On closed timelike curves, cosmic strings and conformal invariance

Reinoud Jan Slagter Univ of A’dam The Netherlands Asfyon, Astronomisch fysisch onderzoek Nederland Two most famous compact objects in GRT i.e. stationary axially symmetric

The Kerr solution The spinning cosmic string

exact sol. with correct phys properties: ►No-rotation: Schwarzschild solution Precursor: ►Asymptotical flat van Stockum solution: unusual behavior ►No CTC’s [ at least hidden] [rotating dust cylinder] ►Problems matching interior to exterior ►Asymptotic conical! ►periodic time coordinate ►CTC’s ?? what3-th TMF is-conf the Torino status 2019 of these objects concerning CTC’s ? Bonner (2008) on CTC’s:

Problematic causality issues ( possible not hidden behind horizon):

A. Extreme BTZ- black hole [(2+1)-dim.] B. Spinning Cosmic strings ►►► U(1) scalar-gauge field C. In general: approaching the Planck scale complementarity by conformal invariance : will causality and locality survive?? [‘t Hooft 2017]

Historical attempts: Gödel, van Stockum cylinder, rotating dust-cylinder, Gott-spacetime,.... [ all unphysical ]

3-th TMF-conf Torino 2019 At the front: There is no black hole interior ? t’ Hooft-conference on blackhole complementarity and AdS [july 2019]

Interesting discussions:

‘t Hooft ; Maldacena ; Strominger Susskind ; Verlinde ; Duff ; Rovelli ..... ER =EPR ? Rebirth of the wormhole solution?

3-th TMF-conf Torino 2019 Why scalar-gauge field?

The abelian scalar(Higgs) field with gauge group U(1) has lived up to its reputation!

1. As order parameter in super conductivity: Ginzburg-Landau model

2. The U(1)-scalar-gauge field in standard model of particle physics (Higgs mech.)

3. The special 휙4 self interacting Nielsen-Olesen vortex solution. Gives insight in class. gauge theories Yang-Mills-Higgs equations for monopoles.

4. Needed in inflationairy model [ horizon-flatness problems solved?]

5. General Relativistic-cosmic string solution

6. Super-massive cosmic strings: can build-up huge mass in the extra-dimension of the bulk spacetime ( warped spacetimes: hierarchy problem solved)

7. ►►► quasar alignment? Quasar-confinement for large red-shift must be of

primordial origin [ Slagter, IJMP D, 2018]

3-th TMF-conf Torino 2019 풅풔ퟐ = −풆−ퟐ풇 풅풕 − 푱 풅흋 ퟐ + 풆ퟐ풇 푳 풅흋ퟐ + 풆ퟐ휸 풅풓ퟐ + 풅풛ퟐ f, J, L and γ functions of (r,z) Note: radiating spacetimes: 풕 → 풊풛, 풛 → 풊풕, 푱 → 풊푱 Asymptotically:

ퟐ푴 푺풋풙풌 ퟐ푴 ퟐ ퟐ 풊 풋 풌 풅풔 = − ퟏ − + 푨ퟎ)풅풕 + ퟒ흐풊풋풌 ퟑ + 푨풊 풅풕풅풙 + ퟏ + 휹풊풋 + 푨풋풌 풅풙 풅풙 풓 풓 풓

Weyl sol: 풅풔ퟐ = −풆ퟐ흍풅풕ퟐ + 풆−ퟐ흍 풓ퟐ 풅흋ퟐ + 풆ퟐ휸 풅풓ퟐ + 풅풛ퟐ ퟐ ퟐ ퟐ 훁 흍 = ퟎ, 휸풛 = ퟐ풓𝝈풓𝝈풛, 휸풓 = 풓 흍풓 − 흍풛 + − ퟐ + − 풎 푹 +푹 −ퟐ풛ퟎ 풎 ퟒ푹 푹 ± ퟐ ퟐ 흍 = 풍풏 + − , 휸 = − ퟐ 풍풏 + − ퟐ ퟐ , 푹 = 풓 + (풛 ± 풛ퟎ) ퟐ풛ퟎ 푹 +푹 +ퟐ풛ퟎ ퟐ풛 (푹 +푹 ) −ퟒ풛 ퟎ ퟎ ퟐ풎 ퟐ풎ퟐ Exterior of a thin uniform rod, density ~ δ, −풛 < 풛 < 풛 . Correct asym form: 흍~ퟏ − + + ⋯ . ퟎ ퟎ 풓 풓ퟐ Schwarzschild sol:

