The Impact of Cosmological Expansion on the Planck Length: A Quantum Perspective

[Jimster41]

I get that Planck length is derived from h, G and c. But as a theoretical measure of length how is it affected or not affected by cosmological expansion?

 

[marcus]

Not affected, as far as anyone knows. In standard cosmology, which nearly everyone uses, those three constants do not change during expansion. Expansion has no perceptible effect on small scale local physics.

Jimster, I can't think of any reason that expansion (continuing expansion of distances between objects which are at rest with respect to the CMB background) SHOULD affect G, or hbar, or c.

It shouldn't have any perceptible effect on a metal meter rod, or a wooden yardstick, either. those are determined by numbers of atoms and by interatomic forces, crystal bond lengths etc. essentially by hbar and c and the unit charge, charge on the electron. and so on.

$$\sqrt{\frac{\hbar G}{c^3}}$$

 

[Jimster41]

Thanks,

I had thought that might be the answer I would get, that they are two different scales somehow.

But then I thought of taking the distance between two astronomical objects that would be expected to show the changing metric and dividing it into Planck lengths (why not)? Then it seems hard to say that none of the Planck lengths, nor the number of Planck lengths, can be affected while the measure of distance is changed.

At the end of the day aren't a Mpc and a Planck length both concerned with scaling of the space dimension of space-time interval? I don't see what allows them to be held as separate.

This question did come to me while following the Bell space-ship paradox conversation recently, when PeterDonis introduced the invariant "expansion Tensor" to the puzzle. This made sense but then I got confused as to how that expansion tensor in an expanding universe is "everywhere congruent" or geodesic(?) and therefore why aren't two stationary locations in an expanding universe Bell spaceships? All of which does connect to the question of just exactly where and exactly why the "Born Rigid" string between those ships breaks and whether or not a rod has a maximum length in an expanding universe. These things all seem related to me.

 

[Chalnoth]

Thanks,

I had thought that might be the answer I would get, that they are two different scales somehow.

It's the difference between the behavior of the system and the rules that the system obeys. The rules set limits on what the system can possibly do, while the behavior is what the system is actually doing.

The Planck length, as marcus noted, is determined by our current best understanding of physical laws, i.e. it's a feature of the rules themselves.

The expansion, on the other hand, is a description of what our universe is doing right now. The physical laws place limits on what the expansion can do, while the expansion itself has no impact on the physical laws.

 

[Jimster41]

It's the difference between the behavior of the system and the rules that the system obeys. The rules set limits on what the system can possibly do, while the behavior is what the system is actually doing.

I think my question was consistent with that:

• Rule: The fundamental unit of length determined from our best understanding of physical laws is a constant - a fixed relation between G, h and c.
• Behavior: The measure of length in the universe is not constant over time.

Unless a provision exists by which the rule allows the behavior, these seem in conflict? I was just trying to understand what that provision or mechanism was.

 

[Chalnoth]

What measure of length, though? Certainly no measure of length that is defined by the physical laws changes with expansion.

If, on the other hand, you are defining your length measure by the distances between far-away objects, then that measure will change with expansion. But that measure is just the distance between objects and doesn't represent anything fundamental.

 

[Jimster41]

Does a length of a Mpc change over time with expansion?
Can I divide a Mpc into n Planck lengths? (I can imagine maybe that isn't legal, though I don't know why, since both are in units of meters)
How many Planck lengths was the Mpc at z? How many at z+1?
If the answer is n+m Planck lengths I get that. But I'm puzzled by the discreteness. If the answer is n Planck lengths then I don't follow? Though I can imagine somehow it works out with c, G, h etc. I don't understand how, but I know those FRLW LCDM equations are subtle.

I'm totally prepared to believe this was a silly question. But I don't believe it just yet.
And I'm not trying to be irritating. This is the question I have. And I can't answer it. And it is bugging me.

 

[marcus]

Does a length of a Mpc change over time with expansion?

No.Can I divide a Mpc into n Planck lengths?​

Yes. That seems like an intelligent question to be asking. And the number n does not change.

 

[marcus]

The whole idea of expansion presupposes some unit of distance that does not change, or some such constant standard of comparison. Whatever their historical origins, human units like Planck length, meter, second, Mpc have been standardized and codified so as to be consistent.

