The DOI matches its Crossref record; 19 of the 19 references it cites were found in a registry or an index; 2 of the 18 sources its claims cite were checked for support.
On the dimensionality of spacetime
Open accessAbstract
Some superstring theories have more than one effective low-energy limit corresponding to classical spacetimes with different dimensionalities. We argue that all but the (3 + 1)-dimensional one might correspond to `dead worlds', devoid of observers, in which case all such ensemble theories would actually predict that we should find ourselves inhabiting a (3 + 1)-dimensional spacetime. With more or less than one time dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers.
Overview
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Written from the paper’s full text.
TL;DR
Tegmark argues that only a (3+1)-dimensional spacetime is plausibly compatible with observers, because other numbers of time or space dimensions undermine predictability, stability, or complexity. This turns the multiple-dimensionality prediction of some superstring/ensemble theories from a weakness into a possible selection-effect strength.
Key findings
- With more or less than one time dimension, the partial differential equations of nature would lack hyperbolicity and well-posed initial-value problems, so observers could not make predictions with finite error bars; only \(m=1\) permits the required predictability.
- With more than three space dimensions, there are no stable classical atoms or planetary orbits, and the hydrogen atom has no bound states; the two-body problem has no stable orbits and a finite annihilation cross section.
- With fewer than three space dimensions, general relativity has no gravitational force for \(n<3\), and \(n=2\) faces topological problems such as two nerves being unable to cross; such worlds are conjectured to be too simple and barren for observers.
- The paper concludes that all other dimensionalities might be “dead worlds,” so ensemble theories may predict that we should find ourselves in (3+1) dimensions with 100% certainty.
Methods
- Conceptual/analytical letter; no empirical data or sample. It reviews known results on spatial dimensions and presents new arguments about time dimensions.
- Uses the classification of second-order linear PDEs by eigenvalue signs—elliptic, hyperbolic, ultrahyperbolic—and their causal/well-posedness properties, including Asgeirsson’s theorem.
- Applies selection-effect/anthropic reasoning: a probability distribution over spacetime dimensionality should be weighted by whether observers can exist in each case.
Limitations
- The paper explicitly states it has not rigorously demonstrated that observers are impossible for other dimensionalities; it only argues that it is far from obvious that any other \((n,m)\) permits observers, placing the burden of proof on critics.
- It suggests possible future exploration of stable structures in \((4,1)\) via short-distance quantum corrections or string-like particles.
- Evident from the text: the argument is qualitative and plausibility-based, with no empirical tests; it focuses mainly on linear PDEs and only briefly asserts that nonlinear terms cannot rescue locally ill-posed problems.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
21 Own interpretation · 11 No full text of the sources · 6 No basis given · 2 Sources not found · 1 Supported · 1 Partly supported · 1 Not checked yet · accuracy not yet measured
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“To compute the theoretically predicted probability distribution for the dimensionality of our spacetime ‡, we clearly need to take into account the selection effect arising from the fact that some of these states are more likely to contain self-aware observers such as us than others.”
§ On the dimensionality of spacetime > Abstract
What the sources say
- Partly supportedcites [3] sourceExperimental — accuracy not yet measured
In “Predictions from Quantum Cosmology” (page not supplied), Vilenkin weights a probability distribution over varying constants of nature by the average number of civilizations in each universe. This supports accounting for observer-selection effects, but the paper does not apply that calculation to a probability distribution for the dimensionality of spacetime. (LLM, conf=0.87)
boostedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“When n > 3, the two-body problem no longer has any stable orbits as solutions [6].”
§ On the dimensionality of spacetime > L70
What the sources say
- Supportedcites [6] Büchel W, 1963, Physikalische Blätter source
The section on planetary orbits for n > 3 says a planet can follow an exact circular orbit, but that orbit cannot remain stable under small disturbances; otherwise the planet spirals away or falls inward. The paper also rules out orbits whose radius oscillates between a minimum and maximum. Thus it supports the claim about stable orbits, not a claim that no orbit of any kind exists. (LLM, conf=0.93)
plainn = 2- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Supported
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and that the Bayesian prior probabilities of quantum-mechanical origin are completely irrelevant.”
§ On the dimensionality of spacetime > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Rests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“that if observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability, then all such ensemble theories may actually predict that we should find ourselves inhabiting a (3 + 1)-dimensional spacetime with 100% certainty, as illustrated in figure 1, and that the Bayesian prior probabilities of quantum-mechanical origin are completely irrelevant.”
§ On the dimensionality of spacetime > Abstract
hedgedstates a possible cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
- Rests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.
In the paper
“and traditional quantum atoms cannot be stable either [5].”
§ On the dimensionality of spacetime > L70
What the sources say
- Not checked — no full text of the sourcecites [5] Tangherlini F R, 1963, Nuovo Cimento source
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“With more or less than one time dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions.”
§ On the dimensionality of spacetime > Abstract
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“On the other hand, giving 'initial' data for an elliptic PDE on a non-closed hypersurface, say a plane, is an ill-posed problem.”
§ On the dimensionality of spacetime > Elliptic equations allow well-posed boundary value problems.
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A localized observer can therefore make predictions about its future.”
§ On the dimensionality of spacetime > Elliptic equations allow well-posed boundary value problems.
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“The situation in general relativity is analogous [5].”
