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.

Reza Mansouri proved that there is no general relativity solution for collapsing compressible fluid?

UPDATE Nov 9, 2020: Reza Mansouri did NOT prove the non-existence of a solution. He only proved that no "uniform" metric, where spatial directions at each point are stretched uniformly, is a solution. More complicated solutions might exist.

---

http://www.numdam.org/article/AIHPA_1977__27_2_175_0.pdf

R. Mansouri: On the non-existence of time-dependent fluid spheres in general relativity obeying an equation of state.
Ann. Inst. Henri Poincare vol. XXVII no. 2 (1977) pp. 175 - 183.

"It is shown that there are no solutions of Einstein's field equations representing a collapsing (or bouncing) fluid sphere obeying an equation of state p = p(ϱ) except for the trivial case p = 0 [identically]."

This interesting paper deserves a blog post of its own. If the result is correct, then Einstein's field equations cannot handle a very simple dynamic process where compressible fluid contracts under its own weight, except in the case where the "fluid" is dust and has the pressure zero.

The classic Oppenheimer-Snyder paper from year 1939 assumes a collapse of dust. The authors of that paper state that the case with a non-zero pressure would be hard to calculate.

Newton's gravity certainly can handle the process of collapsing fluid. Since the low-energy limit of Einstein's field equations is Newton's gravity, the problems in finding a solution must happen with very dense objects like neutron stars.

We have been suspecting in this blog that Einstein's field equations are too restrictive, and no solutions might exist for physically realistic systems.

Fluid in the real world consists of particles or fields. Is the Mansouri result valid if we model the fluid with fields?

Mansouri at the end if the paper suggests that we should use an equation of state where

      p = p(ϱ, t, r)

depends on the time and the radial coordinate. Does such an equation make sense? The pressure, of course, also depends on how deep we are in the fluid sphere, that is, r. But why the density ϱ is not enough to tell us also that factor? The fluid compresses and its density grows with pressure.

https://www.researchgate.net/publication/259392422_There_is_no_Slow_Uniform_Contraction_of_a_Fluid_Sphere_obeying_an_Equation_of_State

In this 1980 paper Reza Mansouri wonders how general relativity "knows" about thermodynamical aspects which may require the pressure to be a function of both the density ϱ and the radius r.

Another way to interpret the result is that Einstein's field equations are too strict and do not have a solution for a realistic problem.

The Christodoulou and Klainerman nonlinear stability theorem

http://www.numdam.org/item/SEDP_1989-1990____A15_0/

D. Christodoulou and S. Klainerman proved that the Minkowski space metric is stable in general relativity with respect to a small perturbation of the initial data (initial values).

The authors state in the paper that:

"it is not even known whether there are, apart from the Minkowski spacetime, any smooth, geodesically complete solution which becomes flat at infinity on any given spacelike direction."

They refer to the theorems of Lichnerowicz and Birkhoff. According to the authors, the theorem of Lichnerowicz implies that a static solution which is geodesically complete and flat at the infinity on any spacelike hypersurface must be flat.

Geodesically complete means that all geodesics extend to all time.

Gödel's cyclic universe

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

In the Gödel metric, there are closed timelike curves.

Suppose that we try to form a realistic model of the universe with the Gödel metric. Then the random distribution of matter makes the metric very complicated.

Are there closed timelike curves also in this random metric? If yes, how can we make the "ends meet", so that everything happens in the exact same way when we have traveled one full cycle? At the first glance, the probability is zero that the ends meet for random initial data.

In de Sitter models of the universe, there are no closed timelike curves. But there are spacelike curves which "run around the universe". Can we make the ends meet for such curves?

Discussion

The existence of any solution for any realistic physical system seems to be an open problem in general relativity.

In an earlier blog post we remarked that if the matter field lagrangian L_M does not conserve energy in a spherically symmetric closed system, then Birkhoff's theorem implies that general relativity does not have any solution for such a system.

In our blog we have suggested that the Einstein-Hilbert action should be replaced with a rubber sheet model which would be more flexible, and the existence of a solution for virtually any reasonable initial data would be intuitively clear.

What do we know about the validity of general relativity?

Empirical data shows that the limit of general relativity, the newtonian gravity, is a very good approximation.

