# Question on the singularity theorem

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I have just started studying Cosmology and we have been asked to prove that in an expanding FRW Universe which obeys the strong energy condition: $$ho + 3P >0$$ Then there must exist a Big Bang singularity. I can see that this condition implies $$ddot{a}/a leq 0$$ which when you plot against t gives an always decreasing rate of expansion over time. This when plotted then shows that if you extend the curve far back enough will cross the t axis at some finite t-value representing a Big Bang singularity. My question is why can't the universe begin with some finite non-zero a value? is there some physical reason that we expect the Universe to begin with a=0?

It does not have to start from the big bang. There are also different universe models such as the big bounce. Where the universe has an infinite past, such that it expands and then contract.

See

https://www.quantamagazine.org/big-bounce-models-reignite-big-bang-debate-20180131/

https://www.wired.com/story/what-if-the-big-bang-was-actually-a-big-bounce/

For more detailed explanations you can look some cosmology textbooks

One of the biggest surprises that General Relativity (GR) has given us is that under certain circumstances the theory predicts its own limitations. There are two physical situations where we expect for General Relativity to break down. The first is the gravitational collapse of certain massive stars when their nuclear fuel is spent. The second one is the far past of the universe when the density and temperature were extreme. In both cases, we expect that the geometry of spacetime will show some pathological behaviour.

The first step towards a mathematical characterisation under which circumstances GR breaks down was achieved in the seminal work of Penrose and Hawking in their singularity theorems). The general structure of the theorems establishes that if on a spacetime ( $>, g_$ ):

• the matter content satisfies an energy condition
• gravity is strong enough in some region
• and a global causal condition is met

then ( $>, g_$ ) must be geodesically incomplete.

The second requirement of the theorem can be stated sometimes by requiring the existence of a closed trapped surface, $cal$ . By this is meant a $C^<2>$ closed spacelike 2-surface such that the two families of null geodesics orthogonal to $>$ are converging. This is the formal description of the intuitive idea that the gravitational field becoming so strong in some region that light rays (and so all the other forms of matter) are trapped inside a succession of 2-surfaces of smaller and smaller area.

The global causal conditions come in different forms. The idea of chronological spacetime is that there are no closed timelike curves. In the other hand, a strongly causal satisfies that for every point $pin>$ there is a neighbourhood $cal$ of $p$ which no non-spacelike curve intersects more than once. Finally, one can require that there is surface $S$ which is any subset of spacetime which is intersected by every non-spacelike, inextensible curve, i.e. any causal curve, exactly once. This surface then is called a Cauchy Surface.

The notion of geodesic incompleteness can be better understood by defining what we mean by geodesic completeness. A geodesic complete spacetime is one where any geodesic admits an extension to arbitrarily large parameter values. Then, a spacetime that is not geodesically complete, must be geodesically incomplete. Geodesic incompleteness describes intuitively that there is an obstruction to free falling observers to continue traveling through spacetime. In some sense, they have reached the edge of spacetime in a finite amount of time they have encountered a singularity.

As a note, these theorems do not show curvature blow-up which is the usual method to show that Black holes or the Big bang have a 'true' gravitational singularity and not just a lose in differentiability (as in shock waves or the tip of a cone).

## Question on the singularity theorem - Astronomy

Institute of Innovative Physics in Fuzhou, Department of Physics, Fuzhou University, Fuzhou, China

Received May 26, 2011 revised June 28, 2011 accepted July 12, 2011

Keywords: General Relativity, Inner Solutions of Hollow and Solid Spheres, Black Hole, Theorem of Singularity

The precise inner solutions of gravity field equations of hollow and solid spheres are calculated in this paper. To avoid space curvature infinite at the center of solid sphere, we set an integral constant to be zero directly at present. However, according to the theory of differential equation, the integral constant should be determined by the known boundary conditions of spherical surface, in stead of the metric at the spherical center. By considering that fact that the volumes of three dimensional hollow and solid spheres in curved space are different from that in flat space, the integral constants are proved to be nonzero. The results indicate that no matter what the masses and densities of hollow sphere and solid sphere are, there exist space-time singularities at the centers of hollow sphere and solid spheres. Meanwhile, the intensity of pressure at the center point of solid sphere can not be infinite. That is to say, the material can not collapse towards the center of so-called black hole. At the center and its neighboring region of solid sphere, pressure intensities become negative values. There may be a region for hollow sphere in which pressure intensities may become negative values too. The common hollow and solid spheres in daily live can not have such impenetrable characteristics. The results only indicate that the singularity black holes predicated by general relativity are caused by the descriptive method of curved space-time actually. If black holes exist really in the universe, they can only be the Newtonian black holes, not the Einstein’s black holes. The results revealed in the paper are consistent with the Hawking theorem of singularity actually. They can be considered as the practical examples of the theorem.

We know that the static solutions of the Einstein’s equation of gravity field with spherical symmetry are the Schwarzschild solutions which include inner and external ones. We consider a static and uniform sphere with radius and constant density, inner pressure intensity is related to coordinate but does not depends on time. By considering static energy momentum tensor of idea fluid, the Schwarzschild inner solution is [1,2].

(1)

here. The metric is finite at the center point of sphere. However, it should be pointed out that in the process of solving the Einstein’s equation of gravity field, what we obtain is actually

(2)

It can be proved based on (2) that the space curvature is infinite at point. In order to avoid the infinity, we let integral constant to be zero directly in the current theory. However, according to the theory of differential equation, integral constant should be determined by the known boundary conditions on spherical surface, in stead of the metric at the spherical center which is unknown. By considering the fact that the volume of sphere in curved space is different from that in flat space, we can prove. Therefore, no matter what the mass and density of solid sphere are, the curvature infinity at the center of sphere is inevitable.

