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. 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 2 / 28
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. H 3 , R 3 or S 3 ), 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. d s 2 = − d t 2 + a 2 ( t )dΣ 2 1 − kr 2 + r 2 � d r 2 d θ 2 + sin 2 θ d φ 2 � dΣ 2 = , where k = 1 for S 3 , k = 0 for R 3 , and k = − 1 for H 3 . 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. 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 3 / 28
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 T ab = ( ρ + p ) u a u b + pg ab , where the observer’s 4-velocity u a is the timelike vector field ∂ t , normalized. The Einstein equation is G ab + Λ g ab = κ T ab , where Λ is the cosmological constant, κ := 8 π G (where G = c = 1). c 4 Next, we consider Λ = 0, since Λ � = 0 reduces to it by the substitution � ρ ρ + κ − 1 Λ → p − κ − 1 Λ . p → 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 4 / 28
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 a 2 + k ρ = 3 ˙ , a 2 κ the acceleration equation ρ + 3 p = − 6 a ¨ a , κ 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. 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 5 / 28
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 . 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 6 / 28
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. 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 7 / 28
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. 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 8 / 28
Solution Fundamental objects Scalar or 4-form? d s 2 = − d t 2 + a 2 ( t )dΣ 2 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 d vol := √− g d t ∧ d x ∧ d y ∧ d z = a 3 √ g Σ d t ∧ d x ∧ d y ∧ d z . To obtain the total mass at t , one integrates the 3 -form ρ d vol 3 , where d vol 3 := √ g Σ t d x ∧ d y ∧ d z = a 3 √ g Σ d x ∧ d y ∧ d z = i ∂ t d vol . 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 . 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 9 / 28
Solution Formulation in terms of finite quantities Friedmann equations, densitized Let’s rewrite the Friedmann equations using the substitution � � ρ = ρ √− g = ρ a 3 √ g Σ , p = p √− g = pa 3 √ g Σ . � The Friedmann equation becomes � � √ g Σ . ρ = 3 a 2 + k κ a ˙ � The acceleration equation becomes p = − 6 a √ g Σ . κ a 2 ¨ ρ + 3 � � Hence, � ρ and � p are smooth, as it is the densitized stress-energy tensor √− g = ( � T ab ρ + � p ) u a u b + � pg ab . 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 10 / 28
Solution Formulation in terms of finite quantities The Einstein equation, densitized √− g is smooth, it follows that the Einstein tensor density Because T ab √− g is smooth too, and the following densitized Einstein equation , G ab √− g + Λ g ab √− g = κ T ab √− g , G ab 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 occurs 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. 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 11 / 28
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. 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 12 / 28
Solution Conclusions FLRW Big Bang Big Bang singularity, corresponding to a (0) = 0, ˙ a (0) > 0. 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 13 / 28
Solution Conclusions FLRW Big Bounce Big Bounce, corresponding to a (0) = 0, ˙ a (0) = 0, ¨ a (0) > 0. 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 14 / 28
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? 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 15 / 28
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 . 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 16 / 28
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