On the dimensionality of spacetime

To compute the theoretically predicted probability distribution for the dimensionality of our spacetime, we need to take into account the selection effect that some states are more likely to contain self-aware observers than others.

“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.”

• cites [3] sourcePartly supported

  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)

  Experimental — accuracy not yet measured

When n > 3, the two-body problem no longer has any stable orbits as solutions.

“When n > 3, the two-body problem no longer has any stable orbits as solutions [6].”

• cites [6] Büchel W, 1963, Physikalische Blätter sourceSupported

The Bayesian prior probabilities of quantum-mechanical origin are completely irrelevant to the predicted dimensionality under the condition that observers require minimum complexity, predictability and stability.

“and that the Bayesian prior probabilities of quantum-mechanical origin are completely irrelevant.”

• AssumptionRests on an assumption: 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, then all such ensemble theories may actually predict that we should find ourselves inhabiting a (3+1)-dimensional spacetime with 100% certainty.

“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.”

• AssumptionRests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.

Traditional quantum atoms cannot be stable in a space with n > 3.

“and traditional quantum atoms cannot be stable either [5].”

• cites [5] Tangherlini F R, 1963, Nuovo Cimento sourceNot checked — no full text of the source

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.

“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.”

Giving initial data for an elliptic PDE on a non-closed hypersurface is an ill-posed problem.

“On the other hand, giving 'initial' data for an elliptic PDE on a non-closed hypersurface, say a plane, is an ill-posed problem.”

A localized observer in a hyperbolic world can make predictions about its future.

“A localized observer can therefore make predictions about its future.”

In general relativity, the situation for n > 3 is analogous.

“The situation in general relativity is analogous [5].”

• cites [5] Tangherlini F R, 1963, Nuovo Cimento sourceNot checked — no full text of the source

Elliptic equations allow well-posed boundary value problems.

“Elliptic equations allow well-posed boundary value problems.”

Worlds with n < 3 are too simple and barren to contain observers.

“worlds with n < 3 are just too simple and barren to contain observers.”

A world with the same laws of physics as ours and with an (n+m)-dimensional spacetime can only contain observers if the number of time dimensions m = 1, regardless of the number of space dimensions n.

“Here, we will present an argument for why a world with the same laws of physics as ours and with an (n+m)-dimensional spacetime can only contain observers if the number of time dimensions m = 1, regardless of the number of space dimensions, n.”

• AssumptionRests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.

If an observer is to make use of its self-awareness and information-processing abilities, the laws of physics must be such that it can make at least some predictions.

“If an observer is to be able to make any use of its self-awareness and information-processing abilities, the laws of physics must be such that it can make at least some predictions.”

An observer in a world with no time dimensions (m = 0) would not be able to make any inferences about other parts of space based on local observations.

“This means that an observer in a world with no time dimensions (m = 0) would not be able do make any inferences at all about the situation in other parts of space based on what it observes locally.”

In quantum mechanics, the hydrogen atom has no bound states for n > 3.

“a study of the Schrödinger equation shows that the hydrogen atom has no bound states for n > 3 [5].”

• cites [5] Tangherlini F R, 1963, Nuovo Cimento sourceNot checked — no full text of the source

Asgeirsson's theorem implies that specifying u in a cylinder determines u throughout the region made up of truncated double cones.

“A corollary of a remarkable theorem by Asgeirsson [13] is that if we specify u in the cylinder in figure 3, then this determines u throughout the region made up of the truncated double cones.”

• cites [13] Courant R, 1962 sourceNot checked — no full text of the source

Asgeirsson's theorem also applies to the ultrahyperbolic case, showing that initial data on a hypersurface containing both space-like and time-like directions leads to an ill-posed problem.

“Asgeirsson's theorem also applies to the ultrahyperbolic case as well, showing that initial data on a hypersurface containing both space-like and time-like directions leads to an ill-posed problem.”

