The substantialist view of spacetime generated a century of confusion and a failed cosmological model. The resolution is simple: Adhere to the core principles of General Relativity.
The big take away is that the universe is not expanding. The big bang never happened. The cosmological redshift is caused by gravitational redshift. The CMB is blueshifted hawking radiation from the horizon. Dark matter, dark energy, and the cosmological constant were placeholders for the distribution of mass. You can view the preprint article if you prefer a pdf version.
These are some extraordinary claims but the amount of evidence is overwhelming. Better yet, everything can be boiled down to a single claim - "Spacetime ain't real."
Spacetime is an abstract apparatus used to measure things. It is a phenomenon that's more similar to currency, speciation, laws, and proper names than it is to mass and energy. The physical world has intrinsic properties, whereas all of the properties of spacetime are assigned to it by humans.
Of course, in physics, even in the emptiest possible regions, zero-point fluctuations fill the void. Radiation, gravity, virtual particles, and the irreducible energy density of quantum fields permeate all of existence. In reality, the vacuum energy belongs to quantum fields, not to spacetime itself.
This means cosmologists have a major problem. "Expanding space" requires space to be a thing that undergoes change overtime.
We don't need space to expand. Dark magic is not only unnecessary but entirely inappropriate for this situation. The observations don't show that space is expanding. They reveal a distance proportional redshift in distant starlight. Amazingly, general relativity already provides a natural way to accommodate the cosmological redshift with gravitational redshift.
\[1 + z = \sqrt{\frac{g_{tt}(\text{observer})}{g_{tt}(\text{source})}}\]Or equivalently in terms of gravitational potential:
\[1 + z = \sqrt{\frac{1 + 2\Phi_{\text{obs}}/c^2}{1 + 2\Phi_{\text{source}}/c^2}}\]Mass density is given by the equation:
\[ \rho = \frac{3m}{4\pi r^3} \]This equation establishes a relationship between the mass, volume, and density of a given region. The laws of physics should take the same form in all coordinate systems. More precisely, all equations describing physical laws must be generally covariant, including the mass density equation above.
This means there is an obvious tension between the foundations of General Relativity and modern cosmology, namely, the assumption of homogeneity. In reality, mass density is a scale dependent property, but there is still a symmetry that can be exploited. There is a natural set of constraints, baked into the mass density equation. We know the smallest scales must have the highest density because they are embedded within the larger scales. This necessarily means that in our universe, mass density is inversely proportional to the given volume, not homogeneous.
In General Relativity, the universe is more accurately understood as a continuous and unified manifold. Gravitational potential increases along \(r\).
Working from the assumption that Hubble's law is caused by gravitational redshift, we find the distribution of mass density to be \(\rho(r)=k/r\)
Spacetime exists as a relational property of mass. No region has zero density. General relativity describes all gravitational phenomena through a single coupling constant.
Mass density follows an inverse relationship with radius:
\[\rho(r) = \frac{k}{r}\]where \(k \approx 0.833 \text{ kg/m}^2\) is a universal constant.
Integrating the density profile yields:
\[m(r) = \int_0^r 4\pi r'^2 \rho(r') \, dr' = \int_0^r 4\pi r'^2 \frac{k}{r'} \, dr' = 2\pi k r^2\]The static, spherically symmetric metric is:
\[ds^2 = -A(r) \, dt^2 + \frac{1}{1-\frac{2Gm(r)}{c^2r}} \, dr^2 + r^2 d\Omega^2\]Einstein's equations with zero pressure give:
\[\frac{d\ln A}{dr} = \frac{2Gm(r)}{c^2r^2(1-\frac{2Gm(r)}{c^2r})}\]Substituting \(m(r) = 2\pi k r^2\) and integrating:
\[A(r) = \frac{1}{1 - \frac{r}{r_h}}\]where the observer-dependent horizon radius is:
\[r_h = \frac{c^2}{4\pi Gk} \approx 1.29 \times 10^{26} \text{ m}\]The formula is always 1 + z, regardless of whether z is positive (redshift) or negative (blueshift). The sign of z itself determines the direction of the shift, not the formula structure.
\[ 1 + z = \sqrt{\frac{A(r_{\text{obs}})}{A(r_{\text{em}})}} = \sqrt{\frac{1 - r_{\text{em}}/r_h}{1 - r_{\text{obs}}/r_h}} \]For \(z>0\) the photon is observed with a lower frequency (longer wavelength) than it was emitted, because it climbs out of the gravitational potential between \(r_{\text{em}}\) and \(r_{\text{obs}}\).
