A static and stable metric solution from first principles
C. Brown
The applicability of Birkhoff's theorem to large scales remains an open question, particularly in the context of non-vacuum solutions. We explore this lacuna by proposing a unique static solution to the Einstein field equations. The solution generalizes the Schwarzschild metric to cosmological scales. Our approach reexamines the cosmological principle through the lens of scale-dependent symmetry, yielding a metric tensor with a radially-dependent g₀₀ component that obviates the need for the FLRW formalism while preserving large-scale homogeneity. The resulting spacetime manifold exhibits a constant affine connection, leading to geodesics with a uniform acceleration, which accommodates the observed hierarchical structure of the universe across multiple scales. The model predicts a cosmological event horizon coincident with the Schwarzschild radius, offering a novel interpretation of the Hubble-Lemaître law in terms of cumulative gravitational redshift along null geodesics. This framework provides a geometrically consistent explanation for the observed redshift-distance relationship without invoking universal expansion. Furthermore, it suggests a reinterpretation of the cosmic microwave background as a manifestation of Hawking radiation associated with the global geometric structure. This framework, while preliminary, presents a promising avenue for further investigation into the nature of cosmological scale transitions and the unification of local and global spacetime structures within the context of general relativity.
Translation: Modern cosmology uses an invalid solution to the Einstein Field Equations and a series of rationalizations to defend it. Within the framework of relativity there is a rule called Birkhoff’s theorem that operates like an if-then statement: if a spacetime is spherically symmetric and empty of matter (a vacuum), then the spacetime must be static and is described by the Schwarzschild solution. And guess what? The universe is spherically symmetric. Doing things correctly enables us to interpret the cosmological redshift as gravitational redshift. The resulting distribution of mass is more accurate than the simple homogeneous distribution assumed by modern cosmology. It also adheres to the principles of relativity, is capable of describing a wide range of phenomonea without invoking magic, and could potentially eliminate the issues that arise with GR on the quantum scale.
Introduction
The cosmological principle can be stated in several ways, but the spirit of it is that the largest scales of the universe should appear the same to all observers, no matter where they are located. Other versions of the principle are often formulated as something akin to "the universe is homogeneous and isotropic on the largest scales." [1] Many sources claim that Albert Einstein introduced the cosmological principle in his 1917 paper "Cosmological Considerations in the General Theory of Relativity" (Einstein, 1917). In the paper, Einstein discusses the difficulties of formulating the universe with general relativity. His intuition was that the universe was infinite and static but the interplay between density, mass, and volume prevented him from identifying a stable solution.
"It is well known that Newton's limiting condition of the constant limit for Q at spatial infinity leads to the view that the density of matter becomes zero at infinity."
Einstein overcame this challenge by assuming the density was "uniformly distributed" throughout the universe.
"If we are concerned with the structure only on a large scale, we may represent matter to ourselves as being uniformly distributed over enormous spaces, so that its density of distribution is a variable function which varies extremely slowly."
"...if we assume the universe to be spatially finite, we are prompted to the hypothesis that p is to be independent of locality."[2]
A few years later, a similar metric solution (now called the FLRW metric) was independently derived by Alexander Friedmann, Georges Lemaître, Howard Robertson, and Arthur Walker. Their approach includes a scale factor, which assumes the universe is either expanding, collapsing, or balanced. The FLRW metric gained prominence after Hubble discovered the cosmological redshift, providing empirical support for the expanding universe.
Today, the Lambda Cold Dark Matter (ΛCDM) model is the prevailing cosmological paradigm. ΛCDM is a collection of fitting parameters that have evolved from various theories and assumptions about the universe's composition, structure, history, and evolution. These include the FLRW metric, dark energy, dark matter, inflation, baryon acoustic oscillations, primordial nucleosynthesis, and more.