On the other hand, according to the current theory, the inner pressure intensity of sphere is [3].

(3)

On the spherical surface we have. To make pressure intensity to be finite at the center of sphere, we have to introduce a constraint condition for spherical radius with

or (4)

here is the Schwarzschild radius. If, pressure intensity will become infinite. In this case, stable solution is impossible and material would collapse towards the center of sphere so that singularity black holes appear. However, if integral constant, pressure intensity (3) and constraint condition (4) will be changed. All calculations based on (3) and (4) about high density celestial bodies in the current astrophysics should be reconsidered.

Let’s first strictly calculate the solutions of gravity field equations of hollow and solid spheres, and then discuss the problems of singularities below.

2. The Strict Inner Solution of Gravity Field of Hollow Sphere

Suppose that the inner radius of hollow sphere is and the external radius is, the gravity mass is. The region with and the region with are vacuum. The region inside two spherical shells with is composed of complete liquid with constant density and pressure intensity. Because material is distributed with spherical symmetry, the metric can be written as

(5)

The solution of the Einstein’s equation of gravity in the region is the well-known Schwarzschild metric with

(6)

In order to determine integral constant, we compare (6) with the Newtonian theory under the asymptotic condition with

, (7)

Here is the static gravity mass of hollow sphere in the Newtonian theory. By comparing (6) with (7), we obtain. So we have the same result for hollow sphere

, (8)

To calculate the metric in the region beneath two spherical shells, the mixing energy momentum tensor of complete fluid is used [1]

, , (9)

The Einstein’s equation of gravity field is

(10)

According to the standard procedure of calculation in general relativity, we obtain

(11)

(12)

(13)

(14)

In which and is an integral constant. We will prove in the next section. (12) minus (13) then multiplied by and considering (11), we obtain

(15)

(16)

here is an integral constant. By considering (11), (12) and (16), we obtain

(17)

On the other hand, by taking the differential of (14) with respect to, we get

(18)

Substituting (14) and (18) in (17), we get

(19)

By considering the relation, (19) can be written as

(20)

(21)

(22)

here is an integral constant. If let in (14) and (21), we reach the result of current theory [1]

(23)

In which constants,. If, we have

(24)

(25)

(26)

The forms of constants are complex, but it is unnecessary for us to write them out. We have

(29)

let in (28), we get

(30)

Therefore, is finite at point. We can write (22) as

(31)

In the region, the metric can be written as at last

In the vacuum region of hollow sphere cavity, the solution of the Einstein’s equation of gravity field is still the Schwarzschild solution

, (33)

Let’s determine the integral constants, , and below.

3. The Calculations of Integral Constants for Hollow Sphere

By considering the continuity of metric tensors on the external spherical surface, according to (8), (14) and (32), we have

(34)

(35)

Similarly, on the internal sphere surface with, according to (14), (33), (35) and (36), we have

(36)

(37)

Let’s prove in curved space now. If space is flat, the relation between mass and volume of hollow sphere is with

(38)

It should be emphasized that is the Newtonian gravity mass. We introduce it by considering the asymptotic relation (7) between the Einstein’s theory and the Newtonian theory of gravity. Substitute (38) in (37), we get. This is just the current calculating result of general relativity.

However, (38) can not hold in curved space. Because there is a length contraction along the direction of radiuswe should have and, so. In the curved space, the volume should be calculated by the following formula

(39)

The integral of (39) is difficult. If the third item in radical sign is neglected, we obtain [1].

If we consider factor in (39), the integral becomes more complex. So in curved space, we have:

(41)

Substitute (41) in (37), we get

(42)

Because, and can be chosen arbitrarily, we have and in general. Therefore, from (35) and (37), we obtain

(43)

Similarly, because, and are arbitrary, we have and in general. In this way, all integral constants are determined. In the region of spherical cavity, we can write (33) as

, (45)

According to the Newtonian theory, the material distributed outside the spherical cavity with spherical symmetry does not affect the gravity field in the cavity. But according to (45), it will have some effect for the cavity.

4. The Singularity of the Inner Metric of Hollow Sphere

According to (45), the metric and curvature has singularity at the point. This is inherent singularity which can not be eliminated by the coordinate transformation. The seriousness of problem is that for any hollow sphere composed of common material, no matter what are its mass and density, singularity always exists at its center. This does not agree with practical observation. It is impossible actually. On the other hand, we consider (14) and let

or (46)

It seems that there is a singularity surface beneath two spherical shells. We now discuss this problem. The real number solution of (46) is

(47)

However, if is really a real number, the following relation should be satisfied

(48)

Let in (46) and considering (34), we have

(49)

In the weak field, the item containing can be neglected. By considering (48) and (49), we get

(50)

We know that even for high density celestial body just as white dwarf, the difference is still very small when we do calculation based on both general relativity and the Newtonian theory. The material density of white dwarf is By using this value in (50), we get, similar to the size of white dwarf. For common galaxy, we have By using this value in (50), we have, which is just the size of galaxies. So (50) can be satisfied for common spheres and (47) becomes

(51)

By developing (51) into the Taylor series, if, we have. That is to say, there is a singularity surface in the cavity. Because the metric of cavity is (33), in stead of (14), there is no singularity surface in the region inside the hollow sphere. If, we have. In this case, there is a singularity surface in the region.