Unless someone specifically says different, that is what is UNDERSTOOD to be the unchanging standard of comparison when we talk about expansion.

There are a few papers where they consider some normal constants like c and G as changing but they are outliers. And they don't walk around incognito. You will see a clear warning in the title or abstract that this is one of those rare papers.

IN ALMOST ALL COSMOLOGY RESEARCH human units like meter second Planck length Mpc are assumed fixed unchanging codified.

Without some accepted constant standard of comparison it would be meaningless to talk about expansion of a certain class of distances as one does in cosmology.. "Expansion relative to what?" people would be asking.

 

[marcus]

Jimster, you seem to be interested in units of measurement, how they are defined etc. You might enjoy reading up on METROLOGY, the science of defining units---with actual operations and instruments, not just blahblahblah. and the science of making measurements in terms of those units as operationally defined.

It's fascinating. They have big international conferences to decide how things like the second, and the meter, and the coulomb of charge, and the ampere of current, are to be defined. And the kilogram of mass. these things are no longer going to be defined in terms of block and rods of metal in somebody's basement, that was the old Arts and Crafts way. Every time they change a definition, everybody has to get on board and they have to have a big highly publicized conference (usually in Paris, you can figure out why) in order to vote on it. Metrology is an absolutely fascinating field.

Have a look at this proposed diagram of the foundations of the unit system. Second (s) depends on "atomic clock" but Meter (m) depends on second AND on the speed of light. Meter does not depend on a platinum alloy rod somewhere in the basement.
And kilogram depends on some other things AND Planck's constant. See the logical arrow? It does not depend on a block of metal in some dry constant temperature room. How can that be? So far this is just proposed, to be taken up at the next big meeting IIRC. I don't think it has happened yet but stay tuned.

 

[Jimster41]

Thanks for all that! What an great diagram. Where did it come from? The one I think I'm interested in is the top one. The "delta v...". What the heck is that one?

 

[Bandersnatch]

The one I think I'm interested in is the top one. The "delta v...". What the heck is that one?

That's not v, that's $\nu$. Denoting frequency. It shows that a second is defined in terms of hyperfine splitting frequency of a Caesium-133 atom.
See here:
http://hyperphysics.phy-astr.gsu.edu/hbase/acloc.html

 

[marcus]

Thanks for all that! What an great diagram. Where did it come from? The one I think I'm interested in is the top one. The "delta v...". What the heck is that one?

I'm glad you liked it. If in fact you are interested in how units are defined (operationally) would it be all right to make a friendly enlargement of the focus of your thread and consider this more generally---other types of units besides length? Tell me if it's not OK and I will stop.

Bander already said that the top dot is the CAESIUM ATOMIC CLOCK---that defines the green s-dot---the SECOND.
So then in the diagram, look at the purple A-dot, that is just to the left of the second. That is the Ampere unit of current.

Here's the thing about defining "A" the ampere unit of current. There is a type of transistor called "Josephson junction" which when you get it cold enough will actually precisely regulate the rate that individual electrons are going thru it by a high pitched signal.
so you can actually measure electric currents in terms of number of electrons going thru per second

So if you have a very accurate clock to serve as a cycles per second frequency standard you can measure electric current by the corresponding frequency in a Jo-junction. Cycles per second correspond to electrons passing per second in the transistor.

This amounts to our using NATURE'S unit of charge, the charge on the electron, as our unit of charge----measuring amount of charge by number of electrons.

And right below the purple A-dot there is the blue m-dot which is the METER. This is now defined using the atomic clock second and nature's unit of SPEED which is c, the speed light travels in a vacuum. We simply declare that light travels exactly 299792458 meters in one second.
so if we can measure time very accurately we can automatically measure length.
one meter is, by definition, the distance light travels in 1/299792458 of one second.

You can see in the diagram that the meter has two arrows coming into it: one from the atomic clock second and one from natural speed constant c.
The definition of meter depends on those two things.

It is analogous to definition of our current unit, Ampere, which depends on two things: the atomic clock second and the natural charge constant e, the charge on the electron.

