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Einstein Equation at Singularities Cent. Eur. J. Phys. • 12(2) • 2014 • 123-131 DOI: 10.2478/s11534-014-0427-1 Central European Journal of Physics Einstein equation at singularities Research Article Ovidiu-Cristinel Stoica1∗ 1 University Politehnica of Bucharest, Faculty of Applied Sciences, Splaiul Independentei nr. 313, sector 6, Bucharest, RO-060042, Romania Received 3 June 2013; accepted 5 December 2013 Abstract: Einstein’s equation is rewritten in an equivalent form, which remains valid at the singularities in some ma- jor cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein’s equation and the metric tensor). PACS (2008): 04.20.-q; 04.70.Bw; 98.80.Bp Keywords: Einstein equations • general relativity • singularities • cosmology • black holes © Versita sp. z o.o. Introduction Let (M; g) be a Riemannian or a semi-Riemannian man- ifold of dimension n. It is useful to recall the definition of the Kulkarni-Nomizu product of two symmetric bilinear The singularities in General Relativity can be avoided forms h and k, only if the stress-energy tensor in the right hand side of Einstein’s equation satisfies some particular conditions. (h ◦ k)abcd := hackbd − hadkbc + hbdkac − hbckad: (1) One way to avoid them was proposed by the authors of [1], who have shown that the singularities can be removed by constructing the stress-energy tensor with non-linear The Riemann curvature tensor can be decomposed alge- electrodynamics. On the other hand, Einstein’s equation braically as leads to singularities in general conditions [2–7], and there the time evolution breaks down. Is this a problem of the Rabcd = Sabcd + Eabcd + Cabcd; (2) theory itself or of the way it is formulated? This paper proposes a version of Einstein’s equation which where is equivalent to the standard version at the points of 1 Sabcd = R(g ◦ g)abcd (3) spacetime where the metric is non-singular. But unlike 2n(n − 1) Einstein’s equation, in many cases it can be extended at is the scalar part of the Riemann curvature and and beyond the singular points. 1 ∗ Eabcd = (S ◦ g)abcd (4) E-mail: [email protected] n − 2 123 Brought to you by | CERN library Authenticated Download Date | 10/4/17 1:56 PM Einstein equation at singularities ∗ is the semi-traceless part of the Riemann curvature. Here where is the Hodge duality operation. It can be obtained by contracting the semi-traceless part of the Riemann ten- 1 sor Sab := Rab − Rgab (5) κ ∗ ∗ n Eabcd = − (FabFcd + Fab Fcd) : (10) 8π is the traceless part of the Ricci curvature. Therefore it is natural to at least consider an equation in The Weyl curvature tensor is defined as the traceless part terms of these fourth-order tensors, rather than the Ricci of the Riemann curvature and scalar curvatures. The main advantage of this method is that there are sin- gularities in which the new formulation of the Einstein Cabcd = Rabcd − Sabcd − Eabcd: (6) equation is not singular (although the original Einstein equation exhibits singularities, obtained when contract- ab The Einstein equation is ing with the singular tensor g ). The expanded Einstein equation is written in terms of the smooth geometric ob- Gab + Λgab = κTab; (7) jects Eabcd and Sabcd. Because of this the solutions can be extended at singularities where the original Einstein where Tab is the stress-energy tensor of the matter, the equation diverges. This doesn’t mean that the singular- 8πG ities are removed; for example the Kretschmann scalar constant κ is defined as κ := , where G and c are the abcd c4 RabcdR is still divergent at some of these singularities. gravitational constant and the speed of light, and Λ is the But this is not a problem, since the Kretschmann scalar cosmological constant . The term is not part of the evolution equation. It is normally used as an indicator that there is a singularity, for example to 1 r Gab := Rab − Rgab (8) prove that the Schwarzschild singularity at = 0 cannot 2 be removed by coordinate changes, as the event horizon singularity can. While a singularity of the Kretschmann Ricci curvature is the Einstein tensor, constructed from the scalar indicates the presence of a singularity of the cur- R gst R scalar curvature R gst R ab := asbt and the := st . vature, it doesn’t have implications on whether the singu- As it is understood, the Einstein equation establishes the larity can be resolved or not. In the proposed equation connection between curvature and stress-energy. The cur- we use Rabcd which is smooth at the studied singularities, abcd vature contributes to the equation in the form of the Ricci and we don’t use R which is singular