Stephen R. Green and Robert M. Wald did not prove "no backreaction" in the FLRW universe

A Simple, Heuristic Derivation of our "No Backreaction" Results

Stephen R. Green, Robert M. Wald

(Submitted on 25 Jan 2016)

"We provide a simple discussion of our results on the backreaction effects of density inhomogeneities in cosmology, without mentioning one-parameter families or weak limits. Emphasis is placed on the manner in which "averaging" is done and the fact that one is solving Einstein's equation. The key assumptions and results that we rigorously derived within our original mathematical framework are thereby explained in a heuristic way."

https://arxiv.org/abs/1601.06789

Let us analyze from a mathematician's point of view what the authors have proved.

https://arxiv.org/abs/1505.07800

We will show that Thomas Buchert, George F.R. Ellis, Syksy Räsänen, et al. (2015) are at least partially right in their criticism of the papers and claims of Green and Wald.

The assumptions

The authors start from an assumption that the real universe fulfills the Einstein equations

       G_ab g_ab + Λ g_ab = 8π T_ab,

where

       g_ab = g^(0)_ab + γ_ab,

and g^(0)_ab is a standard FLRW metric and γ_ab is "small".

T_ab is the stress-energy tensor of the real universe where we live.

Let us analyze the assumptions. The authors assume that:

1. There is an exact solution g_ab for general relativity with the stress-energy tensor T_ab.

2. The solution is "near" a standard FLRW metric g^(0)_ab.

Assumption 1 is something people have tried to prove for 104 years, but have failed.

Assumption 2 is not self-evident either. It might well be that general relativity has a solution, but it is not "near" a standard FLRW metric.

The theorem

The authors define a stress-energy tensor T^(0)_ab where the mass content of the universe is spread evenly and T^0_ab is the stress-energy tensor for some FLRW metric.

They proceed to show that

       G_ab g^(0)_ab + Λ g^(0)_ab - 8π T^(0)_ab
       = T_diff.

is "small" in a limiting case λ -> 0, where λ is a parameter that specifies a whole family of metrics g_ab(λ).

If T_diff = 0, then g^(0)_ab is an exact solution for the averaged stress-energy tensor T^(0)_ab.

What if T_diff is not zero? Since T^(0)_ab is the stress-energy tensor for some FLRW metric, there exists an exact FLRW metric solution for it. Can we prove that this solution is "close to" g^(0)_ab?

Here we would need a way to compare metrics and how "close" they are to each other. In the case of standard FLRW metrics, we might define that the metrics are close if they give almost the same predictions of the future of the universe for an astronomer. Did the authors prove this?

Why the authors did not prove that the backreaction is negligible?

The reason is the assumptions 1 and 2 above.

1. They did not prove that a solution exists at all.

2. They did not prove that if a solution exists, it is "close to" a standard FLRW solution.

Since they did not prove that the assumptions are true, they did not prove that any of the conclusions are true.

Assumption 2 says that the solution is close to an FLRW metric, and the theorem says, among other things, that the solution is close to an FLRW metric. The theorem is, to some extent, circular reasoning

Our remarks above highlight the fact that all current cosmological models may be broken in the sense that they might not approximate any solution of general relativity. It is not the models' fault. The problem is that existence of physically realistic solutions of general relativity is an open problem.

How to embed two stars in the Minkowski space?

If we have just one spherically symmetric star in an asymptotic Minkowski space, then the Schwarzschild exterior metric is an exact solution of the Einstein equations outside the star.

Inside the star we may use the Schwarzschild interior metric.

        ●             ●
   star 1       star 2

What if we have two stars?

Can we find an approximate solution?

Let us write the Schwarzschild metric for a single star

        g_M + g_S,

where g_M is the standard Minkowski metric (-1, 1, 1, 1) and g_S is a "small" deformation of the Minkowski metric. Let T_S be the associated stress-energy tensor.

One may conjecture that

         g_sum = g_M + g_S1 + g_S2,

where g_S1 and g_S2 are the deformations caused by the stars 1 and 2, is an approximate solution of the Einstein equations.

Since the Einstein equations are nonlinear, the simple sum g_sum above probably is not an exact solution. If it were, that would be in the literature. Also, the stars attract each other. The solution cannot be a sum of two static solutions.

If we calculate the stress-energy tensor T_sum for g_sum, using the Einstein equations, then T_sum differs slightly from

        T_S1 + T_S2.