§ On the dimensionality of spacetime > L70
What the sources say
- Not checked — no full text of the sourcecites [5] Tangherlini F R, 1963, Nuovo Cimento source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Elliptic equations allow well-posed boundary value problems.”
§ On the dimensionality of spacetime > Elliptic equations allow well-posed boundary value problems.
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
- Probably not a substantial claim. 6
- It may state two separate claims as one. 8
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures.”
§ On the dimensionality of spacetime > Abstract
hedgedstates a possible cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers.”
§ On the dimensionality of spacetime > Abstract
hedgedstates a possible cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
In the paper
“Since the tunnelling probabilities between these states are negligible, such a theory for all practical purposes predicts an ensemble of classical (n+m)dimensional spacetimes, and the prediction for the dimensionality takes the form of a probability distribution over n and m [1].”
§ On the dimensionality of spacetime > Abstract
What the sources say
- Not matchedcites [1] Albrecht A, 1994
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Rests on an assumption: Many superstring theories have several stable (or extremely long-lived) states that constitute different effective low-energy theories with different spacetime dimensionalities.
- Rests on an assumption: The tunnelling probabilities between different superstring states are negligible.
- Not matched
In the paper
“As was pointed out by Ehrenfest back in 1917 [4], neither classical atoms nor planetary orbits can be stable in a space with n > 3”
§ On the dimensionality of spacetime > L70
What the sources say
- Not checked yetcites [4] Ehrenfest P, 1917, Proc. Amsterdam Acad.
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“There are also inflationary models predicting a universe consisting of parts of exponentially large size having different dimensionality [2].”
§ On the dimensionality of spacetime > Abstract
What the sources say
- Not checked — no full text of the sourcecites [2] source
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“Another interesting difference, which can be shown by an elegant geometrical argument [10], is that particles become less stable when m > 1.”
- Also stated in: § L72 · plain
“When m > 1, this constraint vanishes [10].”
§ On the dimensionality of spacetime > L72
What the sources say
- Not checked — no full text of the sourcecites [10] Deser S, 1984, Ann. Phys. source
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Also stated in: § L72 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For instance, they will be hyperbolic in a metric of the signature (+ ---), corresponding to (n, m) = (3, 1), elliptic in a metric of the signature (+ + + + +), and ultrahyperbolic in a metric of the signature (+ + --).”
§ On the dimensionality of spacetime > Letter to the Editor L73
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A claim is checked only against the cited paper’s full text, never its abstract, and none is available for its sources.
In the paper
“there is no gravitational force in general relativity with n < 3.”
§ On the dimensionality of spacetime > L70
What the sources say
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Even when m > 1, there is no obvious reason why an observer could not, none the less, perceive time as being one-dimensional, thereby maintaining the pattern of having 'thoughts' in a one-dimensional succession that characterizes our own reality perception.”
§ On the dimensionality of spacetime > L70
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Modulo the important caveats in the discussion section, this means that such a world cannot contain any objects that are stable over time, and thus probably cannot contain stable observers.”
§ On the dimensionality of spacetime > L70
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“It has been argued [7] that organisms would face insurmountable topological problems if n = 2: for instance, two nerves cannot cross.”
§ On the dimensionality of spacetime > L70
What the sources say
- Not checked — no full text of the sourcecites [7] Freeman I M, 1969, Am. J. Phys. source
hedgedstates a cause- Asserted: fails — The authors may not put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“there are no space-like or time-like hypersurfaces in the ultrahyperbolic case, and hence no well-posed problems †.”
§ On the dimensionality of spacetime > L74
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“• Less than three space dimensions: insufficient complexity.”
§ On the dimensionality of spacetime > Additional information
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
- Rests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.
The paper relies on this without support.
Many superstring theories have several stable (or extremely long-lived) states that constitute different effective low-energy theories with different spacetime dimensionalities.
“Many superstring theories have several stable (or extremely long-lived) states that constitute different effective low-energy theories with different spacetime dimensionalities, corresponding to different compactifications of the many (e.g.11 or 26) dimensions of the fundamental manifold.”
The paper relies on this without support.
The tunnelling probabilities between different superstring states are negligible.
“Since the tunnelling probabilities between these states are negligible”
The paper relies on this without support.
Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.
“if observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ On the dimensionality of spacetime > L70
“Another problem, emphasized by Wheeler et al [8], is the well known fact (see e.g.[9]) that there is no gravitational force in general relativity with n < 3.”
§ On the dimensionality of spacetime > References
“8. Whitrow C J 1955 Brit. J. Phil. 6 13”
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Metadata
- Authors
- Tegmark, M.
- Affiliations
- Institute for Advanced Study (US)
- Corresponding author
- Tegmark, Max
- ORCID
- Tegmark, Max
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1997-04-01
- Volume
- 14
- Issue
- 4
- Pages
- L69-L75
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2098169780
- Semantic Scholar
- 080c1e8b41755c3bce3675fc605c5f3f05a2801e
- Cited by
- 135
- References
- 17
- Citations by year
- 2026: 8 · 2025: 7 · 2024: 3 · 2023: 9 · 2022: 4 · 2021: 7 · 2020: 4 · 2019: 10 · 2018: 1 · 2017: 10 · 2016: 8 · 2015: 8 · 2014: 6 · 2013: 6 · 2012: 3
- FWCI
- 3.60
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- spacetime dimensionality, superstring theory, observer problem, hyperbolicity, temporal dimension, extra dimensions, gravitational force, ensemble theory
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