# Dimensionless cosmology-getting rid of units

• marcus
In summary, the conversation discusses the concept of dimensionless cosmology, where time, mass, and length are treated as dimensionless quantities without any units. This approach is commonly used in physics and fundamental research, and can be seen as a way to simplify equations and calculations. However, the use of units serves two important purposes: a priori scaling and error checking. While it is possible to work with a dimensionless system, units provide a helpful framework for understanding and communicating about physical quantities.

#### marcus

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dimensionless cosmology---getting rid of units

In cosmology and fundamental physics some papers are written in a system where time, mass, and length are dimensionless. It is not that they are using Planck units---they aren't using any units at all. On usenet recently (sci.physics.research), one of the moderators had this to say

From: John Baez (baez@galaxy.ucr.edu)
Subject: Re: Perturbing fundamental constants
Newsgroups: sci.physics.research
Date: 2003-03-12 22:12:47 PST

In article <BA92DC5B.A835%rbj@surfglobal.net>,
robert bristow-johnson <rbj@surfglobal.net> wrote:

>If length, time, or mass are not dimensionless quantities, then neither are
>c, hbar, nor G, no matter *what* set of units you define.

Right. Ergo, Lodder must have been talking about units where length, time and mass *are* dimensionless quantities.
There is no angel from on high that comes down and punishes you if you decide to use units where hbar, c and G are 1 and are dimensionless.

Nor is there one that comes down and punishes you
if you decide to use units where hbar, c and G are 1 but have dimensions of ML^2/T, L/T and L^3/MT^2.

The angels only get fluttered if you flip-flop and change conventions within a single given calculation.

Apart from that, it's up to you to choose your system of units, including how many independent "dimensions" you want, and what they are.

It's common to work with units where length, time and mass are the 3 basic dimensions. It's common to take the units of these quantities to be the Planck length, Planck time and Planck mass.

It's also common to work with units where everything is dimensionless.

It's also common to work with units where there are more than 3 basic dimensions - for example, people often take temperature and charge as dimensions in addition to length, time and mass.]]

END OF BAEZ QUOTE (sorry about caps, just want to mark it clearly)

You may well know what he's talking about. You pull a journal off the shelf and open it up and at the beginning of the article it says
"c = G = hbar = k = 1" .
That means time, length, mass, and temperature are dimensionless. No units.
And it means the equations in the paper can look funny because they are missing constants you expect. Like it could say "E = m".

In what way, if any, are units indispensible? What are they basically, if one can do physics without them? What would it be like to open a college physics text and see at the beginning something like: "In this text we don't use units and the quantities are dimensionless and the scales are defined by assigning these values to the main constants----c = E9, G = E-9, hbar = E-30,..."
Is that fundamentally any different from setting them equal to one?

One would know, or be told, that in that system length = 1 means, in metric terms, 1.616 centimeters (with uncertainty in the last digit), and mass = 1 translates in metric terms to around 22 grams, and so on. But the homework problems would not need to refer to any system of units and could be in dimensionless terms. Advantages? Disadvantages? Any comment?

We have dimensions for two reasons:

(a) A priori scaling.

By having predefined units for things, one never has to rely on reference material to convey sizes of things.

(b) Error checking.

By including units in our calculations, we will quickly discover a class of mistakes that include things like omitting a factor in a formula, or measuring different objects to different scales. It also provides hints (to the relief of many a physics student) as to the relationship between quantities.

Hurkyl

Amen to that. I totally agree. But let's see how it looks

Originally posted by Hurkyl
We have dimensions for two reasons:

(a) A priori scaling.

By having predefined units for things, one never has to rely on reference material to convey sizes of things.

(b) Error checking.

By including units in our calculations, we will quickly discover a class of mistakes that include things like omitting a factor in a formula, or measuring different objects to different scales. It also provides hints (to the relief of many a physics student) as to the relationship between quantities.

Hurkyl

I believe you are right about (a) and (b) and they are very good things to have in the system. Dimensional reasoning is a great heuristic and a good automatic check, and having assimilated a set of units is a great help to our ability to imagine and communicate. Maybe *maybe* it is possible to get some of these good things to happen with a dimensionless system if one assimilates what it means to assign a definite set of values to the constants. I want to explore the possibility and see how it looks:

If one imitates "High Physics" practice (analogous to walking around naked) and doesn't use any units in ordinary general physics then what happens?

