Singular General Relativity Q Singular General Relativity Is the Application of the a = − Dt Methods of Singular Semi-Riemannian Geometry to R General Relativity

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Singular General 3 Results: Smoothening black hole Relativity singularities How I learned to stop worrying and love Apparently, the black hole singularities are malign, the singularities because some of the components of the metric are divergent (gab → ∞). For the standard stationary Cristi Stoica ([email protected]) black holes, we can ﬁnd transformations of coordinates which make them smooth. This also makes, for Relativity and Gravitation charged black holes, the electromagnetic potential and 100 Years after Einstein in Prague ﬁeld smooth. The metric is put in a form which allows June 25 – 29, 2012, Prague, Czech Republic the evolution equations to go beyond the singularities.

1 Introduction: Do the singularities predict the breakdown of General Relativity?

It is often said that General Rela- Malign singularities Benign singularities tivity predicts its own breakdown, gab → ∞ gab smooth, det g → 0 by predicting the occurrence of singularities. In addition, when trying 3.1 Schwarzschild black hole to quantize gravity, it appears to be perturbatively nonrenormalizable. In Schwarzschild coordinates • Are these problems signs that we should give up General Relativity in exchange for more radical ap- r − 2m r ds2 = − dt2 + dr2 + r2dσ2, proaches (superstrings, loop quantum gravity etc.)? r r − 2m • Are these limits of GR, or of our tools? • What if understanding the singularities also helps where dσ2 = dθ2 + sin2 θdφ2. with the problem of quantization?1 In non-singular coordinates [5] 4τ 4 ds2 = − dτ 2 2m − τ 2 2 The cure for singularities +(2m − τ 2)τ 2T −4 (T ξdτ + τdξ)2 + τ 4dσ2, where (r, t) 7→ (τ 2, ξτ T ), T ≥ 2. 2.1 Singular semi-Riemannian geometry

The methods of singular semi-Riemannian geometry 3.2 Reissner-Nordströmblack hole (see [2, 3] and the appendix) allow us to: • find mathematical descriptions of singularities, In Reissner-Nordströmcoordinates and understand them • replace the singular quantities with regular ones, which are equivalent to them ∆ r2 ds2 = − dt2 + dr2 + r2dσ2 if det g 6= 0 • write non-singular field equations, for r2 ∆ example extensions of Einstein’s equations where ∆ = r2 − 2mr + q2. The electromagnetic potential is singular: 2.2 Singular General Relativity q Singular General Relativity is the application of the A = − dt methods of singular semi-Riemannian geometry to r General Relativity. For this, the metric is required to be benign, but often a metric which is apparently In non-singular coordinates [6] 2 2T −2S−2 2 malign can be put in a form which is benign, by a ds = −∆ρ (ρdτ + T τdρ) S2 coordinate change. + ρ4S−2dρ2 + ρ2Sdσ2, • Malign singularity: g → ∞. Benign singularity: ∆ ab where (t, r) 7→ (τρT , ρS), T > S ≥ 1. g is smooth and det g = 0. The electromagnetic potential is non-singular: 1Usually is considered that when GR will be quantized, this will solve the singularities too, by showing probably that quan- A = −qρT −S−1 (ρdτ + T τdρ) tum fields prevent the occurrence of singularities. Here we will explore the opposite view.

1 1 4 Information paradox where Sab := Rab − 4 Rgab, and Penrose-Carter diagrams for non-rotating and electri- (h ◦ k)abcd := hackbd − hadkbc + hbdkac − hbckad. cally neutral evaporating black holes. In this case we can write the expanded Einstein equation [11]:

(G ◦ g)abcd + Λ(g ◦ g)abcd = κ(T ◦ g)abcd.

The FLRW spacetime is the warped product I ×a Σ, a : I → R, a ≥ 0, with metric

ds2 = −dt2 + a2(t)dΣ2

3 3 3 where usually Σ is S , R , or H [3]. The energy density ρ and pressure density p are singular at the Big-Bang, where a(t) = 0: A. If the singularity is malign, entering information is lost. When accounting for Quantum Mechanics, this 3 a˙ 2 + k 6 a¨ ρ = , ρ + 3p = − . causes in particular a violation of unitarity, if a system κ a2 κ a which is entangled with another system is lost in the singularity. B. If the singularity is benign, the ﬁeld But in terms of the correct densities equations can be extended beyond the singularity, and √ √ the information is preserved [5–8]. ρe = ρ −g, pe = p −g,

5 Einstein equation at singularities the equations become smooth: 3 √ 6 √ Singular semi-Riemannian geometry (see appendix ρ = a a˙ 2 + k g , 3p + ρ = − a2a¨ g . e κ Σ e e κ Σ and [2,3]) introduces generalizations of semi-Riemann- ian spacetimes which admit degenerate metric but Hence, ρe and pe are smooth, as it is the densitized have smooth Riemann curvature Rabcd. But the Ein- stress-energy tensor stein tensor is usually singular. Yet, in 4D we can rewrite the Einstein equation in terms of quantities √ T −g = (ρ + p) u u + pg . which remain smooth at singularities. The resulting ab e e a b e ab equations are equivalent to Einstein’s so long as the These singularities are quasi-regular [9, 10]. metric is regular, but work as well at singularities.