The difference ΔT may contain matter of positive or negative density throughout the universe, maybe some pressure, momentum, and even shear stresses. If it is possible for such matter to exist, then by adding that matter to the spacetime, we would have an exact solution of the Einstein equations.

However, we are looking for an exact solution where the space is a vacuum outside the stars.

As far as we know, no one has found an exact solution for two stars, or proved that a solution exists. Since the solution is not static, there will be gravitational radiation from the stars. That complicates the situation further.

We are not aware of anyone finding an exact solution of two stars embedded in an FLRW universe, either, if there is no matter between the stars. The solution of C. Gilbert requires that each star is alone in its "hole" in the uniform FLRW universe.

Since we do not know if any solution exists, we cannot say that g_sum is an "approximate" solution.

The reasoning above shows that Stephen R.  Green and Robert M. Wald have not proved that they have found an approximate solution for a realistic FLRW universe, because they did not prove that any solution exists at all.

What if we assume that a solution exists? Can we somehow show that g_sum is close to that solution? That looks like a hard task. If we would be able to show that some iterative process of refining the candidate solution converges, then we might be able to prove something about the closeness of g_sum.

Green and Wald in their papers define a family of metrics indexed by λ, and they probably mean that when λ goes to zero, then the limit metric is an exact solution of the Einstein equations. We need to check if the resulting limit metric is trivial. Since no one has been able to prove the existence of exact solutions in the lumpy case, the limit metric probably is trivial.

Constructing exact solutions by using shells of negative or positive mass

If we have a star of a Schwarzschild mass m, then we can "reset" its gravitational field at some distance r, by enclosing it into a spherical shell which carries a negative mass -m.

    (                ●                )

                star m      shell -m

The gravitational field is zero outside the shell.

We can form various exact solutions by embedding the star-shell structure to an ordinary Minkowski space.

We avoid the non-linearity of the Einstein equations by isolating the field of each star, so that the fields do not overlap anywhere.

If we have two stars of a mass m, we can embed each into a shell, and then put a spherical shell of the mass 2m around the whole system. In that way we get an exact solution which mimics Newton's gravity of two stars in much of the space.

But the field is zero in a certain zone, and the use of negative mass makes the solution very unphysical. Our model is not a good approximation of Newton's gravity. Can we improve the model somehow?

Our example suggests that it may be impossible to mimic Newton's gravity with exact solutions of the Einstein equations.

A rubber model of gravity can be made linear?

It may be that the differential equations of the laws of nature have to be linear, so that the existence of solutions is guaranteed. Can we make a rubber model of gravity such that the model is linear at the fundamental level?

The nonlinear behavior of gravity close to a black hole would be a high level phenomenon and would not reflect nonlinearity in the deeper level of the theory.

Cosmological models and dark energy

Observations suggest that the expansion of the universe started accelerating about 5 billion years ago.

The currently observed spatial flatness of the universe cannot be explained with the known matter and dark matter.

These facts suggest that there is dark energy which is accelerating the expansion.

The above facts assume that the FLRW model of general relativity is approximately right for inhomogeneous matter content. But we do not know if general relativity has any solution for such a matter content.

We need to check what exactly is the evidence for an accelerating expansion. If we assume that the spatial metric has been roughly flat for the past 5 billion years, how does the acceleration exhibit itself?

Is it possible to embed a star into an FLRW model?

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1956MNRAS.116..678G&db_key=AST&page_ind=0&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES

C. Gilbert was able to calculate the following scenario in 1956.

We have a standard FLRW universe with a perfectly uniform dust density ϱ, except that at the origin of the spatial coordinates there is an empty, vacuum sphere of a radius r, and at the origin there is a spherically symmetric object with a Schwarzschild mass m.

  ######<---------●--------->######
uniform            m           a
density ϱ

C. Gilbert says that there is a "hole" in the uniform mass distribution.

Earlier, A. Einstein and E. G. Straus had studied this problem.

https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

The FLRW model has a scale factor R(t) (in Wikipedia this is a(t)), and reduced circumference polar coordinates with a space curvature 1 / R_0^2 (in Wikipedia this is k).

C. Gilbert finds out that if

       m = 4/3 π a^3 ϱ,

then we can glue the Schwarzschild solution in the spherical hole smoothly to the FLRW solution in the uniform density zone.

