Short answer: we don’t know—and in modern physics, spacetime is usually modeled as non‑compact, but it doesn’t have to be.
The longer, careful answer depends on what you mean by spacetime, which theory, and which notion of compactness you are using.
1. In general relativity: no default assumption
In general relativity, spacetime is modeled as a 4‑dimensional smooth Lorentzian manifold \((M, g)\).
Compactness is not assumed one way or the other.
Both compact and non‑compact spacetimes are mathematically allowed solutions of Einstein’s equations.
Common examples
- Minkowski spacetime (special relativity):
\[
\mathbb{R}^{1,3}
\] ✅ non‑compact
- Schwarzschild spacetime (around a star/black hole):
✅ non‑compact
- FLRW cosmological models (used in cosmology):
✅ can be compact or non‑compact, depending on spatial topology
So from the standpoint of GR:
compactness is a choice, not a theorem.
2. Spacetime vs space: an important distinction
Physicists often separate:
- Spacetime (4D manifold)
- Spatial slices (3D “space” at a given cosmic time)
These can behave very differently.
Typical situation in cosmology
|
Object |
Compact? |
|
Spacetime (4D) |
Usually non‑compact |
|
Spatial slice (3D) |
Could be compact or non‑compact |
Even if space is compact, time almost never is.
3. Cosmological models: finite but unbounded space
In standard cosmology (FLRW models):
Spatial geometry options
- Positive curvature (\(k = +1\))
Example: 3‑sphere \(S^3\)
✅ compact space
- Zero curvature (\(k = 0\))
Example: \(\mathbb{R}^3\) or torus \(T^3\)
✅ compact or non‑compact
- Negative curvature (\(k = -1\))
Usually non‑compact, but compact quotients exist
Crucially:
Observations constrain curvature, not topology.
A spatially compact universe is still observationally allowed.
4. Why spacetime itself is almost always non‑compact
Even if space were compact:
- Time typically extends indefinitely to the past and/or future
- This makes spacetime non‑compact as a 4D object
Example:
\[
\text{Spacetime} = \mathbb{R} \times S^3
\]
- \(S^3\) is compact
- \(\mathbb{R}\) is not
- Product is non‑compact
This is the standard situation in cosmology.
5. What would compact spacetime mean?
A fully compact spacetime would require:
- Space compact ✅
- Time compact ✅ (e.g. circular time \(S^1\))
Example: \[
S^1 \times S^3
\]
This is mathematically valid, but physically problematic:
- Closed timelike curves
- No global causality
- No well‑posed initial value formulation
So physicists usually avoid compact spacetime even if compact space is allowed.
6. Observational status
What we currently know:
- The observable universe is finite
- That does not imply spacetime—or even space—is compact
- We cannot observe global topology beyond the cosmic horizon
So:
Spacetime may be infinite, finite but unbounded, or something more exotic—we don’t yet know.
7. Summary table
|
Question |
Answer |
|
Is spacetime compact by default? |
❌ No |
|
Is spacetime usually modeled as compact? |
❌ Almost always non‑compact |
|
Can space be compact? |
✅ Yes |
|
Is compact space observationally ruled out? |
❌ No |
|
Is compact time physically favored? |
❌ Generally avoided |
One‑sentence takeaway
In modern physics, spacetime is almost always modeled as non‑compact, although space itself could be compact; the universe may be finite without having an edge—but time keeps spacetime from being compact as a whole.
If you’d like, we can push this further into:
- compactness vs geodesic completeness,
- topology vs curvature,
- Cantor‑like spatial universes,
- or the analogy with removing points from \([a,b]\)—which is very much the right intuition for the questions you’ve been asking.
