## Contradictions between the Black Hole and the Big-Bang Theories, and the Structure of the Universe

Based on the authors’ recent work [1, 2, 3], the main objectives of this post are as follows:

• First, assuming the Einstein theory of general relativity, we demonstrate that the theory of black holes and the theory of Big-Bang are contradicting to each other.
• Second, we introduce a theorem, proved recently by the authors, on the structure and geometry of our Universe, assuming the Einstein theory of general relativity and the principle of cosmological principle. This theorem gives rise to a new theory of the Universe, which is consistent with redshift and CMB observations.

## 1. Conditions for the existence of black holes

Consider a massive body with mass ${M}$ and radius ${R}$. The Schwarzschild radius is defined by ${R_s=2GM/c^2}$. Based on the Einstein field equations, in the exterior of the body, the Schwarzschild solution is given by

$\displaystyle ds^2= -\left(1-\frac{R_s}{r}\right)c^2dt^2+\left(1-\frac{R_s}{r}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2 \qquad \text{ for } r > R, \ \ \ \ \ (1)$

and in the interior the Tolman-Oppenheimer-Volkoff (TOV) metric is

$\displaystyle ds^2= -e^u c^2dt^2+\left(1-\frac{r^2R_s}{R^3}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2 \qquad \text{ for } r < R. \ \ \ \ \ (2)$

The theory of black holes then shows that the massive body is a black hole if and only if ${R=R_s}$. Here we note that if ${R> R_s}$, the massive body is not a black hole. Also, it is impossible to have ${R< R_s}$. Otherwise, the Schwarzschild solution would be valid in the vacuum region ${R< r< R_s}$, where the factor ${1-\frac{R_s}{r}}$ changes sign and the metric becomes nonphysical. In other words, a black hole must be filled with matter in the entire interior.

The above necessary and sufficient condition implies that a massive body with radius ${R}$ and density ${\rho}$ is a black hole whenever

$\displaystyle R^2=\frac{3c^2}{8\pi G \rho}. \ \ \ \ \ (3)$

## 2. Contradictions between Big-Bang and Black-Hole Theories

First, if the Universe began with the big-bang, the energy density of the early universe ${\rho}$ would be so big that the universe would be filled with many black holes of infinitesimal radius ${R}$ defined by (3).

Second, if the Universe were born to a Big-Bang and expanded continuously, then in the expansion process it would generate successively a large number of black holes, whose radii vary as follows:

$\displaystyle r=\sqrt{\frac{R}{R_T}}R,\ \ \ \ R_0

where ${M_T}$ is the total mass in the universe, ${R_0}$ is the initial radius, ${R}$ is the expanding radius, ${r}$ is the radius of sub-black holes, and ${R_T}$ is the radius of the Universe viewed as a black hole.

To see this, we consider a homogeneous universe with radius ${R. Then the mass density ${\rho}$ is given by

$\displaystyle \rho =\frac{3M_T}{4\pi R^3},$

which implies that the mass of a ball ${B_r}$ of radius ${r}$ with this mass density is

$\displaystyle M_r=\frac{4\pi}{3}r^3\rho =\frac{r^3}{R^3}M_T=\frac{r^3}{R^3} \frac{c^2 R_T}{2G}, \ \ \ \ \ (5)$

Recall that the condition for the ball ${B_r}$ to form a black hole is

$\displaystyle M_r =\frac{c^2 r}{2G}. \ \ \ \ \ (6)$

The combination of (5) and (6) implies immediately (4).

In summary, (4) demonstrates clearly that if the Universe were born to a Big-Bang and continuously expands, then it would contain many black holes with smaller ones being embedded in the larger ones. This is not what we observed in our Universe.

## 3. Theorem on Structure of the Universe

The aforementioned contradictions force us to examine the black hole and Big-Bang theories from both the fundamental level and the observational evidences.

It is clear that the large scale structure of our Universe is essentially dictated by the law of gravity, which is based on Einstein’s two principles: the principle of general relativity and the principle of equivalence. Also, strong cosmological observational evidence suggests that the large scale Universe obey the cosmological principle that the Universe is homogeneous and isotropic.

The black theory is a direct consequence of the Einstein theory of general relativity, and is a more trustable theory. Consequently, a careful examination of modern cosmological theories on the structure of the Universe is inevitable. In fact, we have recently shown a theorem on the geometry and structure of our Universe under the assumption of general relativity and the cosmological principle:

Theorem on Structure of our Universe [2, 3]. Assume the Einstein theory of general relativity, and the principle of cosmological principle, then the following assertions hold true:

1.  our Universe is not originated from a Big-Bang, and is static;
2. the topological structure of our Universe is the 3D sphere ${S^3}$ such that to each observer, the corresponding equator with the observer at the center of the hemisphere can be viewed as the black hole horizon;
3. the total mass ${M_{\text{total}}}$ in the Universe includes both the cosmic observable mass ${M}$ and the non-observable mass, regarded as dark matter, due to the space curvature energy; and
4. a negative pressure is present in our Universe to balance the gravitational attracting force, and is due to the gravitational repelling force, also called dark energy.It is clear that this theorem changes drastically our view on the geometry and the origin of the Universe.

Inevitably, a number of important questions need to be addressed for this scenario of our Universe in the following sections.

## 4. Redshift problem

The natural and important question that one has to answer is the consistency with astronomical observations, including the cosmic edge, the flatness, the horizon, the redshift, and the cosmic microwave background (CMB) problems. These problems can now be easily understood based on the structure of the Universe and the blackhole theorem we derived. Hereafter we focus only on the redshift and the CMB problems.

