A Topological Extension of GR: Black Holes Induce Dark Energy

Marco Spaans

University of Groningen

spaans@astro.rug.nl

General Relativity

• Equivalence principle: Einstein’s local interpretation of Mach’s global principle.

• GR not invariant under local conformal transformations → singular metric fluctuations on the Planck scale: Wheeler’s (1957) quantum foam of wormholes.

• GR elegant and succesful, preserve in quantum gravity.

• The Einstein equation Gμν = 8πGN Tμν does not specify local and global topology.

The Equivalence Principle: Macro- versus Microscopic paths

Need to specify what path to take

quantum-mechanically.

Put Differently: The Feynman (1948) Path Integral• A particle/wave travels along many paths as an expression of

the superposition principle: connect any A to any B. .

• ∫ paths eiS leads to a semi-classical world line and a large scale limit for GR. But, how to distinguish paths on LPlanck?

Observers cannot tell paths apart locally due to errors!

• This talk: Choose paths that can be identified despite quantum uncertainty in geometry; paths fundamental and rooted in topology; Spaans 1997, Nuc.Phys.B 492, 526.

↔

Space-time on the Planck scale (10-33 cm) is a fabric woven of many paths: shape → connectivity.

Physics: Multiplicity

• Scrutinize down to Planck scale: strong coupling between observer and observee.

• To identify a path, which is distinct under continuous deformation of space-time, one needs a proper example to compare to.

• Thought: It takes one to know one. So topological fluctuations on LPlanck take the form of multiple copies of any path through quantum space-time (i.e., in 4D).

• Idea: Topological dynamics on LPlanck because no distinct path can (remain to) exist individually under the observational act of counting, thus building up quantum space-time.

Mathematics: Loop Algebra• Use 3D prime manifolds P: T3, S1xS2, S3 to construct a

topological manifold, with the S1 loop as central object: Two distinct paths from A to B together form a loop.

• T3, three-torus (Χ=0→LM), provides topologically distinct paths through space-time. Superposition by Feynman.

• S1xS2, handle (LM), can accrete mass as well as Hawking (1975) evaporate (so BH~wormhole in 4-space). The handle represents Wheeler’s quantum foam.

• S3, unit element (simply connected, large scale limit).

• Minimal set. Physical principle: find the paths that thread Planckian space-time (Feynman+Wheeler). Mathematical principle: topological constructability (primes+homeomorphisms); Spaans 2013, J.Phys 410, 012149

Constructing Space-Time Topology

Define Operators T (annihilation) and T* (creation):

• TP ≡ nQ, n=number of S1 loops and Q is P with an S1 contracted to a point (dimension-1).

• T*P ≡ PxS1 (dimension+1).• [T,T*] = 1 (non-commuting).• NS3 = S3, NT3 = 7T3, NS1xS2 = 3S1xS2.• N≡T*T+TT* (symmetric under T ↔ T*): counting.

• Repeated operations with N allow one to build a space-time with Planck scale heptaplets of three-tori and Planck mass triplets of handles embedded in 4-space.

Equation of Motion: Counting = Changing

• n[T3](m) = 7m

for discrete m=0,1,2,…; topological time t=(m+1)TPlanck and a continuous interval (m+) ≡ (m,m+1).

• n[S1xS2](m+1) = 3n[S1xS2](m) +Form(m+) -Evap(m+) -Merg(m+) →

n[S1xS2](m) = 3m if F,E,M small (drives inflation), 2n[S1xS2](m) = Evap(m+) if F,M and δn[S1xS2] small;

for every Planckian volume, independent of BH location.

• Matter degrees of freedom F,E,M modify the number of handles: GR part of quantum space-time through longevity and transience of all BHs.

• Topologically distinct paths, under homeomorphisms, between any two points:

• Use T3, S1xS2 and S3

• Build a lattice of three-tori, with spacing LPlanck, to travel through 4D space-time as an expression of the superposition principle and attach Wheeler’s handles: Quantum paths that connect points are then globally identified through non-contractible loops.

Multiply Connected Space-Time

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Topological Induction of BHs

• Solution 2θ(m) = Evap(m+), for the number θ of BHs in a time slice of width ~TPlanck: Space-time responds globally to local BHs by inducing Planckian BHs. Wave function of the universe collapses when BH forms.

• Planckian BHs evaporate in about Tev~TPlanck.• Macroscopic BHs θ living longer than a Hubble time

generate a globally stable quantum foam density Λ.• Extension of Einstein gravity and Mach’s principle: Global

topology and geometry determine (changes in) Planck scale topology and the motions of matter, and vice versa.

• Λ = θ mPlanck / Lf3, with Lf the size of the universe when the

first long-lived BH, so with Tev[m']>Tuniv[m], forms: Lf is frozen in when one has a global 4-space topology in the sense of Mach and the quantum foam stabilizes.

Dark Energy

• Cosmological observations suggest an accelerated expansion of the universe, usually attributed to some form of vacuum energy Λ: Einstein’s cosmological constant in the form of dark energy (Riess ea 98, Perlmutter ea

98, etc.), with ρ0~10-29 g cm-3. This while θ~1019 today.• Topological induction then requires the dark energy to

follow the number of macroscopic BHs in the universe. Induced Planckian BHs require an increase in 4-volume

for their embedding: Lf≈2x1014 cm from ρ0 and θ.

• Observed star/BH formation history of universe yields: Λ(z)/Λ(0)~(1+z)-0.36, z<1; a ~30% decline: w ≈ -1.1, not constant, consistent with observations (Aykutalp & Spaans 12). In fact, w = -1.13 ± 10% (Hinshaw ea 12; Planck: Ade ea 13).

Summary

• Use the topological freedom in GR to implement the superposition principle through the notion of globally distinct paths for particles to travel along.

• The S1 loop dynamics of three-tori and handles generate such paths: multiply to identify. Leads to the creation of Wheeler’s Planckian BHs by macroscopic BHs, i.e., a testable form of dark energy.

• What if (many) primordial BHs evaporate until today?• Inflation driven by induced mini BHs: n≥55 e-foldings,

n~ln(Lf/LPlanck)1/2 ~lnθ1/6.• Further scrutiny of the lattice of three-tori is necessary:

fluctuations with ns=1-r ≈ ln(7)/n = 0.96 and macroscopic temporal displacement in the CMB with amplitude 1/14.