Browsing Archive: November, 2015
Posted by on Monday, November 30, 2015,
Today's article is going to be a short one, because we are nearly at Einstein's equations of general relativity, with only one more minor tool needed to make the final leap.
In the last two articles I gave intuitive definitions of both the Riemann tensor and the Ricci tensor, which are used to define the curvature of spacetime. The third element of this trio of curvature measures is the Ricci scalar, which can be thought of as an average of the Ricci tensor, which itself is an average of the R...
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Posted by on Sunday, November 29, 2015,
Having now defined the Riemann tensor, which provides one measure of the curvature of spacetime, the next step towards general relativity is to define two variations on it.
The first object that we can define is called the
Ricci Tensor, and it is closely related to the Riemann tensor that we defined last time. Recall that the Riemann tensor was defined as function
B = R(A,U,V)in which A is the vector being moved, and U and V represent two other vectors that define the two paths along which we a...
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Posted by on Saturday, November 28, 2015,
In the last article I reviewed the problem of having vectors defined in a spacetime where the metric function gives different lengths in different parts of spacetime, and outlined how this could be resolved using connection variables to move vectors around. Although this allows the comparison of two vectors in different parts of spacetime, it creates a new problem.
What happens if we move a vector along two different paths?
Using the covariant derivatives and the connection variables, a vector ...
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Posted by on Friday, November 27, 2015,
In the last article we saw how special relativity is really nothing more than a function that gives the distance between two points in a four-dimensional spacetime, and the requirement that this distance be the same for all coordinate systems we might use. We also saw how, in the most general metric function this distance could vary depending on both time and on position within space. And that leads us to a new problem.
If the distance between two points can vary depending on where in spacetim...
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Posted by on Friday, November 27, 2015,
To me it has always seemed natural to divide the equations of general relativity into five distinct levels, each of which has its own laws and constraints. In my opinion each level must be studied and understood separately before they can be combined into one complete theory.
The foundation of the theory is the spacetime metric, which is the subject of today's article.
And while the name sounds impressive, it is actually a very simple object. A metric is just a fancy mathematical name for any ...
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Posted by on Thursday, November 26, 2015,
In the previous article, I introduced the Minkowski metric as a generalization of the theorem of Pythagoras and stressed the importance of the distance between two points being invariant under a change of coordinates. In fact the entire special theory of relativity is nothing more complicated than the statement that the distance between two points in four-dimensional spacetime, and given by the Minkowski metric, is the same for all coordinate systems. A few readers have asked me to give a few...
Posted by on Wednesday, November 25, 2015,
This week is a very important anniversary in the scientific community, as it was 100 years ago this week that Albert Einstein submitted an academic paper for peer review, creating the General Theory of Relativity.
In the century that followed, the theory has developed from an obscure mathematical exercise that was famously claimed to be understood by only three people in the world, to a tool so useful that no GPS system would work properly without it. Only one hundred years after it was fi...
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Posted by on Wednesday, November 25, 2015,
Exactly one century ago today, Albert Einstein submitted a paper for publication in which he generalized his already successful special theory of relativity. The single paper contained the bulk of the theory of general relativity in a clean finished form, and would forever change not only all of physics, but also our view of nature itself.
It must be said as well that there is some controversy on the authorship, as the brilliant mathematician David Hilbert had submitted a nearly identical theo...
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Posted by on Tuesday, November 3, 2015,
In :
Mathematics
A few days ago I was watching an old re-run of the excellent British television program, QI, and the host of the show made an interesting claim regarding a normal deck of playing cards. He took a new deck, which is ordered by number and suit, and gave it a few shuffles. Afterwards he claimed that no deck of cards had ever before been in that order, as the number of possible orderings was so large that statistically it was (almost) impossible that two randomly shuffled decks could ever be in t...
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I am a theoretical physicist & mathematician, with training in electronics, programming, robotics, and a number of other related fields.
