# Spatial dimensions with non-equivalent dimensional eigenvalues

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?

• Isn't this Egan’s newest one, Dichronauts? – JDługosz Mar 10 '17 at 3:41
• If so, the answer is gregegan.net/DICHRONAUTS/DICHRONAUTS.html . If not, I’m not understanding. – JDługosz Mar 10 '17 at 3:43
• @JDługosz Egan's Dichronauts has two spatial dimensions and two of time in its 4-space. This system seems to involve three spatial dimensions and one time, but one of the spatial dimensions and the time dimension has negative value. The two 4-spaces seem similar to the extent that it is difficult to discern if they are the same or different. Me? I'm puzzled. – a4android Mar 10 '17 at 4:41
• If you could explain a bit the implication of opposit eigenvalues, you may attract more answers (I personally have some recalling of eigenvalues, but no clue on how their sign is related to physical reality) – L.Dutch - Reinstate Monica Mar 10 '17 at 7:12
• Yea, Dichronauts is (+,+,−,−) which is mentioned in the OP’s 3rd paragraph. How he gets a “normal” world line from that makes sense having been familiar with Orthoginal, and I've been meaning to update this with an overview of how that works. Can’t do that here as it wants hard-science, not explainations and concepts. – JDługosz Mar 10 '17 at 7:21

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.

• You should check Egan's dichronoauts webiste, he explored the implcations in details (wrote a book about it). There are way more consequences that what you briefly described. gregegan.net/DICHRONAUTS/DICHRONAUTS.html – Fred Mar 10 '17 at 7:04
• @Fred Thanks, it is interesting. His geometric results are similar to mine, except that he thought on this far more :-) Although he doesn't seem really interested in the preservation laws. – Gray Sheep Mar 10 '17 at 7:21
• Point #4 might be easy to calculate, but Egan points out the ramifications make it challenging because widely separated particles are “close” in this metric. – JDługosz Mar 10 '17 at 7:30
• @Fred I think the gregan.net site is not really about QM. My idea about the "eating" of the space dimensions uses QM. In pure Newton, having instantenous effect on a space dimension wouldn't mean that this dimension collapse. – Gray Sheep Mar 14 '17 at 9:20

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 ?

• I thought the eigenvalue is the difference between space and time dimensions? Anyway, could you add the link to Egan's explanation of the consequences and maybe a short summary? – Ville Niemi Mar 10 '17 at 7:11
• That's not a hard-science answer. And @VilleNiemi, if it were possible to summarise Egan’s page I would have done so already instead of just posting the link (as a comment). I think it’s not possible to explain it in a reasonable blockquote. – JDługosz Mar 10 '17 at 7:24
• @JDługosz I was afraid of that. Still a proper answer would essentially duplicate Egan's explanation and Egan's explanation is almost certainly the best source for answer so... A good answer will be a summary of Egan's explanation, no? – Ville Niemi Mar 10 '17 at 8:19
• There might be other ideas. Or, as Morning Star notes, he could investigate different aspects. I think Egan’s page is prerequisite reading that any math answers can assume. – JDługosz Mar 10 '17 at 8:32
• Greg Egan already did five dimensional and higher universes in Diaspora (1997). – a4android Mar 10 '17 at 11:44