After Einstein [145] developed a framework for the theory of general relativity, Einstein sought field equations, which would correspond to the field version of Newton’s universal law of gravitation. Einstein [146] and independently Hilbert [147] achieved this in November 1915 [148] . However, when Einstein [149] tried to apply his theory of gravitation to the universe as a whole (cosmology), he found that his equations from 1915 appeared to be incompatible with a static mass distribution of constant density. He discovered that a consistent model of the universe could be developed, if he added an additional term to his field equations that contained a constant, λ.

Einstein’s and Hilbert’s field equations from 1915 are:

$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=-8\pi G{T}_{\mu \nu }$(1)

and Einstein’s field equations from 1917 are:

$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+\lambda g_{\mu \nu }=-8\pi G{T}_{\mu \nu }$(2)

$R_{\mu \nu }$ is the Ricci tensor, $g_{\mu \nu }$ the fundamental tensor, R the curvature scalar, $T_{\mu \nu }$ the energy-momentum tensor, G the gravitational constant, the speed of light is 1 and λ is a number, which is called the cosmological constant¹. In order to differentiate between the two sets of equations we call Equation (1) the Einstein-Hilbert field equations and Equation (2) the Einstein field equations. The spherically symmetric solution of the Einstein-Hilbert field equations for the empty space surrounding a non-rotating point mass is called the Schwarzschild solution [150] . The corresponding solution of Einstein’s field equations is called the Kottler solution [151] . If $\lambda >0$, the Kottler metric is also known as the Schwarzschild-de Sitter metric and if $\lambda <0$, as the Schwarzschild-anti-de Sitter metric.

It has long been believed that the gravitational redshift can not explain the observed cosmological redshift. This is true for the Schwarzschild metric [150] , which is widely employed to study the general relativistic phenomenon associated with most stars and planets specifically the earth and sun. If, however, the Schwarzschild metric is applied to type Ia Supernova it does not yield the observed redshift-distance relationship fosterning the belief that the gravitaional redshift can not explain the observed cosmological redshift.

To the contrary we will show that generalizations of the Schwarzschild solution—the Taub-NUT solutions do however lead to the observed redshift-distance relationship and to the observed time dilation. In the following we develop a theory of cosmology based on the gravitational redshift associated with the Taub-NUT solutions to the Einstein field equations of general relativity.

In standard relativistic cosmology there are three distinct possible causes of redshift: Doppler, gravitational and cosmological [24] . We will show that the observed cosmological redshift, z, is due to the gravitational redshift. Consequently, we will conclude there are only two causes of redshift in relativistic cosmology: Doppler and gravitational.

In general the gravitational spectral shift between any two points A and B in space is given by [24] [152] - [154] :

$z=\sqrt{\frac{g_{00}\left(r_B\right)}{g_{00}\left(r_A\right)}}-1$(4)

We assume that $r_A$ in the above equation is a constant meaning that we can let $\gamma =g_{00}\left(r_A\right)$. Consequently, we can drop the subscripts to obtain:

${\left(z+1\right)}^2=\frac{g_{00}\left(r\right)}{\gamma }$(5)

This equation tells us that the gravitational redshift depends on the $g_{00}\left(r_A\right)$. We will explicitely show that the Schwarzschild $g_{00}\left(r_A\right)$ does not lead to the redshift-distance relationship. However, we will also show that the $g_{00}\left(r_A\right)$ of specific generalizations of the Schwarzschild solution, Taub-NUT solutions, do lead to the observed redshift-distance relationship and to the observed time dilation.

In order to emphasize and make clear that we are referring to the gravitational redshift associated with cosmological solutions to the Einstein field equations and not to the well known gravitational redshift associated with the Schwazschild solution, in this work we refer to the left side of the above equation as the cosmological gravitational redshift.

