Well, here we are watching videos and news and meeting on line again. So, today we get some SpaceX Starship and Starlink news as well as follow up with more equations of general relativity on one page and end with the announcement of the closest black hole to us.
First up, we saw that the SN4 Starship has completed its first test firing, which was only 3 seconds, but this end we find in an apparently successful test. The test hop is still maybe something like a week or two away. Go Elon!
Starship SN4 completes 3 second test burn at Boca Chica (Source: @BocaChicaGal) 
News on the Starlink front is that the next batch of Starlink satellites initially set for May 8 has now been postponed to no earlier than May 18. This time the satellites will have some additional sun shades installed so that they won't reflect as much sunlight and therefor keep the night sky a little darker for observatories around the world. Go Elon! That is a good thing but hopefully we still get a few days of string of pearls observations. Time will tell, keep tuned!
Next batch of Starlinks, with sunshade, to launch no earlier than May 18 (Source: www.astronomynow.com) 
Next up we had access to another zoom meeting from UCI. This meeting, which normally would have been conducted with free breakfast as part of the UCI Physical Science Breakfast Lecture series was this time conducted on line, without breakfast. Darn! Anyway it was good to still participate in a lecture this time provided by UCI Professor Rachel Martin on the scientific effort ongoing to develop an antiviral drug for COVID19. Her team, which includes other laboratories around the world, has focused in on a key protease that the virus needs to be able to replicate inside the host cell, but is not used anywhere else by the normal cellular machinery. So, disruption of this protease should not have any side effects and yet would prevent the replication of the virus. Pretty neat. They announced the effort and how it is proceeding so fast that they are going out on a limb without having finance approved by normal funding agencies. So, at the end of the meeting, they had sucked $25 out of my pocket, for a free breakfastless lecture, in order to support the ongoing effort. I think it turned out ok. Good luck and good success, Professor Martin and teams!
Professor Martin and Dean, Professor Bullock at UCI "Breakfast" Lecture series on antivirals for COVID19 (Source: UCI) 
Now with a lot of free time on our hands we can return to our ongoing study of general relativity. Last time we went over how the trajectory of particles around a black hole, for instance, could be calculated by solving the geodesic equation which is based on the connection coefficients from general relativity. Now, I did not go through with the actual solving of those equations and this time choose to look at the other half of Einstein equations that connect the geometry of curved space to the mass and energy that actually cause the curvature. So, here we see, all on one page, the equations of general relativity that connect curvature with mass. This one page is just a collection of the formulas abstracted from Sean Carroll's "Spacetime and Geometry."
Equation 4.44 is the main equation that ties the geometry of spacetime, described by the left hand side of the equation, to the mass and energy density as described by the right hand side. The mass and energy density terms are contained as components in the "T" tensor, whose components for a simple case are just mass density and pressure as shown in Equation 1.111. This simple form of the stress energy tensor only has components on the diagonal, the other 12 components being zero, which then means we can simplify the calculations. So, four of the differential equations will be equal to the components from the "T tensor, the other 12 differential equations will all be equal to zero and will not depend on the initial mass and energy distribution. We see from Equation 3.113 which shows the Riemann tensor being calculated from the connection coefficients, which themselves are derived by derivatives of the metric as shown in Equation 3.1. Finally the other components on the left hand side of Eq. 4.44 are the Ricci Tensor and the Ricci Scalar, which are defined by Equations 3.144 and 3.146, respectively. So, we see there are many equations, each with many components. Luckily for the case of the simplified stress energy tensor, the number of equations reduces to just four equations. But, you cant relax, since just calculating and solving these equations is still very difficult.d I can't really solve them yet, but we still have more time available.
General Relativity equations (Source: abstracted from Sean Carroll, "Spacetime and Geometry") 
You can see from this one page summary of the equations of general relativity that derivatives are the major mathematical operation needed here. Remember than a derivative calculates the change of a variable or function due to a change in some other variable. In general relativity, both changes in time and space are important variables. In the screenshot image below, taken from Sean Carroll's textbook, we see the notation used to describe differentiation. The differentiation is with respect to some variable x super mu, where mu runs over three dimension, one for time and three for space. If you read any general relativity text you will see all of these three different, but identical meaning forms. Of special interest is the "comma" notation, which just means in this example, take the derivative with respect to the variable index mu.

An example of where this comma notation is used is shown in the screenshot below provided by Math Whiz, Dave. I asked Dave how you would go about solving like, Equation 4.44 from the previous figure, so he worked it out, with a little help from Mathematica. Here you can see the solution to Equation 4.44 can be rearranged to be just a bunch of derivatives of the metric. Again, Dave has used the comma notation to show all of the derivative of the metric. Wow, that is pretty neat, thanks for that Dave!
Going through all the math to show Einstein tensor as a function of derivatives of the metric (Source: Math Whiz Dave) 
Ok, just thinking about all of those derivatives has worn me out. One final news item from Megan Ganon of Smithsonian Magazine shows the nearest black hole, HR6819. It is in the southern hemisphere and it makes you wonder how many other small black holes are out there just waiting to be discovered. HR6819 is not the actual black hole but instead refers to an optically visible binary pair of stars that orbit around an object now considered to be a black hole of about 4 solar masses. So even though black holes cannot be seen, their existence is given away by light from infalling material or by observation of other objects like stars that are in orbit around the black hole. In this case, the binary pair orbits the black hole in about 40 days. Hmm, that is really moving! In addition there are probably over a hundred million other black holes, which are the result of dead stars from an earlier epoch still present in just the Milky Way alone.
Artist's impression of the closest black hole (Source: Megan Ganon, Smithsonian Magazine, May 6, 2020) 
Until next time, here from our burrow, stay safe, but it's time to recover more of our freedom,
Resident Astronomer George
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