General relativity breaks down at Planck scale

[mersecske]

Why?

Some measurements confirm this statement?
Or this is a theoretical conclusion?

 

[Drakkith]

I'm not sure, i'd guess its just a theoretical conclusion based on math.

 

[mersecske]

Why, what is the logical deduction?

 

[dpackard]

I too would like to know how it is we know quantum effects become important in this regime. Other than knowing we need QG before we get down to a singularity, what in particular makes us think the Planck scale is selected by nature as the transition scale?

 

[JustinLevy]

I too would like to know how it is we know quantum effects become important in this regime. Other than knowing we need QG before we get down to a singularity, what in particular makes us think the Planck scale is selected by nature as the transition scale?

It is a dimensional analysis argument.

Quantum mechanics gives us a time or length scale for an object of particular energy or momentum.
General relativity gives us a length scale given a mass (or energy).

Therefore we can look at where these length scales are comparible. The Planck units are units defined using G, hbar, c, etc.

 

[pervect]

Wheeler and others believe that the structure of space-time becomes a "quantum foam" on the quantum scale. The now-current Wiki article seems reasonable sane on the topic and has some references.

http://en.wikipedia.org/w/index.php?title=Quantum_foam&oldid=382044471

There is some discussion of the "quantum foam" idea in MTW's "Gravitation" as well. I would say at this point that this is not a theorem so much as a speculation, one that's significant enough to have a reasonable amount written about it, however.

 

[atyy]

This is based on the finding that quantum general relativity, although a good effective quantum theory at low energies http://arxiv.org/abs/gr-qc/9512024, is not perturbatively renormalizable at high energies (ie. short distances).

It has not been ruled out that quantum general relativity is non-perturbatively renormalizable, in which case it would not break down at the Planck scale. http://arxiv.org/abs/0709.3851

 

[JesseM]

Why?

Some measurements confirm this statement?
Or this is a theoretical conclusion?

Theoretical. I don't know if there are any situations where they both make clear predictions that contradict each other, but see here for a discussion of one of the main problems in figuring out how to reconcile them, having to do with the fact that the uncertainty principle would seem to allow for huge uncertainty in energy at sufficiently small scales, but in GR big energies cause significant curvature of spacetime, and my understanding is that physicists only know how to make predictions in quantum field theory if they have a specific known background spacetime. I guess another more general conflict is that quantum field theories treat the other set of forces using a common set of rules, but if you try to apply these rules to gravity you get infinities which can't be "renormalized" as in the case of the other forces.

 

[mersecske]

"quantum foam" means a difficult topology, but what's the problem. GR can describe any kind of topology. If Wheeler states that spacetime is a "quantum foam", this means that he uses GR at that scale.

 

[JustinLevy]

"quantum foam" means a difficult topology, but what's the problem. GR can describe any kind of topology. If Wheeler states that spacetime is a "quantum foam", this means that he uses GR at that scale.

No. You are assuming that there is still a smooth manifold in your statements there, as we need that for GR.

So "quantum foam" doesn't just mean a difficult topology.

Imagine for example a fractal landscape ( http://en.wikipedia.org/wiki/Fractal_landscape ). On very long scales it looks like a smooth flat 2-d surface. But as you zoom in, it appears to have tecture, and you can zoom in infinitely far and it will just be more and more roughness. You can't "zoom in" far enough that it appears smooth. It is fundementally rough, and "emergently" smooth.

This is very far from just a topology issue. The mathematical language of GR, differential geometry, assumes a manifold. Wheeler's point is that quantum mechanics hints that we may have to use a more base concept -- some kind of "pre-geometry" -- to even approach quantum gravity correctly.

For example, some attempts to quantize GR directly have found a fractal spacetime that only in length scales much greater than Planck lengths does a smooth spacetime with 4 dimensions emerge ... on smaller length scales the fractal dimension appears closer to 2. So at least in some approaches, Wheeler's intuition is playing out.

 

[atyy]

For example, some attempts to quantize GR directly have found a fractal spacetime that only in length scales much greater than Planck lengths does a smooth spacetime with 4 dimensions emerge ... on smaller length scales the fractal dimension appears closer to 2. So at least in some approaches, Wheeler's intuition is playing out.

But aren't these are approaches in which GR does not break down at the Planck scale?

 

[bcrowell]

There are two separate questions here: (1) Why do we believe that GR and QM are inconsistent? (2) Why do we expect that inconsistency to manifest itself at the Planck scale?