According to (16), the pressure intensity in the hollow sphere is

If there is a surface with radius inside the hollow sphere on which we have

(53)

The pressure intensity on the surface will become infinite. Therefore, if there exists black hole in hollow sphere, the black hole would be a spherical surface. Substitute (44) and (45) in (53), we get

Because the radii and are arbitrary, from (54) that we may find a proper so that (68) can be satisfied. However, on this spherical surface composed of black holes, space-time has no singularity. That is to say, the surface of space-time singularity does not overlap with the surface on which material collapses. This is incomprehensible. It should be noted that up to now we have no any restriction on the mass and density of hollow sphere. This result indicates that common hollow spheres may be unstable. They may collapse into the black hole of spherical surface!

Similarly, because the internal and external radii are arbitrary, let or, we have and in general. Because the hollow sphere is placed in vacuum without material outside and inside its two surfaces, this result is also incomprehensible. The singularities of hollow sphere are shown in Figure 1 . It is obvious that the results can not be true.

Figure 1 . The singularity of hollow sphere.

5. The Singularities of Solid Sphere’s Metric and Black Holes

According to the present calculation of general relativity, the internal metric of a common solid sphere has no singularity when the radius of sphere is greater than the Schwarzschild radius. According to the strict calculation in this paper, the situation is completely different. The solid sphere is a special situation of hollow sphere when its internal radius becomes zero. The internal metric of solid sphere is still described by (30), but the conditions of boundary are different. On the spherical surface, we have

(55)

(56)

In order to determine in (55), we have to know the relation between and. In curved space, we have

(57)

By substituting (57) in (55), we can decide in principle. We have in general. If suppose, we have

(58)

Because, the boundary condition (55) can not be satisfied. We estimate the magnitude of volume’s change in curved space based on (58). Let and omitting high order items, we have

(59)

For neutron stars, we have and, so. If considering the universe as a uniform sphere, we have and, so. For so-called black hole, we have according to (4) and. For common spheres just as the sum and the earth, is a very small but non-zero quantity.

After is determined, by substituting it into (28) and (29), we can determine and. However, we can not yet determinate and only based on (56). Another condition is needed. By considering the fact that the pressure intensity on the surface of sphere should be zero with, from (16) and (31) we have

(60)

(62)

Now, all integral constants are determined. The internal metric of solid sphere is

(63)

Because and in general, we have and at pointso the infinite of space-time curvature also appears at the center of sphere.

The pressure intensity of solid sphere is also represented by (52), but the integral constants should be represented by (61). In general, we have and, so we have. That is to say, no mater what are the mass and density of sphere, the pressure intensity at the center of sphere can not be infinite. So-called the singularity black holes in which material collapses towards its center are impossible. Meanwhile, the pressure intensity may become negative value at the center of sphere and its nearby region too. On the other hand, if there is spherical surface with radius so that we have

(64)

the pressure intensity may become infinite on the surface. Substitute (61) in (63), we obtain

(65)

Therefore, if black hole exists in solid sphere, it can

Figure 2 . The singularity of solid sphere.

only take the form of a spherical surface. Such solid sphere is not stable for material will collapse to its spherical surface. But according to our common experiences there is no singularity of space-time curvature on the spherical surface. Such result is also uncanny.

The singularity of solid sphere is shown in Figure 2 . Notice that we did not impose any restriction for mass and density, and there are so many strange characteristics for common solid spheres. The results are completely different from the current understanding in general relativity. The theory of singularity black hole in the current astrophysics and cosmology has to be reconsidered.

6. Discussions on the Theorem of Singularity and the Rationality of Current Singularity Black Hole Theory

S. W. Hawking etc. proved the theorem of singularity by means of the method of differential geometry [4]. The theorem was based on three prerequisite conditions. 1. General relativity was tenable. 2. The law of causality was tenable. 3. There were some points in space-time at which material densities were non-zero. The theorem claimed that if theses three conditions were satisfied, singularity inevitably existed in space-time. Hawking etc. considered singularities as the beginning and ending of time. The Big Bang theory was considered as the beginning of time and the black holes were regarded as the ending of time.

We note that the theorem had no restriction on material’s mass and density and did not demand that singularities were embodied in material. That is to say, according to the theorem, singularities may bare in vacuum. In order to avoid this embarrassing situation, Penrose proposed the so-called principle of the universe supervisor. The principle declares that there exists the universe supervisor who prohibit the appearance of bare singularities in vacuum. In other word, due to the existence of the universe supervisors, all singularities will be wrapped in the centers of black holes with great masses and high densities. According to the solutions of the Einstein’s equation of gravity, there exist Schwarzschild black holes with spherical symmetry and the Kerr black holes with axial symmetry and so on. The singularities were hidden in the centers of material. In this way, they can not be perceived directly, and physicists seem tolerate their existence.

The results revealed in the paper are consistent with the Hawking theorem of singularity actually. We can consider them as the practical examples of the theorem. By considering the fact that the volume of hollow and solid spheres in curved space are different from that in flat space, the strict calculation reveals that singularities can not be avoided at the centers of common hollow and solid spheres with small masses and low densities. On the other hand, because the pressure intensity can not be infinite at the center of sphere, material can not collapse towards the spherical center. Also the result shows that the pressure intensity may become negative values at the center and its nearby region.