Instead of saying c = hbar = G = e = k = 1, I am going to assign them other values: powers of a thousand.
Also time = 1 should correspond to 54 milliseconds to give us a convenient degree of precision. Beyond that let's say
c = E9
hbar = E-30
e = E-15
Boltzmann's k = E-18

then it will turn out that
the width of my little finger at the first knuckle is 1
a classic twostep pace is 100
a mile is 100 thousand
a tenpercent shortened minute is 1000
highwayspeed (about 67 mph) is 100
at a cold (1.6) temperature sound speed is 1000
the Earth goes 100 thousand
sealevel gravity norm is 1.76
sealevel pressure norm is 219
the gravity constant G is 1.00E-9
a gallon volume is 1000
the average density of the planet Earth is just a bit over 1
the average density of water at the ocean's surface is 1/5
the frequency (always angular here) of middle D is 100
the power per area of sunlight is 10, or very close thereto

And so on. There is a lot of stuff to use in physics problems and examples---but stated without the use of units.

In what way, if any, are units indispensible? What are they basically, if one can do physics without them? What would it be like to open a college physics text and see at the beginning something like: "In this text we don't use units and the quantities are dimensionless and the scales are defined by assigning these values to the main constants----c = E9, G = E-9, hbar = E-30,..."
Is that fundamentally any different from setting them equal to one?
My QFT text (Peskin&Schroeder) does do that! The very first sentence is "We will work in God-given units, where c=hbar=1."

The problem with not setting them to unity is that you can't eliminate the constants from all your equations then, which is an enormous convenience; less important, you can't do natural comparisons (like momentum to mass to energy) quite as easily.

I think problem (a) isn't an issue; in God-given units it's no harder to remember standard scales. But (b) is an issue... to beginning students I think eliminating units would just confuse the hell out of them.

Dimensions give insight to what pre-physics our universe originated from, to how our universe compares and connects to others, and to where the universe undergoes "symmetry breaking, chaotic inflation, and anthropic perspective" (Max Tegmark, May 2003 Scientific American).

I long ago read an article titled "Dimensions Anyone" written by an MD in the Analog Science Fiction magazine. It was one of those slightly wrongheaded pieces that makes you think deeply about the subject, much more interesting than some pedanticly correct paper with no new ideas. Here's a few conclusions I drew after that thinking;

The natural value of C should be 1 because it is related to the Poincare Group and to hyperbolic space i.e. R(3,1).

The natural value of h-bar should be 1 because it represents one radian of quantum mechanical phase change per radian of plane angle. This means that you can factor through all the classical units that contain the quantity mass to get an essentially quantum mechanical system of units. In this system you have wavenumber instead of momentum and Q.M. frequency instead of energy.

The concept of mass is a holdover from classical physics when we didn't know about such concepts as group velocity and phase velocity in the propagation of states, and as such it can be discarded. Of course h and h-bar don't show up in our equations either, although I retain h-bar to designate angular momentum, but it no longer has the same units.

You can't set e to be 1 because e^2/h-bar*C has to be appr. 1/137.

Something else I believe is important, but I can't prove in any way, is that a rational system of units should have only one scale quantity, e.g. a length or frequency. If you have more than one such quantity, for instance, you may end up dividing by zero.

I should add that e, h-bar and C are all scale invariant but not dimensionless. Futhermore the laws of Electromagnetism (expressed by Maxwell's Equations) are scale invariant but in Electrodynamics the elctron rest mass breaks scale symmetry. Symmetry of scale is one of the most important broken symmetries in Nature and it is only fully obeyed in Electromagnetism. &dot; &times; &prod;

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I was very glad to see your post, continuing this thread on system of units where the main natural constants have value unity.

Originally posted by Tyger
You can't set e to be 1 because e^2/h-bar*C has to be appr. 1/137.

You are in good company here. You would have e be sqrt alpha, or approximately sqrt(1/137). Many would agree.

In Tom's QFT thread in Theoretical Physics here

people seem to like an online QFT textbook by Casalbuoni

http://arturo.fi.infn.it/casalbuoni/lezioni99.pdf [Broken]

Casalbuoni summarizes his units on page 4 and says hbar=c=1
epsilon and mu naught should be one.
&mu;0 = 1,...just trying notation

So for him 1/137.036... = alpha = e^2/4pi

I should not be getting into this swamp of notational detail.

I've had discussions with people. There does not seem to be
agreement on whether e is one or sqrt(1/137) or sqrt (4pi/137).

But the basic idea of developing a set of units that are in harmony with nature seems to be good and to appeal widely to folks----not just your MD :)

Have to run but will get back to this later and maybe edit it so not so scattered.