The formula above looks sensible in the case where the expansion rate of the universe is zero, the dust density very small, and the spatial metric is flat. Then we can start from a uniform FLRW solution and can collect the dust from the hole to the center. The mass-energy of the central mass becomes approximately the right.

A spatially flat universe expanding at a constant rate

However, if the universe is expanding rapidly, the collector first needs to stop the dust in the hole from expanding, for example, using a spring system attached to individual dust particles.

That will give extra energy to the collector.

Does the collector need to spend that extra energy when he pulls the dust particle to the center?

If the spatial metric is flat and grows as

        constant * t,

then the collector feels a force F(r) pulling particles at a distance r away from him. In the newtonian approximation, the force is proportional to r^2, and we can define a potential V(r), such that

       F(r) = dV(r) / dr.

Thus, in this simple, newtonian case, the collector will spend exactly the kinetic energy which he harvested from a dust particle, to win the potential difference V(r) between the particle position and the center.

However, in most cases the universe is not expanding at a constant rate. The expansion slows down because of the gravity of the matter, or may speed up because of a cosmological constant.

Can the collector get the right m at the center?

It is not at all clear that the collector can collect the right mass-energy m at the center, so that the solution of C. Gilbert would be satisfied.

What if we cannot satisfy C. Gilberts solution? If general relativity has any solution at all for the collection process, there must happen something which either prevents the collection process, or which magically puts the right mass-energy to the center, so that C. Gilbert's solution is satisfied.

From our universe we know that some kind of collection of mass has happened. The collection is not spherically symmetric, though.

What could prevent the collection from happening? If the dust distribution is infinitely rigid, then we cannot collect dust. That would be a very strange solution to the problem.

Is there some magical process in general relativity which might provide the extra energy needed to stop the dust in the hole from expanding? That seems unlikely.

We conclude that general relativity probably is not compatible with the elementary process of collecting dust to form a star. That is a major blow to general relativity as a theory of gravity. The problem again seems to be the excessive strictness of the Einstein equations.

The simulations with the FLRW model and galaxy formation probably use a newtonian gravity approximation glued on top of the FLRW model. According to the paper of Räsänen et al. in our previous blog post, the simulations differ by a factor of 2 or so from our empirical observations.

Have the people running the simulations checked if there is any way of making a solution of general relativity from the approximation? A combination of FLRW and Newton is not guaranteed to produce anything like a solution of general relativity.

Let us look at the literature, what other people have written about this.

What does it mean if general relativity has no realistic solutions?

Effective theories

The assumed divergence of the perturbation series of QED is usually explained with a heuristic conjecture that we do not know physics at the Planck scale.

People have coined a term effective theory to describe a situation where a theory works well at some length or energy scales, but may diverge for very short distances.

General relativity

We know that Newton's gravity is very accurate for weak gravitational fields.

The linearization of the Einstein equations describes well binary pulsars, gravitational lensing, and gravitational waves.

If we try to solve the Einstein equations through some iterative approximation method, it may happen that the results are extremely accurate for a few first terms.

If the iteration anyway diverges, what might explain that? Can the divergence come from phenomena at the Planck scale?

A better explanation is that the Einstein  equations treat some quantity as absolutely rigid. The quantity cannot stretch and adapt, so that a solution could be found.

We have seen that Birkhoff's theorem means absolute rigidity with respect to the energy conservation of a spherically symmetric isolated system.

Reza Mansouri's result shows absolute rigidity with respect to an equation of state p = p(ϱ). General relativity simply refuses to comply with such an equation of state, even though the equation is reasonable and might describe a realistic physical system.

A rubber sheet model does not have such rigidity. It is intuitively clear that a rubber sheet model adapts to many types of lagrangians. There is no need for the lagrangian to conserve energy. The equation of state in Reza Mansouri's result would pose no problem for a rubber sheet.

Does the lumpiness of the universe have a large impact on the metric?

UPDATE July 30, 2019: Green and Wald are wrong. See our latest post.

---

https://arxiv.org/abs/1505.07800

Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?

T. Buchert, M. Carfora, G.F.R. Ellis, E.W. Kolb, M.A.H. MacCallum, J.J. Ostrowski, S. Räsänen, B.F. Roukema, L. Andersson, A.A. Coley, D.L. Wiltshire

(Submitted on 28 May 2015 (v1), last revised 15 Oct 2015 (this version, v2))

"No. In a number of papers Green and Wald argue that the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects. In particular, they argue that the effect of inhomogeneities on average properties of the Universe (backreaction) is irrelevant. We show that this latter claim is not valid. ..."


