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Imagine a cubic volume of space, defined today, with edges of the order of the Planck length.

As cosmological time goes forward this volume expands so that the edges of the cube expand with the scale factor. This seems ok.

Equally one should be able to imagine time going backwards so that the edges contract with the scale factor.

But then they would be smaller than the Planck length. If the Planck length is the smallest length that general relativity can deal with is there a contradiction here?

Qmechanic
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John Eastmond
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  • What do you mean by 'time going backwards so that the edges contract with the scale factor'? – Avantgarde Jul 25 '17 at 14:01
  • Are you confirming some type of quantum gravity must have been useful in the earliest big-bang era? – Cosmas Zachos Jul 25 '17 at 15:32
  • Related: https://physics.stackexchange.com/q/9720/2451 and links therein. – Qmechanic Jul 25 '17 at 17:35
  • I assume that physics is time-reversible so that the Planck scale volume must be able to contract as well as expand. However if it does contract then it's size becomes sub-plankian which is beyond GR. – John Eastmond Jul 25 '17 at 18:22

2 Answers2

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You are trying to mix quantum mechanics with general relativity. The two are incompatible theories. The Planck length it's a perfect example of exactly that incompatibility.

Quantum field theory behaves differently on different length scales. At large lengths, quantum effects are borderline nonexistent. At the scale of molecules and atoms, you can treat the quantum mechanics semi-classically, meaning that you don't need to use the full behavior of quantum mechanics to describe the physics. At the atomic scale and below, you need the full treatment of quantum mechanics. And if you go to a short enough length scale, you even need to pull in special relativity. It's the combination of special relativity and quantum mechanics that gives us quantum field theory, but it is also this merging of the two that leads to a length at which all smaller lengths have no useful meaning.

General relativity, on the other hand, has no inherent length scale. The rules of general relativity are the same no matter how large or how small you get. A supermassive black hole whose event horizon is measured in lightyears behaves exactly the same as a black hole the size of quark.

So when you try to apply a general relativity to quantum mechanics, you run into problems like a Planck length that changes with the age of the uninverse. It very much is a contradiction. This is why one of the biggest efforts in fundamental physics today is quantum gravity. We are looking for a new theory that is self consistent and reduces to both quantum field theory and general relativity in their respective limits.

Johnathan Gross
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There is no minimum length in the theory of general relativity. In addition, we have no observational evidence that the theory breaks down at any particular length scales.

What you're talking about (I think) is that it's generally agreed that a singularity in the early universe is unphysical, and a self-consistent quantization of general relativity would show there is no singularity.

But we have no proof of any "smallest length scale" or any reason to think that "a singularity is impossible". I think the "contradiction" you're pointing out is exactly the one that drives the development of quantum gravity.

levitopher
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