Meanwhile, there may be curved surfaces inside the common hollow and solid spheres on which pressure intensities can become infinite so that material will collapse to them. But the space-time curvatures are still finite on the surfaces. The surfaces of space-time singularity do not always overlap with the surfaces with infinite pressure intensities. All these characters can not agree with our practical experiences of common hollow and solid spheres. They are incomprehensible in physics.

According to the current understanding, the black holes exist at the center of Quasars. However, according to the observations of Rudolf E. Schild and Darryl J. Leiter, the center of Quasar 0957 + 561 is a close object, called MECO (Massive Eternally Collapsing Object) [5]. It is not a singularity black hole, and is surrounded by a strong magnetic field. The observation of Rudolf E. Schild was consistent with the calculation and analyses in this paper. That is to say, if there are black holes in the universal space, they can only be the Newtonian black holes, not the Einstein’s singularity black holes!

More essentially, the true world excludes infinites. A correct theory of physics can not tolerate the existence of infinites, especially singularities in daily life’s hollow and solid spheres composed of common material. It is well known that the history of physics is one to overcome infinites. Modern physics grows up in the process to surmount infinites. As revealed in this paper, singularity in general relativity is actually caused by the description method of curved space-time. Physicists and cosmologists should take cautious and incredulous attitude toward the problems of singularity black holes. It is not a scientific attitude to consider singularity black holes as objective existence without any question to them. We should think in deep, whether or not our theory has something wrong. When we enjoy the beauty and symmetry of the Einstein’s theory of gravity, remember that we should not neglect its limitations and possible mistake.

[1] J. R. Oppenheimer and H. Snyder, “On Continued Gravitational Contraction,” Physical Review, Vol. 56, No. 5, 1939, pp. 455-459. doi:10.1103/PhysRev.56.455

[2] Y. L. Zhang, “Introduce to Relativity,” The Publishing Company of Yunnan People, Kunming, 1989, p. 388.

[3] L. B. Feng, X. C. Liu and M. C. Li, “Genarel Relativity,” Jilin Science Publishing Company, Jilin, 1995, p. 109.

[4] S. W. Hawking and G. F. R. Eills, “The Large Scale Structure of Space Time,” Cambridge University Press, New York, 1972.

[5] R. E. Schild, D. J. Leiter and S. L. Robertson, “Black Hole or Meco: Decided by a Thin Luminous Ring Structure Deep within Quasar Q0957 + 561,” Journal of Cosmology, Vol. 6, 2010, pp. 1400-1437.

## 2 Quantum Loophole # 1: Violation of the Energy Conditions

It is well-known that quantum effects can indeed violate classical energy conditions, such as the weak energy condition. In particular, quantum effects can give rise to negative local energy densities. An example of this is the Casimir effect: the electromagnetic vacuum state between a pair of perfectly conducting plates has an energy density of

where L is the plate separation and units in which ℏ = c = 1 are used. This violates both the weak and the averaged weak energy conditions, as an observer between the plates at rest observes a constant negative energy density. Interestingly, the averaged null energy condition is not clearly violated in this case. The only null rays which avoid hitting the plates (and hence their presumably large positive energy density) are those which are parallel to the plates. In this case, T μ ν n μ n ν = 0 , so the averaged null energy condition is marginally satisfied. One might wonder if the violation of the weak energy condition by the Casimir effect is an artifact of the assumption of perfectly reflecting boundary conditions. It has recently been shown [11] that more realistic plates with finite, but sufficiently high, reflectivity can also produce negative local energy density. In all cases, there is an inverse relation between the size of the negative energy region (the plate separation) and the magnitude of the negative energy density.

A second way that quantum effects can create negative energy density is through quantum coherence effects. One can construct quantum states in a quantum field theory in which the local energy density is negative. The simplest example of this is a quantum state for a bosonic field which is superposition of the vacuum and of a two particle state for a particular mode:

where N is a normalization factor and ϵ is the relative amplitude to measure two particles rather than no particles in the state. In Minkowski spacetime, the local energy density is the expectation value of the normal ordered stress tensor operator, : T t t : ,

The only other piece of information that we need is that in general ⟨ 0 | : T t t : | 2 ⟩ ≠ 0 . If we take | ϵ | sufficiently small, then the | ϵ | 2 term in ρ can be neglected, and we can then choose the phase of ϵ so as to have ρ < 0 at a selected spacetime point. This state is essentially a limit of a squeezed vacuum state.

Although the local energy density in states such as that described above can be made arbitrarily negative at a given spacetime point, one finds that there are two important restrictions on the negative energy density, at least for free fields in Minkowski spacetime. The first is that the total energy must be non-negative:

The second is that the energy density integrated along a geodesic observer’s worldline with a sampling function f ( τ ) must obey a “quantum inequality” of the form [12, 13, 14, 15]

where τ 0 is the characteristic width of f ( τ ) and c is a dimensionless constant, which is typically somewhat less than unity. The physical content of these inequalities is that there is an inverse relation between the magnitude of negative energy density, and its duration. An observer who sees a negative energy density of magnitude | ρ m | will not see it persist for a time longer than about | ρ m | − 1 / 4 . This restriction greatly limits what one can do with quantum negative energy. Macroscopic violations of the second law of thermodynamics, which would occur with unlimited negative energy, seem to be ruled out [16] , for example.

