Those of you who have been following physics for the last few decades have no doubt heard about string theory and how it requires that spacetime have 11-dimensions and not the four we are most familiar with (3 for space and 1 for time) . This has been a popular topic of discussion since the 1970s, and has spawned thousands of popularizations and lectures. I have actually written a few of these!

The difficulty is that we never experience even ordinary space directly, and as Einstein noted, space itself is more of a human mental construct than it is a physical feature of our world. When physicists need some additional dimensions to space to model our world, we find ourselves confused as our brains try to fabricate an inner model of what that might mean. The idea of ‘dimension’ is not what you might think. It is just another way of saying that some system you are modeling mathematically needs N numbers to define it uniquely in space and time. For example, economists work in ‘spaces’ where their models use N=4 unique numbers, so their universes are 4-dimensional too!

Space, as a physical idea is just our experience that we only need three numbers to locate a geometric point on Earth. We need its latitude, its longitude and its elevation above the surface of Earth. Two of these numbers are angles, but we can easily convert these angles into meters because the surface of Earth has a fixed radius and forms a sphere. The angles just represents the lengths of arcs on the surface.

Extending this into the solar system and beyond, we only need three numbers to define the location of a planet or a star. If we try to provide four numbers to locate any point in space, that fourth measurement can be replaced in terms of the previous three numbers. Because this minimum number is three, we say that our space is 3-dimensional. Similarly, all of Euclid’s geometry takes place on a flat surface where points are defined by two numbers, so it is a 2-dimensional surface. The figure above shows an example of ordinary 3-d space and its coordinate axis in the Cartesian ‘X, Y, Z’ system. But there are other possible coordinate systems we can use such as spherical (R, theta, phi) or cylindrical (R, theta, h), but they all define points in 3-d space by a unique three-number address. But wait…there’s more!

**Three is not enough. **Our physical world actually requires a time coordinate to define where objects are because, of course, all objects are in motion or are internally changing. Time is a fourth coordinate that helps us keep track of the complete history of an object as it moves through 3-d space. This path is called a worldline. This is why Einstein’s relativity refers to ‘spacetime’ as the fundamental arena in which all events take place. Spacetime is a 4-dimensional object with three coordinates expressed in terms of meters, and a time coordinate expressed in units of time such as nanoseconds, seconds, years etc. Spacetime is also called a manifold because every point in it can be uniquely defined by four numbers. There are mathematically an infinite number of these points. Because we have yet to find any evidence to the contrary, the manifold of all points in our physical universe, call it M, is just the same as the 4-d spacetime manifold proposed by Einstein’s relativity, call it M_{4}, so in short-hand we can write a symbolic sentence: M = M_{4}.

**M _{4} is not big enough: **What you need to keep in mind is that in our 4-dimensional world, we can obviously see that ‘space’ is very different than ‘time’. When physicists talk about our world or our universe having more than 4-dimensions, they also mean that additional coordinates are needed beyond (3-d) space and (1-d) time to keep track of the properties of matter and energy in their new theories. There is nothing about any of these additional coordinates that needs to have the same character as either space or time. All they need to be is numbers defining a 10-dimensional coordinate system. Some physicists, however, do think of these added dimensions as having some kind of space-like attributes.

The way that these extra dimensions are added to our world to create the M of our physical universe is that they represent coordinates in a separate kind of space, call it M_{6}, that adjoins our M_{4}. In string theory, this M_{6} space is vanishingly small. You will not find traces of it in M_{4}, so don’t worry about somehow taking a wrong turn in M_{4} and suddenly ending up in M_{6}. String theory requires that M_{6} be a closed, finite manifold present at **every point** in M_{4}. The size of each of these new dimensions is only 10^{-33} centimeters, which is why these M_{6}s are called ‘compact’ manifolds. Only quantum particles and fields have access to M_{6}, and this access causes particles to have different characteristics.

**So what is M?** The complete specification of our manifold can be written as M = M_{4 }x M_{6}. The address of each point in M is now given by, for example, the 4 coordinates (x,y,z,t) to define their location in M_{4}, but to define the location of this point inside one of these compact M_{6} manifolds you need six additional numbers, which we can write as (a,b,c,d,e,f). The complete specification for points in the M manifold is then (x,y,z,t,a,b,c,d,e,f) or more neatly (x,y,z,t)(a,b,c,d,e,f) where we keep the two manifolds symbolically separate by placing parenthesis to group their respective coordinates.

So the question is, are the coordinates (a,b,c,d,e,f) dimensions of space like (x,y,z), are they more like the dimension of time in M_{4}, or are they something else? As I said earlier, we already know that the time coordinate, t, is not at all like the other three space coordinates in M_{4}, so there is no reason to believe that (a,b,c,d,e,f) should resemble either (x,y,z) or t. In fact, in string theory we already know something about the character of the coordinates in M_{6}.

