# General relativity breaks down at Planck scale

1. Sep 20, 2010

### mersecske

Why?

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

2. Sep 20, 2010

### Drakkith

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

3. Sep 20, 2010

### mersecske

Why, what is the logical deduction?

4. Sep 20, 2010

### 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?

5. Sep 20, 2010

### JustinLevy

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.

Last edited: Sep 20, 2010
6. Sep 20, 2010

### pervect

Staff Emeritus
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.

7. Sep 20, 2010

### 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

8. Sep 20, 2010

### JesseM

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.

9. Sep 21, 2010

### mersecske

"quantum foam" means a difficult topology, but whats 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.

10. Sep 21, 2010

### JustinLevy

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.

11. Sep 21, 2010

### atyy

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

12. Sep 22, 2010

### bcrowell

Staff Emeritus
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.

13. Sep 23, 2010

### mersecske

The quantum foam can be smooth, only the scale is Planckian. I dont 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!

14. Sep 24, 2010

### Naty1

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

I also like Justin's observation
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.

15. Sep 24, 2010

### JesseM

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:
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.

16. Sep 25, 2010

### mersecske

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

17. Sep 25, 2010

### Parlyne

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).

18. Sep 25, 2010

### JesseM

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?

19. Sep 26, 2010

### yogi

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 -

20. Sep 26, 2010

### yogi

21. Sep 26, 2010

### bcrowell

Staff Emeritus
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.

22. Sep 27, 2010

### yogi

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.

23. Sep 27, 2010

### 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.

24. Sep 27, 2010

### JesseM

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?

25. Sep 27, 2010

### JustinLevy

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/Lectures-Transitions-Renormalization-Frontiers-Physics/dp/0201554097

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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.

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