The simple answer to #1 is that nobody has succeeded in producing a theory of quantum gravity. We're not sure why they've failed. Possibly we need to give up some cherished feature of one of the theories, like background independence (cherished by relativists) or unitarity (cherished by quantum theorists).

The answer to #2 is that the Planck scale is the only scale you can construct out of the appropriate physical constants. This is not an absolutely secure argument. For example, if large extra dimensions http://en.wikipedia.org/wiki/Large_extra_dimensions exist, then we could see quantum gravity at the LHC.

 

[mersecske]

No. You are assuming that there is still a smooth manifold in your statements there, as we need that for GR.

The quantum foam can be smooth, only the scale is Planckian. I don't think that quantum foam is a real fractal, because particle scales (due two particle creation and annihilation) define the scale. Or not?

I think that quantum foam can be a fractal just in the case
when something is existed below quarks, what we don't know yet!

 

[Naty1]

I like Ben Crowell's answer #1...His answer # 2 is appropriate for classical relativistic theory.

I also like Justin's observation

So "quantum foam" doesn't just mean a difficult topology.

Quantum foam in fact is usually interpreted to mean the cessation of space and time as we know it...just as smooth waves of water dissolve into a spray..called spindrift at high winds velocities...Below Planck scale you can't even speak about "topology" because space and time dissolve into wild uncontrolled undulations...a type of quantum "ambiguity"

Two places where relativity and quantum mechanics each break down to infinities are the big bang and black holes...neither works at those type singularities and we have no theory so far that does.

 

[JesseM]

The answer to #2 is that the Planck scale is the only scale you can construct out of the appropriate physical constants.

This answer seems incomplete insofar as it doesn't have anything specifically to do with gravity. After all, before physicists had figured out how to quantize classical electrodynamics, no one expected that the characteristic scale of quantum electrodynamics should be the Planck scale, and of course it isn't! I found an interesting paper here which gives a series of basic arguments for the Planck scale should be the scale of quantum gravity, summarized on p. 3:

Lacking real experiments we use thought experiments (Gedankenexperiment) in this note. We give plausible heuristic arguments why the Planck length should be a sort of fundamental minimum - either a minimum physically meaningful length, or the length at which spacetime displays inescapable quantum properties i.e. the classical spacetime continuum concept loses validity. Specifically the six thought experiments involve: (1) viewing a particle with a microscope; (2) measuring a spatial distance with a light pulse; (3) squeezing a system into a very small volume; (4) observing the energy in a small volume; (5) measuring the energy density of the gravitational field; (6) determining the energy at which gravitational forces become comparable to electromagnetic forces. The analyses require a very minimal knowledge of quantum theory and some basic ideas of general relativity and black holes, which we will discuss in section II. Of course some background in elementary classical physics, including special relativity, is also assumed.

For example, argument (4) involves the idea that to probe what's going on in smaller and smaller volumes of space, you need probes of higher and higher frequency and therefore higher energy, and if the volume were as small as the Planck scale the energy density would be so high as to form a Planck-scale black hole.

 

[mersecske]

I think that the main problem about Planck scale and quantum gravity is not the topology but the probability in the quantum theory.

 

[Parlyne]

This answer seems incomplete insofar as it doesn't have anything specifically to do with gravity. After all, before physicists had figured out how to quantize classical electrodynamics, no one expected that the characteristic scale of quantum electrodynamics should be the Planck scale, and of course it isn't!

The constants that Planck scale quantities are derived from are the speed of light, Planck's constant, and Newton's gravitational constant (which also appears in the GR field equations). The presence of Newton's constant is what makes these quantities refer to gravity and why no one should expect quantum electrodynamics to have any relation to the Planck scale (since Newton's constant doesn't show up anywhere in classical electrodynamics).

 

[JesseM]

The constants that Planck scale quantities are derived from are the speed of light, Planck's constant, and Newton's gravitational constant (which also appears in the GR field equations). The presence of Newton's constant is what makes these quantities refer to gravity and why no one should expect quantum electrodynamics to have any relation to the Planck scale (since Newton's constant doesn't show up anywhere in classical electrodynamics).

Still seems like an overly handwavey argument, the fact that you can construct length, distance and energy densities from some fundamental constants which include the gravitational constant doesn't give any clear reason why this should be the characteristic scale of quantum gravity, the arguments in the paper I linked to are more physical. For a different electrodynamics analogy, would any pre-QED physicists have argued that quantum electrodynamics effects don't become significant until we reach the Planck charge?