**Very small:** String theory proposes that they have a size of about the Planck Length, which is 10^{-33} centimeters. You can’t get lost in M6 because you always wind up back at your starting point after you have walked 10^{-33} centimeters!

**Periodic: **Some or all of these extra dimensions allow for the properties of physical quantities such as fields to have some periodic feature to them. An important quantity is the spin of a particle, which can be either multiples of 1 (called bosons) or 1/2 (called fermions). All elementary particles (electrons, quarks) are fermions, and all force-carrying particles (photons, gluons) are bosons.

**Provides a complex topology: **The number of times a quantity ‘wraps’ around a periodic extra-dimension, the more mass it has. When M_{6} is defined by these coordinates, the shapes of the M_{6} manifolds, called their topology, can have holes in them. String theory requires that these shapes be in a class called Calabi-Yau manifolds in order that the particles and fields resemble our world.

**Metric signature: **In special relativity, the distance between points in 4-dimensions, dS, is defined by dS^{2} = dx^{2} + dy^{2} + dz^{2} – c^{2}dt^{2} where, for instance the symbol dx is the difference in the x coordinates between two points dx = x_{2} – x_{1}. Notice that the space coordinates enter with a positive sign and the sole time coordinate enters with a negative sign. Physicists write this as the metric signature (+,+,+,-). In string theory, the extra dimensions are added with positive signs to make the string theory models work correctly to get something like dS^{2} = dx^{2} + dy^{2} + dz^{2} +da^{2} + db^{2} + dc^{2} +dd^{2} + de^{2} +df^{2} – c^{2}dt^{2}. So, the extra dimensions are taken to be space-like even though the maximum distance along any one of these 6 dimensions is only about 10^{-33} centimeters!

That is the mathematics of it, but do the extra dimensions really, really, reaally exist?

The truth of the matter is that we don’t know! If string theory is found to be incorrect, then there is no obvious reason to have a M_{6} at all, and so we are left with the M_{4} universe that we know and love. We cannot investigate the universe at the Planck scale, so the only way to test this idea of extra dimensions is to find that a theory like string theory is correct and accurate. We can then say that, because experiments confirm the predictions of string theory very accurately, there must be something like these M_{6} manifolds with their additional dimensions beyond the four we can directly measure in M_{4}. On the other hand, it may also be that M_{6} is just mental ‘scratch paper’ that humans need in order to mathematically describe the particles and fields in our M_{4} universe, but our universe is actually doing something else entirely!

One important prediction from string theory is that gravity is a force modeled by closed strings in this 10-dimensional manifold. It also has the ability to range far and wide across these additional dimensions while the other three forces are confined to M_{4}. This means that any gravitational interaction takes place in the 10-d arena and we only experience the part of this gravitational field that is in our ‘small’ 4-d universe. In fact, the theory says that this is why gravity is such as weak force because we are only seeing a small part of its full 10-d effects. This allows us to test at least this prediction by string theory.

**Supernova 1987A**: This supernova produced a pulse of neutrinos measured back on earth through the detection of about 12 neutrinos in the Super Kamiokande Neutrino Detector. These pulses arrived within seconds of the optical burst. Neutrinos are like the Miner’s Canaries and their numbers depend very sentitively on the energy of the collapsing supernova core into a neutron star. These 12 neutrinos matched the energy calculations, but only if the core of the star achieved its temperature by gravitational collapse, and with the energy determined by the action of gravity with little or no leakage into other dimensions. The tests were only able to say that the supernova data are consistent with these extra dimensions being smaller than about 1 angstrom (10^{-8} cm). So this result is a bit inconclusive.

**Gravity waves:** In 2017, the LIGO gravity wave detector rang out with the passing gravity wave called GW170817 from the collision between two neutron stars. The added twist was that the collision was also observed by its intense optical outburst. When detailed calculations of the energy in gravity waves and visible light were performed, the energy matched the theory exactly. The theory, based on standard general relativity, does not inlude the diminution of the gravity pulse by leakage of energy outside our M_{4} spacetime. The conclusion was that on this basis, gravity does not act as though there are extra dimensions! In fact, the gravity wave measurements give an observational estimate for the dimensionality of M_{4} of D = 4.02 +/-0.07. There is not much uncertainty to hide 6 more dimensions.

So there you have it. Those extra dimensions are probably space-like but their compact character makes them very different than either the huge space-like character of our familiar 3-dimensions to space.

Stay tuned for my next Blog in two weeks where I will be discussing the laatest Webb Space Telescope data!!!