Spatial dimensions with non-equivalent dimensional eigenvalues

Question

Our universe is described as having three physical dimensions, plus one time dimension, where the eigenvalues for the physical dimensions are all the same, but the eigenvalue for time is opposite (x, y, z, t: +, +, +, -) or (x, y, z, t: -, -, -, +)

A universe where the eigenvalue of time is equal to that of the spatial dimensions (x, y, z, t: +, +, +, +) has been explored in depth by Greg Egan in his Orthogonal series of books.

However, what would happen if the eigenvalues of the physical dimensions were not identical, e.g.: (x, y, z, t: +, +, -, -)? or even, in a 4-physical-dimensional universe, (w, x, y, z, t: -, +, +, +, -)?

In the universe I'm interested in, there is one time dimension with a negative eigenvalue, and 3 or 4 physical dimensions, but one of the physical dimensions has an eigenvalue equivalent to that of the time dimension, while the others have the opposite eigenvalue as is the case in our universe.

How would the physics and chemistry of such a universe be different to our own? How would movement in the negative-eigenvalue spatial dimension work? Would life - or even matter - be able to exist?

Answer 1

In the special relativity, the 4-distance is calculated on this way: $ds^2=dx^2+dy^2+dz^2-dt^2$.

In your world, having a space-like dimension with $-1$ signature, it would be calculated like $ds^2=dx^2+dy^2-dz^2-dt^2$.

It has far consequences:

1. The space won't be isotropic any more. Thus, things would behave differently if you rotate them around the $z$-axis and any linear combination of $x$ and $y$. The laws of the physics would behave differently in the different (space) directions.
2. On the Noether theorem, the symmetries of the Universe have a deep connection to its conservation laws. The isotropy of the space results the conservation of angular momentum. In a non-isotropic Universe, the angular momentum isn't a conserved quantity any more.
3. Any effect will be instant around any direction, for which $dx^2+dy^2-dz^2=0$. It would effectively mean, that moving any point-like particle on such an axis, you get the same system. This would be a new symmetry, which would result a new conservation law, what doesn't exist in our Universe.
4. To calculate, which conservation law is it, is complex but it doesn't require much more math/physics skills as in the high school. Any physics student from around the second year of his studies can do this for you, although it probably wouldn't be easy to find a cooperative one.

Having an additional, conventional space dimension (thus, an 5D spacetime) doesn't change this significantly.

I am not sure, but this "new conservation law" would probably essentially mean, that any point of the space, for which $dx^2+dy^2-dz^2=0$ are the same. If it is correct, then the result is that this negative-signature space dimension "eats" all of the others. Thus, you have essentially an 1D space.

Note: this all depends on if the (non-curved) spacetime of your Universe is still governed by the Special Relativity, as ours.

---

Extension: things would significantly change if you calculate what happens with the gravitation, too. Gravitation changes the geometry of the spacetime, thus the distances wouldn't be calculated like $ds^2=dx^2+dy^2-dz^2-dt^2$, instead you have a tensor (essentially a table) for which

$$ds^2=\underline{dr} \begin{bmatrix} g_{xx} & g_{xy} & g_{xz} & g_{xt} \\ g_{yx} & g_{yy} & g_{yz} & g_{yt} \\ g_{zx} & g_{zy} & g_{zz} & g_{zt} \\ g_{tx} & g_{ty} & g_{tz} & g_{tt} \end{bmatrix} \underline{dr}$$

$g_{n_1 n_2}$ is determined by the mass and impulse distributions, it is essentially the General Relativity analogy of the gravitational field. For small (much lighter as black holes) and slow (much slower as speed of light) you get the Newtonian gravitation from it.

It is possible, that near strongly graviting objects the spacetime would be multidimensional again.

Answer 2

There is no difference betwee time and space dimensions, it s just a shortcut of language we use to describe t in our universe.

Greg egans calls z u. It doesn t matter if you say z/u is a space or time dimension, what matters are the implcation for the space time geometry.

So yes, Dichronauts describes your universe.

So our universe is (+,+,+,-). Greg Egan's Orthogonal describes (+,+,+,+) and Dichronauts describes (+,+,-,-). I guess all the possibilites have been explored!

How about 5 dimensional universes Mr Egan ?