In 1923 George David Birkhoff published a theorem that states any spherically symmetric solution of the vacuum Einstein field equations is necessarily static and locally isometric to an open subset of the Schwarzschild metric.[3] Although Birkhoff's theorem applies to vacuum solutions, and is not considered for large scale object, it is important to mention here as it has been shown to be directly applicable to realistic situations with symmetrical objects such as stars and clusters. [4]
Is Birkhoff's theorem applicable to cosmological scales? This is one of the core questions this article hopes to elevate. If symmetry increases with scale, then the answer should be yes. The intuitive consideration is that the applicability of Birkhoff's theorem is directly proportional to the scale in question, with the largest scales having the most amount of symmetry. This puts the FLRW at odds with Birkhoff's theorem. If Birkhoff's theorem is applicable to the observable universe, then we can only intuit that we should use the Schwarzschild metric to describe the observable universe, and not the FLRW. Unfortunately, it is not that simple, as non-vacuum solutions like the FLRW have additional considerations beyond gravity. [5] However, it certainly seems like a valid reason to elevate Einstein's original intuitions of a static and stable universe, especially if it can be formulated without mysterious forces and phenomena.
II. Methodology
Hubble's law ($H_0$) is often described as a relationship between a galaxy's recession velocity and its distance from the observer. Nevertheless, it is more accurate to describe $H_0$ as a relationship between the frequency of starlight and its relative position to the observer. This distinction is essential because the redshift is directly observable, while recession velocity is inferred. Mathematically, the redshift can be expressed as $z \approx (H_0/c) \times d$, where $z$ is the redshift, $c$ is the speed of light, and $d$ is the distance.
Can Hubble's law be interpreted as gravitational redshift instead of recession velocity? Consider the following:
\[g = \frac{GM}{r^2} \to \frac{4\pi r^3 G\rho}{3r^2} \to \frac{4\pi rG\rho}{3}\] \[H_0 = \frac{GM}{r^2c} \to \frac{4\pi r^3 G\rho}{3r^2c} \to \frac{4\pi rG\rho}{3c}\]Rearranging and solving for $\rho$
\[\rho(r) = \frac{3H_0}{8\pi G} \to \frac{3g}{4\pi Gr}\]A. Formulating the metric
We begin with a peculiar case of the Schwarzschild metric having the mass density distribution described above, with $g$ remaining constant irrespective of $r$. The standard exterior Schwarzschild metric:
\[ds^2 = -\left(1-\frac{2GM}{rc^2}\right)c^2dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 + r^2d\Omega^2\]We can simply express the interior metric as the following:
\[ds^2 = -\left(1-\frac{gr}{c^2}\right)c^2dt^2 + \left(1-\frac{gr}{c^2}\right)^{-1}dr^2 + r^2d\Omega^2\]With this particular choice of distribution, $\rho$ varies with the radius in such a way that the overall gravitational acceleration remains constant, yet $g$ remains constant irrespective of $r$, which is an unusual characteristic in gravitational models. The standard Schwarzschild metric approaches flatness as $r$ reaches infinity, while this modified version exhibits the opposite behavior: the curvature approaches infinity as $r$ nears the Schwarzschild radius, depending on the perspective.
B. Continuity of derivatives
The above metric is a special case of the Schwarzschild metric, which is essentially the same as the standard expression. Gravitational acceleration $g$ is used to show the unique form for cosmological scales but it is important to note that the solutions could be formulated in a number of different ways. The boundary between the exterior and interior can also be drawn anywhere, smoothly matching between one metric and the other.
Matching $g_{tt}$ at $r=R$ Interior metric:
\[1-\frac{gr}{c^2} = 1-\frac{2GM}{rc^2}\]Simplified
\[g = \frac{2GM}{r^2}\]Matching $g_{tt}$ at $r=R$ Exterior metric:
\[1-\frac{2GM}{rc^2} = 1-\frac{gr}{c^2}\]Matching $g_{rr}$ at $r=R$ Interior metric:
\[\left(1-\frac{gr}{c^2}\right)^{-1} = \left(1-\frac{2GM}{rc^2}\right)^{-1}\]Matching $g_{rr}$ at $r=R$ Exterior metric:
\[\left(1-\frac{2GM}{rc^2}\right)^{-1} = \left(1-\frac{gr}{c^2}\right)^{-1}\]For $r < \frac{c^2}{g}$, all components of the metric are continuous. There are no singularities, and the spacetime described by this metric is smooth and well-behaved.
For $r = \frac{c^2}{g}$, the time-time component $g_{tt}$ becomes zero. The radial-radial component $g_{rr}$ becomes infinite, indicating a coordinate singularity (not a true physical singularity).
For $r > \frac{c^2}{g}$, the metric form provided is no longer applicable and the standard exterior solution would be used.
