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Big Bang singularity in the Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime

Cristi Stoica

University Politehnica of Bucharest, Romania The International Conference of Differential Geometry and Dynamical Systems (DGDS-2013), 10-13 October 2013, University Politehnica of Bucharest, Romania The author thanks Professors C-tin Udri¸ ste, V. Balan, G. Pripoae, O. Simionescu, M. Vi¸ sinescu, P. Fiziev, D.V. Shirkov, for support, valuable discussions and suggestions.

Introduction The Big-Bang Singularity in General Relativity

The Big-Bang Singularity in General Relativity

General Relativity tells us that an expanding universe like ours started with a big-bang singularity (Hawking, 1966a; Hawking, 1966b; Hawking, 1967). It is hoped that when GR will be quantized, this will solve the problem of the big-bang singularity too. Loop quantum cosmology obtained significant positive results in this direction (Bojowald, 2001; Ashtekar & Singh, 2011; Vi¸ sinescu, 2009; Saha & Vi¸ sinescu, 2012) In the following, I will show that even without modifying General Rela- tivity, the big-bang singularity doesn’t break the dynamics, if we formulate it in properly chosen variables.

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Introduction Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime

Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime

If S is a connected three-dimensional Riemannian manifold of constant cur- vature k ∈ {−1, 0, 1} (i.e. H3,R3 or S3), I = (A, B), −∞ ≤ A < B ≤ ∞, and a : I → R is a function, a ≥ 0, then the warped product I ×a S is called a Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime. ds2 = −dt2 + a2(t)dΣ2 dΣ2 = dr2 1 − kr2 + r2 dθ2 + sin2 θdφ2 , where k = 1 for S3, k = 0 for R3, and k = −1 for H3. In general the warping function is taken a ∈ F(I), and a > 0. Here we just want to be smooth, and we let it become 0, to include possible singularities.

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Introduction Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime

Einstein equation

The stress-energy tensor of a fluid having a mass density ρ and pressure density p, as seen in the orthonormal frame of an observer comoving with the fluid, is Tab = (ρ + p) uaub + pgab, where the observer’s 4-velocity ua is the timelike vector field ∂t, normalized. The Einstein equation is Gab + Λgab = κTab, where Λ is the cosmological constant, κ := 8πG c4 (where G = c = 1). Next, we consider Λ = 0, since Λ = 0 reduces to it by the substitution ρ → ρ + κ−1Λ p → p − κ−1Λ.

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Introduction Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime

Friedmann equations

By taking the time component and the trace of Einstein’s equation, we get the Friedmann equation ρ = 3 κ ˙ a2 + k a2 , the acceleration equation ρ + 3p = −6 κ ¨ a a, the fluid equation, expressing the conservation of mass-energy, ˙ ρ = −3 ˙ a a (ρ + p) , We see that ρ, and in general also p, become singular for a = 0.

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Introduction Objective and Methods

Objective

In the following, I will present a description of the big-bang singularity in the FLRW spacetime, which satisfies the conditions It is invariant. It is made in terms of geometric objects that remain finite at the singularity. The dynamics is also expressed in terms of finite quantities. General Relativity is not modified.

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Introduction Objective and Methods

What we need to understand

To do this, some remarks are helpful. The metric tensor is a dynamical entity. Therefore, during its evolution, it can become degenerate. When the metric becomes degenerate, some distances between distinct points may become 0, while they are still distinct. We have to reconsider what quantities have geometric and physical meaning at the singularity.

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Introduction Objective and Methods

Distance separation vs. topological separation

On a manifold which is homeomorphic with a cylinder, we can take a metric metric that makes it look like a cone. But the vertex of the cone is actually a circle of the cylinder, and the metric is degenerate on the circle. Similarly, one should not assume that, at the Big Bang singularity, the entire space was a point, but only that the space metric was degenerate at t = 0.