Time dilation in relativity is defined via the proper time, $\text{d}\tau =\sqrt{\text{d}{s}^2}$. In our cosmological theory the proper time is:

$\text{d}\tau =\sqrt{g_{00}}\text{d}t$(6)

We employ Equation (5) to obtain the relationship between time dilation and redshift in our theory of cosmology. We find:

$\text{d}\tau =\sqrt{\gamma }\left(z+1\right)\text{d}t$(7)

Suggestions by Wilson [155] and Rust [156] that light curve broadening should occur in Type Ia Supernova, if the universe is actually expanding, have been observationally confirmed by [104] [157] - [159] . These authors found a time dilation or slowing down of the supernova by the factor of $\left(z+1\right)$. They interpreted this result as evidence that cosmological redshifts are caused by an expanding universe.

The above equation for time dilation in the stationary universe has the same $\left(z+1\right)$ dependency, but it is not associated with cosmic expansion rather it is due to the gravitational redshift. We conclude: the observed light curve broadening can not be used to prove that the universe is expanding. Segal, Andrews and Holushko [160] - [162] came to the same conclusion although their theoretical standpoints are completely different than ours.

Comparing the above equation with observations of time dilation we conclude: $\gamma =1$ and Equation (5) reduces to:

${\left(z+1\right)}^2=g_{00}\left(r\right)$(8)

We remind the reader that we call the left side of Equation (8) the cosmological gravitational redshift.

In our work cosmological solutions to the field equations of general relativity refer to those that lead to the observed redshift-distance relationship and to the observed time dilation.

The solutions to the field equations contain constants. Our objective in this work is to demonstrate that the cosmological gravitational redshift explains the observed redshift-distance relationship and the observed time dilation. So we are not concerned with the physical meaning of the constants. Rather we merely ask: What numerical values must the constants assume so that they lead to the observed redshift-distance relationship and to the observed time dilation. Consequently, this initial formulation of our theory is purely parametric.

To obtain the numerical values of the constants in $g_{00}\left(r\right)$ we will employ Equation (8). The left side of this equation is known from observations, whereas the right side is theoretical and comes from the solutions to the field equations of general relativity. The numerical values of the constants in $g_{00}\left(r\right)$ are obtained by curve fitting the $g_{00}\left(r\right)$ to the observations.

The first two solutions of the field equations that we consider are the most well known and also the simplest, the Schwarzshild and Kottler metrics. For the Kottler metric we have:

$g_{00}=1-\frac{\alpha }{r}-\frac{\lambda }{3}{r}^2$(9)

Inserting this into Equation (8) yields:

${\left(z+1\right)}^2=1-\frac{\alpha }{r}-\frac{\lambda }{3}{r}^2$(10)

The above equation is valid for the Schwarzschild metric too if we let $\lambda =0$.

The zero point of the redshift-distance relationship is: $z=0$ at $r=0$, whereby Equation (8) becomes: $g_{00}\left(0\right)=1$. But, this point does not exist according Equation (10) because $\frac{\alpha }{r}\to \infty$, as $r\to 0$. Thus, neither the Kottler nor the

Schwarzschild metric leads to the observed redshift-distance relationship and they are therefore not cosmological solutions meaning that the gravitational redshifts associated with them are not cosmological. In addition two more well known solutions that are not cosmological are the Kerr solution, which corresponds to a rotating massive body and the Kerr solution with the cosmological constant.

The Schwarzschild metric is not a cosmological solution, however, generalizations to the Schwarzschild metric are cosmological solutions. Taub [163] discovered a solution to the Einstein field equations, which Newman, Unti and Tamburino [164] extended to a larger manifold, whose initials form the “NUT” of “Taub-NUT”. They also proved that the Taub-NUT solutions are a generalization of the Schwarzschild solution. We found four solutions of the Taub-NUT family that are cosmological. Specifically, we applied these Taub-NUT metrics to the supernova from which the cosmological redsifts are derived. Below we give the $g_{00}\left(r\right)$ components of the fundamental tensor for each of the four solutions and by comparing these theoretical $g_{00}\left(r\right)$ with observations of redshift-distance we derive the numerical values of the constants they contain.