I've been reading this short essay of Baez recently, see what you think---
http://math.ucr.edu/home/baez/constants.html

Must comment on this, when I get back. Suspect that Casalbuoni might agree, tends to convert everything to length including energy and momentum:
[Something else I believe is important, but I can't prove in any way, is that a rational system of units should have only one scale quantity, e.g. a length or frequency. If you have more than one such quantity, for instance, you may end up dividing by zero.]

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Hello I'm back again. I wanted to comment on this:
Originally posted by Tyger
Something else I believe is important, but I can't prove in any way, is that a rational system of units should have only one scale quantity, e.g. a length or frequency. If you have more than one such quantity, for instance, you may end up dividing by zero.

Langauge issues are matters of taste and judgement and style and just as important as strictly logical issues. I do not think that one can "prove" that one style is better than another. But it is really interesting to watch peoples perspectives and preferences gradually converge.

For instance, what you say here is in line with what Casalbuoni does in his very nice QFT text. He happens to like the distance which he calls a "fermi" and which is a femtometer=10^-15 meter.

So page 5, he says (and this is a style thing not a logic thing):

"Therefore the approximate relation to keep in mind is
1 = 200 MeV . fermi"

That is his scale-key to the world of quantum fields----he sees everything in fermis. And to demonstrate his approach to scales he quickly calculates the Compton wavelength of the electron without pausing to take a breath or look anything up.

the rest energy is 0.5 MeV so the Compton is one over that, or 1/0.5 MeV. He multiplies both sides of his "approximate relation" by this and gets

Compton = 1/0.5MeV = 400 fermi

So this is the tapemeasure he uses for measuring sizes of Comptons---it is his "inch" so to speak.

And he also points out on page 5 that a fermi is 3.3E-24 second
so that one can measure time in fermis too.
Indeed a second is 3E23 fermis long.

Presumably any quantum field theory book you looked at would be doing something like this, just with non-essential personal differences. Damgo said his text (Peskin and Schroeder, which I haven't lookd at) does something like this. It is called using god-given units :) and must be said with a smilie so as not to offend anyone.

I think a new sense of what units are is going to coallesce in our culture. What physisists and cosmologists do for their own comfort will eventually trickle down into the general culture. Maybe it won't but I has a suspicion it already is. Hope to hear more from you on the subject.

I thought that some things need to be clarified. This is how I see it.

There are three types of quantities in Physics. Scale quantities, i.e. those that vary when we change the scale of our units for length and time, say half or double them. Energies and areas are examples of those. Then there are scale invariant quantitites, their properties are obvious from their name. Action, Angular Momentum, Electric Charge fit that bill. And then there are dimensionless quantities, i.e. pure numbers. In your original post some of the authors mentioned were actually choosing units of 1 for some of the scale invariant quantities, which does simplify calculation and make it easier to develop insight but it doesn't actually make them dimensionless, even though the people doing it may think so!

In the system I most often use h-bar is one radian of quantum mechanical phase per radian of plane angle, but they are certainly two very different kinds of angles, so we can't quite say that it is a dimensionless quantity. Likewise e^2 has the dimensions of quantum mechanical phase times velocity, so it can't be made dimensionless either. That's the square of electric charge. I'm going to do a little write up on a novel way of looking at electric charge that will make the mystery of 1/137 even more mysterious but will maybe help someone develop some new insights.

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## 1. What is dimensionless cosmology?

Dimensionless cosmology is a theoretical approach to studying the universe that removes the use of units in calculations and equations. This allows for a more simplified and precise understanding of cosmological principles.

## 2. How does dimensionless cosmology differ from traditional cosmology?

In traditional cosmology, units such as meters, seconds, and kilograms are used in equations to describe the properties of the universe. In dimensionless cosmology, these units are removed, and the focus is on dimensionless quantities that are independent of units.

## 3. What are the benefits of using dimensionless cosmology?

Dimensionless cosmology allows for more accurate and precise calculations, as it eliminates the potential for errors caused by using different units. It also allows for a more fundamental understanding of cosmological principles, as it focuses on the underlying relationships between quantities rather than their numerical values.

## 4. How does dimensionless cosmology impact our understanding of the universe?

By removing the use of units, dimensionless cosmology allows for a more universal and consistent approach to studying the universe. It also allows for easier comparison and interpretation of data from different sources, as units do not need to be converted. This can lead to new insights and discoveries in cosmology.

## 5. Are there any limitations to using dimensionless cosmology?

While dimensionless cosmology has many benefits, it is not a perfect approach and has its limitations. It may not be suitable for all types of calculations, and some concepts may be more difficult to understand without the use of units. Additionally, it may not be practical or necessary for everyday use, as units are still important for practical applications in the real world.