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Solution Fundamental objects

Scalar or 4-form?

ds2 = −dt2 + a2(t)dΣ2 The fields ρ and p are defined in an orthonormal frame. But when a = 0, the metric is degenerate, and there is no orthonormal frame. The volume form, or volume element, is dvol := √−gdt ∧ dx ∧ dy ∧ dz = a3√gΣdt ∧ dx ∧ dy ∧ dz. To obtain the total mass at t, one integrates the 3-form ρdvol3, where dvol3 := √gΣtdx ∧ dy ∧ dz = a3√gΣdx ∧ dy ∧ dz = i∂t dvol. Hence, it’s natural to rewrite the equations using 4-forms or scalar densities ρ√−g and p√−g, which luckily can be defined in any coordinates/frames. By doing this, the equations become independent on the condition that the frame is orthonormal, and are smooth.

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Solution Formulation in terms of finite quantities

Friedmann equations, densitized

Let’s rewrite the Friedmann equations using the substitution ρ = ρ√−g = ρa3√gΣ,

• p = p√−g = pa3√gΣ.

The Friedmann equation becomes

• ρ = 3

κa

• ˙

a2 + k √gΣ. The acceleration equation becomes

• ρ + 3

p = −6 κa2¨ a√gΣ. Hence, ρ and p are smooth, as it is the densitized stress-energy tensor Tab √−g = ( ρ + p) uaub + pgab.

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Solution Formulation in terms of finite quantities

The Einstein equation, densitized

Because Tab √−g is smooth, it follows that the Einstein tensor density Gab √−g is smooth too, and the following densitized Einstein equation, Gab √−g + Λgab √−g = κTab √−g, is smooth. We don’t have to change General Relativity to obtain it, since the Hilbert- Einstein Lagrangian density R√−g − 2Λ√−g already contains √−g. When deriving the Einstein equation, we should not divide by √−g, because this can be 0. Therefore, the densitized Einstein equation

• ccurs naturally in GR,

allow us to get back to the Einstein equation, if √−g = 0, but is valid also in regimes where the Einstein equation is singular.

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Solution Conclusions

Finite Friedmann-Lemaˆ ıtre-Robertson-Walker Big-Bang

In conclusion, although the Friedmann and Einstein equations are singular at the big-bang, if we rewrite them using tensor densities, we get rid of the infinities.

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Solution Conclusions

FLRW Big Bang

Big Bang singularity, corresponding to a(0) = 0, ˙ a(0) > 0.

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Solution Conclusions

FLRW Big Bounce

Big Bounce, corresponding to a(0) = 0, ˙ a(0) = 0, ¨ a(0) > 0.

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What is the meaning of this?

What is the meaning of this?

We have seen that the Friedmann and Einstein equations can be expressed in terms of quantities that remain finite and smooth at the big-bang. Is this a lucky coincidence, or it reflects something more general?

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General context Definition of singular semi-Riemannian manifolds

Singular semi-Riemannian manifolds

Definition A singular semi-Riemannian manifold is a pair (M, g), where M is a differentiable manifold, and g is a symmetric bilinear form on M, named metric tensor or metric. Constant signature: the signature of g is fixed. Variable signature: the signature of g varies from point to point. If g is non-degenerate, then (M, g) is a semi-Riemannian manifold. If g is positive definite, (M, g) is a Riemannian manifold.

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General context Definition of singular semi-Riemannian manifolds

What is wrong with singularities

1 For PDE on curved spacetimes: the covariant derivatives blow up:

Γcab = 1 2gcs(∂agbs + ∂bgsa − ∂sgab)

2 For Einstein’s equation blows up in addition because it is expressed in

terms of the curvature, which is defined in terms of the covariant derivative: Rd abc = Γd ac,b − Γd ab,c + Γd bsΓsac − Γd csΓsab Gab = Rab − 1 2Rgab Rab = Rsasb, R = gpqRpq Even if gab are all finite, these equations are also in terms of gab, and gab → ∞ when det g → 0.

