# Q: What would life be like in higher dimensions?

The original question was: Assuming we had four (or more) spatial dimensions in which to freely move around, like say a 4+1 dimensional universe, how might one extend our 3+1 dimensional physics to that universe?

Side note: When someone says “3+1 dimensions”, what they mean is “3 regular space dimensions, and one time dimension” which is exactly the situation we live in (apologies to our pan-dimensional readers).

Physicist: Right off the bat, more dimensions means more freedom of movement.  One of the more mundane effects of that is that in 4 dimensional space there’s an extra direction you can move and/or fall over in.  So if you want to build a working bar stool you’d need at least 4 legs instead of just 3.  In fact, in D-dimensional space bar stools need at least D legs, or they’ll fall over.  Just one of the subtle economic effects of higher dimensional living.

You’d also find that in 4 or more dimensions, you’d be able to do a lot of tricks impossible in 3 dimensions, like creating Klein bottles or (equivalently) taping the edges of two Möbius strips together.  Sailing knots could take on stunning complexities.  In fact, they’d need too!  All of the knots that work in 3 dimensions fall apart immediately in 4.

In four dimensions you could make this surface without worrying about it intersecting itself.

Most physical laws are already written in a dimension-free form.  For example, in Newton’s second law, $\vec{F}=M\vec{A}$, $\vec{F}$ and $\vec{A}$ are both vectors, but they can be vectors in any number of dimensions.  So you can use $\vec{F}=M\vec{A}$ for objects on a line (1-D), on a table-top (2-D), in space (3-D), or whatever (whatever-D).

There are some laws that are usually written in a 3-D form, but that’s generally a matter of convenience more than necessity.  For example, we talk about the “angular momentum vector”, which is defined to be perpendicular to the plane of rotation.  It’s convenient because in three dimensions there’s always exactly one perpendicular direction to a plane, whereas in 4 dimensions (for example) there are 2.

In 3-D we can formulate laws about spinning things in terms of the one direction that isn’t spinning (h), the “axis of rotation”. But we can always formulate laws in terms of the two directions that are spinning, regardless of dimension.

This is pretty easy to fix and generalize, it just becomes a little more difficult to work with.  All that said, while our physical laws can be generalized to any number of dimensions, the manifestation of those laws are wildly different.  So, living in higher dimensions would be pretty alien.

Based on our understanding of gravity (gained from studying this podunk universe), gravitational force should drop by $\frac{1}{R^{D-1}}$, where D is the dimension and R is the distance between the objects in question.  It so happens that because of the nature of orbits, a stable orbit can only exist in 2 or 3 dimensions.

The “effective potential” representing the balance between the gravity and centrifugal forces of an orbiting object.  Orbits can be stable in 2 and 3 dimensions. In all other dimensions planets and moons will always either spiral in or fly away.  Shown here is the potential energy from gravity and the centrifugal force combined.  If there’s a “cup” you can form a “bound orbit” in it.

In 4 or more dimensions orbits are always unstable, and in 1 dimension the idea of an orbit doesn’t even make sense.

Most physicists consider light to be native to only 3 dimensions, because light is an EM wave and it’s direction of propagation is perpendicular to both its Electric and Magnetic fields.  (Fun fact: the direction that light points is called the “Poynting vector“, after John Henry Poynting.  Life’s funny.)  In 4 or more dimensions this direction isn’t unique, and in two dimensions there’s no direction at all.  However, you can express EM waves just in terms of “E” in any dimension without problem.

Assuming light can exist in higher dimensions, it would behave very strangely.  Sound waves too.  In odd dimensions other than 1 (3, 5, 7, …) waves behave the way we normally see and hear things: a wave is formed, it moves out, and it keeps going.  However, in even dimensions, and 1 as well, (1, 2, 4, 6, …) waves “double back” on themselves.  You can see this in ripples on the surface of water (2-D waves).  Ripples are more complex than just a ring; the entire circle within the ripples is disturbed.