For \(z<0\) the photon is observed with a higher frequency (shorter wavelength), having fallen deeper into the gravitational potential between \(r_{\text{em}}\) and \(r_{\text{obs}}\).
Mass within horizon: \(M = 2\pi k r_h^2\)
Horizon area: \(A = 4\pi r_h^2\)
\[\sigma_h = \frac{M}{A} = \frac{k}{2} \approx 0.416 \text{ kg/m}^2\]Hawking temperature at horizon:
\[T_H = \frac{\hbar kG}{ck_B} \approx 1.42 \times 10^{-30} \text{ K}\]From the oberver's perspective, the 1+z manifests as a blue shift:
\[1+z = \sqrt{\frac{\delta}{r_h}}\] \[1+z = \frac{T_{CMB}}{T_H} \approx 1.91 \times 10^{30}\]For \(T_{\text{CMB}} = 2.725\) K, the UV cutoff is:
\[\delta = r_h \left(\frac{T_H}{T_{\text{CMB}}}\right)^2 \approx 3.48 \times 10^{-35} \text{ m} \approx 2.15 \, l_P\]The above calculation is very sensitive to the method used to calculate it. High precision is needed, but the result is quite remarkable. The emitted radiation is exceptionally cold but at that perspective the 1+z becomes immense, providing exactly what is needed to obtain the \(2.725K\) and the UV cutoff, with no adjustable parameters. Everything can be fine-tuned through the coupling constant \(k \approx 0.833 \,\mathrm{kg/m^2}\)or \(H \approx 2.33 \times 10^{-18}\).
The vacuum catastrophe refers to the ~10120 discrepancy between the theoretical vacuum energy density predicted by quantum field theory and the observed value from cosmology. In QFT, vacuum energy density is defined as the energy of the quantum vacuum per unit volume. It is calculated by summing the zero-point energies of all quantum fields for all possible modes. Our methodology requires us to abandon the substantialist perspective and redefine it. Here, vacuum energy density is not a fundamental property of space itself, but rather the minimum gravitational energy density for a given volume.
\[\rho(r) = \frac{k}{r}\]where $k \approx 0.833 \, \text{kg/m}^2$ is a universal constant. This density profile yields a total mass function:
\[m(r) = \int_0^r 4\pi r'^2 \rho(r') \, dr' = 2\pi k r^2\]The metric associated with this mass distribution is:
\[ds^2 = -\frac{1}{1-\frac{r}{r_h}}dt^2 + \frac{1}{1-\frac{2Gm(r)}{c^2r}}dr^2 + r^2 d\Omega^2\]where $r_h = \frac{c^2}{4\pi Gk} \approx 1.29 \times 10^{26} \, \text{m}$ defines a coordinate horizon. This horizon radius relates directly to the Hubble constant:
\[H = \frac{c}{r_h} \approx 72 \, \text{km/s/Mpc}\]The scale-dependent density at the horizon is:
\[\rho(r_h) = \frac{k}{r_h} \approx 6.48 \times 10^{-27} \, \text{kg/m}^3 = \frac{2}{3}\rho_c\]where $\rho_c = \frac{3H^2}{8\pi G} \approx 9.71 \times 10^{-27} \, \text{kg/m}^3$ is the critical density.
The scale-dependent density provides a natural regularization mechanism through two physical cutoffs:
\[\begin{align} \text{IR cutoff} &: r_h \approx 1.29 \times 10^{26} \, \text{m} \\ \text{UV cutoff} &: \delta \approx 2.15 \times l_P \approx 3.48 \times 10^{-35} \, \text{m} \end{align}\]where $l_P$ is the Planck length. The UV cutoff emerges from the CMB temperature relation.
In quantum field theory, the vacuum energy density scales as $\rho_{vac} \sim \frac{1}{l_P^4}$, giving:
\[\rho_{vac,QFT} \approx 6.79 \times 10^{252} \, \text{kg/m}^3\]The ratio between QFT prediction and observation is $\rho_{vac,QFT}/\rho(r_h) \approx 10^{279}$.
The cosmological constant problem is resolved by making $\Lambda$ scale-dependent:
\[\Lambda(r) = \frac{8\pi G\rho(r)}{c^2} = \frac{8\pi Gk}{c^2 r}\]The values at the two cutoffs are:
\[\begin{align} \Lambda(l_P) &\approx 9.62 \times 10^8 \, \text{m}^{-2} \\ \Lambda(r_h) &\approx 1.21 \times 10^{-52} \, \text{m}^{-2} \end{align}\]This gives a ratio $\Lambda(l_P)/\Lambda(r_h) \approx 8 \times 10^{60}$, explaining the cosmological constant problem without fine-tuning.
