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A tip on speed of light units

December 8, 2011

Mohan, mdashf


A major advantage of Physics in terms of speed of light unit is we do not have to deal with huge numbers if we are to make a down to the pen and paper calculation. This is possible because speed of light is a constant even though a large number. So you can compute everything as if speed-of-light=1 and the numbers and the related parameters such as what are know as Lorentz factors can be dealt like small numbers, especially the rapidity factor beta. For gamma you can expand the gamma in terms of beta, beta always runs from 0 to 1. Then gamma will be a binomial expansion. The binomial expansion is credited to Newton, what a man for Physics and Mathematics at the same time.

To get gamma you can either expand to a small number of power of beta if the beta is small or if it is close to 1, which is when people say “we are dealing with an ultra-relativistic particle here” you can do a trick. Chose a good power of beta, large enough so that you do not incur round off errors in your binomial expansion, which is not a mathematical problem as much as it may sound like, nonetheless it is, but take help from your Physics intuition if you are to enjoy certain privilege, you do not have to carry everything like a computer does. This is ofcourse dependent on the physics problem. {the equations you have to solve, not the mathematical rigor which can take your valuable time for the same problem to be solved easily, Physicists often work in a reverse direction, they solve all their problems through some kind of invisible union and create methods and respurces that are unified to a great deal, so they can kick your ass if need be and you won’t have any one to complaint to}

That {} means you have to browse through some of the literature and only useful textbooks and have in mind what it is that you are looking for and not waste too much time in getting inspired about mathematics all over again. The first time around you did in highschool is enough to have made you a physicist or a related expert so you must have by now made yourself comfortable what it is that you have to keep in mind, again there are hidden but unified ways physicists do this, hence there is often so much hue and cry about physics and you go home with the ideas that these people are basically very arrogant. call 911 if you think so.

Now the thing in my mind would be while trying to find out gamma is to truncate gamma at a power of beta. If I chose at say 11, then depending on what are the actual binomial numbers I am expanding I may end up at say 22, thats the power of beta. But binomially you may have to use formulas only for beta-power=11. Certain things have been so formulated they take care of themselves once you make suitable changes. There are well defined expressions in mathematics that give you the gamma therefore at a chosen power truncation. Suppose for my case beta=0.99, then I have to put in my values before or after calculus and get a desired value for my gamma or their functions. If beta=1, life is easier because all you have to do is “sum” all your binomial coefficients. This is where you can match your expansion carefully {you have to be very very careful eg about what powers you have on your actual expression before expansion, a short function say with power -3/2 and you have to chose the right summation theorems. But for Physics the cases might be easier, nature does not invent mathematics for the sake of claiming pleasure, what works, works, howsoever simple it may be, not less at-least, more comes from the actual situations being cmplex in nature, and every good physicist can tell you a bit about such.

Let me not procrastinate therefore and tell you, once you have your betas expanded you can make as many differentiations you want and requisitely put your beta values and sum the coefficients to get your desired results.

The article I am writing is titled “speed of light unit” I just gave you a bonus. Here is what you need to remember when doing speed of light calculations. You have to be consistent in the units, when you are doing the calculations you can put in every equation see “classically” a c=1. The equations became simpler. The difference between c=1 and beta=1 is the latter is a unitless fraction always the denominator is c=3.0×10^8 m/s {or any unit as you chose} The numerator is always in the same unit as the denominator, so when you take the ratio it always resides inside [0, 1]. So its units are hidden although canceled out if you were consistent. But for c=1 it is not a physical variable, its a physical trick, its units are indeterminate as all you have to do is keep in your mind that it has been removed for a while but once the calculations are on which is what comes up without warning you, you have to be not only consistent with the equation but you have to be consistent with every single variable you have in your calculation. Its fun. Basically you can do so because it is a constant and you can apply ALL mathematical operations that you can that works on constants, whose units need not bother you.

But when you have to calculate/compute to find the result of your equations you have to be sure about two things

1. how many c’s are needed by that particular variable

2. What is the “classical” unit of the variable. eg mass has a classial unit of GeV/cc, momentum has a classical unit GeV/c and so on

[nobody told you this, they told you SI and cgs units etc, speed of light units, but not classical units, a classical unit is a unit where you can use anything, cgs, SI etc, as per your need in the situation, so you do not have to worry about the GeV or whatever units but be consistent about it in a chosen segment, the number of c’s must be classically satisfied]

SO one additional point 3. Energy has no c in it, it’s a GeV, momentum has 1 c, GeV/c, mass is 2 c, GeV/cc.

Compared to energy momentum is 1-c down, compared to momentum mass is 1-c down but compared to energy, mass is 2-c (=c^2=cc) down. You can make a list of all other variables, probably I have never had a need to go beyond. So all you have to look for is if you are expressing E=energy in eV, you better look for and multiply or leave as it is to momentum 1-c. So if you express momentum in eV you know that somewhere either you have multipled a c, or you have to, if you do not excesses will show up as 10^-9, 10^-8, 10^12 and so on. That is a good way of finding out if there is an error or not. It is called order of magnitude in Physics. Your orders must be very close to each other unless you know something is supposed to be off by say 10^-5 so you call it a first order smallness and reject the higher orders sich as its square and cubes or a large number 100 into this first order factor. Once you have done this all, in this way you will see some beuaty and you can take them all and put all the ugly factors inside parentheses and declare on twitter you have discovered something that other suckers were missing.

Thats all I had to say.

{Tonight on CBS you can expect two important things coming, stay tuned}

Here are two sites you might find useful

Summing binomial coefficients

Computing factorials

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