What is the simplest evidence that there are more than 4 dimensions?

All of the 'evidence' is of the theoretical kind. Attempts to create a quantum theory for gravity have led in many different directions since the mid-1970s, and the most heavily worked theories involve objects called 'strings' that exist in a 10 or 11-dimensional space, of which only four correspond to our normal, infinite 3-space dimensions and 1-dimension of time. This 4-dimensional object is the familiar 'spacetime' that forms the basis of special and general relativity. The additional 6 or 7 dimensions according to some theories can be 'very large' but most propose that they are far-smaller than the size of an atomic nucleus. The exact sizes depend on parameters like the 'string tension' which sets the scale for string physics. All fundimental particles are loops of this string with a physical scale in centimeters set by the string tension parameter.

The additional 6 or 7 dimensions exist at each point in the 4-dimensional spacetime, and by most theories exist as compact geometrical objects. For example, the simplist of these would be a spherical 6 or 7-dimensioanl hypersphere. Other topilogies are also possible such as toroids (donuts!) and objects with progressively more holes in them. According to some string theories, the number of holes, called the genus of the topology, are related to the number of fundamental families of particles. For instance, in the Standard Model, we have three generations of quarks and three generations of leptons. Also, the specific geometry of these compact spaces controls the kinds of symmetries we see among the fermions (matter particles) and the bosons ( force carriers). For example, a simple sphere allows for a quantum number that is an integer according to the number of times it can 'wrap' around the surface: 2pi, 4pi, 6pi etc. Supersymmetry allows these geometries to describe both the matter particles (electrons, quarke, neutrinos) and force particles (photons, gluons etc). String theory, and the specific kinds of particles families and symmetries we see in the Standard Model require that these compact spaces be of a particular kind called Calabi-Yau manifolds. Unfortunately, there are literally billions upon billions of these geometries: each a diffferent realization of what our physical world could contain in terms of alternate 'Standard Models'.

Applications of these string theory ideas have also led to the so-called 'brane' theory in which our 3-dimensional space is embedded in a 10-dimensional volume called The Bulk. All of the Standard Model particles and fields are string loops the begin and end in our brane, however gravity exists in the full Bulk.We only see a portion of the full gravitational force in our brane which explains why it appears so weak. Each 3-dimensional brane is separated from potentially many others by the typical scale of these extra dimensions, believed to be something like the 10^-33 centimeters.But again, this scale is set by the string tension so it needs to be experimentally determined through the detection of extra space dimensions, if possible.

Because gravity is described in terms of closed string loops, it is the only force that can extend beyond our 3-D brane. Physicists probe the extra dimensions to space by exploring gravity and its inverse-square law at smaller and smaller scales. If it departs from the familiar inverse-square relation at some scale, this would imply the effect of one or more extra-dimensions. According to a seminal paper by Arkani–Hamed, Savas Dimopoulos and Gia Dvali (called the ADD paper), there is a relationship between the scale of these N, extra dimensions and the gravitational force law at different scales. 



R = 10^(30/N - 17)  centimeters



This leads to the following:

For N=1 extra dimension, R=10 trillion centimeters.
For N=2 extra dimension, R=0.01 centimeters (100 microns)
For N=3 extra dimension, R=0.0000001 centimeters (10 Angstroms).
For N=4 extra dimension, R=3 x 10^-10 centimeters.
For N=5 extra dimension, R=1 x 10^-11 centimeters.
For N=6 extra dimension, R=1 x 10^-12 centimeters.
For N=7 extra dimension, R=2 x 10^-13 centimeters (nuclear-scale).

For n=1, the departures would be seen at interplanetary distances, but we know from careful studies of planetary motion that there is no detection at this scale. For n=2 extra dimensions we are at the 100 micron scale, and the gravitational force law should change from inverse-squafre to an inverse-fourth-power law. Many traditional means for detecting gravity forces at these scales are technically difficult, but not impossible. For n=3 extra dimensions, we are at the scale of atomic physics. By the time we get to seven extra dimensions, the ADD results predict departures from Newtonian gravity at nearly the nuclear scale.

To probe scales where N>3 dimensions could exist we have to look at individual particle interactions in the quantum world and look for gravity-dependent differences. These searches have been conducted at a variety of accelerator laboratories around the world including FermiLab and the Large Hadron Collider. A particular feature of these searches is the production of a single very energetic “mono-object” that does not balance the transverse momentum carried by anything else emerging from the collision (as would be required by momentum and energy conservation). Examples of such objects are particle jets, very energetic photons or heavy W and Z vector bosons. Such collisions only appear to be imbalanced, however, because the emerging jet or boson is balanced by a graviton that escapes detection.

So, how are we doing in observing extra dimensions?

According to a summary of experimental studies ca 2012 by the official Particle Data Group, There is no evidence for N=2 extra dimensions above a scale of 30 microns, and an updated limit in 2014 place a limit of smaller than 1 micron. So far, no evidence has turned up for N>3 large dimensions at the scale of atomic or nuclear interactions.

Return to Dr. Odenwald's Gravity page at the Astronomy Cafe Blog.