Unit Analysis


We are used from Newtonian mechanics tothink about distances, time intervals and velocities in terms of units likemph, m, km, s, m/s etc. However, in special relativity the scale where therelativistic phenomena become visible is very different. Because this scale isdetermined by the speed of light, it would greatly simplify our calculationsand reasoning if we were to employ a system of units adapted to this constant.




The speed of light in the vacuum isapproximately


c300 000 000 m/s

This is an inconveniently long number. Weeither need a larger unit for distance, or a smaller unit of time. For manyreasons, the second seems to be a more adequate unit of time. Instead, we canintroduce a new unit of distance, defined as the distance covered by light inone second.


 We'll call this the light−second:


1 light−second=1 cs300 000 000 m

Notice that the light-second is a unit fora dimension of distance.


Using our new unit, we can re-write c as:


c=1 light−second/second



It is a consequence of special relativitythat there are no velocities greater than the speed of light. This makes thespeed of light into a natural scale for measuring velocities. You have alreadyseen that γ was defined in terms of v/c. This suggests that it would be auseful simplification to express all velocities in units of c. For example,v=0.1 c instead of writing v=30 000 000 m/s.

狭义相对论的结果是,没有比光的速度更大的速度。这使得光的速度成为自然的尺度来测量速度。你已经看到,γ是用VC定义的,这表明用C单位表示所有速度是很有用的简化,例如,v0.1 C,而不是写V30 000 000 ms

Then we would re-write γ as:


where v is measured in units of c.


The utility of these units will become muchclearer in Lecture 4.


Natural Units


In our previous discussion of dimensionlessanalysis, we have tried to motivate the choice of time and distance asfundamental dimensions. At the same time, because the speed of light is auniversal constant, it could suggest us that we can also express all thedistances in terms of the speed of light and some time unit. For example, ameter could be defined as


So now our two yardsticks are c and s. Butthere is a fundamental difference between them: c has a well defined physicalmeaning, while s still needs to be defined in terms of some arbitrary numericalvalue. A second is defined as the "the duration of 9 192 631 770 periodsof the radiation corresponding to the transition between the two hyperfinelevels of the ground state of the caesium 133 atom" (SI Brochure, Section2.1.1.3). Because we wanted to preserve the historical value of the second, wehad to introduce that messy number.

所以现在我们两个尺度是C,但他们之间有着根本的区别:C有明确的物理意义,同时还需要在一些任意的数值来定义。第二个定义为“时间的9 192 631,对应的辐射的770个周期的过渡之间的两个超精细能级的基态铯133原子”(SI Brochure,第2.1.1.3)。因为我们想保留第二个的历史价值,所以我们不得不引入这个混乱的数字。

Because the speed of light is free of anyarbitrary values, it's called a natural unit. You should be careful, becausewhen using natural units some physicists like to write c=1, but this is not areal equality, but a misnomer. c is a dimensionful quantity ([c]=L/T) while 1is dimensionless.

因为光的速度没有任何任意的值,所以它被称为自然单位。你可要小心了,因为当采用自然单位一些物理学家喜欢写C = 1,但这不是真正的平等,但用词不当。C是一个dimensionful数量([C] = L / T),1是无量纲。             

We will continue our discussion of naturalunits in the future lessons. Our goal is to show how we can employ otherfundamental constants to develop a system of natural units.