Gravitational lensing and the waves observed by LIGO prove that the linear limit of general relativity works well.

Binary pulsars lose energy in waves at the rate predicted by general relativity. This shows that the theory is on the right track for strong gravitational fields outside neutron stars.

General relativity works very well outside heavy bodies, that is, in the vacuum. What about inside heavy bodies, like a neutron star or a forming black hole?

The interaction of the lagrangian L_M and the metric is important inside heavy bodies. If general relativity handles the interaction in a wrong way, that might have measurable implications for star collapses into a neutron star or a black hole.

Are there solutions of general relativity for any realistic physical problem?

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

The dictatorial strictness of the Birkhoff equation bans any matter field lagrangian L_M which would create or lose energy. The solution outside the spherically symmetric system is a static Schwarzschild metric with a constant gravitating mass M, regardless of what spherically symmetric physical process happens at the origin.

The strictness hints that there actually might only exist solutions of general relativity in artificial, highly symmetric setups.

For example, can we glue together two Schwarzschild solutions and obtain a solution for the two body problem? We certainly can solve such an initial condition problem in newtonian mechanics. If we would have a rubber sheet model of general relativity, the existence of a solution is obvious, though the existence may be hard to prove.

Let us consider a geometric problem with a smooth 2D manifold (surface) in the euclidean 3D space.

Suppose that the Ricci scalar curvature in the manifold is zero everywhere but at two circular areas, where it is a constant R_0.

If we would have just one circular area with R_0, then the solution is a cone (?) or a tube where the circular area functions as the cap.

Can we somehow glue together two cones, so that the Ricci scalar is zero at the gluing point? It looks hard, except in a few symmetric cases.
         _______
        (_______)

Above is a solution: a pipe with two half-sphere caps. Can we add still a third circular area with R_0? That looks hard.

Einstein's field equations imply that the Ricci scalar curvature R is zero everywhere where the stress-energy tensor T is zero. That greatly restricts the geometry of the solution.

In a rubber model, there would probably not be such a strict restriction on R.

Two static masses held at a constant separation

Has anyone calculated the general relativity solution for two masses which are kept static and at a constant distance by electric repulsion? Or alternatively, are kept at the distance by a rigid rod?

Can we glue together two individual solutions, such that the Ricci scalar curvature is exactly what is required by the energy of the static electric field, or by the mass and the pressure in the rod?

The newtonian solution gives a hint. Suppose that we shoot an expanding ball of dust at a velocity v from the middle point between the masses. Does the dust ball volume keep growing at the same speed as it would in a Minkowski space, or is it "focused" or "defocused" as it grows? The newtonian approximation is linear. The behavior of the dust ball is obtained by summing the effect of the fields of the two masses. Since the Ricci scalar is zero for each field independently, it is zero for the linear sum.

If the distance of the two masses is of the order of the Schwarzschild radius r_s of each mass, then the problem is harder. Gravity is not linear in this case. Could it be that the dust ball gets focused or defocused in such an extreme gravitational field?

Is Birkhoff's theorem compatible with a wormhole?

Suppose that we are able to feed more energy to the spherically symmetric system through a symmetric wormhole which opens there at the origin. Then we could increase the gravitating mass M of the system. Birkhoff's theorem would be broken.

Are there solutions for the whole universe?

Suppose that the universe is approximately a finite balloon and is expanding or contracting. Birkhoff's theorem is a symptom that general relativity is not flexible in finding solutions. Are there any solutions for the case where the mass distribution is not uniform?

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255525/

A survey paper by Alan D. Rendall lists a large amount of research into the existence of solutions of general relativity. At a first glance, it seems to be an open problem if a global solution for the universe exists in any realistic case.

Are there realistic solutions for an asymptotically Minkowski space?

There seems to be a lack of non-existence theorems, too.

The appearance of singularities in a collapse might be interpreted as a non-existence theorem, if we ban solutions with singularities. Apparently, it is an open problem if singularities form in a realistic collapse.

The role of pressure in the Einstein gravity versus a rubber sheet model

The perpetuum mobile of yesterday's blog post is spoiled by the large gravity of the pressure in the fluid. See the note which we added to the top of the post.

In a rubber sheet model, incompressible fluid in the vessel is able to keep the spatial metric stretched. Then the mass M, which originally produced the spatial stretch and the depression in the rubber, can be moved away.

