Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime

Abstract

For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis t comma x right-parenthesis element-of left-parenthesis 0 comma normal infinity right-parenthesis times double-struck upper T Superscript German upper D"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t down-arrow 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t \downarrow 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q overTilde Subscript 1 Baseline comma midline-horizontal-ellipsis comma q overTilde Subscript German upper D Baseline element-of double-struck upper R"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace t equals 1 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lbrace t = 1 \rbrace</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , as long as the exponents are “sub-critical” in the following sense: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript StartLayout 1st Row upper I comma upper J comma upper B equals 1 comma midline-horizontal-ellipsis comma German upper D 2nd Row upper I greater-than upper J EndLayout Endscripts left-brace q overTilde Subscript upper I Baseline plus q overTilde Subscript upper J Baseline minus q overTilde Subscript upper B Baseline right-brace greater-than 1"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo form="prefix">max</mml:mo> <mml:mstyle scriptlevel="1"> <mml:mtable rowspacing="0.1em" columnspacing="0em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>I</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>J</mml:mi> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:munder> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo> ~

Topics & Concepts