A tip on speed of light units

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 known as Lorentz factors can be dealt like small numbers, especially the rapidity factor beta.

There is other definitions for rapidity factor? I think its just in terms of hyper-trigonometric functions, thats all.

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 there are 2 cases; ( — one ) you can either expand to a small number of power of beta; if the beta is small or ( — two )  if it ( — beta ) 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 ( — more than necessary ) 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 of-course 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 resources 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 above ” … ” 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 ( — enough maths ) in high-school ( — and it ) 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 ( — solve problems ), 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. ( Call 9-11 if you think Physicists are arrogant, thats all I can tell you)

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 ( — is 22 ). 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 ( — ultra-relativistic as noted above), then I have to put in my values before or after calculations 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 ( — we do), what works; works, howsoever simple it may be, not less at-least, more comes from the actual situations being complex in nature, and every good physicist can tell you a bit about such complexity.

Let me not procrastinate therefore and tell you, once you have your betas expanded you can make as many differentiation you want and as per your requisition 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 ( — “classically” as defined below ) a c=1. The equations become simpler. The difference between c=1 and beta=1 is the latter is an unit-less fraction; where always: the denominator is c = 2.99 x 10 (to the power 8) meters/s ( — in SI unit ) ( — or another value as per any unit you chose ). The numerator is always in the same unit as the denominator, so when you take the ratio it ( — must ) always resides inside [0, 1].

So its ( — beta’s ) 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 ( — c’s ) 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 ( — c ) 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 ” classical ” unit of (GeV/c)/c, momentum has a ” classical ” unit GeV/c and so on. (– classical defined below )

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/c)/c.

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{to the power 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 multiplied a c, or you have yet to, if you do not; excesses will show up as 10 {to the power – 9 }, 10 { to the power – 8 }, 10 { to the power 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 { to the power – 5 } so you ( — can ) call it a first order smallness and reject the higher orders such as its square ( — because that will be orders of magnitude smaller, fractions of orders of  0.00001 are included but fractions of order 0.0000000001 are neglected ) and cubes.

Once you have done this all, in this way you will see some beauty 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.