In Einstein's gravity, pressure cannot keep the spatial metric stretched. Moving of the mass M is prevented by the infinite gravity which the infinite pressure would cause in the fluid.

In a rubber model, we might simulate the Einstein gravity by attaching strings to the rubber sheet and the mass M so that moving M would necessarily straighten the rubber sheet. If the sheet cannot straighten => M cannot move.

Even though we have so far failed to construct a perpetuum mobile in the Einstein gravity, we still face the strange strictness of Birkhoff's theorem: it requires the matter field lagrangian L_M to conserve energy. A rubber sheet model would allow L_M to create or lose energy - the rubber sheet is not a dictator to other fields.

A perpetuum mobile which is based on the gravity of photons?

UPDATE July 11, 2019: This perpetuum mobile does not work for the following reason. Instead of the pipe configuration, imagine that the vessel and the fluid are transparent. We send a massive spherical photon shell M up from the center of the vessel.

Then only the topmost photons do not observe any pressure in the fluid. The photons at the bottom of the shell do observe the fluid already contracted. They see approximately the final, spatially flat metric of the vessel. In particular, the bottom photons see the full gravity of the pressure in the final metric of the vessel.

The photons, on the average, see half of the final gravity caused by the pressure. If we would move the massive shell very slowly upward, the result would be roughly the same.

The photons cannot escape the change of the metric which the photons themselves create.

The construction of a perpetuum mobile in the Einstein equations turns out to be hard. We need to check what types of lagrangians does the conservation of the ADM mass allow. It looks like ADM handles a pressurized vessel, after all, and the coupling between gravity and the matter fields is minimal for a pressurized vessel. What kinds of matter field lagrangians would be nonminimal?

We also need to study the differences between the Einstein model and rubber models. If we fill the pressurized vessel with incompressible fluid, then we cannot move the mass M at the center at all. Any movement would raise the pressure infinite and the infinite gravity of the pressure prevents M from moving from the center. The infinite gravity would also stop the locally measured time inside the vessel.

In a rubber model, we are able to move the center mass M. It just has to climb uphill from the depression in the rubber sheet and over the incompressible fluid. The pressure in a rubber model is able to stretch the spatial metric.

---

Our earlier pressurized vessel thought experiment was based on very slow movements. The perpetuum mobile did not work because the added pressure created a strong gravitational field which we had to fight when we raised the big mass M.

But can we fool nature? Let us put a very strong pipe through the pressurized vessel.

Let us for now make the unnatural assumption that the fluid in the vessel is very lightweight.

We use a bunch of photons in a very strong box as the mass M which stretches the spatial metric. The photons are inside the pipe in the middle of the vessel.

The volume of the vessel has grown because of the mass M of the photons. We filled the vessel to the brim with the fluid which is almost incompressible.

Let us then open the sides of the photon box. The photons escape at the speed of light.

The pressure in the vessel starts to build up and grows very high. But the photons do not know anything of the changed gravitational field of the vessel because they are escaping at the speed of light. For the photons, the vessel might as well be pressureless.

The photons escape. They lose the exact same energy when climbing up from the gravitational well which they gained when we packed them into the box.

We are left with a vessel with a huge pressure in it. The pressure can do a lot of work which came out of nothing. Conservation of energy is broken.

Analysis of the requirement of "lightweight" fluid

Let us analyze the impact if the fluid has a considerable mass. What if the mass of the extra fluid spoils the perpetuum mobile?

The start configuration of the experiment is that we have a diffuse gas of photons at the infinity and some extra fluid at the infinity.

At the origin we have the vessel full of fluid and the box for the light.

We use a pulley P to lower the extra fluid down from the infinity and gain some energy E.

The newtonian approximation tells us that the photons lose the same energy E when they climb back to infinity, compared to the case where we would not have lowered the extra fluid.

The non-zero mass of the fluid does not change the fact that we have a perpetuum mobile.

The exterior Schwarzschild metric is strange in a Minkowski space - the Big Bang comes to the rescue

The Schwarzschild metric says that the radial coordinate r has for local observers stretched by a factor

       1 / sqrt(1 - r_s / r)

relative to the global Schwarzschild coordinates. There r_s is the Schwarzschild radius of the mass M which is at the origin of polar coordinates.