 

[yogi]

The answer to #2 is that the Planck scale is the only scale you can construct out of the appropriate physical constants. This is not an absolutely secure argument. For example, if large extra dimensions http://en.wikipedia.org/wiki/Large_extra_dimensions exist, then we could see quantum gravity at the LHC.

You can also get a set of dimensions by combining e, c and G. I believe these were first formulated by Irish Physicist George Johnson Stoney. The dimensions are different than Planck units - one of the reasons I have always considered Planck units as more or less metaphysical numerology - it all started as a dimensional analysis without physics and it still has no physics -

 

[yogi]

For a another take
Chronos posted this on another thread http://arxiv.org/abs/gr-qc/0601097

 

[bcrowell]

You can also get a set of dimensions by combining e, c and G. I believe these were first formulated by Irish Physicist George Johnson Stoney. The dimensions are different than Planck units - one of the reasons I have always considered Planck units as more or less metaphysical numerology - it all started as a dimensional analysis without physics and it still has no physics -

The Planck scale includes Planck's constant, and that's why it's expected to be the scale at which quantum gravity effects become strong. Stoney's units could be fine for some other purpose, but they don't make use of Planck's constant.

 

[yogi]

The Planck scale includes Planck's constant, and that's why it's expected to be the scale at which quantum gravity effects become strong. Stoney's units could be fine for some other purpose, but they don't make use of Planck's constant.

If the interpretation of the constants G c and something else combined is to reveal some deep property about the quantum universe - why would any set of constants be any better than any other set without some supporting physics. e and h are related through alpha

My own subjective opinion is that Planck units have misled theorists - its hard to find a book where they are not elevated to the status of profound significance - trying to fit a theory within the confines of a dimension we can never hope to explore is a step in the wrong direction.

 

[JustinLevy]

This is dimensional analysis. It is powerful, and yes it has limits in how strong of statements it can make.

The question is merely: at what scale do quantum corrections become important? Look at the length scale where the gravitational interactions become on the scale of the quantum length scale.

Yes this argument is a dimensional analysis argument. GIVEN our current theories of gravity and quantum mechanics, THIS IS the scale at which the quantum corrections should become important. Yes, this DOESN'T mean we've obtained detailed understanding of what happens at this scale, or detailed understanding of what the actual theory of quantum gravity is, merely from this argument. No one is claiming that.Dimensional analysis is not so limited as some of the people in this thread seem to think. Either that, or they think people are claiming way more than they actually are when we talk about classical GR breaking down at the Planck scale.

 

[JesseM]

The question is merely: at what scale do quantum corrections become important? Look at the length scale where the gravitational interactions become on the scale of the quantum length scale.

Yes this argument is a dimensional analysis argument.

What do you mean by the "scale" of "gravitational interactions"? And does "the quantum length scale" refer simply to combining various constants to get the Planck length, or does it refer to some more physical idea like a statement that some physical quantity becomes significant at that length scale?

 

[JustinLevy]

What do you mean by the "scale" of "gravitational interactions"? And does "the quantum length scale" refer simply to combining various constants to get the Planck length, or does it refer to some more physical idea like a statement that some physical quantity becomes significant at that length scale?

It sounds like your questions are on the concepts of dimensional analysis itself. Dimensional analysis is quite powerful when used appropriately, and just like statistical mechanics, can cut through all the "un-needed details" to give you very useful broad results.

These techniques are used for all kinds of things. While engineers may not always learn where they come from, they use dimensional analysis in many classes. Looking up pressure curves for non-ideal gases, often use scaled pressures and temperatures, etc. and the gases, despite being non-ideal, are now all described almost identically on a graph. Consider Reynolds number, for fluid dynamics etc.

While not a book "on" dimensional analysis, this book is pretty good and actually pauses to discuss dimensional analysis a bit. It is one of the (many) tools used to discuss phase transitions.

Lectures On Phase Transitions And The Renormalization Group
Nigel Goldenfeld
https://www.amazon.com/dp/0201554097/?tag=pfamazon01-20

---------------
To help "bridge the gap", I'll attempt to make the dimensional analysis "more physical" sounding to you here.