There seems to be an ongoing debate of the relevance of galaxy clusters on the large scale (> 100 megaparsecs) structure and development of the universe.

Syksy Räsänen et al. argue, that the impact on the metric might be relevant and even be the sole explanation for the accelerated expansion. No dark energy would be needed.

https://arxiv.org/abs/1506.06452

Stephen R. Green and Robert M. Wald have introduced a model which, they claim, shows that the effect of lumpiness is negligible.

One would think that by now there would be numerical simulations which decide the impact that lumpiness has on the universe.

This is an interesting dilemma. Which of the camps is right?

Does there exist a perturbed solution of an FLRW universe at all?

We know that the standard FLRW universe is an exact solution of the Einstein equations. It has a perfectly uniform mass-energy distribution.

If we start from the age where the cosmic microwave background was born, then the differences in mass-energy density were of the order 1 / 100,000.

Does there exist a perturbed FLRW solution which would have the characteristics of the early universe?

Since the Einstein equations are very strict, even a small perturbation might make the FLRW solution to diverge, so that there is no solution at all.

The question is not just what magnitude corrections does the lumpiness cause in the standard FLRW model - the question is if a solution exists at all.

The camp of Syksy Räsänen et al. has observed the fact that the corrections might blow up.

In this blog we have suspected that the Einstein equations are too strict, so that no solution exists at all for realistic mass distributions. A symptom of that would be that corrections to a known symmetric solution would blow up when we try to perturb it moderately.

Stephen R. Green and Robert M. Wald, if they are right, have to present a mathematical proof that the 1 / 100,000 perturbation does not make the corrections blow up - that is equivalent to proving that a solution of the Einstein equations exist. Mathematically, this is a very hard task. Has anyone made progress on this? Could the technique of Christodoulou and Klainerman work in an FLRW universe, too?

The stability of QED versus the stability of general relativity

https://www.physicsforums.com/insights/struggles-continuum-part-6/

Freeman Dyson observed in 1952 that the perturbation series of quantum electrodynamics probably diverges, even though the partial sums of the first terms give very accurate predictions for natural phenomena.

In general relativity, the so-called post-newtonian approximation gives accurate results for binary pulsars. We do not know if LIGO uses a similar technique.

If a numerical approximation series converges, then the limit might be an exact solution of the Einstein equations.

By studying numerical approximation algorithms we may get heuristic information about the existence of a solution for the Einstein equations. Divergence of an approximation series may be a symptom that no solution exists. However, QED shows that the series may appear to converge even though it is divergent.

We need to check what approximation methods Räsänen, Wald, etc. use and do the methods appear to converge.

It is possible that the FLRW model combined with some approximation method produces accurate results. Then we would have a practical model. To show that the model really is a result of general relativity, we need to show that general relativity has a solution and that the model approximates that solution.

We have suggested in this blog that general relativity should be replaced with a more flexible rubber sheet model of gravity. Then the existence of solutions might be very easy to prove. Furthermore, the rubber model might show its validity by predicting the properties of neutron stars better than general relativity.

What is a singularity like in general relativity?

We do not know if general relativity has a solution for a realistic collapsing star. If there exists a solution, it is not known if a singularity forms in a realistic collapse.

For a spherically symmetric collapse of perfect dust we have a solution in general relativity, and a singularity forms at the center if we extend the spacetime as far as we can.

https://en.wikipedia.org/wiki/Lemaître_coordinates

Let us define coordinates using particles freely falling from an infinite distance to the singularity. The "time" coordinate is the proper time of a falling particle. The "radial" coordinate is the time delay between successive falling particles. These are the Lemaitre coordinates introduced in 1932.

The proper time of each particle ends in a finite time when it arrives at the center.

"time"
  ^                              singularity
  |                             #
  |                    #
  |          #              freely
  |#                   ^   falling particle
  |                     | 
   ------------------------------------> "radius"

The hash symbols mark the line of the singularity, above which we cannot define the time coordinate.

The line of the singularity is spacelike. A particle which comes to the line cannot linger at the line but disappears completely.

A falling body stretches into a long line of particles before the particles one at a time disappear into the singularity. (Does the pressure grow at all among the particles or does the spatial volume stay constant or even grow for a freely falling dust ball?)

The light cone of particles within the Schwarzschild radius is such that when the proper time of the particle advances, the particle inevitably bumps into the line of the singularity.

