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
quantummechanically.
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 spacetime 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.
Multiply Connected Space-Time
• 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.
▄
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.