Universe in a Black Hole with Spin and Torsion

Nikodem Popławski

General Relativity and Gravitation: A Centennial Perspective Pennsylvania State University State College, PA, 6/8/2015

Licensed under Public domain via Wikimedia Commons

Big Bang

Cosmic Microwave Background

Problems of general relativity General relativity describes gravity as curvature of spacetime. • Singularities: points with infinite density of matter. • Incompatible with quantum mechanics. We need quantum

gravity – a dream of theoretical physicists. It may resolve the singularity problem.

• Field equations contain the conservation of angular momentum without intrinsic angular momentum (spin).

• Spin is quantum-mechanical: extending GR to include spin of elementary particles is a natural first step towards quantum gravity. Simplest extension: Einstein-Cartan theory.

Problems of big-bang cosmology • Big-bang singularity: resolution from quantum gravity?

• What caused the big bang? What existed before? • Horizon problem: different regions of the Universe have not

contacted each other because of large distances between them, but they have the same physical properties (the Universe is homogeneous and isotropic).

• Flatness problem: the Universe is nearly flat. The initial conditions of the Universe must have been fine-tuned.

• What is dark energy? What is dark matter? What happened to antimatter? What does time flow in one direction?

Problems of big-bang cosmology

• Inflation (extremely rapid, exponential expansion of the early Universe) solves the flatness/horizon problems, and predicts the observed spectrum of CMB perturbations. It does not solve the singularity problem and does not explain the big bang.

• Bouncing cosmologies avoid the big-bang singularity, and also explain the flatness/horizon problems.

• What caused a bounce and inflation? Why did inflation end?

• Simple solution: Einstein-Cartan theory.

Image credit: One-Minute Astronomer

We can see the Universe expanding: galaxies look redder as they speed away (just as sirens sound lower pitched).

The 2-dimensional surface of the balloon is an analog to our 3-dimensional space. The 3-dimensional space in which the balloon expands is not analogous to any higher dimensional space. Points off the surface of the balloon are not in the Universe in this simple analogy. The Universe may be finite or infinite.

Explains what happens in black holes (regions of space from where nothing can escape). Every black hole becomes a doorway (Einstein-Rosen bridge) to a new, growing universe on the other side of its boundary (event horizon).

Black holes may be wormholes A child universe in a black hole is invisible for observers outside the black hole (which exists in a parent universe) because the black hole forms in infinite future for such observers. For observers in the child universe, the bridge looks like a white hole (the opposite of a black hole). The motion of matter through a bridge is one-directional and can define the arrow of time.

To form a bridge, we need a mechanism which avoids a singularity in a black hole.

Past

Future

General relativity: Matter tells spacetime how to curve, spacetime tells matter how to move. Quantum mechanics: Elementary particles possess an intrinsic angular momentum (spin) which does not represent rotation.

Gravity with spin requires curvature and torsion Curvature – “bending” of spacetime by energy and momentum. Torsion – “twisting” of spacetime by spin. Twisting a thin rod is less apparent than bending. Effects of torsion are important only at extremely high densities (in black holes and in the very early Universe). Torsion in Einstein-Cartan theory manifests itself as a repulsive force which opposes gravitational attraction and prevents singularities (points of infinite density).

A new universe in a black hole forms because of torsion

Black holes with torsion

Mathematical realizations of spherically symmetric, vacuum solutions of GR (& ECSK): • Schwarzschild black hole Final stage of collapse of most massive stars in GR (singular). • Schwarzschild white hole Cannot form physically (singular).

• Einstein-Rosen bridge (wormhole) Final stage of collapse of most massive stars in ECSK (regular). GR & ECSK favor different mathematical solutions.

E = mc2

Gravity with spin: Einstein-Cartan-Sciama-Kibble theory

Simplest theory of spacetime with torsion.

ECSK gravity • Einstein equations: Ricci tensor (curvature) is proportional to the energy and momentum density. Bianchi identities give the conservation law for the energy and momentum. • Cartan equations: Torsion tensor is proportional to the spin density. ECSK in vacuum reduces to GR and obeys the equivalence principle. Cyclic identities give the conservation law for the spin angular momentum. GR: only Einstein equations. Conservation laws for the energy and momentum without spin.