1] Length scale in classical GR
for black hole of energy E, on order of
L_{gr} = \frac{GE}{c^4}

2] Length scale from quantum mechanics
Due to quantum wavelength
\Delta x \Delta p \ge \frac{\hbar}{2}
to contain in a box of size L requires an energy on order of
E = \frac{\hbar c}{L_{qm}}

these two become comparable at:
Length scale = Planck length = \sqrt{\hbar G/c^3}


Reworded to sound more physical:
If we have a huge black hole, quantum mechanics doesn't have much to say... GR should hold fine. However, for very small length scales, GR could claim there is a black hole, but quantum mechanics says the very wave nature of particles prevents it from being confined in such a small volume. This is the scale at which quantum effects (wave nature of particles) make the assumptions of GR (classical fields and particles) become incorrect enough that we wouldn't be approximating the real result with GR anymore. We'd need a quantum theory of gravity.

-----------
Test your knowledge:
Classical electrodynamics says there cannot be any stable arrangement of charges. Quantum mechanics disagrees with the assumptions of classical electrodynamics, and luckily for us allows atoms to form. Without solving Schrodinger's equations, using dimensional analysis, can you find what length scale classical electrodynamics breaks down for an electron interacting with a proton, and therefore what size you would expect atoms to be?

 

[JesseM]

It sounds like your questions are on the concepts of dimensional analysis itself. Dimensional analysis is quite powerful when used appropriately, and just like statistical mechanics, can cut through all the "un-needed details" to give you very useful broad results.

These techniques are used for all kinds of things. While engineers may not always learn where they come from, they use dimensional analysis in many classes. Looking up pressure curves for non-ideal gases, often use scaled pressures and temperatures, etc. and the gases, despite being non-ideal, are now all described almost identically on a graph. Consider Reynolds number, for fluid dynamics etc.

But when you say "dimensional analysis", do you mean there aren't even any rough physical arguments, it's purely a matter of shuffling various constants to reach some physical conclusion?

To help "bridge the gap", I'll attempt to make the dimensional analysis "more physical" sounding to you here.

1] Length scale in classical GR
for black hole of energy E, on order of
L_{gr} = \frac{GE}{c^4}

2] Length scale from quantum mechanics
Due to quantum wavelength
\Delta x \Delta p \ge \frac{\hbar}{2}
to contain in a box of size L requires an energy on order of
E = \frac{\hbar c}{L_{qm}}

these two become comparable at:
Length scale = Planck length = \sqrt{\hbar G/c^3}


Reworded to sound more physical:
If we have a huge black hole, quantum mechanics doesn't have much to say... GR should hold fine. However, for very small length scales, GR could claim there is a black hole, but quantum mechanics says the very wave nature of particles prevents it from being confined in such a small volume. This is the scale at which quantum effects (wave nature of particles) make the assumptions of GR (classical fields and particles) become incorrect enough that we wouldn't be approximating the real result with GR anymore. We'd need a quantum theory of gravity.

OK, this seems like a more physical argument (just to be clear, when you talk about the energy needed in QM I assume you mean something like the minimum depth a potential well of width L would need in order for there to be at least one bound state for a photon, i.e. if L is the wavelength then a bound state which fits about one wavelength in the well would have energy approximately equal to E=hf=hc/L according to Planck's equation, so the potential must be at least that deep for there to be a bound state). Perhaps a physicist would make a purely dimensional argument when talking to an audience of physicists who would be assumed to know a way to translate this into a more physical argument, but what I question is whether it's meaningful to use a purely dimensional argument if you don't have a more physical argument in mind. How would you tell "good" dimensional arguments from "bad" ones (like a hypothetical argument saying that quantum gravity effects should apply to any particle with a mass smaller than the Planck mass, or that quantum electrodynamics effects only become important for charges smaller than the Planck charge) if you don't have recourse to more specific physical arguments?

Test your knowledge:
Classical electrodynamics says there cannot be any stable arrangement of charges. Quantum mechanics disagrees with the assumptions of classical electrodynamics, and luckily for us allows atoms to form. Without solving Schrodinger's equations, using dimensional analysis, can you find what length scale classical electrodynamics breaks down for an electron interacting with a proton, and therefore what size you would expect atoms to be?

Well, if you divide Planck's constant by (c*some mass) you get something with units of length, but in purely dimensional terms how do you decide whether to use the mass of an electron or the mass of a proton, since the two differ by three orders of magnitude? Again it seems like you need some sort of physical argument, even if it's one a physicist would find obvious enough to just leave it implicit...

 

[Passionflower]

It is a dimensional analysis argument.

Quantum mechanics gives us a time or length scale for an object of particular energy or momentum.
General relativity gives us a length scale given a mass (or energy).