ퟐ 풎 = 풛ퟎ, 풕풓풂풏풔풇풐풓풎풂풕풊풐풏: 풓 = 흆 − ퟐ풛ퟎ흆풔풊풏휽, 풛 = 흆 − 풛ퟎ 풄풐풔휽 ퟏ ퟏ ퟏ + − + − 흆 = 풛ퟎ + 푹 + 푹 , 풄풐풔휽 = 푹 − 푹 ퟐ ퟐ풛ퟎ ퟐ

Kerr solution: 푱 ≠ ퟎ [see any textbook] 3-th TMF-conf Torino 2019

푟2 Papapetrou sol: 푑푠2 = 퐹 푑푡 − 푊푑휑 2 − 푑휑2 − 푒휇 푑푟2 + 푑푧2 퐹

1 훼2 − 훽2푟2 퐹 = 푊 = − 푧 푧 2 2 3/2 훼 푐표푠ℎ − 훽 푠𝑖푛ℎ 푧 + 푟 푧2 + 푟2 3/2 푧2 + 푟2 3/2 1 훽푧 3 훽푧3 1 Has the correct asymp form 퐹 ≈ + − + ⋯ . However no term ~ (no mass) 훼 훼2푟3 2 훼2푟5 푟 Lewis-Van Stockum sol (rotating dust cylinder ). ■ Not correct asymp. form ■ Manifestly CTC’s Semi infinite line-mass [SILM]:

2 2 휓 = 푐1푙푛 휖 푧 − 푧1 + (푧 − 푧1) +푟 + ln 퐶

2 휀(푧 − 푧1 1 훾 = 2푐1 푙푛 + + ln (퐸퐶) 2 2 2 2 ((푧 − 푧1) +푟

1 Ricci-flat; if 퐸 = then lim 훾 = 0 then C can be transformed away except for 푐1 = 1 퐶 푟→0 1 Flat for 푐 = 0 and 푐 = 1 1 2

E,C constant and 휀 = ±1 ; 푐1 푟푒푙푎푡푒푑 푡표 푡ℎ푒 푚푎푠푠

Infinite line-mass [ILM]: 3-th TMF-conf Torino 2019 Why are axially symmetric spacetime [with U(1) –Higgs] so interesting?

A. “Kerr “-spacetime: ► 풅풔ퟐ = − 풅풕 − 푱풅흋 ퟐ + 풃ퟐ풅흋ퟐ + 풅풓ퟐ [+풅풛ퟐ] There is no structure in z-direction: so suppress

B. (2+1)- dim spacetime: local flat, but CTC’s for b < J

■ Einstein: ( wire-approx) angle deficit and J=const.

ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ■ planar “gravity” 풅풔 = − 풅풕 − 푱ퟎ풅흋 + ퟏ − ퟒ푮흁 풓 풅흋 + 풅풓 transf to Minkowski: ...... + 풓ퟐ풅흋 ퟐ ퟎ < 흋 < ퟏ − ퟖ푮흁 ퟐ흅

■ massive spinning point-source (“cosmon”) mass density 흁 and intrinsic spin 푱ퟎ 퐉 ■ CTC’ for 퐫 < ퟎ [can one confine the source within a small enough region?] 퐬 (ퟏ−ퟒ퐆훍) 풕풕 ퟐ 풕풊 풊풋 ퟐ ■ distributional enery momentum 푻 ~ퟒ푮흁휹(풓) 푻 ~푱휺 흏풋휹(풓) ■ if one tries to hide the string by 풕 → 풕 − 푱흋 : helical time coordinate: CTC’s everywhere!

C. where do we need these properties? Quantum gravity!! Planar gravity fits in very well! 푅휇휈= 0 but we have point-masses! [quantized version: ‘t Hooft 2002,..] mass is a local topological defect proportional to the wedge cut out the 2-plane. 3-th TMF-conf Torino 2019

“Cosmon”: z suppressed

When “lifting-up” to (3+1): ► infinite line-mass no longer tenable [Geroch et al.1987] ► the self-gravitating cosmic string necessary with 2 푒2 parameters 훼 = and 휂 훽

흏푼 ퟏ 푮 = 휿ퟐ 푻 푫 푫흁휱 − ퟐ = 0 휵흁푭 − 풊풆[휱(푫 휱)∗-휱∗ 푫 휱 ] = ퟎ 흁흂 4 흁흂 흁 흏휱∗ 흁흂 ퟐ 흂 흂

► The only physical acceptable mass distribution will be the U(1) scalar-gauge field.