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General context Definition of singular semi-Riemannian manifolds

What are the non-singular objects? (Stoica, 2011a)

Some quantities which are part of the equations are indeed singular, but this is not a problem if we use instead other quantities, equivalent to them when the metric is non-degenerate. Singular Non-Singular When g is... Γcab (2-nd) Γabc (1-st) smooth Rd abc Rabcd semi-regular Rab Rab

• |det g|

W , W ≤ 2

semi-regular R R

• |det g|

W , W ≤ 2

semi-regular Ric Ric ◦ g quasi-regular R Rg ◦ g quasi-regular

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General context Definition of singular semi-Riemannian manifolds

(V,g) V*

u u+w w

(V●,g●) (V●,g●)

V●=V/V○ u●

(V , g) is an inner product vector space. The morphism ♭ : V → V ∗ is defined by u → u• := ♭(u) = u♭ = g(u, ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . V • := im ♭ ≤ V ∗ is the image of ♭. The inner product g induces on V • an inner product defined by g •(u♭

1, u♭ 1) := g(u1, u2), which is the inverse of g iff det g = 0.

The quotient V • := V /V ◦ consists in the equivalence classes of the form u + V ◦. On V •, g induces an inner product g •(u1 + V ◦, u2 + V ◦) := g(u1, u2). (Stoica, 2011b)

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General context Definition of singular semi-Riemannian manifolds

The Koszul object

The Koszul object is defined as K : X(M)3 → R, K(X, Y , Z) := 1 2{XY , Z + Y Z, X − ZX, Y −X, [Y , Z] + Y , [Z, X] + Z, [X, Y ]}. In local coordinates it is the Christoffel’s symbols of the first kind: Kabc = K(∂a, ∂b, ∂c) = 1 2(∂agbc + ∂bgca − ∂cgab) = Γabc, For non-degenerate metrics, the Levi-Civita connection is obtained uniquely: ∇XY = K(X, Y , )♯.

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General context Definition of singular semi-Riemannian manifolds

The covariant derivatives (Stoica, 2011a)

The lower covariant derivative of a vector field Y in the direction of a vector field X: (∇♭

XY )(Z) := K(X, Y , Z)

The covariant derivative of differential forms: (∇Xω) (Y ) := X (ω(Y )) − g•(∇♭

XY , ω),

∇X(ω1 ⊗ . . . ⊗ ωs) := ∇X(ω1) ⊗ . . . ⊗ ωs + . . . + ω1 ⊗ . . . ⊗ ∇X(ωs) (∇XT) (Y1, . . . , Yk) = X (T(Y1, . . . , Yk)) − k

i=1 K(X, Yi, •)T(Y1, , . . . , •, . . . , Yk)

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General context Definition of singular semi-Riemannian manifolds

Semi-regular manifolds. Riemann curvature tensor (Stoica, 2011a)

A semi-regular semi-Riemannian manifold is defined by the condition ∇X∇♭

Y Z ∈ A•(M).

Equivalently, K(X, Y , •)K(Z, T, •) ∈ F(M). Riemann curvature tensor: R(X, Y , Z, T) = (∇X∇♭

Y Z)(T) − (∇Y ∇♭ XZ)(T) − (∇♭ [X,Y ]Z)(T)

Rabcd = ∂aKbcd − ∂bKacd + (Kac•Kbd• − Kbc•Kad•) Is a tensor field. Has the same symmetry properties as for det g = 0. It is radical-annihilator. It is smooth for semi-regular metrics.

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The mathematics of singularities Examples of semi-regular semi-Riemannian manifolds

Examples of semi-regular semi-Riemannian manifolds (Stoica, 2011a; Stoica, 2011c)

Isotropic singularities: g = Ω2˜ g. Degenerate warped products (f allowed to vanish): ds2 = ds2

B + f 2(p)ds2 F.

FLRW spacetimes are degenerate warped products: ds2 = −dt2 + a2(t)dΣ2 dΣ2 = dr2 1 − kr2 + r2 dθ2 + sin2 θdφ2 , where k = 1 for S3, k = 0 for R3, and k = −1 for H3.

