How to understands natural units

[fet2105]

I am reading a textbook on special relativity and the author says at one point that he switches to natural units where c=1. The way he quickly explains this is by arguing that if c=3 x 10^8 m/s then we are working in units where seconds = 3 x 10^8m . . . . This has always baffled me and I never got a sufficient explanation of how we can just let c=1. Can anyone enlighten me?

 

[ghwellsjr]

I am reading a textbook on special relativity and the author says at one point that he switches to natural units where c=1. The way he quickly explains this is by arguing that if c=3 x 10^8 m/s then we are working in units where seconds = 3 x 10^8m . . . . This has always baffled me and I never got a sufficient explanation of how we can just let c=1. Can anyone enlighten me?

It might make more sense to think in terms of seconds and light-seconds. Then 1 light-second equals 3 x 10^8m. In any case, you can use any unit for time and then the distance unit is how far light travels during that unit of time. This really becomes significant when you throw away your rulers and use your laser rangefinder to measure distance.

 

[Dale]

If distance is measured in light years and time is measured in years then c=1.

 

[DrGreg]

...or to put it another way, c = 1 light-year per year or 1 light-second per second.

 

[fet2105]

Well, that isn't so bad . . . Thanks guys!

 

[D H]

A more radical point of view is that in natural units, c=1, period. Not one light years per year, not one light second per second, just a unitless one. Per this point of view, insisting on seeing c as having dimensions of length/time is "unnatural".

In special relativity, time and distance are different aspects of the same thing. For example, one way to look at the Lorentz transformation is that it is a hyperbolic rotation in space-time. This is perhaps a trick that happens to work if one views time and distance as having different units. It is anything but a trick if views time and distance as being different aspects of the same thing.

By way of analogy, look at how Americans customarily measure mass and force. US customary units have the pound mass as the unit of mass and the pound force as the unit of force. This means one has to resort to F=kma to represent Newton's second law. That k is the constant of proportionality that relates the fundamentally different quantities of force, mass, and acceleration. The metric system uses F=ma. The constant of proportionality has vanished. It's still there, hiding, but it's numeric value is one. The key question is whether that constant of proportionality is a unitless one or is the dimensioned quantity one Newton / (one kilogram * one meter/second²). The modern view is that it's a unitless one. Per this modern point of view, force and mass times acceleration are different aspects of the same thing. A system of units that views force as something distinct from the product of mass and acceleration is archaic and inconsistent.

Those customary units of the pound mass and the pound force are an archaic set of units that are fundamentally inconsistent with respect to Newtonian mechanics. The metric system is a consistent set of units, but only with respect to Newtonian mechanics. With respect to modern physics, it too is an archaic and inconsistent set of units. From the perspective of special relativity, time and distance are different aspects of the same thing. Energy, mass, and momentum are also different aspects of the same thing. The speed of light must necessarily be a unitless one to express these relationships in their proper form.

 

[fet2105]

In response to DH, when you say that time and distance are different aspects of the same thing . . . is that "same thing" space time? So, for the spacetime interval s^2 = -(ct)^2 + d^2 we are justified in saying that s^2 = -t^2 + d^2 because distance and time are two different aspects of the same thing? . . . So if this is correct then what are the units of time now? light meters?

 

[BruceW]

yes, s^2 = -t^2 + d^2 in natural units, since c=1. And for the 'dimensions' of time... well if you have c=1 and \hbar = 1 then you must have length and time both having the same 'dimension' as 1/mass. And since we define energy as having dimensions of mc^2, and c=1, that means energy and mass have same 'dimension'. So time has same 'dimension' of 1/energy. I talk about energy here because the particle physicists often like to talk about things in terms of electron-volts. But instead, if we further say that the gravitational constant G=1, then we have Planck units and time is 'dimensionless'.

Sorry about all the scare-quotes around the word 'dimension'. It is because I mean dimension as in dimensional analysis, not dimension as in the spatial dimensions. In dimensional analysis, 'dimension' can mean stuff like energy or whatever (not just length). I've been using the word 'dimension' instead of 'units' because I think 'units' refers to things like electron-volts, meters, e.t.c. while 'dimension' refers to things like energy,length, e.t.c.

 

[fet2105]

Got it, I think that I officially understand natural units. Thanks.

 

[jfizzix]

"My Car goes 40 rods to the hogshead, and that's the way I likes it!"- Abe Simpson

That's about 10 1/2 feet per gallon lol

 

[BruceW]

hehe, outdated units are awesome :)