So compactifying one of the space coordinates runs into problems. Now: ► Lifting procedure via holographic principle ? ► use conformal invariant Higgs gravity model !! [Slagter, Dustin, JHEP, 2019] 3-th TMF-conf Torino 2019 In (2+1)-dimensional gravity: The Banados-Teitelboim Zanelli (BTZ) black hole: ퟏ 푬풊풏풔풕풆풊풏: 푮 = − 품 풍 = length scale where curvature sets in 흁흂 풍ퟐ 흁흂

푟2 1 2 2 2 2 2 푑푠퐵푇푍 = 8퐺푀 − 2 푑푡 + 2 2 2 푑푟 − 8퐺퐽푡푑휑 + 푟 푑휑 푙 16퐺 퐽 푟 2 + 2 − 8퐺푀 푟 푙 There is an inner and outer horizon and ergocircle ►M=-1/8G, J=0: global 푨풅푺ퟑ

►풍 → ∞ ∶ Killing horizon and surface gravity: Hawking temperature

2퐺퐽2 퐽 ►Coord transf: 푟2= − 8퐺푀푟 2 [푟 ≠ ] 푀 2푀

2 2퐺 푑푠2 = − 8퐺푀푑푡 − 퐽푑휑 + 푑푟 2 + 8퐺푀푟 2푑휑2 푀

Just the spinning particle solution! p 3-th TMF-conf Torino 2019 Remember: cosmic string solution asymptotically:

푑푠2 = −푒푎1 푑푡2 − 푑푧2 + 푑푟2 + 푒−2푎1(푘푟 + 푎 )2푑휑2 2 푎푖 integration const, but k [mass] determined by field eq., so by the string variables

We already noticed: there is an obscurity by defining mass M by surface charges associated to the 2 killing vectors And: One cannot ignore the interior of the spinning “object” and it will not consists of “ordinary matter”

3-th TMF-conf Torino 2019 Motivations for Conformal Invariant [CI] Gravity 1. Mainly quantum-theoretical: opportunity for a renormalizable theory with preservation of causality and locality [alternative for stringtheory?] note: “formulating GR as a gauge group was not fruitful”, so “add” CI to gauge

2. Formalism for disclosing the small-distance structure in GR

Note: “there seems to be no limit on the smallness of fundamental units in one particular domain of physics, while in others there are very large scales and time scale” consider: local exact CI, spontaneously broken just as the Higgs mechanism

3. CI can be used for “black hole complementarity” and information paradox [ related to holography [‘t Hooft 1993, 2009]

4. Alternative to dark energy/matter issue [Mannheim, 2017]; Construct traceless 푇휇휈 [needed for CI: particles massless] and use spontaneous symmetry breaking! 5. Explore issues such as “trans-Planckian” modes in Hawking radiation calculation and the nature of “entanglement entropy” [ER=EPR?] Example: warped 5D model: dilaton from 5D Einstein eq [Slagter, 2016] Some results of Conformal Invariance ► CI in GR should be a spontaneously broken exact symmetry, just as the Higgs mechanism

품 (풙) = 흎 풙 ퟐ품 (풙) ► One splits the metric: 흁흂 흁흂 품 흁흂 the “unphys. metric”

treat 흎 and scalar fields on equal footing!

► CI is well define on Minkowski: null-cone structure is preserved. ► If 𝑔 휇휈 is (Ricci?) flat: 휔 is unique (QFT is done on flat background!) ퟏ 1 ► If 𝑔 is non-flat: additional gauge freedom: 품 → 휴ퟐ품 , 흎 → 흎, Φ → Φ, … … . . 휇휈 훀 Ω [no further dependency on Ω, ω] ퟐ SO: can we generate 품 흁흂 = 훀 휼흁흂? I will present 2 examples (see next)

► conjecture: avoiding anomalies we generate constraints which will determine the physical constants such as the cosmological constant ►Consider conformal component of metric as a dilaton (ω)with only renormalizable interactions. ► Small distance behavior (ω→0) regular behavior by imposing constraints on model ► Spontaneously breaking: fixes all parameters (mass, cosm const,…) [‘t Hooft, 2015]