Quantum inequalities have been proven under a variety of conditions to hold in curved spacetime [17, 18, 19, 20] , as well as in flat spacetime. In particular, if the sampling time τ 0 is small compared to r , the local radius of curvature, then the flat space form, Eq. ( 8 ), is approximately valid in curved spacetime as well. The inequalities basically say that the local energy density cannot be vastly more negative than about − 1 / r 4 . This fact has been used to place severe restrictions on some of the more exotic gravitational phenomena which negative energy might allow, such as traversable wormholes [21] , or “warp drive” spacetimes [22] .

The key question remains: can quantum violations of the energy conditions avoid the singularities of classical relativity? In at least some cases, the answer is yes. An example of this was given many years ago by Parker and Fulling [23] , who constructed a non-singular cosmology using quantum coherence effects to avoid an initial singularity. These authors explicitly constructed a quantum state which violates the strong energy condition and in which the universe would bounce at a finite curvature, rather than passing through a curvature singularity. Furthermore, the bounce can be at a scale far away from the Planck scale. This example shows that quantum effects can avoid an initial cosmological singularity, but leaves open the question of whether the singularity is necessarily avoided by quantum processes.

The case of the black hole singularity is technically more difficult to study, and no explicit construction analogous to the Parker-Fulling example in cosmology has been given. However, several authors have discussed the form which non-singular black holes might take. Frolov, Markov, and Mukhanov [24] , for example, have discussed the possibility that the Schwarzschild geometry might make a transition to a deSitter spacetime before the r = 0 singularity is reached.

Most of the work on quantum singularity avoidance has been in the context of a semiclassical theory, where matter fields are quantized but gravity is not. This theory should break down before the Planck scale is reached, at which point one would need a more complete theory. It is not clear that one can get generic singularity avoidance in this theory far away from the Planck scale. One can give a simple argument for this: In Planck units, quantum stress tensors typically have a magnitude of the order of ⟨ T μ ν ⟩ ∼ 1 / r 4 , whereas the Einstein tensor is of order G μ ν ∼ 1 / r 2 , where r is the local radius of curvature. The backreaction of the quantum field on the spacetime geometry is large when ⟨ T μ ν ⟩ ≈ G μ ν , which is when r ≈ 1 , that is, at the Planck scale. Of course, this argument does not always hold, as the Parker-Fulling example shows. However, the reason that Parker and Fulling were able to get a bounce well away from the Planck scale is twofold: Their example requires negative pressure, but not negative energy density (violation of the strong but not the weak energy condition), and their model contains a massive field, introducing a new length scale. Thus, in their example, the violation of the appropriate energy condition is not characterized by 1 / r 4 . However, the quantum inequalities seem to suggest that one cannot get such large violations of the weak energy condition, and that local negative energy densities in curved spacetime are likely to be of order − 1 / r 4 .

It should be noted that it is possible to violate energy conditions at the classical level with nonminimally coupled scalar fields, and this fact has been used by Saa, et al [25] to construct nonsingular cosmologies with such fields as the matter source. Thus if there are such nonminimal fields in nature, all of the discussion of quantum violation of the energy conditions may be moot.

## Contents

In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes) in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.

Other ways in which singularities occur is by degeneration of manifold structure. The presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.

### Algebraic curve singularities Edit

Historically, singularities were first noticed in the study of algebraic curves. The double point at (0, 0) of the curve

are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves.

It was then a short step to define the general notion of a singular point of an algebraic variety that is, to allow higher dimensions.

### The general position of singularities in algebraic geometry Edit

Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system. One can say that the extrinsic meaning of a singular point isn't in question it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes). This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).

At about the same time as Hironaka's work, the catastrophe theory of René Thom was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the stable phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed the visible is the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a catastrophe theory supposed to account for discontinuous change in nature.

### Arnold's view Edit

While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory as propagated by Christopher Zeeman caused a reaction, in particular on the part of Vladimir Arnold. [2] He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.

### Duality Edit

An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.

The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate and these fields are linked.

## Do black holes have a back door?

In the 2014 science fiction film Interstellar, a group of astronauts traverse a wormhole near a black hole called Gargantua. A recent study by researchers at the Institute of Corpuscular Physics in Valencia suggests that matter might indeed survive its foray into these space objects and come out the other side. Illustration: A realistic accretion disc gravitationally lensed by a rotating black hole. Credit: Double Negative artists/DNGR/TM & © Warner Bros. Entertainment Inc./ Creative Commons (CC BY-NC-ND 3.0) license. One of the biggest problems when studying black holes is that the laws of physics as we know them cease to apply in their deepest regions. Large quantities of matter and energy concentrate in an infinitely small space, the gravitational singularity, where space-time curves towards infinity and all matter is destroyed. Or is it?

A recent study by researchers at the Institute of Corpuscular Physics (IFIC, CSIC-UV) in Valencia suggests that matter might in fact survive its foray into these space objects and come out the other side.

Published in the journal Classical and Quantum Gravity, the Valencian physicists propose considering the singularity as if it were an imperfection in the geometric structure of space-time. And by doing so they resolve the problem of the infinite, space-deforming gravitational pull.

“Black holes are a theoretical laboratory for trying out new ideas about gravity,” says Gonzalo Olmo, a Ramón y Cajal grant researcher at the Universitat de València (University of Valencia, UV). Alongside Diego Rubiera, from the University of Lisbon, and Antonio Sánchez, PhD student also at the UV, Olmo’s research sees him analysing black holes using theories besides general relativity (GR).

