The Geometry of Einstein's Universe--Post #9

Looking at the picture to the left, I am sure most readers of this blog have seen this photo with the equation written on the board and the familiar face of Albert Einstein. As a joke, many people on the Internet have Photoshopped it, eliminating the equation and adding whatever graffiti they deemed more interesting or humorous. I found myself laughing at a few of the artworks myself.

What was actually happening here is Mr. Einstein was asked by the photographer to write something on the board indicative of his current theories and work. I am sure also that most heavy-duty mathematicians or people knowledgeable with physics recognize the seemingly cryptic question Mr. Einstein has written on the chalk board and understand it fully. (Note the huge question mark on the right side.)

Since I am neither a mathematician nor a physicist, it was not readily apparent to me. I had to do a lot of reading about Einstein's General Theory of Relativity plus get a rough idea of Riemann hyper-geometry before I really knew what was being asked here. By the way, that book I mentioned in my last blog entry was of little help in understanding this application of hyper-geometry. I actually ended up buying an expensive book on Riemann manifold geometry that is huge in size and mind-boggling in terms of its depth into the subject. I hope to save you the agony of trying to read such a book by simplifying what you need to know out of it.

If you are now like I was before all that, you have no clue what a "tensor" is and haven't the foggiest what the large letter R represents. I will explain all that in a moment. First, let me give you some background on why Einstein wrote a second treatise on relativity.

You see, the first treatise (Special Theory of Relativity) could not seem to get along with Newton's explanation (mathematically) of how gravity worked. If you were a good physics student in high school, then you are familiar with Newton's gravitation equation:


The G is the gravitational constant.  The masses of two interacting objects is represented by the letter m, and r is the radius distance between them measured from their centers.

With one exception, this equation worked well for explaining the orbital paths and behavior of objects in our solar system, including Earth, but it wasn't working for explaining the orbital path of Mercury. This was for good reason, as Einstein pointed out in the development of his General Relativity.

In Special Relativity Einstein showed the logic of light being the super speed of the cosmos. His theory precludes anything from traveling faster. Yet, according to Newton's gravitational law, gravity could cause an object to move faster than light. Take for example if our Sun was to suddenly vanish--Newton's physics would have Earth being affected by that before the light waves from that event could reach us. According to Newton, Earth would fly out of its orbit instantaneously. This could not happen, Einstein thought. Thus, Einstein had to come up with a completely different law explaining gravity. It involves warping something called the the space-time fabric.

As it turned out, the measurement of that "warp" could only be found in the mathematics of Riemann. I won't go into all the mathematics of how this works at this point, but I still owe the reader an explanation of the question posed by the equation and its components shown in the above photo.

Roughly speaking when something gets "warped" there is a tensor field associated with that warping. The warping is dependent on two associated vectors that combine into something called a "tensor". One vector of the tensor is the "force" that is placing the stress on what is being warped; and the other is a vector whose magnitude is determined by the size of the area being affected as well as the directional vector (orientation) of that area that is under stress.

According to Einstein's General Theory of Relativity, gravity creates a tensor field that extends into a 4th dimensional space. Riemann came up with a brilliant tool for determining exactly what that warping looks like. It involves what he called a "tensor metric".  It works equally as well with something in a two-dimensional space as it does with hyper-space.

Einstein, of course, was interested in showing how Riemann's metric worked in four-dimensional space.  Mathematically, this metric is a 4 x4 matrix that has 16 components in it. The 16 components are derived from 4 projections for each of the 4 coordinate components of the 4-axis coordinate system of a four-dimensional space.   Don't try to visualize such a coordinate system in your mind; we humans cannot do so.  Our brains are wired such that we cannot form a mental picture of anything past three dimensions. Fortunately, however, we can create the geometric system and its associated math computations despite this deficiency.

As it turns out in writing down these 16 components within our matrix, six of these components are redundant; so you end up (after simplifying) with only 10 components that completely describe the "warping" at every point in the four-dimensional space. Such was the genius of Riemann.

Riemann died at the young age of 39.  Shortly thereafter, another man named Ricci amplified our ability to work with the Riemann tensor metric by putting together a remarkable mathematical system called "tensor algebra". In the photo of Einstein shown above, you see a large R with two small subscripts (j and k). The R is the tensor. In this case, it is a particular tensor called the "Ricci curvature tensor".  On the right side of the equation you see a large ZERO. It is not the letter "O". The subscripts j and k are the variables representing the  numerical values for each of the components in the tensor matrix.

Einstein was using Ricci's tensor math to ask a very important question. According to the Ricci tensor, if curvature (the amount of warping) is zero there is no distortion and the space geometry (called the "manifold") is "flat". In such a case, Newton's math should work and Euclidean geometry should be sufficient. However, if there is some curvature, then the geometry is not flat and we must apply some form of hyper-space geometry for any of our physical calculations.

Einstein, however, believed there was a warping into a four dimensional space; and therefore he could not follow Newton and Euclid. In my next blog entry, I will cover Einstein's final curvature equation that explained gravity, and was the final product of combining the mathematical tools of Riemann and Ricci.

Einstein's publication of this gravitational theory was not accepted by many at first.  It wasn't until an English astronomer named Eddington published his astronomical data and photos of a solar eclipse confirming this predicted warping that the entire physics community accepted the validity of Einstein's curvature equation. It has been confirmed many times since by a number of scientific events and measurements.







 
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Comments (2)

  1. razzlematazzle

    Wow , I now no who to go to if I ever need help with my maths and sciences haha. After reading your blogs, I have learned many things that I never knew before. This makes me excited to follow my path into sciences and learn all the other infinite theories there are out there. Thats what I like about sciences, you never stop learning.

    I read your comment you left on my blog about Newton. It made me pretty happy, I appreciate the things you wrote. I liked how you didn’t tell me I was wrong about the things I have or have not done or thought about unlike most people. Your kids are really lucky to have a father like you, I wish my parents were more like that. Anyways, I pretty much just wanted to say Thank You.

    August 17, 2010
  2. kaylacarroll

    We are reading about the Einstein from when we are kids and there are a lot of things that we can learn through their life which includes bestessays review writing services as well and they are most genius people but sometime they also suck in their life so we should learn from them.

    June 23, 2017