If r is large, then the above is approximately

       1 + 1/2 * r_s / r.

If we integrate the extra length

        1/2 * r_s / r

from, say, 100 r_s to the infinity, the integral diverges logarithmically.

That is, there is an infinite extra distance which we must stretch from the Minkowski space, if we embed just 1 kilogram of mass into it. It is strange if just 1 kg of mass would have such infinite consequences for the metric.

Making a rubber sheet model for the metric is hard, or impossible, if a small mass causes an infinite change in the spatial metric.

Fortunately, we do not live in a Minkowski space but in an expanding FLRW universe where the mass density is roughly constant at the scale of 1 billion light years.

We may consider the 1 kg mass as a local dense spot in the mass distribution of the universe. Let us assume that the whole universe is full of thin gas or dust, and the 1 kg mass was born from a collapse of a spherical cloud of some 10^26 cubic meters of that matter. Then we probably (?) can ignore the Schwarzschild geometry farther away than 3 * 10^8 meters from that 1 kg of mass.

In our rubber model in an earlier blog post, we assumed that the rubber sheet is firmly attached to a circular frame. We can take the radius of the frame as 3 * 10^8 meters in this particular case.

For the Sun, the radius of the frame would be 300 light years. That is still tiny, compared to the size of the visible universe.

We need to replace the rubber sheet model with a rubber balloon model. A dense mass deforms the balloon. The deformation energy is roughly the same as the gravitational binding energy of the dense mass.

The visible universe seems to have a flat spatial metric. We must take the balloon to be very large, so that its spatial curvature is not obvious? We would only see a small part of the huge expanding balloon.

A better Einstein-Hilbert action formula?

Does spatial Ricci curvature work as the energy density?

In the text below we are looking for solutions for a static setup. We do not try to handle a dynamic configuration yet.

We have to find an action formula where positive pressure is able to cause positive Ricci curvature in the spatial metric.

The integral of the local spacetime volume over

       R_s + L_M

is one candidate. There R_s denotes the Ricci curvature of the spatial metric.

There is an obvious drawback in R_s. If we have a round mass on the sheet, then R_s is negative around the mass, because the rubber somewhat resembles a saddle surface there. Having a negative energy density somewhere does not sound good. Also, R_s might not be a realistic measure of the deformation energy.

How to determine the temporal metric?

How to restrict the temporal metric? One can make the action integral as small as one wishes by slowing down the local flow of time. Einstein was able to restrict the slowdown of time by including the temporal metric in the calculation of the Ricci curvature.

If the lagrangian would be of a rubber sheet, then the vertical position z plays the role of the slowdown of the local time: the smaller the z, the slower the local flow of time.

In the rubber model, z is not an independent variable but is determined by the spatial metric (= the stretching of the rubber sheet). In the variation of the action integral, z is not among the parameters we can vary. Only the spatial metric parameters (usually g_11, g_22, g_33) take part in the variation.

A (non-equivalent) alternative to z is to use the thickness of the rubber sheet to measure the pace of time. The volume of rubber stays constant unless a very large pressure is applied to it. The pace of time would be determined by the spatial stretching of the rubber. However, if we have a long straight steel bar resting on a rubber sheet, then the tension of the rubber is almost constant close to the steel bar. The depression in the rubber is deeper close to the bar. Obviously, the thickness of the rubber is not a good measure for the pace of time.

The natural coordinate system C and the elastic energy density E(g)

If we paint a coordinate grid in the unstretched rubber, we get a global spatial coordinate system C. We can define the spatial metric with respect to those global coordinates. Then the metric tells how much the rubber sheet is stretched at that point and we can calculate the deformation energy. The deformation energy of a single weight on the rubber sheet is spread throughout space, while the term R in the Einstein-Hilbert action is nonzero only inside the mass.

Let us denote the elastic energy density of the rubber sheet by

       E(g),

where g is the spatial metric in the global coordinate system C. Then the action is the integral on

       E(g) + L_M

over the spacetime volume.

The temporal metric is determined by the spatial metric. If the action is completely analogous to a real newtonian mechanics rubber sheet model, then we know that energy is conserved.

Let

       sqrt(g_11) = 1 + s.