In this section we compare the theoretical cosmological gravitational redshift on the right side of Equation (8), with the observed redshift-distance diagram from Vincenzi et al. [165] , which consists of 1829 Type Ia supernova. The data contains the observed relationship between spectral shift, z, and distance modulus, μ. We employ:

$r={10}^{\left(\frac{\mu }{5}+1\right)-9}$(11)

to convert μ to r, the distance of a supernova in Gpc.

The Vincenzi et al. data [165] contains the errors in the redshift, z, but in our analysis we employ ${\left(z+1\right)}^2$, which is the cosmological gravitational redshift, so we need to compute the error in this quantity. Expansion of ${\left(z+1\right)}^2$ is:

${\left(z+1\right)}^2=z^2+2z+1$(12)

If the uncertainity in z is $\delta z$, then the uncertainity in $z^2$ is $\sqrt{2}\delta z$ and the uncertainity in 2z is also $\sqrt{2}\delta z$ [166] . Thus the uncertainity in ${\left(z+1\right)}^2$ is $2\sqrt{2}\delta z$, which is significantly larger than $\delta z$.

The part of the Taub-NUT solution that interests us is: [167]

$g_{00}\left(r\right)=-\frac{r^2-2\alpha r-n^2}{r^2+n^2}$(13)

n is called the NUT parameter. As mentioned above we do not attempt to give α and n in our cosmological theory a physical meaning. In this work we are concerned only with their numerical values. Assuming the validity of the Big Bang theory [163] Taub applied his metric to cosmology. In contrast we employ this metric to compute the cosmological gravitational redshift and show that it leads to the observed redshift-distance relationship.

Curve fitting the above equation to the left side of Equation (8) leads to numerical values of the constants: $\alpha =219.822$ and $n=31.204$. Figure 1 shows that Equation (8) along with the above equation leads to the observed redshift-distance relationship.

This metric is actually referred to as the Charged Lorentzian Taub-NUT metric. As mentioned above we do not attempt to give the constants in the above equation a physical meaning in our cosmological theory. In this work we are concerned only with their numerical values of the constants. So the word “charged” in the name of the metric certainly in no way implies that the universe is charged. So in order to avoid confusion in this work we drop the word “charged”.

Following Abbasvandi [168] :

$g_{00}\left(r\right)=-\left(\frac{r^2-2\alpha r-n^2+4n^2{g}^2+e^2}{r^2+n^2}-\frac{3n^4-6n^2{r}^2-r^4}{l^2\left(r^2+n^2\right)}\right)$(14)

Curve fitting the above equation with Equation (8) leads to agreement between theory and observation except for a small constant difference at $r<0.5$ Gpc. However, if we demand $g_{00}\left(0\right)=1$ as mentioned above, we obtain the constraint:

$e=\frac{\sqrt{3n^4-4n^2{g}^2{l}^2}}{l}$(15)

Curve fitting these last two equations to the left side of Equation (8) leads to the numerical values: $\alpha =219.822$, $n=-31.204$, $l=2.56554\times {10}^8$ and $g=1$. Figure 1 shows the agreement between theory and observation. This agreement also holds for $r<0.5$ Gpc.

Following Mann [169] :

$g_{00}\left(r\right)=\frac{l^{-2}{\left(r^2+s^2\right)}^2+\left(\kappa +4l^{-2}{s}^2\right)\left(r^2-s^2\right)-2\alpha r^{{}^{}}}{r^2+s^2}$(16)

s is the NUT charge and the cosmological constant is: $\lambda =-\frac{3}{l^2}$. We were not able to curve fit this equation due to lack of convergence. However, when we added the condition: $g_{00}\left(0\right)=1$, which led to the equation:

$\kappa =\frac{l^2-3s^2}{l^2}$(17)

We were able to obtain a fit with the constants: $\alpha =219.822$, $s=31.204$ and $l=7.251\times {10}^7$. The values of $l$ lead to: $\lambda =-5.706\times {10}^{-16}$. Figure 1 depicts this fit.