In even dimensions (like the 2-D surface of water), waves propagate in a more complex way than we’re used to.  Instead of a simple pulse, you get an “area filling” wave.

If you set off a firecracker in 3, 5, 7, etc. dimensions, then you’ll see and hear the explosion for a moment, and that’s it.  If you set of a firecracker in 4, 6, 8, etc. dimensions, then you’ll see and hear the explosion intensely for a moment, but will continue to see and hear it for a while.  For light the effect would be fairly subtle, except for extremely long-distance effects, like somebody reflecting a bright light off of the moon.  You probably wouldn’t notice the effect day-to-day.  However, it would ruin the experience of sound.  In 4 dimensional space the firecracker, even in open air, would sound like thunder; loud at first, and leading into a drawn out boom.  It may not even be possible to understand people when they speak.

All the fundamental particles should still exist, but how they interact would be pretty different.  Which elements are stable, and the nature of chemical bonds between them, would be completely rearranged.  Some things would stay the same, like electrons would still have two spins (up or down).  But atomic orbitals, which are determined by spherical harmonics (which in turn are more complicated in higher dimensions), would generally be able to hold more electrons.  As just one example (for our chemistry-nerd readers), you’ll always have 1 S orbital in every energy level, but in 4 dimensions you’ll have 4 P orbitals in each energy level, instead of the paltry 3 that we’re used to.  This messes up a lot of things.  For example, in 4 dimensions Magnesium would be a noble gas instead of a metal.  Every element after helium would adopt weird new properties, and the periodic table would be longer left-right and shorter up-down.

So, while the laws of physics are actually the same, if you lived on a four-dimensional Earth in a four-dimensional universe you’d find that (among other things): your bar stool may need an extra leg, Earth wouldn’t be able to orbit anything, you’d never be able to hear anything crisply, and the periodic table of the elements would be seriously rearranged.

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### 22 Responses to Q: What would life be like in higher dimensions?

1. Andrew says:

GREG EGAN DIASPORA SPOILER!

Ever read Greg Egan’s Diaspora? I’m wondering how accurate his depiction of a 5 + 1 universe was. He mentioned that there would be no atoms, since no orbits are stable, including those of the electron around the nucleus.

2. steve says:

What about string theory with all of its extra dimensions? Do string theorists have a way to account for all the weird stuff that doesn’t happen but should?

3. The Physicist says:

The extra dimensions of string theory are much smaller than the smallest particles. So, nothing has “room” to move in the other directions. It’s a little like being stuck in a pipe that’s as big as you are: there may be other dimensions, but you can only move back and forth in one direction.

4. Gaurav Bhattacharjee says:

You didn’t mention one thing : the effect of gravity.
The more the dimensions, the more the force of gravity changes with respect to time. This means that if two or more subatomic particles collide, they will form, so called, micro black holes because of the increase in the strength of gravity . So, there will be no electrons, protons, neutrons nor atoms. Only photons and neutrinos will be present.
So, there will be no life in higher dimensional spacetime, as there will be no atoms to clump and form matter.

5. Gaurav Bhattacharjee says:

Even if you are ignoring the increased strength of the gravitational interaction between elementary and subatomic particles, life in higher dimensions would not be possible. Planets won’t exist, because they will be swallowed in by the star around which they are orbiting due to the increased strength of gravity.

6. The Physicist says:

It’s not quite as straight forward as gravity just being stronger. In higher dimensions gravity increases more sharply at small scales (even in 3 dimensions it increases very sharply), but that doesn’t immediately mean that atoms would collapse into black holes. In fact, since the units are different you can’t really draw any conclusions about the gravitational constant in 4-D given the gravitational constant in 3-D. That is, in 3-D, the units of G are m3/kg/s2. In 4-D it’s m4/kg/s2.