Then the elastic energy density due to the stretch s is something like

       k s^2 / (1 + s).

Modify the Einstein-Hilbert action so that also positive pressure contributes besides L_M?

The lagrangian would then be something like

        R + L_M + L'_M,

where L'_M means the "derivative" of L_M when the spatial metric is varied. Positive pressure would act like a positive mass density.

A problem with ad hoc lagrangians like this is that they probably do not conserve energy and allow a perpetuum mobile.

Let us check if someone has investigated a gravitational "source" of the type above.

Earlier alternative theories to general relativity

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

At a first glance, none of the theories listed in Wikipedia explicitly specifies positive pressure of a source of gravity. Pressure acting as a source could be a consequence of their definitions, though.

None of the theories listed in Wikipedia explicitly claims to be a rubber sheet model. How do they show that energy is conserved, then?

https://arxiv.org/abs/1603.07655

The model of Tenev and Horstemeyer is equivalent to general relativity. They have a very stiff and thin rubber plane whose elastic energy is in the bending of the plane.

Birkhoff's theorem seems to be incompatible with many lagrangians

Birkhoff's theorem states that the metric outside a spherically symmetric system has to be the static Schwarzschild metric.

Birkhoff's theorem should more precisely be formulated like this: if the Einstein field equations with a lagrangian L_M have a solution for a given spherically symmetric system, then the metric is static outside the system.

Birkhoff's theorem can be derived from the Einstein field equations in one page. One assumes a spherically symmetric metric which depends on the radius r and the time t. One can then show that the metric has to be independent of the time t in the vacuum zone surrounding the system.

Let us think of this. If we would have a lagrangian L_M which does not conserve energy, but creates more mass-energy into the system with increasing time, then the total mass of the system would increase with time and the metric outside it should change with time.

This means that if we assume general relativity & a lagrangian which does not conserve energy, then there is no solution for the Einstein equations.

General relativity seems to "know" that all lagrangians L_M must conserve energy. This is suspicious. Other force fields in physics do not "know" such things.

If the perpetuum mobile, which we sketched in our previous blog post, works, then general relativity itself does not conserve energy together with any lagrangian which allows the building of pressurized vessels. This would imply that general relativity is inconsistent with all realistic lagrangians L_M.

We do not know if classical electromagnetism conserves energy, because the self-force of a charge on itself is not well understood. This implies that it is not known if any realistic physical lagrangian is consistent with general relativity.

The proofs of the conservation of the ADM mass and the Komar mass probably assume that building a pressurized vessel is not possible. The proofs are probably correct if we assume just pointlike particles with Coulomb static electric and gravitational forces between them. Then the lagrangian probably does not allow the building of a vessel.

What is the problem with general relativity? If we think of a rubber sheet model, there we can affect the metric in the surrounding vacuum through operations which we do inside the system in the center. We can use pressure to increase the tension of the rubber in the surroundings of the system. No perpetuum mobiles are possible in a rubber sheet model. A consistent theory of gravitation probably must not imply Birkhoff's theorem.

What would a consistent theory of gravity look like?

Suppose that we would have a rubber sheet model where the weight of a mass can make a pit into the rubber and stretch it, but even an infinite pressure would be unable to maintain that pit if we remove the mass. Then the pressure can do as much of work as we wish, and we trivially have a perpetuum mobile in the model. Clearly, a consistent theory of gravity must allow pressure to stretch the spatial metric of spacetime.

The Einstein-Hilbert action calculates the local spacetime deformation energy from the Ricci scalar curvature at the spacetime point. How does that differ from a lagrangian for a rubber sheet model? In rubber, there is tension which is determined by the environment. Is it necessary that a consistent theory of gravitation has to include such a tension field? The concept of tension in spacetime sounds like an aether theory, which might be bad.

In the pressurized vessel thought experiment, it is the tension of the rubber sheet which resists the increase of the spatial volume of the vessel. It is hard or impossible to calculate the energy without a tension field.

Another question is if the tension has to extend past the wall of the vessel, to the environment outside. Probably yes. A steel ring (= the wall of the vessel) is not attached to the rubber sheet but rather it slides on slippery rubber without friction.

If we just have a weight resting on the rubber, the tension which it causes extends to the whole rubber sheet.