DESI turns redshift separations into comoving distances via the flat-space rule \[ \Delta s_{\text{flat}} = c\,\Delta z / H(z). \] For the static-gravity metric with horizon \(r_h\) the proper radial distance is \[ \Delta s_{\text{proper}} = \frac{\Delta s_{\text{flat}}}{\sqrt{1-D_M/r_h}}, \] which modifies the Alcock–Paczyński ratio to \[ F_{\text{model}}(z)= A\,\frac{D_M(z)}{D_H(z)} \sqrt{1-\dfrac{D_M(z)}{r_h}}. \]
r_h = 4180 MpcA = 0.922Table 1 of DESI 2024 VI (arXiv 2404.03002 v2); \(F_{\text{DESI}}=D_M/D_H\).
| z | FDESI | Fstatic | |Δ| (%) | FΛCDM | |Δ| (%) | Winner |
|---|---|---|---|---|---|---|
| 0.510 | 0.649 | 0.590 | 9.1 | 0.594 | 8.5 | ΛCDM |
| 0.706 | 0.839 | 0.881 | 5.0 | 0.877 | 4.6 | ΛCDM |
| 0.930 | 1.214 | 1.256 | 3.5 | 1.245 | 2.5 | ΛCDM |
| 1.317 | 2.011 | 2.014 | 0.1 | 1.988 | 1.2 | Static |
| 2.330 | 4.661 | 4.651 | 0.2 | 4.546 | 2.5 | Static |
| Mean |Δ| (static) | 3.6 % | Mean |Δ| (ΛCDM) | 3.9 % | |||
r_h = 4180 Mpc and A = 0.922.A or vary \(r_h\) if you explore alternative density profiles.The formulation above necessitates a horizon for all observers. This perspective generates an enormous 1+z very near the horizon, which in turn generates Hawking radiation. The calculator below determines the temperature of this radiation after it has been blue shifted from the geometry.
Note:This calculation requires high precision. Unfortunately, not all calculation methods yield the same results. The result can vary by several planck lengths depending on the software. Regardless, the results are quite compelling.
Observed Temperature (K): -
Redshift Factor (1+z): -
Hawking Temperature (K): -
Horizon Radius r_h (m): -
Effective Radius r (m): -
Mass (kg): -
Volume (m³): -
Density (kg/m³): -
Schwarzschild Radius (m): -
Our density profile, \(\rho(r) = \frac{k}{r}\), yields a uniform gravitational acceleration: \[ a = 2\pi Gk \approx 3.49 \times 10^{-10}~\text{m/s}^2 \] which is comparable to the MOND critical acceleration scale \[ a_0 \approx 1.2 \times 10^{-10}~\text{m/s}^2. \]
In this field, circular velocities grow with radius as \(\ v \propto \sqrt{r} \), predicting a rising rotation curve — in contrast to the Newtonian expectation of \(\ v \propto \frac{1}{\sqrt{r}} \).
Rising rotation curves are very common, especially in in dwarf and low surface brightness galaxies. More work needs to be done in this area, but rising curves should viewed as a step in the right direction.
Conventional physics views gravitational potential \(\phi\) as being radially dependent, but it is most often treated as a property that decreases along \(r\). In the real universe, mass is distributed in a way that potential increases with \(r\). Consider a star embedded within a cloud of dust. In order to determine the \(\phi\) for a given observation, the observer's location in relation to the star and the dust are also required.
In the visualization below, the \(\phi\) increases as the light travels outward and more dust is included.
