Arguments
This topic has been studied in a few articles. Usually, these studies extrapolate what we know and show that the laws of physics that support the stability of our world are (presumably) only possible in such a configuration, although they do not completely rule out other possibilities.
For example, it is pointed out that in Newtonian gravitation, the existence of stable orbits of planets, around a spherical star (of 3-dimensional volume whose surface is a 2-sphere) results from a law of gravitation decreasing in r², which is understandable because the action of gravitation is represented by an isotropic flow emanating from the spherical star of mass M, crossing the 2-spheres of radius r surrounding the star. This surface being equal to 4πr², the flux crossing a constant surface, is “diluted” on the surface of these 2-spheres in 1 / r².
In relativity the relationship is more complex because it is a 4-dimensional global spatio-temporal geometry that is defined, but in a stationary weak field (far from sources generating the “gravitational field”), Newtonian gravitation represents an efficient approximation.
With 4 dimensions of space, by transposing this, we would have a “hyper-star”, hyper-spherical of hyper volume in 4 dimensions, of “hyper-mass” HM, but whose law should, according to the same principle, decrease in 1 / r3 .
This is because the hypersurface of the hypersphere, delimiting the hypersphere, in the “hyperspace-time” with a 5 dimensions signature {-, +, +, +, +}, (if we consider a single temporal component, associated with the four of space), is a 3- sphere.
In a four dimensional “Newton-like mechanics” , this configuration would not produce stable orbits, where the planet would remain long enough at the same distance from the star, star itself stable during this time, as requested for the emergence of life.
Would it be in this hypersurface of signature {-, +, +, +}, that we, humans, (who are three-dimensional beings in space), would live ?
In general relativity, if it is the case, the related phenomenology in this 4-dimensional sub-manifold of the 5-dimensional manifold (a brane?), would be described by the geometry of this sub-manifold.[1]
Anyway, the argument has its limits, because do we really know what physical representation and which experiences should be conducted for grasping the paramters of such “hyper mass”of an “hyper star” and what kind of field it would generate, especially in classical mechanics. [2] .
In relativity, we must find geodesics in the geometry of the “hyperspace-time”, that comply with the criteria of stability, compatible with our emergence.
It is also argued that the existence of stable atoms, as we know them, would be impossible. This would ruin the possibility of a world as we know it.
But, in a 5-dimensional “hyper-spacetime”, we do not really have any ideas on the representation of the associated phenomenology! See the chapter “Can we avoid an anthropomorphic approach?”.
Comments on these arguments
The 3 + 1 dimensional configuration (three of space and one of time) results from a Newtonian approach. In relativity, this is not the case. The structure of space-time is not (3 + 1) but 4 and foliating it into (3 + 1) has no physical character (it’s totally arbitrary).
Therefore, as developed in other pages of this site [3], the null coordinate approach (Newmann-Penrose formalism), taking into account the fundamental role of light which gives the hyperbolic structure to our universe, would give a more physical representation than foliation (3 + 1).
Signature of the metric in general relativity
The hyperbolic structure of the metric of the general relativity is inherited from that of the special relativity metric which is (-, +, +, +), the time coordinate being associated with the “minus” sign and the three space coordinates with the “plus” sign, in Minkowski’s representation because, locally, special relativity applies (except on singularities where the theory is not valid).
Comments, on this topic.
In general relativity, a local basis defined by the tangent vectors at the global coordinates defined on the manifold, may present a different signature. For instance, inside the horizon of black holes, the four global coordinates can be simultaneously spacelike.
However, a local Minkowski base, which can be defined at a point, will still have the signature (-, +, +, +). The hyperbolic structure of space-time is well preserved as the Sylvester’s theorem guarantees it. Let us not forget that the coordinates are arbitrary. The signature (-, +, +, +) is valid in the form of Minkowski. In the Newmann-Penrose formalism, since the local basis has 4 null vectors, the signature would rather be written (0, 0, 0, 0).