Therefore we can look at where these length scales are comparible. The Planck units are units defined using G, hbar, c, etc.

I think it more dificult. Does spacetime have a length scale at all? I think the answer is no, except for static spacetimes. Do you agree?

I think key in this discussion should be the background independence issue.

 

[JustinLevy]

JesseM,
Your beginning questions seem to be misunderstanding dimensional analysis. I'm very sorry, but I don't feel I'm well spoken enough to give a compelling statement of dimensional analysis. Maybe someone else here can.

Your final question though is very good, and shows one of the limits of dimensional analysis: when there are enough parameters to construct multiple scales of the dimension you are intersted in, which are appropriate? This is not always an easy question. But your intuition is guiding you in the correct direction ... some other physical argument must be used (or dimensional analysis just can't help us here). In this case, the two particles are interacting, so from a rough physical argument the appropriate mass would be their reduced mass.

---

I think it more dificult. Does spacetime have a length scale at all? I think the answer is no, except for static spacetimes. Do you agree?

I think key in this discussion should be the background independence issue.

Huh? I'm not sure what you mean here. I'm not talking about the length scale of spacetime all by its lonesome, I'm talking about the length scale of gravitational interactions. Sort of like asking what the length scale of the electric field is without any context, verse asking what the length scale of the electric interaction is in that last post.

Your second comment seems to be getting into more of: why does GR and quantum mechanics conflict so much? ie. why is it so hard to quantize GR? That is a separate issue.

The issue here is: on what scale can we expect the classical GR to no longer make good approximations ignoring quantum effects? This can be answered purely with dimensional analysis. The actual details of quantum gravity, and the details of the correction values or geometry or whatever, are a separate issue.

This really is like the classical electrodynamics + quantum problem above. We can predict the scale at which classical electrodynamics "breaks down". We can predict the scale of atoms purely from dimensional analysis, without needing to actually solve the quantum equations. Dimensional analysis is a very useful tool when used appropriately.

 

[JesseM]

JesseM,
Your beginning questions seem to be misunderstanding dimensional analysis. I'm very sorry, but I don't feel I'm well spoken enough to give a compelling statement of dimensional analysis. Maybe someone else here can.

I don't see how it could have been a misunderstanding since I didn't make any positive claims, I was just asking a question about what you meant by "dimensional analysis", since I'm not too familiar with the term (I've only seen 'dimensional analysis' used in simpler contexts like making sure both sides of an equation have the same dimension). If you didn't mean to imply that one could reach physical conclusions just by playing with constants and without any physical arguments (explicit or implicit), that's fine with me, but then that would suggest that bcrowell's argument lacked the needed physical argument to justify the conclusion that the Planck scale should be the scale of quantum gravity. On the other hand, if you did mean to suggest this, then it seems to me the answer to my initial question should just be "yes". Or perhaps you think I am making a false dichotomy between "physical arguments" and "pure shuffling of constants", I don't see what the third alternative would be though.

 

[JustinLevy]

Or perhaps you think I am making a false dichotomy between "physical arguments" and "pure shuffling of constants"

Yes, more along these lines. I have not attempted to explain dimensional analysis outright, as I don't feel I could do it justice. I don't want to accidentally make it sound like hog-wash to you because of my poor choice in wording. I'm still hoping someone else will take a stab at it, and I'd learn too. For if I can't explain it well to another, then I clearly don't fully understand it myself. Right?



EDIT:
Fine, I'll take a stab at it. _please_ someone else come in and try to add to this.

Dimensional analysis goes like this:
Take a problem, and list the relevant parameters. We can arrange these to get different scales of say length (or energy, or whatever we're trying to solve for), and possibly some dimensionless numbers as well.

To simplify, let's say we can get one scale for length. The point is the answer _must_ be of the form:
L_answer = L_scale * dimensionless number

If we were able to get a dimensionless number from the parameters (let's call it "a"), then the answer must be of the form:
L_answer = L_scale * function(a)

THIS is the main point of dimensional analysis. It allows us to obtain the form of the answer without even needed to solve for anything!

The form of a relation is often enough to make some useful broad results. An interesting example Goldenfeld gives in his book (that I mentioned a couple posts back), with my paraphrasing since I don't have the book here: If you're told the area of a right triangle can be specified in terms of the hypotenuse and the angle it makes with one side, prove the pythagorean theorem.

Since this is a simple situation, we could of course use our geometry knowledge and calculate that function and use more geometry to prove it. But in this problem we can prove the phythagorean theorem purely from the functional form we get from dimensional analysis.