If there is no material falling into the singularity, then a falling observer will see nothing on his way to the singularity. The space is "empty". Only the curvature of spacetime remains as a memory of the fallen matter.

The frozen star versus the singularity model

In the frozen star model, the proper time of a falling observer ends already when he reaches the event horizon. Or, since in that model the time of an outside observer is the canonical time coordinate, the falling observer will move ever slower, never quite reaching the horizon.

In the singularity model, the proper time of the observer ends somewhat later, at the line of the singularity. There is no obvious canonical time coordinate in this case.

Are naked singularities bad?

Since the light cones point to the line of the singularity, nothing at the line of the singularity can affect anything elsewhere in the diagram above. The singularity is very well behaved in that sense. It is a veiled singularity because it cannot affect anything around it.

If matter just disappears in a singularity, then even a "naked" singularity is ok, because its behavior is well defined. It is like things falling off the edge of a table. The singularity of the table edge is not veiled. One can go as near the edge as one wants and return back. If one falls off the edge, then one cannot disturb other things on the table any more. There is no indeterminism in this.

It turns out that singularities are not that bad as mathematical objects, provided that matter just disappears in them. If the singularity would be timelike, then matter could stay in the singularity. A naked singularity might have an unknown effect on its environment, and that would be a problem for physical theories.

Is unitarity broken in a singularity?

Unitarity means that we can reverse time and calculate the development of a physical system back in time, starting from a suitable hypersurface of spacetime.

A singularity devours information. Does it break unitarity?

If we require the hypersurface in the diagram above to avoid the line of the singularity, then we can calculate back in time, and unitarity holds.

One might want to use a hypersurface of a constant time in the diagram above. But the concept of a constant time is coordinate dependent. Why should we try to use a hypersurface which is not wholly in the well-defined spacetime, below the line of the singularity?

We conclude that a singularity does not break a reasonable definition of unitarity.

Should a theory of quantum gravity ban singularities?

Our analysis above did not reveal any reason why a theory of quantum gravity should do away with singularities.

A "popular science" image of a singularity is that matter is there squeezed into an infinite density, and we do not know how matter behaves under such conditions. Therefore, quantum gravity should somehow prevent the infinite density from happening.

Our analysis above tells a very different story: there is no matter at all in the singularity. There is no infinite density.

What about uncertainty relations? If a particle falls into a pointlike singularity, do we know its position and momentum simultaneously too precisely?

Does electromagnetic radiation gain or lose energy when the universe expands or contracts?

Let us consider a conical singularity where the space dimension is S_1, that is, a circle, and the the time dimension is along the height of the cone.

time

  ^

  |      /\

  |     /  \

  |    /    \

        -----> space

Suppose that at some time t_0 we have a standing electromagnetic wave in S_1, such that the maximum E and B are E_0 and B_0.

If we let the spatial dimension S_0 contract to a point, what happens to the standing wave? Does its energy decrease, stay the same, or increase?

The converse development happens in a de Sitter universe. What happens to a classical electromagnetic wave as the universe expands? A common claim is that "each photon" loses energy when its wavelength increases, and therefore the energy of the wave decreases as the universe expands.

The common claim does not take into account the fact that the number of photons may vary as the universe expands. The number of photons is constant in a Minkowski space, but that does not mean it stays constant in an expanding universe.

The energy of a classical wave depends on the strength of the electric field E and the magnetic field B in it. How does a classical wave respond to an expanding spatial dimension?

What about a massive particle? Its wave function stretches as the universe expands. Should we normalize the wave function, so that there is just one electron also in the future?

Since the electromagnetic radiation energy density in the early universe was large and now it is low, the energy of an electromagnetic wave cannot increase much as the universe expands. It might be that the number of photons stays the same.

An observer in an expanding universe is, in a sense, accelerating away from the source of an electromagnetic wave. The measured energy of a wave depends on the observer. If all inertial observers are accelerating away from the source, then we may say that the wave has lost energy.

In a contracting universe, inertial observers are accelerating toward the source, and will see the energy ever higher.

In the diagram above, if we have a standing electromagnetic wave of, say, 5 wavelengths in S_1, then as the time progresses, S_1 grows shorter. The wavelength probably grows shorter at the same rate.

There is no problem with the uncertainty relation since the energy of a photon grows as S_1 becomes shorter.