The Kerr-Taub-NUT metric is a solution to the vacuum Einstein-Maxwell equations, which is locally analytic. We obtain the $g_{00}$ from Miller [170]

$g_{00}\left(r\right)=a^2\frac{\sin {\left(\theta \right)}^2}{\Sigma \left(r\right)}-\frac{\Delta \left(r\right)}{\Sigma \left(r\right)}$(18)

with:

$\Sigma \left(r\right)=r^2+{\left(l+a\cos \left(\theta \right)\right)}^2$(19)

and

$\Delta \left(r\right)=r^2-2\alpha r-l^2+a^2+e^2$(20)

Curve fitting leads to an agreement between theory and observation for $r>0.2$ Gpc. At radial distances less than this value, a small constant deviation occurs. So we demand that the condition $g_{00}\left(0\right)=1$ be fulfilled. This leads to the equation:

$l=\frac{\left(-a^2-e^2-a^2\cos \left(2\theta \right)\right)\sec \left(\theta \right)}{2a}$(21)

Curve fitting now leads to complete agreement between theory and observation over the entire range of the observational data with the constants: $a=171.068$, $\theta =-412.148$, $\alpha =219.822$ and $e=93.8984$. Figure 1 shows this agreement.

Physical theories are based on assumptions. Different theories are based on different assumptions. Big Bang Theory is based on the Cosmological Principle, that is on the assumptions of homogeneity and isotropy. They lead to the Friedman-Lemaitre-Robertson-Walker (FLRW) metric.

Our theory is not based on the cosmological principle rather is based on the Taub-Nut solutions to the field equations of general relativity. They are spatially homogeneous. The fundamental tensors employed in our cosmological theory are not the metric of our entire universe for all spacetime, in the sense that the FLRW metric claims to be, rather they are the metrics associated with the celestial sources from which we obtain the observed redshift-distance relationship. These metrics are a generalization of the Schwarzchild metric [164] .

In our theory non-relativistic matter and the CMB are not included (more on this circumstance below). This differs from Big Bang cosmology, where the deceleration parameter and consequently the redshift-distance relationship which depends upon it, is determined by the average density of matter and energy in the universe. In our cosmology to the contrary the redshift-distance relationship is determined by well known solutions to the field equations of general relativity, whereby the average density of matter and energy in our universe play absolutely no role.

In Big Bang cosmology dark energy is an unknown form of energy required to explain the acceleration of the expansion of space. In our theory of cosmology there is no expansion and therefore no accelerated expansion and therefore no need to introduce the concept of dark energy as it is understood in ΛCDM cosmology. Consequently, the vacuum catastrophe mentioned in the introduction does not exist in our theory. This circumstance is to be expected because [171] noted that the proposed existence of dark energy comes about because of the assumption of the homogeneity of the distribution of matter in space, which is not an assumption of our cosmological theory.

In addition the other fundamental problems of the Big Bang theory: horizon, magnetic monopole, flatness and the prediction that the universe should contain equal amounts of matter and anti-matter also do not exist in our theory.

In the Big Bang theory to a good approximation the redshift is [24] :

$z=\frac{H_0}{c}r+\frac{1}{2}{\left(\frac{H_0}{c}\right)}^2\left(1+q_0\right)r^2$(22)

where H₀ is Hubble’s constant and q₀ is the deceleration parameter. It follows that the gravitational redshift would be:

${\left(z+1\right)}^2={\left(\frac{H_0}{c}r+\frac{1}{2}{\left(\frac{H_0}{c}\right)}^2\left(1+q_0\right)r^2+1\right)}^2$(23)

The values obtained by curve fitting this equation with Equation (8) do not lead to agreement between observation and theory. We conclude: the Big Bang theory and our theory are incompatible with each other.