E = mc2

What is torsion? • Tensors – behave under coordinate transformations like products of differentials and gradients. Special case: vectors. • Differentiation of vectors in curved spacetime requires subtracting two infinitesimal vectors at two points that have different transformation properties. • Parallel transport allows to bring one vector to the origin of the other one, so that their difference would have a meaning.

E = mc2

A δA

δx δAi = -Γi

jk Aj δxk

Affine connection

What is torsion? • Curved spacetime requires geometrical structure: affine connection ¡½

¹º

• Covariant derivative rºV

¹ = ºV¹ + ¡¹

½ºV½

• Curvature tensor R½

¾¹º = ¹¡½¾º - º¡

½¾¹ + ¡½

¿¹¡¿¾º - ¡

½¿º¡

¿¾¹

E = mc2

1

2

3

4

Measures the change of a vector parallel-transported

along a closed curve:

change = curvature × area × vector

What is torsion?

• Torsion tensor – antisymmetric part of affine connection

• Contortion tensor

GR – affine connection restricted to be symmetric in lower indices

ECSK – no constraint on connection: more natural.

E = mc2

Measures noncommutativity of parallel transports

Theories of spacetime Special Relativity – flat spacetime (no curvature) Dynamical variables: matter fields

General Relativity – (curvature, no torsion) Dynamical variables: matter fields + metric tensor

ECSK Gravity (simplest theory with curvature & torsion) Dynamical variables: matter fields + metric + torsion

E = mc2

More degrees of freedom

ECSK gravity • Riemann-Cartan spacetime – metricity r½g¹º = 0

→ connection ¡½¹º = {½

¹º} + C½¹º

Christoffel symbols contortion tensor • Lagrangian density for matter

Metrical energy-momentum tensor Spin tensor

Spin tensor ≠ 0 for fermions.

Total Lagrangian density (like in GR)

E = mc2

ECSK gravity • Curvature tensor = Riemann tensor + tensor quadratic in torsion + total derivative • Stationarity of action under ±g¹º → Einstein equations

R{}¹º - R

{}g¹º /2 = k(T¹º + U¹º)

U¹º = [C½¹½C

¾º¾ - C

½¹¾C

¾º½ - (C

½¾½C

¿¾¿ - C

¾½¿C¿½¾)g¹º /2] /k

• Stationarity of action under ±C¹º

½ → Cartan equations

S½¹º - S¹±

½º + Sº±

½¹ = -ks¹º

½ /2

S¹ = Sº¹º

• Cartan equations are algebraic and linear • Contributions to energy-momentum from spin are quadratic

E = mc2

Same coupling constant k

T. W. B. Kibble, J. Math. Phys. 2, 212 (1961) D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964)

ECSK gravity • Stationarity of action under metric → Einstein equations Curvature = k · (energy-momentum density)

• Stationarity of action under torsion → Cartan equations

Torsion = k · spin density

• Cartan equations are algebraic and linear • Contributions to energy-momentum from spin are quadratic

E = mc2

Same coupling constant k

ECSK gravity • No spinors -> torsion vanishes -> ECSK reduces to GR • Torsion significant when U¹º » T¹º (at Cartan density) For fermionic matter (quarks and leptons)

½ > 1045 kg m-3

Nuclear matter in neutron stars

½ » 1017 kg m-3

Gravitational effects of torsion negligible even for neutron stars. Torsion significant only in very early Universe and in black holes

E = mc2

ECSK gravity • Field equations with full Ricci tensor can be written as

R¹º - Rg¹º /2 = £º¹

Canonical energy-momentum tensor • Belinfante-Rosenfeld relation

£¹º = T¹º + r¤½(s¹º

½ + s½º¹ + s½

¹º) /2 r¤½=r½- 2S½

• Conservation law for spin

r¤½s¹º

½ = (£¹º - £º¹)