At the point of singularity, the energy of a photon in the wave has grown infinite. We may say that the solution is not defined at the point. If we cut the point off from the diagram, then there might be no problems with physics. Of course, if the Planck length is a demarcation line for new physics, then something unexpected might happen.

Conclusions

A singularity might not be as bad a beast as one may think, without first analyzing what really happens.

In general relativity,  the frozen star model stops the extension of the spacetime manifold at the forming horizon. It freezes the proper time within the star to the point when the horizon is forming.

The big flaw of the frozen star model is that how does the manifold inside the star know when to stops extending forward in time?

If we do not freeze, then at least for a spherically symmetric collapse, we have to stop extending the manifold at the singularity. Our analysis above reveals that that might be an acceptable solution.

The Cauchy problem for the universe: which initial values have a solution?

https://en.wikipedia.org/wiki/Cauchy_problem

The Cauchy problem is to solve a partial differential equation for the given initial values (Cauchy data). The values are given on a hypersurface of spacetime.

The Dirichlet problem is a similar problem where the values of a single unknown function are specified on the boundary of a volume in R^n.

Cyclic time

If the cyclic time Gödel universe can be defined for varying initial values, then a Cauchy problem is to find a solution where the universe magically returns to its original configuration after one cycle of time.

Intuitively, it is hard, or impossible, to find initial values which would have a solution. A problem is that if entropy grows at the start, how can we return the entropy back to the low value? If a black hole forms in the universe, how can we return it back to ordinary matter after a cycle?

Cyclic space

What is the difference of cyclic time to the "cyclic space" of a de Sitter universe? Is it easy to find initial values for some time point t_0 after the Big Bang, such that there exists a solution of the universe after t_0. Or a solution before t_0? Is it easy to find initial values which have a solution?

Building a solution for a chosen metric g

Let us start building a solution for general relativity this way:

1. first choose some metric g at some "moment of time", then 

2. calculate the stress-energy tensor, and then

3. place in the spacetime the required mass density, momenta, pressure, and shear stresses.

After that, we can use some foliation and a time step to develop the solution back in time and forward in time.

Do we get a physically realistic solution for general relativity that way? What could go wrong?

The first problem is finding realistic matter which can fulfill the conditions of the stress-energy tensor.

The second problem is if there exists a physically realistic history from the Big Bang to the configuration we found.

If all chosen metrics g require "exotic" matter or an unrealistic history, then general relativity does not have any realistic solution at all.

The standard Schwarzschild interior and exterior metric require exotic matter, that is, incompressible fluid. Furthermore, the metric is static, asymptotically Minkowski, and cannot arise from the Big Bang.

We already noted that Birkhoff's theorem bans lagrangians L_M which do not conserve energy.

The formation of singularities can be interpreted as having no solution?

It is not known if a singularity can form in a realistic collapse of matter in general relativity.

For an artificial problem of a spherically symmetric collapse of dust, a singularity is inevitable if we choose to extend the spacetime maximally, so that we calculate the development also behind the horizon. The familiar Penrose diagram shows the singularity.

We could interpret the result in the way that general relativity does not have a solution for the collapse of the dust. If we ban singularities in a solution, then there is no solution.

What is the ultimate reason why a singularity forms? The reason probably is the equivalence principle, which implies that a freely falling observer will see the dust fall with him at all times.

In newtonian gravity, the dust would collapse into a point. There would be a singularity in newtonian gravity, too.

Reza Mansouri (1979) calculated a more realistic model where a fluid sphere has the pressure a function of mass-energy density p = p(ϱ) only. Mansouri's conclusion is that general relativity has no solution for the collapse of such a fluid sphere.

Suppose that we introduce a new theory of static electricity. The theory predicts that a singularity forms when an electron meets a positron. We would suspect that the theory is wrong. In general relativity, 104 years have taught people to accept a singularity.

https://pdfs.semanticscholar.org/8dd0/dfafef0c53c428fdc3b58f8099aafcf7d089.pdf

Albert Einstein himself did not approve of a singularity but tried to argue in his 1939 Einstein cluster paper that a singularity cannot form.

The formation of a singularity (in an artificial, highly symmetric setup) may be one of the symptoms that general relativity does not have any solution for a dynamical system. For a static system we have the Schwarzschild solution.

A related question is if a singularity can form in a rubber sheet model of gravity. If it obeys the equivalence principle, then dust can collapse into a point, and put an infinite strain on the rubber.