5.1 Densitized Einstein equation On 4D semi-regular spacetimes the Riemann curvature Rabcd is smooth. Hence the Einstein tensor density G det g is smooth too, being:

kst lpq Gab det g = gkla n Rstpq. We can write a densitized version of the Einstein equation: Gab det g + Λgab det g = κTab det g, 6 Weyl curvature hypothesis For regular metric it is equivalent to Einstein’s equation, but works at singularities too [2]. The Weyl curvature hypothesis was proposed by R. Penrose, to account for the high homogeneity and low entropy characterizing the Big-Bang. It states that at 5.2 Expanded Einstein equation the Big-Bang singularity, the Weyl curvature tensor A quasi-regular spacetime is a 4D semi-regular mani- Cabcd = 0. fold with smooth Ricci decomposition At a quasi-regular Big-Bang singularity, Cabcd is 1 1 smooth. At the singularity it vanishes, because it lives Rabcd = R(g ◦ g)abcd + (S ◦ g)abcd + Cabcd in T •M, which has dimension ≤ 3 at singularities [12]. 12 2

2 7 Dimensional reduction 1 Gab = Rab − Rgab and Quantum Gravity 2 These quantities are not usually defined even if gab are ab Different results suggest that a dimension < 4 may act finite, since g → ∞ when det g → 0. like a dimensional regulator for QFT and for QG. In lower dimension, the Weyl tensor vanishes, and the lo- A.2 The idea of a solution cal degrees of freedom (gravitational waves and gravi- Use non-singular quantities, equivalent to the singular tons) disappear, allowing QG to be renormalizable. ones for non-degenerate metric g. Benign singularities undergo a dimensional reduc- Singular Non-Singular When g is... tion, and may be the needed dimensional regularizers Γc (2nd) Γ (1st) smooth for QG: ab abc Rd R semi-regular • g is independent on some directions. abc abcd √ ab p W • − det g → 0, reducing the contribution of lower Rab Rab | det g| ,W ≤ 2 semi-regular W distance Feynman integrals. R Rp| det g| ,W ≤ 2 semi-regular • The admissible fields live in lower dimension spaces: Rab Ric ◦ g quasi-regular R Rg ◦ g quasi-regular rank g = dim T M = dim T •M < 4. p p• p But how to define them?

• For quasi-regular singularities Cabcd = 0. A.3 Degenerate metric and The quantities det g and Cabcd vanish as approaching covariant contraction singularities. But QG needs them to decrease with the (V,g) V* scale. They may do this: As the energy approaches the UV limit, the num- ber of the particles in Feynman diagrams increases. u+w w (V●,g ) If particles are benign singularities, this means lower u ● det g and Cabcd. We conjecture: this gives the needed regularization [13].

V●=V/V○ ● (V●,g )

u● (V, g) is an inner product vector space. The mor- phism [ : V → V ∗ is defined by u 7→ 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 • [ [ A Singular semi-Riemannian an inner product defined by g•(u1, u1) := g(u1, u2), Geometry which is the inverse of g iff det g 6= 0. The quotient V • := V/V ◦ consists in the equivalence classes of the • A singular semi-Riemannian manifold (M, g) is a form u + V ◦. On V •, g induces an inner product • differentiable manifold M with a symmetric bilinear g (u1 + V ◦, u2 + V ◦) := g(u1, u2). form g, named metric, on the tangent bundle TM. • Once we have defined the reciprocal inner product The metric g may be indefinite, and may be degenerate g•(ω, τ), we can define covariant contraction for • (i.e. det g = 0 at some points). It may have constant 1-forms from V . Then we define it for tensors with • or variable signature. two covariant indices living in V . • Malign singularity: gab → ∞. Benign singularity: g is smooth and det g = 0. T (ω1,..., •,..., •, . . . , ωk) A.1 The problem • We extend these definitions to a singular semi-Rie- The geometric quantities can’t be defined: mannian manifold (M, g). 1 c cs i π Γ ab = g (∂agbs + ∂bgsa − ∂sgab) ◦ • • 2 0 T ◦pM (TpM, g) (V •, g ) 0 [ T d d d d s d s p M [ ] R abc = Γ ac,b − Γ ab,c + Γ bsΓ ac − Γ csΓ ab ◦ • ◦ π ∗ i • s pq 0 T pM Tp M (T pM, g•) 0 Rab = R asb,R = g Rpq