• Cyclic identities

R¾¹º½ = -2r¹S

¾º½ + 4S¾

¿¹S¿º½ (¹, º, ½ cyclically permutated)

ECSK gravity • Bianchi identities (¹, º, ½ cyclically permutated)

r¹R¾¿º½ = 2R¾

¿¼¹S¼º½

• Conservation law for energy and momentum

Dº£¹º = Cº½

¹£º½ + sº½¾Rº½¾¹/2 Dº=r{}

º

Equations of motion of particles

Spin fluids • Field equations can be written for full Ricci tensor

R¹º - Rg¹º /2 = £º¹

Canonical energy-momentum tensor

• Conservation law for energy and momentum & spin: macroscopic fermionic matter is a spin fluid:

s¹º½ = s¹ºu½ s¹ºuº = 0

£¹º = c¦¹uº - p(g¹º - u¹uº) ² = c¦¹u¹ s2 = s¹ºs¹º /2

Four-momentum Pressure Energy density density

Spin fluids

• Dynamical energy-momentum tensor for a spin fluid with random spin orientation

Effective energy density Effective pressure

• Spin fluid of fermions with no spin polarization

E = mc2

F. W. Hehl, P. von der Heyde & G. D. Kerlick,Phys. Rev. D 10, 1066 (1974)

Fermion particle number density

Cosmology with torsion • A closed, homogeneous and isotropic Universe

Friedman-Lemaitre-Robertson-Walker metric (k = 1)

• Friedmann equations for scale factor a

Conservation law

E = mc2

a

O

A

Spin-torsion coupling generates

gravitational repulsion

Cosmology with torsion

E = mc2

• The early Universe – ultra-relativistic matter described by temperature T:

• Universe in a black holes begins when an event horizon forms:

• Universe contracts until

Cosmology in a black hole

• Universe then bounces and expands. If no particle production, it expands back to the initial size after which a new contraction begins – oscillatory universe.

Density parameter Number of causally disconnected regions

• Flatness and horizon problems can be solved.

If quantum effects in the gravitational field near the bounce produce enough matter (the total energy does not change), dark energy takes over at later times and a universe in a black hole expands forever. Otherwise, such a universe contracts back and starts over.

Cosmology in a black hole • Particle production by quantum effects in strong gravitational field: 1st Friedman eq. Particle production (2nd Friedman eq.) Particle production rate

A system of two nonlinear, first-order, ordinary differential equations. • Finite number of bounces (minimum scale factor at a bounce and

period of a contraction-expansion cycle increase).

E = mc2

Inflation without scalar field • Near a bounce: To avoid eternal inflation: • During an expansion phase, near critical value of particle

production rate:

Exponential expansion (inflation) which lasts about

E = mc2

Cosmic inflation (exponential expansion of the very early Universe) uses hypothetical fields to explain why the present Universe: • Looks the same (at largest scales) in all directions, • Has the same properties at all points in space, • Is nearly flat. • Has the observed patterns in the Cosmic Microwave Background radiation.

Cosmology with torsion also explains the first three phenomena. In addition, it: • Suggests what existed before the Big Bang, • Removes a singularity at the Big Bang, replacing it with a Big Bounce, • Explains what happens in black holes, • Predicts that our Universe is closed – can be tested (density parameter). • Can derive inflation without scalar fields. Effects of torsion on CMB – work in progress: • Find the equivalent scalar potential and calculate the power spectrum.

There are still questions which need to be answered: Dark energy, dark matter, antimatter (torsion may explain the observed asymmetry between matter and antimatter, Nature of Multiverse.

How to test that every black hole is a doorway to another universe? To boldly go where no one has gone before.

Since all stars rotate, all black holes rotate. A universe in a rotating black hole inherits its axis of rotation as a preferred direction, which may be supported by the observed motion of galaxy clusters and chirality of spiral galaxies. CMB Torsion predicts that elementary particles are spatially extended (10-27 m) which may be observed.

Thank you to: University of New Haven Chad Peterson

Dan Delgado

Antonio Di Vita

UNH 2015 Summer Undergraduate Research Fund Shantanu Desai