“ In quantum field theory we work on a flat background. Then ω is unique. On non-flat background: sizes and time stretches and become ambiguous” Some results of Conformal Invariance ► Dilaton field 휔 need to be shifted to complex contour (Wick rotation) to ensure that 휔 has the same unitary and positivity properties as the scalar field. [for our 5D model: 휔 has complex solutions! ]

► In canonical gravity: quantum amplitudes are obtained by integration of the action over all components of 품흁흂. Now: first over 흎; and then over 품 흁흂 ; then: constraints on 품 흁흂 and matter fields

5 4 푖푆 푑 푊 푑 휔 푑 𝑔 휇휈 … . . 푒 [𝑔 휇휈still inv. under local conv. trans. ] S gauge fixing constraints.

푹 ퟔ ►Vacuum state would have normally R=0; now: 푹 → − 휵흁휵 휴 휴ퟐ 휴ퟑ 흁 so the vacuum breaks local CI spontaneously Nature is not scale invariant, so the vacuum transforms into another unknown state.

►Conjecture: conformal anomalies must be demanded to cancel out → all renormalization group β-coeff must vanish → constraints to adjust all physical constants! ►Ultimate goal: all parameters of the model computable ( including masses and Λ ) black hole complementarity

►It was believed that information would disappear in the central singulatity ► In-falling particle entangled. Firewall? CI can do better: local breaking of exact CI

Distinction between infalling and outside observer: they experience a different 흎

“the infalling observer passing the horizon experience 품흁흂 and mass M, The outside observer experience Hawing radiation and shrinking mass of the hole: the disagree about dilaton field (in a dynamical setting)”

►One could say that 휔 is “unobservable” ►Adjust 휔, when a singularity is encountered ( hide behind horizon) . ►When a local observer encounters a singularity, it,s clock will slow down an infinite time. ►the two observers disagree about the vacuum state of 휔

So CI offers a handle for quantum gravity

Further reading: anti-podal identification crossing the firewall [restores time reversal symmetry!!]

3-th TMF-conf Torino 2019 Connection with 5D Warped Spacetime Consider on a 5D warped spacetime [NOT yet CI] [Slagter, Found of Phys, 2016]

풅풔ퟐ = 퓦(풕, 풓, 풚)ퟐ 풆ퟐ 휸(풕,풓)−흍(풕,풓) −풅풕ퟐ + 풅풓ퟐ + 풆ퟐ흍(풕,풓)풅풛ퟐ + 풓ퟐ풆−ퟐ흍(풕,풓)풅흋ퟐ + 횪풅풚ퟐ

U(1) scalar-gauge field on the brane + empty bulk. Gravity can propagate into the bulk. 5D: ퟓ ퟓ ퟐ ퟒ ퟒ 푮흁흂 = −휦ퟓ 품흁흂 + 휿ퟓ휹 풚 [− 품흁흂횲ퟒ + 푻흁흂]

On the brane: ퟒ푮 = −휦 ퟒ품 + 휿ퟐ ퟒ푻 + 휿ퟒ 푺 − 퓔 흁흂 풆풇풇 흁흂 ퟒ 흁흂 ퟓ 흁흂 흁흂

ퟏ − 휦 (풚−풚 ) From 5D: 풆 ퟔ ퟓ ퟎ 퓦 = (풅 풆휶풕 − 풅 풆−휶풕)(풅 풆휶풓 − 풅 풆−휶풓) 휶 풓 ퟏ ퟐ ퟑ ퟒ

ퟏ 횽 = 휼푿 풕, 풓 풆풊풏흋, 푨 = [푷 풕, 풓 − 풏]훁 흋 흁 흐 흁 흏푽 1 푫흁푫 횽 = ퟐ 4훻휇퐹 = 𝑖휖 훷(퐷 훷)∗−훷∗퐷 훷 Scalar-gauge field eq.: 흁 흏횽∗ 휇휈 2 휈 휈

One could say that the “information about the extra dimension” translates itself as a curvature effect on spacetime of one fewer dimension!! Warped 5D spacetime conformally revisited

We rewrite our metric 2 2 2 2 푑푠 = 휔(푡, 푟) 푊(푦) 𝑔 휇휈 + 푛휇푛휈Γ(푦)