3 A.4 Covariant derivative The usual definition of warped product requires (B, g ) and (F, g ) to be non-degenerate, and f > 0. • The covariant derivative ∇ Y can’t be defined. We B F X Here we allow it to be f ≥ 0, including by this possible can use instead the Koszul form: 1 singularities. K(X,Y,Z) := {XhY,Zi + Y hZ,Xi − ZhX,Y i 2 • If (B, gB) and (F, gF ) are radical-stationary and • −hX, [Y,Z]i + hY, [Z,X]i + hZ, [X,Y ]i} df ∈ A (B), the warped product manifold B ×f F is ] • For non-degenerate metric ∇X Y = K(X,Y, ) . radical-stationary. • The Christoffel’s symbols of the first kind are • If (B, gB) and (F, gF ) are semi-regular, and df ∈ 1 •1 Γ = K(∂ , ∂ , ∂ ) = (∂ g + ∂ g − ∂ g ) A (B), the warped product manifold B ×f F is semi- abc a b c 2 a bc b ca c ab • The lower covariant derivative: regular. [ • Example of warped product: the Friedmann- (∇X Y )(Z) := K(X,Y,Z), for any Z ∈ X(M). A singular semi-Riemannian manifold is radical- Lemaˆıtre-Robertson-Walker spacetime. stationary if K(X,Y, ) ∈ A•(M) := Γ(T •M). • Covariant derivative of covariant indices: References [ (∇X ω)(Y ) := X (ω(Y )) − g•(∇X Y, ω) [1] C. Stoica. Tensor Operations on Degenerate Inner (∇X T )(Y1,...,Yk) = X (T (Y1,...,Yk)) Pk Product Spaces . arXiv:gr-qc/1112.5864, December − i=1 K(X,Yi, •)T (Y1,,..., •,...,Yk) 2011. • A singular semi-regular manifold is a radical- [ • [2] C. Stoica. On Singular Semi-Riemannian Manifolds. stationary manifold so that ∇X ∇Y Z ∈ A (M). • Isotropic singularities. A manifold (M, Ω2g˜), where arXiv:math.DG/1105.0201, May 2011. g˜ is a non-degenerate metric, and Ω a smooth function, [3] C. Stoica. Warped Products of Singular Semi- is semi-regular. Riemannian Manifolds. arXiv:math.DG/1105.3404, May 2011. A.5 Curvature [4] C. Stoica. Cartan’s Structural Equations for Degen- erate Metric. arXiv:math.DG/1111.0646, November Let (M, g) be radical-stationary. 2011. • The lower Riemann curvature operator is: [ [ [ [ [5] C. Stoica. Schwarzschild Singularity is Semi- RXY Z := ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z Regularizable. arXiv:gr-qc/1111.4837, November • The Riemann curvature tensor is: 2011. R(X,Y,Z,T ) := (R[ Z)(T ) XY [6] C. Stoica. Analytic Reissner-Nordstrom Singularity. R = ∂ Γ − ∂ Γ + (Γ Γ − Γ Γ ) abcd a bcd b acd ac• bd• bc• ad• Phys. Scr., 85:055004, November 2012. arXiv:gr- • The Ricci tensor: Ric(X,Y ) := R(X, •,Y, •) qc/1111.4332. • The scalar curvature s := Ric(•, •). • If (M, g) is semi-regular, the Riemann curvature [7] C. Stoica. Kerr-Newman Solutions with Analytic Sin- gularity and no Closed Timelike Curves. arXiv:gr- is a smooth tensor field, but the Ricci and scalar cur- qc/1111.7082, November 2011. vatures may be singular. [8] C. Stoica. Globally Hyperbolic Spacetimes with Sin- gularities. arXiv:math.DG/1108.5099, August 2011. A.6 Singular warped products [9] C. Stoica. Big Bang singularity in the Friedmann- The warped product of two singular semi-Riemannian Lemaitre-Robertson-Walker spacetime. arXiv:gr- manifolds (B, gB) and (F, gF ), with warping function qc/1112.4508, December 2011. f ∈ F (B) is the singular semi-Riemannian manifold [10] C. Stoica. Beyond the Friedmann-Lemaitre-Robertson- ∗ ∗ Walker Big Bang singularity. arXiv:gr-qc/1203.1819, B ×f F := B × F, πB(gB) + (f ◦ πB)πF (gF ) , March 2012. where πB : B × F → B and πF : B × F → F are the [11] C. Stoica. Einstein equation at singularities. arXiv:gr- canonical projections. qc/1203.2140, March 2012. The inner product on B ×f F takes the form [12] C. Stoica. On the Weyl Curvature Hypothesis. arXiv:gr-qc/1203.3382, March 2012. ds2 = ds2 + f 2ds2 . B F [13] C. Stoica. Quantum Gravity from Metric Dimensional Reduction at Singularities. arXiv:gr-qc/1205.2586 , May 2012.

Cristi Stoica [email protected]