• cites [13] Courant R, 1962 sourceNot checked — no full text of the source

The predictability requirement is only met by a small class of PDEs, essentially those which are hyperbolic.

“Although this predictability requirement may sound modest, it is in fact only met by a small class of PDEs, essentially those which are hyperbolic.”

The fundamental Green's function of the Poisson equation ∇2φ = ρ is r^{2-n} for n > 2.

“the fundamental Green's function of the Poisson equation ∇ 2 φ = ρ, which gives the electrostatic/gravitational potential of a point particle, is r 2-n for n > 2.”

If initial data for a hyperbolic PDE is specified on a hypersurface that is not space-like, the problem becomes ill-posed.

“In contrast, if the initial data for a hyperbolic PDE is specified on a hypersurface that is not space-like, the problem becomes ill-posed.”

Within a field theory, an observer should be able to compute field values at more distant spacetime points from nearby field values with non-infinite error bars.

“Specifically, within the framework of a field theory, it should, by measuring various nearby field values, be able to compute field values at some more distant spacetime points (ones lying along its future world line being particularly useful) with noninfinite error bars.”

Hyperbolic equations allow well-posed initial-value problems.

“Hyperbolic equations, on the other hand, allow well-posed initial-value problems.”

More or less than one time dimension gives insufficient predictability.

“• More or less than one time dimension: insufficient predictability.”

• AssumptionRests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.

More than three space dimensions gives insufficient stability.

“• More than three space dimensions: insufficient stability.”

• AssumptionRests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.

The case (n, m) = (1, 3) is mathematically equivalent to the case where (n, m) = (3, 1) and all particles are tachyons with imaginary rest mass.

“Since a mere minus sign distinguishes space from time, the remaining case (n, m) = (1, 3) is mathematically equivalent to the case where (n, m) = (3, 1) and all particles are tachyons [14] with imaginary rest mass.”

• cites [14] Asgeirsson L, 1936, Math Ann. sourceNot checked — no full text of the source

Given the other laws of physics, it is not implausible that only a (3+1)-dimensional spacetime can contain observers that are complex and stable enough to be able to understand and predict their world to any extent at all.

“Our conclusions are graphically illustrated in figure 1: given the other laws of physics, it is not implausible that only a (3+1)-dimensional spacetime can contain observers that are complex and stable enough to be able to understand and predict their world to any extent at all, for the following reasons.”

• AssumptionRests on an assumption: Observers can only exist in a world exhibiting a certain minimum complexity, predictability and stability.

The multiple dimensionality prediction of some superstring theories is a strength rather than a weakness, since it eliminates the discrete fine-tuning problem of explaining why compactification singled out only a (3+1)-dimensional spacetime.

“Viewed in this light, the multiple dimensionality prediction of some superstring theories is a strength rather than a weakness, since it eliminates the otherwise embarrassing discrete fine-tuning problem of having to explain the 'lucky coincidence' that the compactification mechanism itself happened to single out only a (3 + 1)-dimensional spacetime.”

In the (1,3) case, an observer would be unable to make any predictions because well-posed problems require data to be specified in the non-local region outside the lightcones.

“Also in this case, an observer would be unable to make any predictions since, as described in more detail in [15], well-posed problems require data to be specified in the non-local region outside the lightcones.”

• cites [15] sourceNot checked — no full text of the source

Nonlinear terms cannot make ill-posed linear PDE problems well-posed in a larger neighbourhood.

“When PDEs give ill-posed problems even locally, in a small neighbourhood of a hypersurface (where we can generically approximate the nonlinear PDEs with linear ones), it is obvious that no nonlinear terms can make them well-posed in a larger neighbourhood.”

Application of the weak anthropic principle may give the narrowest probability distribution possible for dimensionality.

“Thus, although application of the so-called weak anthropic principle [16] does in general not appear to give very strong predictions for physical constants [17], its dimensionality predictions may indeed turn out to give the narrowest probability distribution possible.”

• cites [16] Tegmark M, 1996Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [17] Carter B, 1974Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.