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Einstein’s equation on semi-regular spacetimes

Einstein’s equation on semi-regular spaces (Stoica, 2011a)

On 4D semi-regular spacetimes Einstein tensor density G det g is smooth. At the points p where the metric is non-degenerate, the Einstein tensor can be expressed by: G ab det g = gklǫakstǫblpqRstpq, where ǫabcd is the Levi-Civita symbol. Therefore, Gab det g is smooth too, and it makes sense to write a densitized version of Einstein’s equation Gab det g + Λgab det g = κTab det g, where κ := 8πG c4 , G and c being Newton’s constant and the speed of light. In many cases, the densitized Einstein equation works even with Gab √det g. It is not allowed to divide by det g, when det g = 0.

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Einstein’s equation on semi-regular spacetimes

What is the meaning of this?

In (Stoica, 2011a; Stoica, 2011c), it is developed the geometry of manifolds endowed with metrics which are not necessarily non-degenerate everywhere. A special type of singularities, named semi-regular, are shown to have nice properties. They admit covariant derivatives or lower covariant derivatives for the fields that are important, and the Riemann curvature tensor Rabcd is smooth. They satisfy a densitized Einstein equation. In the case of stationary black holes, the singularity r = 0 is due to a combination of the fact that the coordinates are singular (just like in the case of the event horizon), and the metric is degenerate. In (Stoica, 2012f; Stoica, 2012a; Stoica, 2013a) it is shown how we can remove the coordinate singularity, making gab finite, and analytic at r = 0. In (Stoica, 2012d) it is shown that black hole singularities are compatible with global hyperbolicity, which is required to restore the conservation of information.

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Einstein’s equation on semi-regular spacetimes

What is the meaning of this?

In (Stoica, 2012b), a class of semi-regular singularities having good behavior is identified. Let’s call them quasi-regular. In (Stoica, 2013b), it is shown that quasi-regular singularities satisfy the Weyl curvature hypothesis (Penrose, 1979). A large class of cosmological models with big-bang that is not necessarily isotropic and homogeneous is identified. In (Stoica, 2012e; Stoica, 2012c) is shown that semi-regular and quasi-regular singularities are accompanied by dimensional reduction, and connections with various approaches to perturbative quantum gravity are presented.

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Thank you!

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Singular General Relativity My papers in Singular General Relativity

• On Singular Semi-Riemannian Manifolds, arXiv:1105.0201
• Warped Products of Singular Semi-Riemannian Manifolds, arXiv:1105.3404
• Schwarzschild Singularity is Semi-Regularizable, Eur. Phys. J. Plus (2012)

127: 83, arXiv:1111.4837

• Analytic Reissner-Nordstrom Singularity, Phys. Scr. 85 (2012) 055004,

arXiv:1111.4332

• Kerr-Newman Solutions with Analytic Singularity and no CTCs, To appear

in U.P.B. Sci. Bull., Series A, arXiv:1111.7082

• Spacetimes with Singularities, An. S

¸t. Univ. Ovidius Constant ¸a (2012)

• vol. 20(2), 213-238, 1108.5099
• Beyond the Friedmann-Lemaitre-Robertson-Walker Big Bang singularity,
• Commun. Theor. Phys. 58(4) (2012), 613-616, arXiv:1203.1819
• Einstein equation at singularities, arXiv:1203.2140
• On the Weyl Curvature Hypothesis, Annals of Physics, Vol. 338, Novem-

ber 2013, Pages 186-194, arXiv:1203.3382

• Quantum Gravity from Metric Dimensional Reduction at Singularities,

arXiv:1205.2586

• An Exploration of the Singularities in General Relativity, J.I.N.R., Dubna,

arXiv:1207.5303

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References

Ashtekar, A., & Singh, P. 2011. Loop quantum cosmology: a status report. Classical and Quantum Gravity, 28(21), 213001–213122. arXiv:gr-qc/1108.0893. Bojowald, M. 2001. Absence of a Singularity in Loop Quantum Cosmology. Physical Review Letters, 86(23), 5227–5230. arXiv:gr-qc/0102069. Hawking, S. W. 1966a. The occurrence of singularities in cosmology. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 294(1439), 511–521. Hawking, S. W. 1966b. The occurrence of singularities in cosmology. II. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 295(1443), 490–493.

