Gravity: a "limp" rubber membrane is a model

Now that we understand QED better, let us return to the problem of ultraviolet diverging loop integrals in quantum gravity.

What kind of a material has a Green's function which diverges very easily? The Fourier decomposition should have large terms for large |k|?

                  #

                  #========= sharp hammer

                  v

       _____     _____    "limp"

                 \/              rubber membrane

                

Let us have a "limp" rubber membrane whose elastic energy grows slower than quadratic. The force resisting stretching is sublinear.

Let us hit the membrane with a sharp hammer. That will produce a very deep pit. In the Fourier decomposition, there will be lots of waves with a large |k|.

Such a classical system, of course, does not produce infinite waves or an infinite energy. Hammers are always blunt. Destructive interference very efficiently wipes out high |k|.

What is the analogy for gravity? We should have an attraction between gravitons, or waves in the limp rubber membrane.

If the elastic energy is quadratic, or the force resisting stretching is linear, then waves in the membrane do not interact. The wave equation is linear.

In a limp rubber membrane, waves can help each other to get to a lower energy state by interference. The energy of constructive interference is less than in the linear case. There is an attractive force between wave packets.

Let us place weights on the membrane. Together they can lower themselves to a lower vertical position than in the linear case. This is like the steepening gravity potential in general relativity.

We found a pretty good model for gravity, and for other fields which are prone to ultraviolet divergences. Let us study how this classical model avoids divergences.

We do not need to place weights on the rubber membrane? The limpness is enough

       ___        ____      ____   membrane

              \•/           \•/

         weight      weight

   

   ---------------------------------

                  Earth

Actually, it might be that we do not need weights at all in this model. If particles are waves stretching the membrane, and a wave can get to a lower energy state by entering an area where the membrane is stretched more, then we have an attractive force. This makes the model much simpler and more beautiful.

Graviton diagrams

Feynman integrals in this model may have a really large amount of high |k| contribution. They are prone to ultraviolet diverge.

      graviton   -----------------------

                                     |

      graviton   -----------------------

Does a Feynman diagram like the above make sense?

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