General relativity breaks down at Planck scale

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?
JustinLevy said:
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 blackhole, quantum mechanics doesn't have much to say... GR should hold fine. However, for very small length scales, GR could claim there is a blackhole, 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?
JustinLevy said:
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...

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.

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.

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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.

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:

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.

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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"

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 occured 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