Specifically, in this work he has applied geometric structures similar to those of a crystal or graphene layer, not typically used to describe black holes, since these geometries better match what happens inside a black hole: “Just as crystals have imperfections in their microscopic structure, the central region of a black hole can be interpreted as an anomaly in space-time, which requires new geometric elements in order to be able to describe them more precisely. We explored all possible options, taking inspiration from facts observed in nature.” Antonio Sánchez (left) and Gonzalo Olmo. Image credit: Universitat de València. Using these new geometries, the researchers obtained a description of black holes whereby the centre point becomes a very small spherical surface. This surface is interpreted as the existence of a wormhole within the black hole. “Our theory naturally resolves several problems in the interpretation of electrically-charged black holes,” Olmo explains. “In the first instance we resolve the problem of the singularity, since there is a door at the centre of the black hole, the wormhole, through which space and time can continue.”

This study is based on one of the simplest known types of black hole, rotationless and electrically-charged. The wormhole predicted by the equations is smaller than an atomic nucleus, but gets bigger the bigger the charge stored in the black hole. So, a hypothetical traveller entering a black hole of this kind would be stretched to the extreme, or “spaghettified,” and would be able to enter the wormhole. Upon exiting they would be compacted back to their normal size.

Seen from outside, these forces of stretching and compaction would seem infinite, but the traveller himself, living it first-hand, would experience only extremely intense, and not infinite, forces. It is unlikely that the star of Interstellar would survive a journey like this, but the model proposed by IFIC researchers posits that matter would not be lost inside the singularity, but rather would be expelled out the other side through the wormhole at its centre to another region of the universe.

Another problem that this interpretation resolves, according to Olmo, is the need to use exotic energy sources to generate wormholes. In Einstein’s theory of gravity, these “doors” only appear in the presence of matter with unusual properties (a negative energy pressure or density), something which has never been observed. “In our theory, the wormhole appears out of ordinary matter and energy, such as an electric field” (Olmo).

The interest in wormholes for theoretical physics goes beyond generating tunnels or doors in space-time to connect two points in the universe. They would also help explain phenomena such as quantum entanglement or the nature of elementary particles. Thanks to this new interpretation, the existence of these objects could be closer to science than fiction.

## Physics question: was the universe 'in' the singularity?

Matt wouldn't agree with him that the statement 'something cannot come from nothing' was true and the caller just could not wrap his head around the fact that holding that position did not mean Matt felt something could come from nothing.

It's a common problem, right? People tend to think that if you say their statement is false, you are declaring the opposite statement true.

But it got me thinking about the big bang, and 'where' all the matter/energy of the universe was at that point.

Is it correct to say that "prior" to the big bang all the matter and energy of the universe was in the singularity?

That something has never come from nothing because all that there is has literally always been here, either as the matter and energy of the universe as we know it, or as the super-condensed quantum weirdness in the singularity.

Is it correct to even say 'in' a singularly? Would ɺs the singularity' be more appropriate?

Or do you think this is trading one confusing discussion for another, even more confusing discussion?

EDIT: i do recognize that there is no reason to assume everyone/anyone here is a cosmologist, obviously, so i am looking at this more from the debate between atheist and theist angle.

I'm really surprised it wasn't obvious in my post, but I'm an atheist.

Goes to show when you write something, you cant tell how someone is going to take it.

"Where did the stuff in the universe come from?" Is a very common question asked by theists in a debate. I thought "it didn't come from anywhere, it was always here" might be more accepted by theists than a discussion on what 'nothing' means, and wanted to get your opinions.

Is it correct to say that "prior" to the big bang all the matter and energy of the universe was in the singularity?

We don't know. This is an open question in science and we are only fairly confident about what probably occurred fractions of a second after the Big Bang. Our methods of observation are limited in what they can tell us and it may be that what actually happened is impossible to observe enough to determine.

That something has never come from nothing because all that there is has literally always been here, either as the matter and energy of the universe as we know it

That rule is an observation of how the universe currently operates and refers to a closed system. Indications are that the universe as we know it is not a closed system so it may well not apply.

Or do you think this is trading one confusing discussion for another, even more confusing discussion?

I think it is trading in religious speculation for scientific speculation. We really don't understand everything about how the universe came to be and conceptually that isn't a problem for science. It is an area of study which people are working on.

The singularity is just a short way of saying "our math doesn't work here". Keep that in mind.

Is it correct to say that "prior" to the big bang all the matter and energy of the universe was in the singularity?

It's probably not correct to talk about anything being 'prior' to the big bang. Because you put it in quotes, I suspect you knew that, but it makes me wonder why you asked the question. As far as we can tell, time began with the big bang. You should also put "singularity" in quotes, it's short hand for 'we don't know'.

You might want to read the wiki articles on zero energy universe and brane theory.

I think this is a proper science, or science terminology question and would be best answered in a science sub.

In, of, was, as. Singularities break down our understanding of space and time.

Yeah, I don't know why Matt refrained from accepting as a generally intuitive proposition that something cannot come from nothing. But the phrasing of that is not good. It is basically just saying all causes must actually exist. It's a tautology.

In terms of the singularity, the word refers to some state of affairs in which we have zero ability to make any inferences. In this context it refers to a state of affairs in which matter is literally on top of itself and notions of space, time, and causation do not apply.

Been watching PBS's Space Time videos recently. These are great at showing just how weird, non-intuitive this physics is. E.g. Space and time don't really exist.

But no it makes no sense to speak of anything prior to the singularity.

Frankly, I am not well-versed enough in the actual physics to speak authoritatively on the subject, but the way I view it is that the Universe was the singularity in the nanoseconds before inflation.

