One sign is different — the four dimensions of spacetime are not four dimensions of space
This universe is said to be four-dimensional. It is a different thing from the four dimensions treated in What happens only in four dimensions.
Minkowski spacetime: ds2 = −c2dt2 + dx2 + dy2 + dz2
What differs is one sign, and nothing else. This article follows what that one sign changes, through five measurable quantities.
- Change the sign and rotation stops being rotation — the eigenvalues leave the circle for the real axis
- The "unit sphere" becomes a hyperbola — and on the phrase "put in an imaginary number"
- Distance degenerates — the distance between two distinct points can be 0
- Velocities do not add — c lies beyond infinity
- Why four dimensions — the smallest dimension in which gravity can propagate
- Why space is three-dimensional — the dimension in which waves leave no tail
- What this article can and cannot say
Change the sign and rotation stops being rotation
Take the smallest possible form: two dimensions, with one of the signs flipped.
Minkowski: −t2 + x2 → the transformation is a boost ( coshφ sinhφ ; sinhφ coshφ )
The forms look alike. It seems that the trigonometric functions have merely become hyperbolic ones. Take the eigenvalues and the difference comes out.
| Angle / rapidity | Eigenvalues of the rotation | |λ| | Eigenvalues of the boost | |λ| |
|---|
The eigenvalues of a rotation are always on the unit circle (|λ| = 1, complex). The eigenvalues of a boost are on the real axis (|λ| ≠ 1).
Changing one sign moves the eigenvalues from the circle to the real axis. That is what the difference is, algebraically.
What follows immediately
The "unit sphere" becomes a hyperbola
Draw the set of points satisfying ds² = 1. In the Euclidean case, a circle. In the Minkowski case, a hyperbola. Now connect the two continuously.
Somewhere in between, it degenerates once. At the moment the second component of the metric is 0, the set is neither a closed curve nor a hyperbola, but two straight lines. For something closed to open up, it must pass through this one point.
On the phrase "put an imaginary number in the time direction"
There is a way of writing it that Minkowski himself used. Set x₄ = ict and —
Formally, it becomes Euclidean four dimensions. "Three real directions and one imaginary direction" comes from here.
This way of putting it is correct. But it is shallow.
Shallow means that it hides something. What it hides can be named.
| What gets hidden | What it is |
|---|---|
| That directions split into three kinds | In the Euclidean case, a rotation can carry any direction to any other. In the Minkowski case there are timelike, lightlike and spacelike, and none can be carried into another. Put in i and this distinction disappears from the formula |
| The light cone and causality | ds² = 0 is meaningful only because the signs are mixed; written in Euclidean form, that set becomes invisible (§03) |
| That it cannot be carried over to curved spacetime | In general relativity the metric varies from place to place. You cannot embed i into a general metric. Which is why this notation fell out of use |
That said, the operation of "putting in an imaginary number" is still an important tool. As the Wick rotation replacing t by −iτ, it connects field theory with statistical mechanics. The continuous deformation in the figure above is the smallest form of that operation.
Being useful as a formalism and representing the structure are different things.
ict is the former, not the latter.
Distance degenerates — this is the real difference
In Euclidean space, the distance is 0 only for the same point. That looks obvious. It is not.
Minkowski: ds2 = 0 ⇔ two distinct points joined by light
In Minkowski spacetime the "distance" between two distinct points can be 0.
The set of those zeros is the light cone.
That the distance is degenerate is the reason time is time.
And this degeneracy is what creates causality. The sign of ds² divides the relation between two points into three.
| ds² | Name | Meaning | Can a rotation carry it over |
|---|---|---|---|
| < 0 | timelike | joinable by a signal slower than light. Can stand in a causal relation. "Before and after" is fixed | No carrying between the three kinds. A Lorentz transformation preserves the light cone |
| = 0 | lightlike | joined exactly by light. The boundary of causality | |
| > 0 | spacelike | joinable by no signal at all. Whether they are "simultaneous" depends on who is looking |
Time is not a fourth axis. It is one sign that differs in a quadratic form.
And as a result directions split into three kinds, between which the symmetry cannot carry anything — that is the substance of the time direction being special.