7. Gaurav Bhattacharjee says:

Okay. Thanks

8. Jason Goodman says:

I think we need to worry about the existence of atoms and molecules. Electrons are bound to nuclei because the electric force permits stable orbits, just like gravity. (There’s quantum weirdness here, but it doesn’t change the basic picture.) In higher dimensions, you’ve shown that gravitational orbits could not exist: electrical orbits might not either. To work this out properly, you’d have to re-derive quantum electrodynamics in extra dimensions, which I can’t even begin to do, but if you naively assume that gravity and electricity always obey the same inverse-power law, atoms and molecules could not exist above N=3.

So to heck with how many legs a stool would need: you might not be able to hold a stool together in the first place.

Because the strong nuclear force has a very different dependence on distance, I’d guess it doesn’t suffer from this problem. So atomic nuclei may exist, drifting freely in space, surrounded by a sea of unbound electrons. Not a nice universe.

9. Will says:

Be a good place to harvest raw materials though.

10. Locutus says:

I know the term “3+1″ is used to mean 3d plus time, but isn’t time a “space” of its own? In other words, what is the difference between 3+1 and 4d, because wouldn’t 4d include time?

Love Long and Prosper

11. theSpleen says:

Could you perhaps explain the thing about even-D (and 1-D) waves being much different from odd-D waves? It seems interesting, but I really don’t understand it…

Also, I would be very interested to know some of the math behind why higher-dimensional orbits can’t exist!

12. The Physicist says:

Time is similar to space, but pretty different. There’s a post here that’s far more specific.

13. Constructing a theory of electromagnetism in four dimensions of space will be difficult. Hell, the only higher dimension where it might be possible, excluding three, is seven. This has to do with the mathematical properties of rotation.

That is, only the two-dimensional, three-dimensional, and seven-dimensional space has a rotation described by a “division algebra” — the complex field, the quaternions, and the octonians, respectively.

I suppose, however, that one way to construct electromagnetism in “N”-dimensions is by a well-known construction of the magnetic field which seems to be dimensionally-invariant (not the Kaluza-Klein construction). That is, to assume the four-dimensional Coulomb’s law and Lorenz invariance. I am not sure where this would leave, but I can guess that the resultant equations would not resemble the canonical Maxwell’s equations in any sense.

14. StringTheoryDropout says:

Octonions are very strange things: quaternion multiplication isn’t commutative, but octonion multiplication isn’t even associative!

It seems to me the better path is the gauge theory approach. That is, in however many dimensions you want, local gauge invariance of charge conservation implies the existence of a magnetic vector potential (conventionally named A). Note that for all they may have told you in E&M that A wasn’t real, it was just a notional thing whose curl was B, that’s actually wrong: the Aharonov-Bohm effect offers experimental proof that what electrons actually feel is A, not B (in accord with their classical Hamiltonian, as used in nonrelativistic quantum mechanics).

Then the field strength tensor (F, which is the proper Lorentz-invariant expression of both E and B together) is dA (the exterior derivative — div, grad and curl all at once), and all of Maxwell’s four equations at once are written d*F = 4pi *J (those stars are Hodge dual operators, not multiplication) for charge density (source term) J. There seems little reason that couldn’t generalize directly to any dimension you like (well, at least 1+1). F wedge F would no longer have enough rank to be an energy density, but supergravity folks often resort to things that look like F wedge F wedge F wedge A, which would be an energy density for 6+1 dimensions…

15. There is a book called Flatland by Edwin A. Abbott that considers how life would be different with more or fewer dimensions. It’s old, but still good.

16. szefunio says:

> electrons would still have two spins (up or down)
That is questionable. In 4D objects will rotate around two axises simultaneously (planets would have 4 geographical poles for example, geographical coordinates would consist of 3 numbers) and electron probably will have 2 spins simultaneously, so 4 different values of spin will be possible (1/2, 1/2), (1/2, -1/2), (-1/2, 1/2), (-1/2, -1/2).

17. Alyster says:

I don’t know if this should be formalized as its own question, but can you please explain how the extra dimensions in string theory are “smaller”? The way I conceptualized extra dimensions is through the linear algebra approach of n-spaces, which from my understanding do not have limits. Does this mean that there are “bounds” to the extra dimensions in string theory?