We believe this conclusion is important and we strengthen it with the following: Using the Taub-NUT solution specifically Equation (8) and Equation (13) along with the above equation we are led to the equation:

$-\frac{r^2-2\alpha r-n^2}{r^2+n^2}={\left(\frac{H_0}{c}r+\frac{1}{2}{\left(\frac{H_0}{c}\right)}^2\left(1+q_0\right)r^2+1\right)}^2$(24)

Solving the above equation for α we obtain:

$\alpha \left(r\right)=\frac{1}{2r}\left(n^2+r^2\right)\left(-\frac{n^2}{n^2+r^2}+\frac{r^2}{n^2+r^2}+{\left(1+\frac{H_0}{c}r+\frac{1}{2c^2}{H}_0\left(1+q_0\right)r^2\right)}^2\right)$ (25)

But the Taub-NUT solution tells us that α is a constant and is not a function of r. Again we conclude: the Big Bang theory and our theory are incompatible with each other. We suspect this incompatibility means that both theories can not be correct, that is at least one of the two theories is false. Finally, we note if we assume $q_0=0$ in the above equations, we obtain the same results and come to the same conclusions.

We note that it has been questioned whether the standard theory of cosmology actually incorporates the cosmological principle “faithfully” [172] . In addition we also note that there are models of cosmology, which are related to the cosmological principle or the Copernican principle [173] - [176] .

In the theory of the expanding universe, the CMB is the radiation left over from the Big Bang. Clearly, our theory of cosmology demands that the CMB must have a different origin. This task however has already been accomplished by the many scientists, who have discussed its origin without Big Bang cosmology. First Guillaume [177] [178] calculated that the temperature of interstellar space from the presence of starlight to be 5.6 ˚K and Eddington [179] 3.1 ˚K while Regener [180] using the energy density of cosmic rays found it to be: 2.8 ˚K, which is very close to the measured value of: 2.72548 ˚K [181] . Nernst [182] calculated the temperature of intergalactic space to be: 0.75 ˚K and Finlay-Freundlich [183] [184] calculated $1.9\text{K}\le T\le 6.0\text{K}$ for its temperature.

All of the above calculations were made without employing the notion of a Big Bang. Born [185] was the first to realize that these temperatures mean that the electromagnetic waves emitted would fall in the radio region. No one looked for these electromagnetic waves and they (the CMB) were serendipitously discovered by Penzias and Wilson [186] . Following Kellermann’s [187] suggestion one can spreculate how the history of cosmology might have been very different, if radio astronomers had looked for and found the CMB based on the above calculations and Born’s insight. In fact, because these values were more accurate than those initially predicted by proponents of the Big Bang (Alpher, Herman and Gamow [188] - [191] ), Assis and Neves [112] concluded that the CMB provides evidence for a non-expanding universe rather than for an expanding one. Other non-Big Bang explanations for the origin of the CMB are: [128] [192] - [196] .

The concept of a non-expanding universe is not at all new. In fact, historically, it was the first theory of physical cosmology. Starting with Olbers [197] and continuing with Einstein, DeSitter, Lense, Lanczos and Nernst [149] [198] - [201] it dominated up until the 1920’s. The discovery of cosmological redshifts by Slipher 1915 eventually caused a change of thought.

The cosmological redshift-recessional velocity relationship being an assumption has opened the door to a variety of possible explanations for the origin of cosmological redshifts (see [202] for a review). One of these explanations was that the observed cosmological redshift is due to the gravitational redshift [203] .

Many of these explanations come under the broad term tired-light hypothesis. Starting with Zwicky [204] [205] and continuing with Hubble and Tolman [110] [206] . A non-expanding universe explanation for the cosmological redshift is also found in many other publications: [115] [162] [183] - [185] [196] [207] - [221] .

Our approach is not related to any of these other explanations. It differs from them in that in our theory of cosmology the origin of the observed cosmological redshift is the gravitational redshift. This interpretation of redshift agrees with the work of [222] - [225] although his approach to cosmology is completely different than ours.