and causes the R tensor ab and the scalar curvature. In the proposed equa- singularity of the Kretschmann scalar. tion, the curvature contributes in the form of the semi- A second reason to consider the expanded version of the E traceless and scalar parts of the Riemann tensor, abcd Einstein equation and the quasi-regular singularities at S (4) and abcd (3), which are tensors of the same order and which it is smooth is that at these singularities the Weyl R have the same symmetries as abcd. curvature tensor vanishes. The implications of this feature R The Ricci tensor ab is obtained by contracting the tensor will be explored in [8]. E S abcd+ abcd, and has the same information (if the metric is It will be seen that there are some important examples of non-degenerate). One can move from the fourth-order ten- singularities which turn out to be quasi-regular. While E S R sors abcd + abcd to ab by contraction, and one can move singularities still exist, our approach provides a descrip- back to them by taking the Kulkarni-Nomizu product (1), tion in terms of smooth geometric objects which remain g but they are equivalent. Yet, if the metric ab is degener- finite at singularities. By this we hope to improve our gab R gst E S ate, then and the contraction ab = ( asbt + asbt ) understanding of singularities and to distinguish those to g E S become divergent, even if ab, abcd, and abcd are smooth. which our resolution applies. E S This suggests the possibility that abcd and abcd are more The expanded Einstein equations and the quasi-regular fundamental that the Ricci and scalar curvatures. spacetimes on which they hold are introduced in section1. This suggestion is in agreement with the following obser- They are obtained by taking the Kulkarni-Nomizu prod- electrovac F vation. In the case of solutions, where ab is uct between Einstein’s equation and the metric tensor. In the electromagnetic tensor, a quasi-regular spacetime the metric tensor becomes de- generate at singularities in a way which cancels them and 1 1 st s makes the equations smooth. Tab = gabFst F − FasFb 4π 4 The situations when the new version of Einstein’s equa- 1 c ∗ ∗ c tion extends at singularities include isotropic singularities = − (FacFb + Fac Fb ) ; (9) 8π (section 2.1) and a class of warped product singularities 124 Brought to you by | CERN library Authenticated Download Date | 10/4/17 1:56 PM Ovidiu-Cristinel Stoica (section 2.2). It also contains the Schwarzschild singu- This cancellation allows us to weaken the condition that larity (section 2.4) and the FLRW Big Bang singularity the metric tensor is non-degenerate, to some cases when (section 2.3). it can be degenerate. It will be seen that these cases include some important singularities. 1. Expanded Einstein equation and 1.2. A more explicit form of the expanded Ein- quasi-regular spacetimes stein equation 1.1. The expanded Einstein equation To give a more explicit form of the expanded Einstein equation, the Ricci decomposition of the Riemann curva- e.g. An equation which is equivalent to Einstein’s equation ture tensor is used (see [9–11]). n whenever the metric tensor gab is non-degenerate, but By using the equations (8) and (5) in dimension = 4, is valid also in a class of situations when gab becomes the Einstein tensor in terms of the traceless part of the degenerate and Einstein’s tensor is not defined will be Ricci tensor and the scalar curvature can be written: discussed in this section. Later it will be shown that the 1 proposed version of Einstein’s equation remains smooth in Gab = Sab − Rgab: (14) various important situations such as the FLRW Big-Bang 4 singularity, isotropic singularities, and at the singularity This equation can be used to calculate the expanded Ein- of the Schwarzschild black hole. stein tensor: We introduce the expanded Einstein equation Gabcd := (G ◦ g)abcd (G ◦ g)abcd + Λ(g ◦ g)abcd = κ(T ◦ g)abcd: (11) 1 = (S ◦ g)abcd − R(g ◦ g)abcd (15) 4 = 2Eabcd − 6Sabcd: If the metric is non-degenerate then the Einstein equa- tion and its expanded version are equivalent. This can be The expanded Einstein equation now takes the form seen by contracting the expanded Einstein equation, for instance in the indices b and d. From (1) the contraction 2Eabcd − 6Sabcd + Λ(g ◦ g)abcd = κ(T ◦ g)abcd: (16) in b and d of a Kulkarni-Nomizu product (h ◦ g)abcd is st s t s s 1.3. Quasi-regular spacetimes hˆac : = (h ◦ g)asct g = hacgs − hat δc + hsgac − hscδa s = 2hac + hsgac: (12) We are interested in singular spacetimes on which the expanded Einstein equation (11) can be written and is smooth.
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