Obviously, there really is no "before" since there was no space-time yet, but conceptually, if you think about it as "no inflation yet" and "inflation has begun", the Universe existed in both of those conditions.

edit -- and this question might be more appropriate for r/askscience or r/space.

Lawrence Krauss is doing an AMA. So, that would be a good place to ask your question.

Matt wouldn't agree with him that the statement 'something cannot come from nothing' was true and the caller just could not wrap his head around the fact that holding that position did not mean Matt felt something could come from nothing.

Oh shit nigga, I remember Eric. Matt ended up having a full-on debate with the guy on a stream. You can watch it here (2:33:46).

What it means is that everything in the universe, including the space was in one single 'location' (hence 'singularity'). Think of it like every single point in spacetime being so much closer together than they are now that you can't actually tell which is which.

"Where did the stuff in the universe come from?" Is a very common question asked by theists in a debate. I thought "it didn't come from anywhere, it was always here" might be more accepted by theists than a discussion on what 'nothing' means, and wanted to get your opinions.

Yeah, usually this point is arrived at after someone asks "then where did god come from" because when the theist says "oh, he's eternal" or whatever, the classic response is "if he can be eternal then why can't the universe be eternal?" and that's kind of how we get to the point of arguing over whether the universe has a beginning in the first place.

It's a common problem, right? People tend to think that if you say their statement is false, you are declaring the opposite statement true.

yeah, I think it's a confusion between the Law of Excluded Middle, and the Excluded Middle logical fallacy. One is a law of logic and a statement of fact, either A or (not A) is true. But just because a given proposition is true or false, that does not mean that a person's position or opinion on the proposition is true or false.

Or do you think this is trading one confusing discussion for another, even more confusing discussion?

Probably. I would argue that it's largely irrelevant and shouldn't be dragged into a theistic conversation. We are talking about cutting edge physics, and physics is generally poorly understood by those who are not experts in the field.

At the beginning, there was everything, and expanded.

If you had a time machine and tried to travel backwards "toward" the Big Bang, then as the Universe contracted around you, spacetime would dilate, and seconds would expand into millennia, and you would never reach your destination. From our frame of reference, the moment of the Big Bang looks like an infinitesimally small slice of time, but from that frame of reference, it did not.

What we know of as the Big Bang isn't so much of a "start" of something (which implies something before it), as it is the Universe being in a very mysterious, unknowable configuration.

Is it correct to even say 'in' a singularly? Would ɺs the singularity' be more appropriate?

Yes, the latter. The singularity is the thing that expanded to become the universe.

But it got me thinking about the big bang, and 'where' all the matter/energy of the universe was at that point. Is it correct to say that "prior" to the big bang all the matter and energy of the universe was in the singularity?

That something has never come from nothing because all that there is has literally always been here, either as the matter and energy of the universe as we know it, or as the super-condensed quantum weirdness in the singularity.

Is it correct to even say 'in' a singularly? Would ɺs the singularity' be more appropriate?

As I understand it the hypothesis of the initial singularity is that the universe was the singularity.

The initial singularity was the gravitational singularity of infinite density thought to have contained all of the mass and space-time of the Universe before quantum fluctuations caused it to rapidly expand in the Big Bang and subsequent inflation, creating the present-day Universe.

In other words the singularity expanded to become the universe. The process is described in the Wikipedia article on the chronology of the universe.

"Where did the stuff in the universe come from?" Is a very common question asked by theists in a debate. I thought "it didn't come from anywhere, it was always here" might be more accepted by theists than a discussion on what 'nothing' means, and wanted to get your opinions.

Well you could point out that (according to the hypothesis of the initial singularity) that the singularity had all of the mass of the universe. Not matter, not stuff, but mass. You could point out that mass and energy are equivalent and can be converted into one another. Finally you could point out that the chronology of the early universe includes steps: Hadron epoch Lepton epoch Photon epoch Nucleosynthesis Matter domination and Recombination which describe the hypothesis of exactly how the mass from the singularity transformed into the "stuff" of the universe.

As for "it has always been here" . well one hypothesis is that time does not pass at a singularity, so there was no time before the Big Bang. This concept defies description by our language, but perhaps you could try to describe it as follows: both statements "the universe was a singularity for all time before the Big bang" and "the universe was a singularity for no time at all" are true (in this hypothesis) and they do not contradict one another.

Going by the naive big bang model, where we just assume physics keeps working in ways that don't qualitatively change the answer: The singularity is a single moment in time where space is a single point. Before that moment, there doesn't have to be anything. At that moment, all the energy and matter of the universe is at/in that single point of space.

The moment that the naive big bang model says the universe is a singularity is defined as t=0 in all cosmological models of the singularity. Realistically, our lack of an understanding of cosmological inflation means we don't know what goes on before t=10 -34 seconds. The inflation era - a period of ludicrously fast expansion of the universe - lasted for at least 10 -32 seconds, but it could theoretically have lasted forever, and still be continuing somewhere far beyond the visible universe.

Inflation doesn't conserve energy, by the way, it multiplies it exponentially. (Which means it can't get something from nothing, but it can make an atom's worth of energy into a galaxy's worth in under 10 -34 seconds).

But assuming inflation lasted the minimum amount of time, our lack of understanding of quantum gravity still means that anything we say about what goes on before t=10 -36 seconds is probably wrong. The concept of space might be fallacious at that point.

From an ontological perspective, though, regardless of what the answer to the above is, how the universe got a certain starting energy density is exactly as big a mystery as why the laws of physics are what they are. A mathematical model must have boundary conditions to go with its rules. And you can ask where the rules come from just as readily as where the boundary conditions come from.