Let's try a quick one. A box has upward initial velocity v, and free falls with gravitational acceleration g. How high does it go?
The only parameters we have are v,g. The only way to obtain a length scale from this is: v^2/g.
So the answer must be of the form:
h = (v^2/g) * dimensionless number

actually solving we find:
mgh = 1/2 mv^2
h = (1/2) v^2/g

With dimensional analysis: We can't get the exact number. We can't get the exact details. But we can get the correct form and the scales that matter to the problem. If we do obtain this form somehow, then we in some sense have obtained a "universal" relation to relate all situations, if we just find the releveant scales in these systems. This is why reynold's numbers, scaled temperature and pressure, etc. work so well.

This next part is the piece that I'm extra worried I can't explain all that well. If we did a good job in choosing the relevant parameters, then the dimensionless number out front should be of order 1. If it was many magnitudes off, then that usually means our choice of parameters wasn't all that good for what we got _wasn't_ the relevant scale. I'm sorry I can't explain this part better. Play with lots of examples to help build up an understanding/intuition is all I can suggest unfortunately.

I said the first step is "choose relevant parameters". So some physical understanding goes into the problem. A lot is hiding in the word "relevant". But after that, it really is just, as you put it, "purely a matter of shuffling various constants".

What I showed previously was me trying to make the dimensional analysis argument feel more physical by relating it to some calculations. This was to help your intuition of why this length scale is important. But as far as dimensional analysis is concerned, none of those calculations needs to be done. There is only one way to build a length scale with the relevant parameters of gravity and quantum mechanics ... the Planck Length.

Again, to stress, dimensional analysis cannot tell us what happens to spacetime and gravity at the Planck scale. It can only tell us this is the length scale at which quantum effects become on the order of the classical predictions (sort of like in the classical electrodynamics example with the atom).

Hopefully that helped. I'm still hoping someone else could add to this, as I still need to learn this better myself.

 

[yogi]

In Engineering, dimensional analysis is exactly what Jesse described - by keeping track of the units you can check the vaidity of both sides of an equation, and quickly spot an error - some of the literature regarding Planck units has appropriated the term "dimensional analysis' - but it is actually a case of "dimensional inference"

 

[JustinLevy]

In Engineering, dimensional analysis is exactly what Jesse described - by keeping track of the units you can check the vaidity of both sides of an equation, and quickly spot an error - some of the literature regarding Planck units has appropriated the term "dimensional analysis' - but it is actually a case of "dimensional inference"

We teach that to physics students as well. But that is just the lowest level of using dimensional information. Often physicists just refer to that as "checking units". It is a useful tool to find where an error occurred in a calculation.

What I described above though: learning the form of the answer and learning the important scales, I believe are what is fully meant by dimensional analysis. At least there are some professors at my university that use it as such, as well as that book I mentioned earlier (which didn't discuss general relativity at all, it was a book on phase transitions).

 

[JesseM]

Thanks for the explanation Justin. I think you may be right that I was posing a false dichotomy, the notion of "dimensional analysis" you're describing does involve some physical intuitions such as the choice of "relevant" parameters, and also general ideas like the notion that quantum gravity should have a characteristic length scale (whereas it doesn't have a characteristic mass scale), but it comes short of more detailed physical arguments.

 

[yogi]

Thanks for the explanation Justin. I think you may be right that I was posing a false dichotomy, the notion of "dimensional analysis" you're describing does involve some physical intuitions such as the choice of "relevant" parameters, and also general ideas like the notion that quantum gravity should have a characteristic length scale (whereas it doesn't have a characteristic mass scale), but it comes short of more detailed physical arguments.

The same 3 constants (G, C and h) also lead to a unit of mass - are you saying its ok to ignor the mass that falls out of the combination as meaningless - but not the length

 

[JesseM]

The same 3 constants (G, C and h) also lead to a unit of mass - are you saying its ok to ignor the mass that falls out of the combination as meaningless - but not the length

No, not saying the Planck mass is meaningless, but it would be a mistake to think that quantum gravity is needed anytime you are analyzing a system with less mass than the Planck mass (though you do if the mass is compressed down to around the Planck length), whereas quantum gravity would be needed anytime you're analyzing interactions on the scale of the Planck length or Planck time--that's exactly what I meant when I said you still needed some basic physical intuitions to do dimensional analysis, even if you don't need detailed physical arguments.