C. Physical meaning
Traditionally, the Schwarzschild metric has been used to describe the spacetime surrounding a single, spherically symmetric, non-rotating mass. However, stars themselves are not point masses. The sun is said to made up of some $10^{57}$ atoms. If the Schwarzschild metric is broadly applicable to stars, then it should also be broadly applicable to cosmological scale objects.
As the scale of anything increases, objects and systems exhibit increased symmetry due to the averaging out of smaller, localized asymmetries. At larger scales, irregularities and deviations present at smaller levels often become less significant, leading to a more uniform and symmetric appearance or structure. This principle can be observed in various contexts, from physical to biological systems, where larger structures tend to display more regular patterns and symmetric features, while the details of smaller-scale variations are overshadowed by the overall coherence of the larger-scale system.
All objects in the universe exist within a gravitational hierarchy. Similar to Russian Matryoshka dolls, stars inhabit galaxies, which themselves gather into clusters. These clusters unite to form superclusters, which are part of even larger structures like filaments, arcs, walls, and rings, each nested within increasingly larger objects. As the scale increases, so does the symmetry of subsequent structures.
The metric outlined above fits this pattern with a great deal of accuracy. According to the proposed distribution, the mass density of galaxies and small clusters would be expected to fall in the range of $10^{-24} - 10^{-21}$ kg/m³ and larger structures in the $10^{-24} - 10^{-27}$ kg/m³ range. The overall structure aligns with the cosmological principle, even if it does not align with the traditional view of cosmological homogeneity. This is because symmetry is preserved by considering each scale independently from other scales.
III. Results
The final relationship of H0, when interpreted as gravitational redshift.
\[H_0 = \frac{2gr}{c}\]
The formulation above reflects a gravitational redshift that is directly proportional to the distance of light travel, the distribution of mass, and the subsequent curvature of the metric.
When H0=72km/s Mpc the resulting relationship between r and ρ is found.
As the chosen radius increases, the density at r decreases at a rate that is inversely proportional to the radius. The Schwarzschild radius r s occurs at 4153 Mpc (13.54 billion light years). In this case, rs is also the observable limit (due to gravitational redshift).
The resulting density distribution more accurately reflects the distribution in our universe than simple homogeneous models like the FLRW. The physical structure of our universe is hierarchical. Stars, black holes, planets, and smaller objects have the highest density of all. Galaxies, clusters, and super- clusters have a much lower density, typically less than 10-21kg/m3. The critical density of the observable universe is quoted to be 9.9-27kg/m3, which would align with the density of the observable universe using this metric. [9]
Another point of interest that comes from this approach is the return of the Newtonian term for gravitational acceleration g. As stated earlier, g remains constant as r increases. A consequence of this is that all objects/regions in the observable universe should have a minimum g (when considered in relation to other cosmological objects). The value of g still depends on the distribution of mass within a region, especially on smaller scales with high asymmetry, but we should still expect a smoothing effect from the symmetry that comes with larger scales. This uniform g presents an opportunity for modifying the equations used in dynamic systems, such as the predictions of galactic rotational curves. While this relationship falls outside the scope of this paper, it is worth noting that the MOND constant is calculated to be 1.2×10-10m/s2. [10]
Finally, with the above formulation, the observable radius occurs at the Schwarzschild radius, which opens the door to a new use for Hawking radiation [11], and potentially, an alternative explanation for the Cosmic Microwave Background Radiation CMBR.
Hawking radiation is a phenomenon that results from virtual particle pairs forming at the event horizon of black holes. One particle falls into the black hole while the other escapes as radiation, leading to a decrease in the mass of the black hole over time in the form of radiation. The temperature equation for Hawking radiation can be expressed with mass, radius, or gravitational acceleration.
The temperature is inversely proportional to the radius. However, there are two perspectives for this radiation. Historically, the equation has been used to describe the temperature of black hole. However, from the black hole's perspective the radiation is blueshifted.
For a coordinate singularity with a radius equal to 1.28e+26 meters, the temperature is calculated to be 1.42e-30 K. For the CMBR to be interpreted as Hawking radiation emanating from the observable limit, a factor of change to the order of 1.92e+30 is needed to get from1.42e-30 K to 2.72 K. Due to the asymptotic nature of spacetime at the event horizon, this factor of change can be obtained by adjusting the observable radius in relation to the point of emission. If accurate, 1.92e+30 would represent the maximum factor of change due to all of the energy in the observable universe, potentially placing an upper bound to relativistic effects.