| Horizon radius | \(r_h = \frac{c^2}{4\pi Gk}\) |
| Hubble parameter | \(H = \frac{4\pi Gk}{c}\) |
| Constant acceleration | \(a = 2\pi Gk\) |
| UV cutoff | \(\delta \approx 2.15 \, l_P\) |
| IR cutoff | \(r_h \approx 13.6 \text{ Gly}\) |
| Coupling constant \(k\) with mass density | \( k = \rho(r) \times r \) |
| With mass function | \( k = \frac{m(r)}{2\pi r^2} \) |
| With horizon radius | \( k = \frac{c^2}{4\pi G r_h} \) |
| With Hubble parameter | \( k = \frac{H c}{4\pi G} \) |
| With constant acceleration | \( k = \frac{a}{2\pi G} \) |
| With Hubble-horizon product | \( k = \frac{H^2 r_h}{4\pi G} \) |
| With surface density at horizon | \( k = 2\sigma_h \) |
| With density at horizon | \( k = \rho(r_h) r_h \) |
| With critical density | \( k = \frac{2}{3} \rho_c r_h \) |
| With Hawking temperature | \( k = \frac{T_H c k_B}{\hbar G} \) |
| With CMB temperature | \( k = \frac{T_{CMB} c k_B \sqrt{r_h/\delta}}{\hbar G} \) |
| With redshift | \( k = \frac{-z c^2}{2\pi G r} \) |
| With metric component | \( k = \frac{c^2(1 - 1/A(r))}{4\pi G r} \) |
| With orbital velocity | \( k = \frac{v^2}{2\pi G r} \) |
| With scale-dependent \(\Lambda\) | \( k = \frac{\Lambda(r) c^2 r}{8\pi G} \) |
| With \(\Lambda\) at Planck scale | \( k = \frac{\Lambda(l_P) c^2 l_P}{8\pi G} \) |
| With \(\Lambda\) at horizon | \( k = \frac{\Lambda(r_h) c^2 r_h}{8\pi G} \) |
| With CMB-cutoff relation | \( k = \frac{\sqrt{\delta/r_h} c k_B T_{CMB}}{\hbar G} \) |
| With cutoff geometry | \( k = \frac{c^2 \sqrt{\delta/r_h}}{4\pi G \sqrt{r_h}} \) |
| With horizon thermodynamics | \( k = \sqrt{\frac{c^4}{32\pi^2 G^2 r_h^2}} \) |
| With \(\alpha\) parameter | \( k = \frac{\alpha c^2}{4\pi G} \) |
| With metric ratio | \( k = \frac{c^2 (r/r_h)}{4\pi G r} \) |
| With vacuum energy density | \( k = \rho_{vac}(r) r \) |
| With gravitational potential | \( k = \frac{c^2 \Phi(r_h)}{2\pi G r_h} \) |
| With speed of light-Hubble | \( k = \frac{c^3}{4\pi G H} \) |
| With CMB-Hubble complex | \( k = \frac{c T_{CMB} k_B \sqrt{\delta/r_h}}{\hbar G H \sqrt{r_h}} \) |
The frequency of starlight decreases at a rate that is directly proportional to its distance of travel. Cosmologists assume this is caused by a dark energy that causes space to expand at an accelerating rate.
Gravitational redshift.
All gravitational boundaries are arbitrary. This is an extremely important part of GR, even though it is a bit abstract. Gravitational potential is dependent on the observer's location with respect to the emitting object and the distribution of mass between, meaning it's a relative phenomenon. The amount of gravity an object has is not a property of that object. There is no such thing as an object in general relativity. A distant star has more gravitational potential because from our perspective, the star is embedded within a sea of other stars, which causes a gravitational redshift that is proportional to the distance of light travel.
The distribution required to achieve Hubble's law is ρ(r)=k/r, where k=0.833 kg/m². This distribution creates an observer-dependent horizon at around 1.29×10²⁶ meters (13.6 billion light years).
Very near the horizon (1-3 Planck lengths), Hawking radiation is produced out of the geometry. The emitted temperature is very cold, around 1.4×10⁻³⁰ kelvin, but the 1+z for that region is extremely high ~1.9×10³⁰. Since this radiation comes directly from the geometry and not out of a gravitational well, it's blue shifted to the 2.725 kelvin we observe. You can use the CMB calculator to see the results for yourself.
At the horizon, the surface density is σ = c² / (8πGRₛ) = 0.417 kg/m² = 1/2×k. Mass enclosed by a sphere: M = 0.417 kg/m² × 4πR² = 1.668πR² kg. Volumetric density at distance r: ρ = 0.834/r kg/m³. This means density decreases inversely with distance from any point.
"Statistically homogeneous" is a nonsensical term. What they really mean to say is that mass density decreases at a rate that is inversely proportional to the given scale, or, in other words ρ(r)=k/r, where k=0.833 kg/m².
Because locally, stars and planets are dramatically more dense than their immediate surroundings. The local topography overwhelms the underlying gradient. The Hubble shift doesn't become noticeable until around 1 megaparsec; at that distance, it begins to overtake the local effects.
This model comes directly from the core postulates of general relativity. The distribution alone accounts for the redshift, so there is no need for expansion, inflation, or dark energy.
All observations support this model. It's the interpretations and assumptions that are different. The coming SHOES project may offer some methods to distinguish between the two models, but it's hard to compete against a magic model that makes adjustments based on the observations. The best you can aim for is a tie.