Directory

General

I have finished taking Professor Charles Dyer's General Relativity I and II (PHY483 & PHY484). I had hoped throughout the course to return here often and leave twitterings (and I make no reference to the odious social abyss) on my thoughts. I had hoped, also without regard for my sense of duty, to post my musings in LaTeX. A Little bit of Professor Charles Dyer: Professor Charles Dyer, or simply Dyer to those who had met with him, is a figure to incite respect. Having risen to the pinnacle of his position and seen his day, he makes his well-found ideas known to those who listen. Many can attest to his ability to draw in bystanders who would not listen.

Christoffel Symbols in General & Flat Space

Here I derive the Christoffel Symbols of flat space geometry.

Geometrics

Here I derive a useful identity and begin to derive the covariant derivative (in an inefficient way looking back).

Here I give a presentation of gravity waves as a project for PHY484. I was accompanied by a two hour presentation to the class.

Here is my equation sheet put together hastily for my relativistic electromagnetism course (studied L&L Classical Field Theory book) the exam was a day before my birthday, and my final exam of my undergraduate. Yay! I only had two days to study for it.

Reading

I'm reading a number of books on stochastic processes on manifolds, and am still very interested in gravity waves. I am interested in theories of quantum gravity.

Ideas & Digressions

• I visited Dr. Charles Dyer to discuss some troubling concepts in regards to spacetime singularities. I had several misgivings that stemmed from thought on my own time. I began by recalling the Schwarzchild spacetime and the spacetime singularity at r=0. The coordinate singularity at r=2m is nothing by a simple transformation. It is the spacetime singularity that is interesting. I note that this r=0 singularity forms a mere point in 3-space thus having dimension 0, with 0 quantities describing it. When we foliate it with time we bring this to 1 dimension. We consider that this singularity arises from the most simple of symmetries. We then pursue the next simple of singularities we have that this singularity in 3-space ought to be of dimension 1. Hence when we foliate it with time we have a singularity of dimension 2. Here we must consider the possible shapes of the singularity. We request that it be bounded for some very good reasons. Physics ought to deal with bounded quantities. Consider the planar wave. This is an infinite concept and yet impossible physically since as it passes across a manifold the distribution of matter ought to change it, and it would no longer be a planar wave. With a bounded singularity of dimension 1 in 3-space we then must have some line similar to a ball of yarn. This line must be infinite in extent (still spatially bounded). Here we must stipulate that it be a circle that intersects with itself, otherwise it would fill the entire (bounded) space, much like a ball of yarn. The circle singularity in 3-space, when foliated with time then becomes a cylindrical singularity in (1+3)-space. Already this is hard to work with because any point near this cylinder becomes impossible to located, since a simple rescaling of the cylinder (which extends in the time dimension to infinity) can change the relative location of the particle.

• I now go back to the Schwarzchild spacetime and ask the question of whether simultaneity is a general consequence of singularities. That is, since the observer who crosses the r=2m line has only one future (that which hits the singularity), is there a corresponding result for all singularities. The answer is no and that it relies on the structure of the singularity. The schwarzchild singularity spans a spacelike slice. This is why simultaneity is void here. All different observers are spacelike separated on the singularity. So now when we return to extending the simplest bounded singularities to 2 dimensions in 3-space, we find that it foliates the analog of a thin 2- sphere, or a spherical egg shell. Here we have no way to deal with this singularity. I ask the question of maximal extent: Should not all geodesic lines end and begin at either a spacetime singularity or infinity? While this must be true, how can this be in an egg shell unless there is indeed a singularity at the centre of the shell also. But then how does this unfold in our progression of simplest singularities of bounded structure? This is where my questions turn towards a more philosophical nature. Indeed this often happens in physics. Undoubtedly my grandchildren will understand this and more. I still find gravitational radiation a more interesting topic. In 1973 it was unbelieved that one could observe gravitational lensing and yet it was 6 years later. The same shall hold true for gravitational waves. All forms of general relativity admit, and indeed, predict waves to exist. It is now just a matter of creating sensitive enough detectors and finding the right frequencies of detection. I believe this shall be a key to understanding the earliest universe before recombination, for which electromagnetic radiation gives no clues (it is opaque).