Cristi Stoica (h o l o t r o n i x @ g m a i l . c o m) Big Bang singularity in the FLRW spacetime October 11, 2013 28 / 28

References

Hawking, S. W. 1967. The occurrence of singularities in cosmology. III. Causality and singu- larities. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 300(1461), 187–201. Penrose, R. 1979. Singularities and time-asymmetry. Pages 581–638 of: General relativity: an Einstein centenary survey, vol. 1. Saha, B., & Vi¸ sinescu, M. 2012. Bianchi type-I string cosmological model in the presence of a magnetic field: classical versus loop quantum cosmology approaches. Astrophysics and Space Science, 339(2), 371–377. Stoica, O. C. 2011a. On Singular Semi-Riemannian Manifolds. Arxiv preprint math.DG/1105.0201, May.

Cristi Stoica (h o l o t r o n i x @ g m a i l . c o m) Big Bang singularity in the FLRW spacetime October 11, 2013 28 / 28

References

arXiv:math.DG/1105.0201. Stoica, O. C. 2011b. Tensor Operations on Degenerate Inner Product Spaces. Arxiv preprint gr-qc/1112.5864, December. arXiv:gr-qc/1112.5864. Stoica, O. C. 2011c. Warped Products of Singular Semi-Riemannian Manifolds. Arxiv preprint math.DG/1105.3404, May. arXiv:math.DG/1105.3404. Stoica, O. C. 2012a. Analytic Reissner-Nordstr¨

• m Singularity.
• Phys. Scr., 85(5), 055004.

arXiv:gr-qc/1111.4332. Stoica, O. C. 2012b. Einstein Equation at Singularities. Arxiv preprint gr-qc/1203.2140, March.

Cristi Stoica (h o l o t r o n i x @ g m a i l . c o m) Big Bang singularity in the FLRW spacetime October 11, 2013 28 / 28

References

arXiv:gr-qc/1203.2140. Stoica, O. C. 2012c. An Exploration of the Singularities in General Relativity. Seminar held at JINR, Dubna, May. arXiv:gr-qc/1207.5303. Stoica, O. C. 2012d. Spacetimes with Singularities.

• An. S

¸t. Univ. Ovidius Constant ¸a, 20(2), 213–238. arXiv:gr-qc/1108.5099. Stoica, O. C. 2012e. Quantum Gravity from Metric Dimensional Reduction at Singularities. Arxiv preprint gr-qc/1205.2586, May. arXiv:gr-qc/1205.2586. Stoica, O. C. 2012f. Schwarzschild Singularity is Semi-Regularizable.

Cristi Stoica (h o l o t r o n i x @ g m a i l . c o m) Big Bang singularity in the FLRW spacetime October 11, 2013 28 / 28

Singular General Relativity My papers in Singular General Relativity

• Eur. Phys. J. Plus, 127(83), 1–8.

arXiv:gr-qc/1111.4837. Stoica, O. C. 2013a. Kerr-Newman Solutions with Analytic Singularity and no Closed Time- like Curves. To appear in U.P.B. Sci. Bull., Series A. arXiv:gr-qc/1111.7082. Stoica, O. C. 2013b. On the Weyl Curvature Hypothesis.

• Ann. of Phys., 338(November), 186–194.

arXiv:gr-qc/1203.3382. Vi¸ sinescu, M. 2009. Bianchi type-I string cosmological model in the presence of a magnetic field: classical and quantum loop approach. Romanian Reports in Physics, 61(3), 427–435.

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