Discussion
There are two obvious sources for redshifted light, the Doppler effect and gravity. If the former is the source of the cosmological redshift, an expanding metric must be used to account for the linear relationship between recession velocity and distance. Expansion is a package that needs inflation, dark energy, and a host of other complex theoretical phenomenea to formulate a coherent theory.
Gravity is not considered because the prevailing view is that there is not enough mass to account for Hubble's law with gravitational redshift alone, and/or gravitational redshift would not align in the linear relationship. Both of these views are objectively false.
A much better approach is to formulate the density distribution directly from measurements of H0. Doing so results in a MORE realistic density distribution than ΛCDM without invoking expansion. The observable radius and density at the observable radius remain unchanged. The Schwarzschild solutions can be generalized to formulate a cosmological solution to the EFE using the density distribution. It turns out, that we can use the old Newtonian gravitational acceleration for the distribution of mass. The Schwarzschild radius and observable radius are equivalent. The singularity is merely a coordinately singularity. However, we can expect blue shifted Hawking radiation due to the extreme metric curvature near the event horizon.
Not only does this model provide us an avenue for eliminating the need for dark matter, dark energy, inflation, and expansion, it opens the door to resolving singularities that arise due to general relativity on the quantum scale. This cosmological model places an upper bound to gravitiational curvature and a lower bound to energy exchange.
I understand it all sounds crazy, but just do it yourself. Here's a step by step that's easy to replicate.
- Assume H0 is caused by gravitational redshift.
- Settle on a value for H0, formulate an equation with mass density instead of mass.
- Find the required distribution of mass needed to account for H0.
- The resulting distribution is a uniform g, with a value that depends your value of H0. (72km/s = 3.5e-10m/s2)
- The amount of gravitational redshift is now proportional to the distance of light travel.
- Find the observable radius.
- Find the Schwarzschild radius (for all values of g, the rs is the observable radius).
- Find the temperature of emitted Hawking radiation. (lowest possible energy exchange?)
- Find the amount of blue shift needed to produce the CMBR. (upper bound to gravitational curvature?)
- Compare g to the MOND constant and DE (of the same order of magnitude for both!).
- Think long and hard about Birkhoff's theorem.
The model is simple, coherent, and accurate. The arguments are compelling. And all you need is General Relativity.
References and Additional Information
- Wald, R. M. (1984). General Relativity. University of Chicago Press, p. 91-92.
- Einstein, A. (n.d.). Cosmological Considerations in the General Theory of Relativity. In: The Collected Papers of Albert Einstein. Princeton University Press. https://einsteinpapers.press.princeton.edu/vol6-trans/433
- Birkhoff, G.D. (1923). Relativity and Modern Physics. Cambridge, MA: Harvard University Press.
- Goswami, R., & Ellis, G.F.R. (2011). Almost Birkhoff theorem in general relativity. Gen Relativ Gravit, 43, 2157–2170. https://doi.org/10.1007/s10714-011-1172-z
- Schleich, K., & Witt, D. M. (2010). A simple proof of Birkhoff's theorem for cosmological constant. Journal of Mathematical Physics, 51(11), 112502. https://doi.org/10.1063/1.3503447
- Lopez, A. M., Clowes, R. G., & Williger, G. M. (2022). A giant arc on the sky. Monthly Notices of the Royal Astronomical Society, 516(2), 1557-1572. https://doi.org/10.1093/mnras/stac2204
- Bond, J., Kofman, L., & Pogosyan, D. (1996). How filaments of galaxies are woven into the cosmic web. Nature, 380, 603–606. https://doi.org/10.1038/380603a0
- Lopez, A. M., Clowes, R. G., & Williger, G. M. (2024). A big ring on the sky. Journal of Cosmology and Astroparticle Physics. https://doi.org/10.48550/arXiv.2402.07591
- WMAP Science Team. (n.d.). What is the Universe Made Of? https://wmap.gsfc.nasa.gov/universe/uni_matter.html
- Milgrom, M. (2015). MOND theory. Canadian Journal of Physics, 93(2), 107-118. https://doi.org/10.1139/cjp-2014-0211
- Hawking, S. (1974). Black hole explosions?. Nature, 248, 30–31. https://doi.org/10.1038/248030a0