Why we need a new metric in GR

Two metrics are mainly used in general relativity to describe most of the observable universe. 

At cosmological scales where universe appears to be isotropic and homogeneous everywhere, FLRW metric is used to calculate equations of motion. ΛCDM model based on this metric explains successfully many observations in cosmology like cosmic redshift, luminosity versus redshift relation of Type 1a supernovae, stretching of supernovae light curves due to time dilation, Cosmic Microwave Background (CMB) and many more (1–7). One basic observation in cosmology where ΛCDM model has not been really successful is the galactic mass and angular size relation evolution with redshift. It has been observed by multiple studies that angular sizes for a given mass or luminosity do not match the predicted values unless a high degree of size or luminosity evolution is assumed (8–16) (some of these studies measure surface brightness which is a function of luminosity and size). Strong arguments have been put against such unrealistic size evolution (14). There is also some tension in the value of Hubble parameter measured using two different methods and so far there is no satisfactory explanation for this (17). Apart from this, many other problems remain in ΛCDM model like flatness problem, horizon problem, CMB anomalies, unknown (dark) energy responsible for accelerated expansion and unknown(dark) matter responsible for the formation of large-scale structures. Since existence and exact nature of dark energy or dark matter has not been confirmed yet despite the best efforts, it remains as the biggest unsolved problem of currenht time (5; 18–26). 

At scales much smaller where the universe is not homogeneous but isotropic around a central mass, the Schwarzschild Metric is used to calculate motion inside gravitational systems like solar systems, galaxies, and galaxy clusters etc. Many phenomena like deflect of light by the sun and gravitational redshift of light; which were not possible to explain accurately using Newtonian gravity, are explained successfully using this metric within the framework of general relativity (1; 2; 7). Another problem which remains to be solved yet is the anomalous virial velocities of stars at the outer edges of galaxies as luminous matter seems insufficient to explain the observed velocities. Again it is postulated that some invisible (dark) matter is responsible for the additional acceleration needed to boost the velocities, though MOND is also able to explain this anomaly phenomenologically (19; 27–40). 

Apart from these puzzling observations there are conceptual problems in mathematical models built based on both the metrics. 

First problem is the boundary at which the metric representing the space-time continuum (field) of the universe changes from Schwarzschild metric to FLRW metric. When solving equations of motion around a central mass we use Schwarzschild metric which is static and centrally symmetric. On the other hand to represent all the inter-galactic space which is apparently expanding due to presence of non-vanishing average density of matter, we use FLRW metric which is not static. But how does field near a massive star merge into the field of expanding space throughout the universe? There have been attempts in the past to reconcile the two fields. This was first attempted by McVittie who used a spherically space-time supposedly describing a point particle immersed in FLRW background (41). However the generally accepted answer to this problem was given by Einstein and Straus where a model describing a massive point particle surrounded by a spherically symmetric vacuum region (vacuole) embedded in FLRW universe was presented (42). The boundary of Einstein-Straus vacuole is an expanding two sphere with it’s center being at rest with respect to the Hubble flow. It has been shown that the Einstein-Straus model demands a very strict spherical symmetry (43), that it is not a robust model and is rather an exceptional and isolated situation. Hence, its suitability for solving the interplay between cosmic expansion and local physics is doubtful (44). It is also argued that this model cannot provide realistic model for the small structures in our universe. For a single solar mass the vacuole radius is almost 400 light years which is two orders of magnitude larger than the average distance between the stars in our galaxy. Another theoretical problem with this model is it’s dynamical instability due to equal and opposite pulls (gravitational pull due to central mass and cosmological expansion of space) at the boundary (45; 46). 

The reason it’s so difficult to reconcile the two metrics is that despite primarily being description of empty space and time, both have completely different assumptions about vacuum. The starting point for defining Schwarzschild metric is the assumption that Ricci tensor is zero in vacuum, which means that stress energy tensor is also zero in vacuum. Now one would expect that vanishing stress-energy tensor means absence of matter or energy which should also mean flat space-time. But that is not what happens in Schwarzschild solution. It is assumed that the stress-energy tensor is zero everywhere except inside the central mass, which somehow gives rise to space-time curvature outside the mass even though the stress-energy tensor is zero there. FLRW metric on the other hand is constructed assuming that Ricci tensor and also stress-energy tensor is non-zero in the vacuum between galaxies. This (different assumptions about vacuum ) is the cause of friction between the two models which cannot be reconciled. We can never really describe a universe using both metric at the same time. We have to either assume that stress-energy tensor vanishes in vacuum or it does not. 

If we take the former assumption then FLRW metric cannot be used to describe the inter-galactic space which is mostly vacuum. And if FLRW metric does not describe the intergalactic space then we have no way to explain the cosmic redshift as photons supposedly gain their wavelength traveling through ever expanding vast emptiness of inter-galactic space. On the other hand if we take the latter assumption then Schwarzschild solution is not valid and we would need a new solution to describe the force of gravity.

It is clear we need a new metric which represents space-time continuum of entire universe and at all scales. The one in which the vacuum is treated consistently, be it inter-galactic space or space inside galaxies or even inside atoms and molecules.