Einstein the fastest possible speed in our universe is that light (which is approximately 3 x 10⁸ ms^-1) and is the same for everyone. Let's say there are two person, one at rest while the other one travelling at a very high speed with respect to the first person. Both of them will measure the speed of light as 3 x 10⁸ ms^-1 as it is not relative to the observer. Let’s call it cosmic speed limit. Even speed of gravity is slower than that light.

                We all know that it takes 8 min for light from sun to reach our earth.  Then we can say that gravity from sun take more than 8 min to reach earth. Then how will earth start moving tangentially immediately? So even if sun disappears the earth will be moving in its orbit for more than 8 min.

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                Einstein said that heavy objects such as planets and stars caused curve in space-time, considering space-time like a trampoline. Now let’s take the curve caused by our sun in the fabric of space-time. Planets follow the path formed by the curve caused by the sun.

                Now look at what will happen if sun suddenly disappears, with the new understanding of gravity proposed by Einstein. If sun disappears, the gravitational disturbance caused by this incident will form a wave, similar to that of ripple in a lake when a pebble is dropped in it. When this gravitational wave passes the earth we will find a sudden change is our orbit and only then the planet will start moving tangential to the orbit.

Spacetime

Space-time is defined as a mathematical model in which space and time are interwoven into one single continuum. The space-time of our universe in interpreted to that of a Euclidean space, with 4 dimensions, space consisting of 3-dimensions and time is taken as the fourth dimension. A unique position in a unique time in a space-time is defined as an event. Event is the basic concept of space-time.

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          All of us are familiar with the concept of planes i.e. 2-D plane, 3-D plane etc. A point in a 2-D plane is denoted as (x,y) and that in a 3-D plane is given as (x,y,z). Here x, y and z can be identified as the unique distance between the point and x-axis for x, y-axis for y and z-axis for z.

          An event in space-time can also be defined similarly as (x,y,z,t),  x, y and z here refers to the unique position of the point in the space while t denotes the unique time of the event. This way the coordinates specify where and when event occur. However the question which arises now is why do we need to include time and make it complex?

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          In classical mechanics, the non-relativistic model, time is considered as an independent and is treated as universal with constant passage irrespective of the motion of the observer. However in relativistic model, time is considered to be inseparable from the three dimensions of space as it has been noticed that time slows at higher speeds of a reference frame with respect to another reference frame. This phenomenon is referred as time dilation.

          However the number of dimensions required for analyzing our universe is a question among scientists. Speculative string theory predicts the number of dimensions to be 10 or 26, with M-theory predicting 11 dimensions, 10- spatial and 1- temporal). The widely accepted model is that of 3- spatial and 1- temporal, which is space-time. Increase in number of dimensions from four would have significant difference only in sub-atomic level.