※ The view in this section has the same structure as the form treated in another article. The light cone is the set ds² = 0 — the critical surface separating two regions. The boundary itself determines the structure.
Velocities do not add — c lies beyond infinity
For rotations, angles add. Turn 30° and then 30° and you have 60°. For boosts, rapidities add too.
What adds is rapidity, not velocity. Repeat the same boost and this is what happens.
| Number of boosts | Rapidity φ | Velocity v/c |
|---|
c is not "fast". It lies beyond an infinite rapidity.
That is why it cannot be reached — not a prohibition, but a consequence of the group being non-compact (§01).
Where the compactness of the rotation group gives "come round and return", here there is "no far end".
Why four dimensions — the smallest dimension in which gravity can propagate
So far the story has been about the differing sign. The number of dimensions itself also carries meaning.
The curvature of spacetime is written by the Riemann curvature tensor. The number of its independent components is fixed by the dimension alone, and it splits into the Ricci curvature (the part matter determines directly) and the Weyl curvature (the rest).
| Dimension | Riemann | Ricci | Weyl |
|---|
In three dimensions, the component count of Riemann (6) and of Ricci (6) coincide.
That is, curvature is completely determined by matter. Where there is no matter (Ricci = 0) it is necessarily flat, and gravitational waves cannot exist.
Only in four dimensions do the 10 Weyl components remain. It can be curved with no matter present — that is where gravitational waves live.
※ Two dimensions is a different matter: Riemann has only 1 independent component there (curvature is exhausted by a single scalar). The two-dimensional row of the table is a naive count of the components of a symmetric tensor.
Why space is three-dimensional — the dimension in which waves leave no tail
There is one more place where the number of dimensions matters: how waves propagate.
Emit a short wave at the origin and observe at a distant point. What is left after the wavefront has passed?
| Spatial dimension | Wavefront arrival | Amount left after passage |
|---|
In two dimensions the disturbance keeps going after the wavefront has passed (17.7% of the peak). In three dimensions it goes quiet once the front has passed (2.2%).
This is Huygens' principle, and it holds only when the spatial dimension is odd.
What this article can and cannot say
| Content | |
|---|---|
| can say | One difference of sign changes all of the position of the eigenvalues, the shape of the unit sphere, the degeneracy of distance, the composition of velocities, the freedom in the curvature and the tail of waves |
| can say | "Putting an imaginary number in time" is formally correct, but hides the splitting of directions into three kinds, and the light cone |
| can say | In three dimensions Riemann = Ricci, so the vacuum is flat. Only in four dimensions do the 10 Weyl components remain (the count is exact) |
| cannot say | This is not an explanation of "therefore the universe is 3+1 dimensional". What is listed above is "what would be a problem if it were not 3+1", not a reason it has to be 3+1 |
| cannot say | The difference between the two sign conventions (−+++ or +−−−). The physics is the same, but a difference does show up in the treatment of spinors — not touched on here |
| cannot say | Curved spacetime. This article is entirely about flat Minkowski spacetime, and does not enter general relativity |
There is no new mathematics in this article
The eigenvalues, the Wick rotation, Huygens' principle and the component count of the Weyl tensor are all known. What was done was to line them up again from the single point that one sign is different, and nothing more.
Sources and reproduction
| Item | Kind | Source / tool |
|---|---|---|
| The Minkowski metric and the ict notation | classical | Minkowski (1908) |
| Huygens' principle holding in odd dimensions | theorem | classical (fundamental solution of the wave equation) |
| Riemann = Ricci in three dimensions | theorem | classical (the Weyl tensor vanishes identically for n ≤ 3) |
| Eigenvalues of rotations and boosts | computed on this machine | eigenvalues of 2×2 matrices |
| Continuous deformation of the set ds² = 1 | computed on this machine | second component of the metric from +1 to −1 |
| Velocity and rapidity | computed on this machine | v/c = tanh φ |
| Counting the curvature components | computed on this machine | n²(n²−1)/12 / n(n+1)/2 / n(n+1)(n+2)(n−3)/12 |
| The tail of waves (1, 2, 3 dimensions) | computed on this machine | spherically symmetric wave equation by finite differences. Observation point r = 6, 2400 grid points |