The paradox of incompressible spatial metric - does it make a perpetuum mobile possible?

UPDATE July 10, 2019: The pressurized vessel perpetuum mobile does not work - energy is conserved. The extra pressure in the vessel, when the mass is removed from the vessel, causes a gravitational pull which offsets the gain from the extra elastic energy E in the fluid. The gravitation from the mass-energy E is small, but the gravitation from the positive pressure in the vessel is large.

---

Our previous blog post observed that the interior Schwarzschild metric has the spatial part of the metric independent of the pressure.

The temporal part of the metric does change with pressure, but that does not help if we are trying to squeeze more incompressible fluid into a spherical vessel.

The thought experiments of our previous posts have tacitly assumed that using enough pressure we can change the spatial metric of spacetime. If that is not possible, many strange things may occur, including a perpetuum mobile.

Suppose that we have a dense body of mass M attached to a very strong frame which surrounds a considerable volume of space around M. The mass M has deformed the spatial metric like in the Schwarzschild exterior solution. Let us build a perfectly rigid grid G which models the local spatial metric some distance s away from M. The grid is not Minkowskian (cartesian) but slightly distorted. We attach G and M to the strong frame.

Now, use a strong force to move M suddenly. The metric of spacetime can only update at the speed of light. At first, the metric stays constant at G. There should be no problem in moving M. But when the spatial metric tries to update at G, it cannot do so because pressure cannot change the metric. The force on the grid G grows infinite. We can use this infinite force to generate mechanical energy as much as we wish.

A pressurized spherical fluid vessel with a uniform extra mass m of weights in the fluid

Suppose that we have a vessel whose wall is very rigid, and the fluid inside is almost incompressible. We add a uniform density of weights, with a total mass m to stretch the spatial metric inside the vessel and fit more fluid into it.

Let the locally measured mass-energy of the fluid be L at the start of the experiment. L contains rest mass as well as some elastic energy from the pressure that gravity exerts on the fluid.

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

Let r_g be the global Schwarzschild coordinate radius of the sphere. The Schwarzschild radius is r_s.

According to Wikipedia, the Gaussian curvature radius R of a two-dimensional spatial slice fulfills

       R^-2 = r_s / r_g^3.

The larger the mass, the shorter the curvature radius and the more the volume within a radius r_g in the global Schwarzschild metric. The pressure does not affect the curvature at all.

We then slowly remove the extra mass of weights m, keeping the locally measured mass-energy density within the vessel always uniform throughout the volume.

The volume of the vessel decreases.

The rigid wall starts exerting a pressure on the fluid inside. The fluid contracts to fit in the vessel, and the fluid contains at the end of the process a considerable extra amount E of elastic energy.

The metric inside the vessel is at all times close to the Schwarzschild interior metric. The mass m is slowly replaced by E, but E can be much less than m.

The extra pressure P inside the vessel causes extra gravity, according to the Komar mass formula.

We gained an amount V of potential energy when we lowered the weights m into the liquid. How much extra work W we now need to spend to lift the weights out? We need to win the extra gravity exerted on m by E and by the extra pressure P. 

We assume that r_g is much larger than r_s. The work W for m to win the gravity of E and P is much less than E.

Note July 10, 2019: The extra pressure can make a much larger gravitational field than the elastic energy E itself. This is what spoils the perpetuum mobile in this case.

Is it a surprise that general relativity might allow a perpetuum mobile?

ADM and several others have proved that the mass-energy of a closed system cannot change if it lives in an asymptotic Minkowski space.

https://arxiv.org/pdf/1010.5557

Hans C. Ohanian remarked in his paper that ADM and others do not handle "non-minimal" couplings between gravity and other force fields.

If there is a complex relationship between the value of the lagrangian L_M and the spacetime metric in the Einstein-Hilbert action, then the coupling is non-minimal, and energy may not be conserved.

Birkhoff's theorem proves that the gravitational mass of a closed spherical system cannot change. We have to check what assumptions Birkhoff uses about the couplings.

Rubber sheet models of general relativity conserve energy. We need to study what is wrong with the Einstein-Hilbert action. It is very unlikely that nature would break conservation of energy in an asymptotic Minkowski space.

In rubber sheet models, pressure does affect the spatial metric inside a fluid sphere, at least in some configurations.