Diffeomorphism invariance invalidates expansion via the Hole Argument. Thus, spacetime has no ontological standing and ΛCDM is not compatible wtih General Relativity. The cosmological redshift is caused by gravitational redshift and the CMB is blueshifted Hawking radiation from the observer-dependent horizon.
Challenge a century of cosmic orthodoxy with a concise, provocative tour through the evidence. This book invites curious readers to look again at redshift, relic light, and the models we use to tell the story of everything.
The Hole argument is a famous thought experiment, originally developed by Einstein and later refined by philosopher John Earman in the 1980s. In short, the argument states that if you have a solution to the EFE for an arangement of mass, the spacetime points of the manifold must remain diffeomorphically invariant. In other words, it is the mass that encodes gravitational curvature, not spacetime.
The FLRW metric assumes a global cosmic time, a preferred slicing, and comoving distances that evolve with a scale factor $a(t)$. These are gauge choices and not diffeomorphism-invariant. This means the FLRW metric is incompatible with General Relativity.
A simple $\rho \propto \frac{1}{r}$ relationship is enough to reproduce the cosmological redshift from gravitational redshift on all scales. Astronomers and cosmologists have ignored this possibility because they draw artifical boundaries around stars and galaxies.
Working directly Hubble's law I've listed the mass, density, and gravitational redshift for various cosmological scales below.
On large scales the observable Universe is well described by a static, spherically symmetric geometry tied to an observer-dependent horizon at radius \(r_h\). A scale-dependent mass profile \[\bar\rho(r)=\tfrac{3k}{2r},\qquad m(r)=2\pi k r^2\] yields constant radial acceleration \(a=2\pi Gk\) and a redshift law \[1+z=\sqrt{\tfrac{1 - r_\text{obs}/r_h}{1 - r_\text{em}/r_h}},\] with metric coefficient \[A(r)=(1-r/r_h)^{-1}.\] With the observer at the origin \((r_\text{obs}=0)\), this reduces to \[1+z=(1-r/r_h)^{-1/2},\] and the small-\(r\) limit recovers Hubble’s law with \[H_0=\frac{c}{r_h}=\frac{4\pi Gk}{c}.\]
\[\bar\rho(r)=\frac{3k}{2r},\qquad m(r)=2\pi k r^2.\]
Static, spherically symmetric line element: \[ ds^2 = -\frac{c^2 dt^2}{1 - r/r_h} + \frac{dr^2}{1 - r/r_h} + r^2 d\Omega^2. \] Redshift between emitter and observer: \[ 1+z=\sqrt{\frac{1 - r_\text{obs}/r_h}{1 - r_\text{em}/r_h}}. \] With \(r_\text{obs}=0\): \(1+z=(1-r/r_h)^{-1/2}\), so \(z\simeq r/(2r_h)\Rightarrow v\simeq H_0 r\).
\[r_h=\frac{c^2}{4\pi G k},\qquad H_0=\frac{c}{r_h}=\frac{4\pi G k}{c},\qquad a=\frac{G m(r)}{r^2}=2\pi G k.\]
\[\sigma_h=\frac{M}{A}=\frac{m(r_h)}{4\pi r_h^2}=\frac{k}{2}.\]
In this framework, the observed 2.725 K Cosmic Microwave Background arises from Hawking radiation emitted just inside the horizon and blueshifted on its way to us.
Hawking temperature at the horizon: \[T_H = \frac{\hbar G k}{c k_B}.\]
For emission a small distance \(\delta\) inside the horizon \((r_\text{em} = r_h - \delta,\ \delta \ll r_h)\): \[1+z = \left(1-\frac{r_\text{em}}{r_h}\right)^{-1/2} = \sqrt{\frac{r_h}{\delta}}.\]
The observed temperature is then \[T_\text{obs} = T_H \sqrt{\frac{r_h}{\delta}}.\]
Setting \(T_\text{obs} = T_\text{CMB} = 2.725\ \text{K}\) gives \[\delta = r_h\!\left(\frac{T_H}{T_\text{CMB}}\right)^2.\] With \(k \simeq 0.833\ \text{kg/m}^2\), this yields \(\delta \approx 3.48\times10^{-35}\ \text{m} \approx 2.15\,\ell_P.\)
Enter the emission distance from the horizon \(\delta\) (meters):
Because the horizon radius equals the Schwarzschild radius of the enclosed mass, both forms of the redshift law coincide near the horizon.
The observed BAO ruler at 147 Mpc arises naturally in this framework, without invoking an expanding universe.
1. Tolman temperature law: \[T(r)\sqrt{A(r)} = T_0,\quad A(r)=(1-r/r_h)^{-1}.\] Setting the recombination temperature \(T_{\rm dec}\approx 3000\ \text{K}\) gives the decoupling redshift \(1+z_* \approx 1100.\)
2. Null geodesics: Radial light paths satisfy \[\frac{dr}{dt}=c\,(1-r/r_h).\]
3. Proper sound horizon: With sound speed \(c_s = c/\sqrt{3},\) the proper distance to the decoupling shell is \[r_s=\frac{c}{\sqrt{3}}\,t_*=\frac{2 r_h}{\sqrt{3}}\ln(1+z_*).\]
4. Observed BAO scale: Including the redshift factor, the observed length is \[\lambda_{\rm obs} = r_s(1+z_*) \approx 147\ \text{Mpc},\] in precise agreement with the measured BAO feature.
From the mass profile \(m(r)=2\pi k r^2\), the gravitational acceleration is scale-independent:
\[a=\frac{G m(r)}{r^2}=2\pi Gk.\]
Circular velocities then follow
\[v(r)=\sqrt{a r}=\sqrt{2\pi Gk\,r},\]
which predicts a gently rising curve \(v\propto \sqrt{r}\). This behavior matches the observed prevalence of rising rotation curves in dwarf and low-surface-brightness galaxies, and the acceleration scale \(a=2\pi Gk\) is numerically comparable to the MOND critical value \(a_0\sim 10^{-10}\,\text{m/s}^2\).