Case where there are two global coordinates simultaneously timelike.
This case is usually considered as not compatible with the existence of life.
Keeping in mind what was said previously about the local metric which also applies in this strange case, let us point out that in relativity, there are solutions, like the Kerr and Kerr-Newmann space-times [4] , where, although the Minkowskian local signature of the metric remains {-, +, +, +} everywhere, in some regions there may exist simultaneously two global timelike coordinates (in this case t and φ in the spherical coordinates (t , r, θ, φ), generating a metric signature for these coordinates (t, r, θ, φ) of the type {-, +, +, -}.
If this case is rare, it nevertheless exists, and its impossibility has never been demonstrated. This, pointed out by B. Carter [5], results in a flagrant violation of causality, with all its consequences. There are worldlines between 2 events A and B where A could have been the cause of B and B the cause of A, we can go back in time and many other temporal paradoxes are possible [6].
However, it should be emphasized that these worldlines are not geodesic, they require an interaction with a phenomenon other than gravitation: for instance, the ejection of matter by a rocket causing an acceleration by reaction.
Thus, according to the definition given of general relativity, we can either consider that these solutions are not to be considered if we consider that only geodesics are described by general relativity [7], or that they are to be considered if one accepts other types of worldlines than the geodesics.
In the latter case, the spacetime of general relativity then serves as a “base” and the local spacetime where non-gravitational phenomena can exist and interact with bodies in spacetime is a “fiber”.
These examples show that the phenomenology described by general relativity is not compatible with the description of a universe with 3 dimensions of space and one of time since, in relativity, time and space are not physical, they are only shadows of spacetime, as stated in Plato’s allegory of the cavern.
Can we avoid an anthropomorphic approach?
This criticism has a more general character, it is certain that we seek to determine if other conditions could give the same phenomena as those which one observes. This implies a great effort to our mind! One is aware of the effort already necessary for understanding the concept of spacetime in relativity which destroyed those of space and time that we supposed inherent to our mind and that of indeterminism in quantum mechanics that ruins the basis of classical physics. Imagining more dimensions would be a step further!
If the weak anthropic argument [8] confirms us on the existence of conditions, (and also specifies limits), allowing to achieve what we observe, which is a truism, it does not say anything about possibilities which would be very different, but which structurally could give something of the same type, in a more or less evolved way.
The universe and its existence, like ours, is a subject where many mysteries will likely remain forever.
[1] In some theories, there are “branes” that have dimensions smaller than that of the space containing them. Here space is taken in the general sense which can contain dimensions of the time type. Let’s not forget the null type dimensions.
[2] If we do not understand very well what a hyper-mass could be in Newtonian mechanics, this does not pose a problem in relativity: The energy-momentum tensor Tμν would be defined for μ, ν varying from 0 to 4. But, let us keep in mind that even though the mathematical formalism is straightforward in relativity, its physical representation and the associated physical tests are problematic.
[3] A detailed description can be found in: http://www.astromontgeron.fr/SR-Penrose.pdf
[4] A rigorous analytical solution was found by Kerr in 1963 to the problem of rotating black holes, well after Schwarzschild’s solution for static black holes which dates from 1916. Note that the problem seemed simple, however, since a rotating black hole is totally defined by 2 parameters: its mass M and its angular momentum J. If the black hole is charged (which is unlikely in cosmology) a third must be added. parameter: the electric charge E. In this case the black hole also has a magnetic moment.
[5] Global Structure of the Kerr family of gravitational fields. Brandon. Carter. Phys. Rev. Vol. 174. Number 5,25 october 1968. A free translation in French is available in: http://www.astromontgeron.fr/A_Carter-68-F.pdf
[6] See a detailed analysis in: http://www.astromontgeron.fr/Trous-noirs-de-Kerr-M2-JF.pdf.
[7] This is the strict definition of the theory of general relativity which deals only with gravity. But, nothing prevents to consider the other hypothesis.
[8] “Argument” is more appropriate than “principle”.