**Does Einstein’s equation define a spacetime at equilibrium?**

Einstein’s equation derived from the **extremal** principle defines a **spacetime** deemed stable (or metastable?) where the elements of matter, energy, follow geodesics.

For instance, in cosmology, in a universe made up of matter (dust, modelized as a perfect fluid), each grain of this dust follows a geodesic, different from that of the others, of space-time.

This very theoretical and ideal case (resulting from an extreme simplification) seems to meet the stability criterion, which could also be metastable because in this ideal case we do not see what could disturb it (in the absence of a perturbation external to gravitation).

It should be noted, however, that in a metastable state, the slightest perturbation, could lead to a catastrophic « evolution » (collapse or exponential expansion) unless it leads meanwhile to a new stable state, but it can be assumed that if this other stable state existed, Einstein’s equation should have given it.

It would therefore seem that a stable system should not emit gravitational waves, since this radiation, as we pointed out in another page on this site dealing with the inertia of the universe, is an ** « inertial » reaction to a disturbance (by a process other than gravitation) of this equilibrium. **

If the equilibrium is not disturbed no gravitational waves are emitted!

**Does the notion of stability of a spacetime make sense?**

Of course, it is necessary to define what the **stability criterion **applies to. The usual analysis **derived from classical mechanics** is based on a foliation of spacetime in time and space and assumes initial conditions on a initial hypersurface from which evolution will result according to Einstein’s equation and « time »: this is the idea of the ADM method.

But in a covariant approach**, where spacetime is considered as an indivisible entity, does it make sense to speak of stability since spacetime is the « whole** » (including space and time, as appearances, « in all their space and time extension, in Newtonian language »)?

In fact, we can say that it is not, since we consider the whole: The whole **exists, **and is well defined, whatever it may be. This problem of the stability of spacetime is **incidental**, since it simply refers to « internal » properties.

Let’s not forget that Einstein’s equation refers to spacetime and that the extremum criterion applies to a spacetime entity (Hilbert’s action using the Ricci scalar for the model in a vacuum).

So this paradox can result from a poor analysis of the problem. This does not exempt us from considering other arguments, but let us keep in mind that this is not necessarily the fundamental problem.

**Experiences detect gravitational waves.**

The standard cosmological model is a theoretical model, because the universe is far more complex and diverse than that. It would be necessary to describe exactly the constitution of the universe with all the parameters of matter-energy (the value of the energy-momentum tensor of Einstein’s equation at each point) to realistically describe the dynamics of this universe. this seems impossible, at least today.

We therefore assume that the ideal model describes the dynamics « at the first order » and we will treat the deviations of this model by a perturbative method. But can this justify the emission of gravitational waves?

These gravitational waves seem to be the manifestation of the existence of a perturbation process, which in this approach, looks not be taken into account in the Einstein’s equation.

Indeed, these gravitational waves, the **inertial reaction **of the entire universe to a perturbation, result of the evolution of the universe towards a new state of equilibrium, taking into account this perturbation, when these waves will have reached the whole spacetime.

It should be remembered that the state of equilibrium being global to the universe, following a disturbance, even a small one, it is the entire universe that is involved and which must be reconfigured in its entirety.

Hence the fact that this **inertia is enormous** (it is that of the entire universe) as evidenced by the « rigidity » of the universe (k.c^{4}/G) which is included in the equation giving the amplitude of gravitational waves

This is why we can speak **of the inertia of the universe **(the resistance to a change of one state of equilibrium towards another).

This process, which should be **transient**, may last forever for an open geometry of the universe which is infinite (passing from one state of equilibrium to another, under the action of this perturbation). Its end is when the new state of equilibrium is reached: In terms of information and causality, the information about the perturbation was communicated to the whole universe (which may be infinite) by gravitational waves, which allowed the new stable configuration.

The quantum fluctuations, which gave rise (via the inflationary phase) to the power spectrum of the Cosmic Microwave Background Radiation (CMB) are the source of a multitude of primordial perturbations that will persist and be amplified, would account, at least, for a part of the dynamics of the universe.

In fact, this multitude of instabilities produces a universe that is not homogeneous and isotropic, but of a large-scale « sponge » structure. The matter is concentrated along filaments, leaving large vacuum regions between them.

**Is the stability of the universe compromised?**

So considering the universe as stable, as Einstein’s equation defines it, is a first-order approximation that can be seriously questioned, by the amplifying nature of these perturbations by gravitation, for example the collapse into black holes of overdensity clouds of matter or supernovae.

We have considered only the primordial quantum fluctuations that played a structural role (formation of large structures and their components), but it is possible that other perturbative, non-gravitational interactions, some of which can no longer be present today, also exerted an influence.

**Will the acceleration of expansion lead to a state of (internal) equilibrium?**

Will the dilution of matter-energy, (supposed in the standard cosmological model), similar as what inflation has done to the curvature of space, allow us to reach a new state of equilibrium?

Note that the « metastable » nature of the extremum of Einstein’s equation could be invoked in this regard.

Today, the expansion is not large enough to overwhelm the gravitation that binds galaxies in clusters and a fortiori the components of these galaxies (stars, gas, etc.), but, in the distant future, it is predicted that this expansion will be such that even atoms, or even their nuclei, will be torn apart.