The most fundamental problem is the redshift problem. Observations clearly show that light coming from a remote galaxy is redshifted, and the farther away the galaxy is, the larger the redshift. In modern astronomy and cosmology, it is customary to characterize the redshift by a dimensionless quantity ${z}$ in the formula

$\displaystyle 1+z=\frac{\lambda_{\rm observ}}{\lambda_{\rm emit}}, \ \ \ \ \ (7)$

where ${\lambda_{\rm observ}}$ and ${\lambda_{\rm emit}}$ represent the observed and emitting wavelenths.

There are three sources of redshifts: the Doppler effect, the cosmological redshift, and the gravitational redshift. If the Universe is not considered as a black hole, then the gravitational redshift and the cosmological redshift are both too small to be significant. Hence, modern astronomers have to think that the large port of the redshift is due to the Doppler effect.

However, due to black hole properties of our Universe, the black hole and cosmological redshifts cannot be ignored. Due to the horizon of the sphere, for an arbitrary point in the spherical Universe, its opposite hemisphere relative to the point is regarded as a black hole. Hence, ${g_{00}}$ can be approximatively taken as the Schwarzschild solution for distant objects as follows

$\displaystyle -g_{00}(r)=\alpha(r)\left(1-\frac{R_s}{\tilde r}\right),\qquad \alpha(0)=2, \qquad \alpha(R_s)=1, \qquad \alpha'(r) <0,$

where ${\tilde{r}=2R_s-r}$ for ${0\leq r is the distance from the light source to the opposite radial point, and ${r}$ is the distance from the light source to the point. Then by the gravitational redshift formula:

$\displaystyle \lambda_{\rm observ} = \frac{\lambda_{\rm emit}}{\sqrt{-g_{00}(r)}},$

we derive the following redshift formula, which is consistent with the observed redshifts:

$\displaystyle 1+z=\frac{1}{\sqrt{\alpha(r)(1-\frac{R_s}{\tilde r})}}= \frac{\sqrt{2R_s-r}}{\sqrt{\alpha(r)(R_s-r)}} \qquad \text{for } 0 < r

## 5. CMB problem

In 1965, two physicists A. Penzias and R. Wilson discovered the low-temperature cosmic microwave background (CMB) radiation, which fills the Universe, and it has been regarded as the smoking gun for the Big-Bang theory. However, based on the unique scenario of our Universe we derived, it is the most natural thing that there exists a CMB, because the Universe has always been there as a black-body, and CMB is a result of blackbody equilibrium radiation.

## 6. PID-cosmological model

We have demonstrated that the right cosmological model should be derived from the new gravitational field equations [1], taking into consideration the presence of dark matter and dark energy:

$\displaystyle R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R = -\frac{8\pi G}{c^4} T_{\mu\nu} -\nabla_\mu\nabla_\nu \phi. \ \ \ \ \ (9)$

In this case, based on the cosmological principle, the the metric of a homogeneous spherical universe is of the form

$\displaystyle ds^2=-c^2dt^2+R^2\left[\frac{dr^2}{1-r^2}+r^2(d\theta^2+\sin^2\theta d\varphi^2)\right],$

where ${R=R(t)}$ is the cosmic radius. The new gravitational field equations provide then the following PID-cosmological model:

$\displaystyle R^{\prime\prime}=-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}+\frac{\varphi}{8\pi G}\right)R \qquad \text{ with } \varphi=\phi'', \ \ \ \ \ (10)$

$\displaystyle (R^{\prime})^2=\frac{1}{3}(8\pi G\rho +\varphi )R^2-c^2, \ \ \ \ \ (11)$

$\displaystyle \varphi^{\prime}+\frac{3R^{\prime}}{R}\varphi =-\frac{24\pi G}{c^2}\frac{R^{\prime}}{R}p, \ \ \ \ \ (12)$

$\displaystyle (R')^2 \phi'=0, \ \ \ \ \ (13)$

supplemented with the equation of state:

$\displaystyle p=f(\rho ,\varphi ). \ \ \ \ \ (14)$

Note that only two equations in (10)-(12) are independent.

Also, the model describing the static Universe is the equation of state (14) together with the stationary equations of (10)-(12), which are equivalent to the form

$\displaystyle \varphi =-8\pi G\left(\rho +\frac{3p}{c^2}\right), \ \ \ \ \ (15)$

$\displaystyle p=-\frac{c^4}{8\pi GR^2}. \ \ \ \ \ (16)$

Now we consider a perturbation of the steady state solution, which gives

$\displaystyle \rho +\frac{3p}{c^2}+\frac{\varphi}{8\pi G} \rightarrow \varepsilon,$

which implies ${R''\not=0}$, and consequently, ${R'\not=0}$. By (13), ${\phi'=0}$, which implies that ${\varphi=0}$. Then by (12), ${p=0}$, which contradicts with (16). In other words, this simple argument tells us that the dual field ${\varphi}$ plays a role of balancing, preventing any perturbation of the combined quantity:

$\displaystyle \rho +\frac{3p}{c^2}+\frac{\varphi}{8\pi G}.$

In a nutshell, we have shown that the steady state solution of (10)-(14) is physically stable.

## References

[1] Tian Ma and Shouhong Wang, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34:2(2014), 335-366; see also arXiv:1206.5078v2.

[2] Tian Ma and Shouhong Wang, Astrophysical dynamics and cosmology, Journal of Mathematical Study, 47:4(2014), 305–378.

[3] Tian Ma and Shouhong Wang,  Mathematical Principles of Theoretical Physics, Science Press, Beijing, 524pp., August, 2015.