Down and to the right from the blue meter-dot there is the orange KELVIN dot standing for K the unit of temperature. It depends on another of nature's fundamental physical constants---the Boltzmann constant which relates heat energy to temperature. I'll leave off here for now. It's more complicated. There are a lot more arrows going into the K-dot. It takes more machinery to define the temperature unit, it depends on having defined more other quantities.

But that is the general idea of that proposed diagram. I don't recall where I found the diagram---it was in some article about the proposed next generation of the metric system. this is something they have to vote on at the next big meeting in Paris, if they haven't already met and voted.

The big issue in that meeting will actually be the definition of the kilogram. This is where there is controversy: there are several competing schemes for defining it being proposed by different groups.
It might involve counting silicon atoms in a crystal.
Or it might involve Planck's constant h, as shown in this proposed diagram. This would involve a device called the "watt balance" which can define force in terms of current and voltage.
(Planck's h gives us a natural unit of force, assuming we can measure speed and length already, and force can be measured using standards of current and voltage. I should have stopped this post two paragraphs ago : ^) Metrology is sophisticated elegant and seductive. : ^D

EDIT: In answer to your question, I found a link that gives the diagram in context:
https://en.wikipedia.org/wiki/Proposed_redefinition_of_SI_base_units

It turns out the next big meeting is set for 2018. The wikipedia article is "Proposed redefinition of the SI base units"

 

[Jimster41]

I was able to figure out most of those mysel. But thank you. It's a great gestalt. And I will look for a good read on it.

Seeing Avogadro's number off by itself has caused me to add a note to self to review the origin of that.

The part about the cesium clock, I looked right past that obvious one thinking it was connecting to expansion...only because I'm still so confused about where expansion and the changing scale factor is described or accounted for in the fundamental dimensions, which as mentioned before should cover the rules of any and all behavior? So, where in that diagram is "proper distance" permitted to change with time. Is it only in the clock? That's all it can be right?

What is it that happens physically to a single photon sitting in an empty parsec's worth of space that causes it's wavelength to stretch in the time interval t, If all the physical things in that parsec's worth of space otherwise stay the same and there is nothing else involved? Is the expansion a non-physical thing that all physics are invariant w/respect to or is there something fundamental that changes with that tick of the clock that makes that photon stretch? If it can only be the clock how does the cesium atom splitting frequency manifest the changing proper distance of the universe.Hmm, as I read more into that cesium atom article... I don't think "splitting freq" means literally "splitting" right (that would just be way too easy) but...what does that mean?Lots to chew on. Much appreciated.

 

[marcus]

I never heard tell that the expansion of distances between objects at CMB rest would have any effect on anybody's refrigerator, or atomic clock, or car, or laptop or anything. Cosmic scale distances are not FUNDAMENTAL. They just change (very gradually) over time in accordance with the dynamics specified by GR. We have no right to expect distances NOT to change, what would keep them the same? Our 9th grade geometry teacher? Miss McPhee? Strong character as she is, she does not have that much clout with nature.

Crystal bond lengths keep lengths of pieces of metal from changing, or sizes of big rocks. But distances between widely separate clusters of galaxies have nothing comparable binding them.

I also never heard that photons exist as "particles" apart from interactions. You could say the field exists and it can be subject to the stretching of distance, but there is no little ball or marble. The two-slit experiment says as much. The field behaves like a field, until there is an interaction, an emission an absorption, a scattering. At least that is a useful mathematical model. Continuous media, discretely interacting. Quantum theory has some basic mystery. but it is not a mystery that distance expansion causes field wavelengths to be stretched out

 

[Chalnoth]

I also never heard that photons exist as "particles" apart from interactions. You could say the field exists and it can be subject to the stretching of distance, but there is no little ball or marble. The two-slit experiment says as much. The field behaves like a field, until there is an interaction, an emission an absorption, a scattering. At least that is a useful mathematical model. Continuous media, discretely interacting. Quantum theory has some basic mystery. but it is not a mystery that distance expansion causes field wavelengths to be stretched out

Photons are quantum particles. They're not conceptually any different from electrons or protons, in terms of their particle/wave nature, except for the fact that electrons and protons have mass, and thus don't always travel at c.