Then a new state of equilibrium « sterile, icy and sinister » could result, unless, in the meantime, other discoveries refute these hypotheses.

**Are there configurations of matter-energy that do not allow us to find an extremum by Einstein’s equation?**

That is, **does Einstein’s equation always have a solution,** regardless of the matter-energy configuration and other constraints that would apply? Could this lack of a solution, as in mathematics, be solved by « algebraic extensions » in the equations, and what would this produce? [1]

But is this lack of a solution conceivable? We are entitled to ask the question, and if so, in this case, since no « stable » state in its internal structure would be possible, gravitational waves would be an integral part of the phenomenon and would be concomitant with the dynamics of the system.

This is another possibility to consider that would explain the gravitational waves that we are observing.

**Is this paradox the result of an erroneous analysis of the problem?**

All this considered, let us remember, as we have pointed out, that it is perhaps the approach to the problem that is erroneous, even though the arguments pointed out can tell us about the internal structure of this spacetime.

But, as long as we don’t have a clear idea of what the stability of a spacetime could be, in its **entirety,** and not in its internal structure which, even if it does not meet a criterion of stability, could it really make sense?

If it seems that this criterion of stability looks to make sense for us, who are not (apparently?) spacetime creatures, it is because we apprehend this spacetime in our foliated « 3D x 1D » dimensionality, in terms of initial condition and temporal evolution.

Let us remember that the universe, as spacetime,** has neither past nor future** (therefore no evolution) because it is the whole, including space and time as appearances.

**Where does the universe get its space-time character?**

The following problem then arises, the universe being a spacetime, given the elements that constitute it, planets, stars, galaxies, clusters, etc., which define its geometry to which, in turn, these same elements couple (which generates a non-linear problem), from where does it get this spacetime nature?

**Is the spacetime nature substantial to objects?**

Is it the **objects mentioned in the previous paragraph that are of a spacetime nature, even if this attribute is so tiny that we do not perceive it, that confer on the spacetime its attribute then amplified at the level of the universe? **

**In this case the **transmission of the attribute would be **substantial.**

**Does the spacetime attribute result from relational structure between objects?**

Or, is it the combination and relation (nonlinear gravitational interaction: generation of geometry and coupling with this geometry) between objects that would not be of a spacetime nature that generates this spacetime attribute: the generation of the spacetime attribute would **be relational .**

**Discussion of these criteria**

Note that in both cases, it is not only the physical constitution of the object (the star for example) that accounts but also the interaction (in this case gravitation) that is attached to it. That must then be included in the attribute of the object: this modifies the substantial character of the object by conferring, on its substance, a “potential” relational capability.

Let us recall that substantial existence is (in general) a rather a metaphysical concept, existence being generally defined by its action (on others), per Leibnitz « **What does not act does not exist**« . This rather would select **relational generation **which is quite common in physics, even with restrictions that we will discuss.

But, let us stress that if objects, even those endowed with gravitational interaction, do not have a relational property, this implies a **transcendence** for a **new character, spacetime, to emerge**.

Note that, in general relativity, there are solutions without objects: « empty » solutions, De Sitter for instance, we can wonder if these are physical solutions, although satisfying Einstein’s equation, they appear rather as basic states defining classes of solutions to the limits.

The case of single objects (Schwarzschild, Kerr, for instance) is interesting: Indeed, it allows us to define a spacetime with which it is coupled (self-coupling). This results from the self-interaction between the geometry created and the object that creates it.

This seems to prove that the object possesses, in its substance, the relational character, even though it is clear that relativity takes on its full interest for multiple bodies.

This example would lead us to declare that spacetime attribute to the universe is therefore transmitted by objects in which, apart their sustantial attribute (matter), a »relational » property (interaction as a field) exists.

In quantum theory, the substance (matter) attribute is the quantum mechanics and the quantum field theory the relational attribute.

**What about the human mind?**

But as far as human mind is concerned, it looks like that we are not be spacetime beings, because the gravitation that governs spacetime looks not have a (sensible) effect on our mind structure.

For our body, it is different, as gravitation rules out some constraints for keeping us alive in and adapted for living and moving on a gravitational field (that of Earth) [2].

Nevertheless, it would be interesting to question whether a complex a sophisticated structure such as ours would possibly emerge without a gravitational field?

We know the trouble and disturbances that astronauts experience in micro gravity, obviously because we are not adapted to it, but the question is worth to be considered. In this case, gravitation should be taken (in which way) into account in our attributes as, without it, we should not exist….

**Notes**

[1] For instance, the extension of real numbers to complex numbers to find solutions to certain algebraic equations.

[2] For our material and spiritual structure, it is the electromagnetic interaction that is dominant, because if gravitation intervenes on our skeleton, muscles, tendons and organs, among others, for the sizing of our bones that must support our weight, of our muscles and tendons to move and the organs to ensure vital functions, in a gravitational field. It is the electromagnetic interaction that ensures the rigidity of the bones, the resistance of the muscles and tendons, etc. As for the vital functions, their chemistry, which involves the outer layers of atoms, is a matter of electromagnetic interaction.