We don't know. Certain theories/hypotheses, like String Theory and Eternal Inflation, vastly reduce the number of necessary rules and boundary conditions. Mathematical realism eliminates all ad-hoc rules and boundary conditions, but it has problems with explaining why we end up in a universe like this.

If you think you can give theists a satisfying answer, sorry that reality doesn't meet your expectations, I suppose. The problem with religion is that it gives an unlikely non-answer god's origins would be a greater mystery than the existence of just some nonliving mathematical rules.

Is it correct to say that "prior" to the big bang all the matter and energy of the universe was in the singularity?

Sort of, in one form or another. Even the boundaries of the Universe itself were contained within the Singularity. Consider the fact that an atom is mostly empty space. If you remove the empty space from between all of the subatomic particles, even the quarks themselves, you can force the Universe into an incredibly dense point-moment of space-time, something that borders on infinite density at t=0. All of our equations tend to place time in the denominator, so t=0 is an asymptote of sorts, in that our current math can get us infinitely close, but not actually there in a way that our math works. And this problem is compacted by the fact the fundamental forces don't yet exist at this time, so long story made short, we can't depend on the usual math to help us understand.
Since our Universe was in a different state, it behaved differently then, meaning the Laws themselves were different for at least some passage of time. But as the Universe expanded out and cooled, and the fundamental forces separated into their distinct manifestations, you saw differentiation of energy and mass being given to different types of quarks, resulting in the condensation of matter. From there, the first stars were born, and the rest is history as they say.

However, they expect that roughly equal amounts of matter and antimatter were formed from this condensation event, and when they come into contact, they mutually annihilate, releasing a ton of energy. If you and an antimatter you high fived in outer space, the resulting explosion would make our greatest nuclear explosions look like pop rocks. And the reason that enough matter still exists in the Universe to make up what we see today is, from what I've been told, something to do with the decay rate of matter and antimatter, that they aren't equal, and that because antimatter is far less stable than actual matter, enough of it had decayed away to where matter won out.

If this is something that interests you, I definitely recommend taking a course in Astronomy if you're in college.

Long story short, the Big Bang is a singularly unique singularity which is mathematically no different than the massively expanded universe in the far, far future. Because they are the same, one infinitely expanded universe becomes the infinitely small start of the next.

The mathematics he uses to demonstrate this comparison is called conformal geometry, a math that remains consistent despite working with the cosmically infinite be it infinitely huge or small. Conformal geometry has some advantages, apparently, in that it "squashes" the infinities at the beginning and end of the universe into quantised concepts. It also has some advantages because it allows the Big Bang to occur without the need for Inflation in the very early moments of the universe.

In fact, one of the main theoretical arguments in favour of CCC is that it overcomes some issues with Inflation which the cosmic background radiation map presents, namely some ripples which should not exist with Inflation. CCC explains those ripples as the gravity outcomes of collisions between super massive blackholes or the final 'pops' as blackholes eventually evaporate in the previous universe. Gravity, the mysterious non-force force, traverses from one universe to the next and makes itself apparent in the ripples of the microwave background. He makes predictions about what those ripples would looks like and has a few people supporting him with claims that they see the suspected ripples.

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## Creationist Implications

The Borde-Guth-Vilenkin theorem does not bring into question anything related to the long ages that are called for by the Big bang theory. According to this theory the universe is approximately 14 billion years old. Although what is actually proved by the theorem is that there must be a beginning to any universe that on average is inflationary. So there seem to be two points of discussion within creationism regarding the BGV theorem. One point is of contention, while the second is one that creationists, either supporting YEC or OEC, can fully support.

1. The universe and Earth are old as opposed to young.
2. A universe with inflation has a beginning.

While the former, point 1, is a rich area of debate and disagreement, the latter, point 2, is actually one that creationists can agree with. Creationists can utilize point 2 in order to bridge the gap from the world to natural theology. There can be a two-step approach. Working backwards from point 2, creationists can argue for the existence of God as opposed to trying to prove the age of the universe and Earth. This challenges the popular arguments of atheists. The kalam cosmological argument is particular way to philosophically argue for a space-less, timeless, beginning-less, metaphysically necessary personal being, and all-powerful cause to the beginning of the universe. When expounded upon the result is what looks awfully familiar to theologians as God. This is an argument brought back into the fold of philosophy of religion in the late 20th and early 21st century, and defended successfully in academic debates, books and papers by William Lane Craig. The kalam cosmological argument specifically demonstrates that what follows logically from the premises is that there must be a transcendent cause to physical space-time reality. The BGV theorem can be used as a powerful empirical evidence of a beginning of any universe which on average has a positive expansion rate, of which the universe presently observed does. In establishing a beginning to the universe, and therefore argument for the existence of God the creationist can be comfortable within the the scientific findings of the BGV theorem specifically, and the Big bang theory more generally. Especially when interacting with atheists or even agnostics upon the existence of something like God.

The beginning of the universe needs some length of history. Usually the emphasis by creationists is the age of the universe and Earth first and then argue about existence of God later. This line of argumentation however assumes God in the process and takes an anachronistic stance by positing in their worldview presentation an already existent God. The philosophical context of the kalam cosmological argument demonstrates the necessary existence of God and from this sophisticated standpoint can the creationist then construct chronological theories regarding the actual history of the cosmos having already dealt with what logically should come first, namely a necessary cause to time itself.