Relativistic pedantry

I must say first off, that I teach math and computer science, and was never qualified to teach physics. But I am interested in physics, and got drawn into in a physics discussion about how time does not stretch or compress in the visible world, and this is why in most of science, time is always the independent variable, stuck for most practical purposes on the x axis.

In the macroscopic world, time and mass are pretty reliable and so close to Einstein’s formulas (or those associated with the Special and General Theories of Relativity) at the macroscopic level that we prefer to stick to simpler formulas from classical mechanics, since they are great approximations, so long as things move well below the speed of light.

I am not sure (is anyone?) about how time is influenced by things like gravity and velocity (in particluar, the formulas stating how time is a dependent varable with respect to these things), but I remember an equation for relative mass, which doesn’t use time that would provide some insight into relativity:

\displaystyle{m(v) = lim_{v \to c^-} \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} = \infty}

Here, the independent variable is velocity, and it is evident that even for bodies that appear to move fast (on the scale of 10 to 20,000 km/h), it doesn’t have much impact on this equation. Rest mass and relative mass are essentially the same, and a body would have to move at nearly the speed of light for the mass of the moving body to change significantly. Indeed, as velocity v gets closer to the speed of light c, mass shoots up to infinity. I understand that Einstein stated that nothing can move faster than light, and this is supported by the above equation, since that would make it negative under the radical.

It does not escape my notice that velocity is supposed to depend on time, making the function m(v(t)), but time warps under things like high velocity also (as well as high gravity), so that time depends on … ? This is where I tell people to “go ask your physics prof” about anything more involved.

Sattelites move within the range of 10,000 to 20,000 km/h, hundreds of kilometres above the Earth’s surface. My assertion that there is not much change here in relativity terms. But this is still is large enough to keep makers of cell phones up at night, since not considering Einstein equations in time calcluations can cause GPS systems to register errors in a person’s position on the globe on the order of several kilometres, rendering the GPS functions on cell phones essentially useless.

My companion was trying to make the latter point, where I was thinking much more generally. We stick to classical mechanics, not because the equations are necessarily the correct ones, but instead because they are simple and lend a great deal of predictive power to the macroscopic world around us.

Android App-O-Rama

Here is some commentary regarding some Android apps I’ve been using on and off. Rating is from 0-5:

  • Basic music app (4): Great for playing whatever you have, but the interface lacks flexibility, and the interface is stymied by the small input screen of the cell phone, like all the other apps.
  • SoundHound (5): I was listening to a tune I forgot the name of  in a noisy Tim Horton’s restaurant. By the time I could get out my android and switch on to SoundHound, the song was almost over. I got the last few seconds coming out of the nearest speaker. It correctly identified the song as “Torn” by Natalie Imbruglia. Fascinating concept. I wonder now if it would work with less popular songs.
  • Songza (2): Selections are not too tied to genre. I would like them more tied to genre. This is because most of my preferences in music hover around classical and jazz. Some of their evening themes, to take an example could be adapted to any genre if only I could be allowed to prefer one: “Bedtime”, “Love and Romance”, “Unwinding”. Rap and hip hop own way too much of this app, and folk and rock own most of the rest. The genre selections such as folk are good, if you are into that. They seem to really know their 1960s folk.
  • Any calculator app (4 and down): Namely AndieGraph, Algeo, and RealCalc. Buttons are small and are barely pressable without making some kind of error. This is especially true of Andie Graph, which is an attempt to mimic the TI-83+ Caclulator (requires a ROM, which I transferred from my actual TI-84+).
  • Proprietary apps which are too big for what they do: Starbucks (3) – Crashes frequently, Adobe Acrobat (3) – seems to take up tons of installation space and updates also take up valuable ROM real estate.
  • Maps – If you don’t have GPS enabled or it is not part of your package, then this pre-installed app is a waste, and on my cell phone (a Samsung) it can’t be uninstalled.

Accelerometers on an android phone

I don’t know too much about cell phones, and I am somewhat intrigued by the recent generations of phones with their own open-source operating system, known as androids. An android exists on my Samsung cell phone. The phone is a common one, nothing special in itself. I can download apps, such as one for a graphing calculator. But one thing I found intriguing was an app called an AndroSensor, which among other things, gives the output of the built-in accelerometer.

Accelerometers, I am guessing, are the hardware responsible for “knowing” which way to orient the display whenever I hold the phone at strange angles with respect to myself. Knowing which way is down enables the cell phone to always present to me text and images in what I perceive to be a perpetually upright orientation.

Over the ‘net, there seems to be mixed reviews as to the accuracy of these devices. You can gauge the accuracy of your accelerometer by first taking a “snapshot” of your acceleration stats at any moment if your AndroSensor allows it (I had to enable it through the settings), then look carefully at the filename it saves it as so you can find it in your file area using your file app.

Luckily, it’s the first file on my file list, and I click on it. The accelerometer breaks down the acceleration into the three component vectors which we will call ax, ay, and az. And of course, these can be expressed as a Cartesian vector, which in my case is: \vec{a} = [1.2558\ 4.9033\ 7.9679]. This makes its magnitude: |\vec{a}| = \sqrt{1.2558^2\ +\ 4.9033^2\ +\ 7.9679^2} = 9.4396 m s-2.

You might guess that this is quite a deviation from the famous 9.8 m s-2 you were taught in high school science class. But I get other accelerations at any moment from my accelerometer by sitting in the same place, at my desk, in front of the same PC. What I did was sit for a couple of seconds while my android recorded its stats on to a comma-separated volume (CSV file), and I did some simple calculations on a spreadsheet. My file is not strictly a CSV file as advertised as semicolons, not commas, separate the fields. I needed to read in the file as “external data” in Excel. This was a little easier to do in Open Office, as I could set the delimiters right away by going through the “Open” dialog like any other spreadsheet file.

My AndroSensor also has a “record” function to record data repeatedly over time. Now, I am not meaning to take a scientific sample or anything, but none of the 21 accelerations I took were the same, and there was wide variation. Some were slightly greater than 9.8 m s-2, while others were well below 9.7 m s-2. The average resultant acceleration I obtained was 9.6952 m s-2.

I find it hard to believe that an android can generate that much variability. But even to be this accurate, you need to place it on a flat, heavy table that isn’t likely to be influenced by movement. I used my home office desk. If  you hold it in your hand, the variations in measurements get a little more strange due to involuntary hand movement, and you can get accelerations greater than 10 and less than 9.4. And that’s if you are stock still for the duration of the measurement.

If you take the value of 9.6952 as being the gravitational constant for the Greater Toronto Area, it still seems far off. Wolfram’s Alpha widget tells me that the GTA has g = 9.80678 m s-2. And since g is weaker at high altitudes, the widget tells me that g = 9.76321 m s-2 at Mount Everest in Nepal. It would seem that for a g value of 9.6952, I would need to be several kilometers above Mount Everest. So, I think that on a first blush, my accelerometer is ok for orienting my display but maybe not OK for getting the gravitational constant in my locale.

The gravitational constant would have only applied to all points on Earth if the Earth were a perfect sphere of uniform composition to the core and had a smooth surface with no irregularities. The Earth is none of these things, and the accepted value of g = 9.8067 m s-2 is changeable due to differences in composition and altitude, which affect both g‘s magnitude and direction. It is not always exactly 9.8067, and gravity doesn’t always pull exactly toward the Earth’s centre. But we are talking about small changes, not something like 9.6952, which seems way off.