← real solution. ↓ ↓ dilaton “unphysical metric” [Bondi-Marden.] 2 휕 휔2−휕 휔2 (휕 −휕 − 휕 )휔 + 푟 푡 = 0 푡푡 푟푟 푟 푟 휔

← solution: 휔2 < 0 needed : integration over complex contour[‘tHooft..] and ω has same unitary and positivity prop as Φ ퟏ ퟏ write the action conformal invariant [ i.e. : 품 → 휴ퟐ품 흎 → 흎 휱 → 휱 ] 흁흂 흁흂 휴 휴 ퟏ ퟏ ퟏ 푨 = 풅ퟒ풙 −품 − 휱휱∗ + 흎 ퟐ 푹 − 품 흁흂(흏 흎 흏 흎 + 푫 휱 푫 휱 ∗)] − 푭휶휷푭 ퟏퟐ ퟐ 흁 흂 흁 풗 ퟒ 휶휷 ퟏ − 푽(휱 , 흎 ) − 휿ퟐ휦흎 ퟒ ퟑퟔ ퟒ 1 Note: * CI broken by mass term via 푽 휱 , 흎 휔2 = − 휅2휔 2 6 4 * we take Λ=0 ퟏ ퟐ ퟐ ∗ ퟐ ퟒ * Newton’s const hidden in 푽 휱 , 흎 , so 푽 휱, 흎 = 휷휼 휿ퟒ횽횽 흎 + 흀횽 ퟖ re-appears when CI is broken Warped 5D spacetime conformally revisited Field equations rewritten[ Slagter,2019]

1 1 퐺 = 푇 (휔) + 푇 (Φ,푐) + 푇 (퐴) + 𝑔 Λ 휅2휔 4 + 휅4푆 + 𝑔 푉(Φ , 휔 ) − ℰ 휇휈 휔 2 + Φ Φ ∗ 휇휈 휇휈 휇휈 6 휇휈 푒푓푓 4 5 휇휈 휇휈 휇휈

1 휕푉 1 1 휕푉 훻 훼휕 휔 − 휔 푅 − − Λ 휅2휔 3 = 0 퐷훼퐷 Φ − Φ 푅 − = 0 훼 6 휕휔 9 4 4 훼 6 휕Φ ∗ 휈 𝑖 ∗ ∗ 훻 퐹휇휈 = 푒 Φ 퐷휇Φ − Φ 퐷휇Φ Calculate Trace: rest term as expected: 2 2 1 휕 푃2−휕 푃2 [16휅2훽휂2푋2휔 2 − 휅4 푟 푡 푒8휓 −4훾 ] 휔 2+푋2 4 5 푟2푒2

휇 4 휇 Bianchi: 훻 ℰ휇휈=휅5 훻 푆휇휈 so (3+1) spacetime variation in matter-radiation on brane can source KK modes

(휔 ) 2 훼 2 훼 푇휇휈 = 훻휇휕휈휔 − 𝑔 휇휈훻훼휕 휔 − 6휕휇휔 휕휈휔 + 3𝑔 휇휈휕훼휔 휕 휔

(Φ ,푐) ∗ 훼 ∗ ∗ ∗ 훼 ∗ 푇휇휈 = 훻휇휕휈ΦΦ − 𝑔 휇휈훻훼휕 ΦΦ − 3 𝒟휇Φ(퐷휈Φ) +(퐷휇Φ) 퐷휈Φ + 3𝑔 휇휈퐷훼Φ(퐷 Φ)

1 푇 (퐴) = 퐹 퐹훼 − 𝑔 퐹 퐹훼훽 휇휈 휇훼 휈 4 휇휈 훼훽 New: Some applications

We will consider now two examples of the “un-physical” metric 𝑔 휇휈

A. Bondi-Marder spacetime [ suitable for our scalar-gauge model]

I. With the contribution from projected Weyl tensor [Slagter ,ArXiv:gr-qc/171108193] II. Without [ Slagter, Phys Dark Universe,2019]

B. Spinning Cosmic String [Slagter, Dustin, JHEP, 2019 ]

Stationary axially symmetric solutions: Kerr solution. CTC’s hidden behind the horizon Where are the others?

Weyl, Parapetrou, van Stockum, ..... All are physically unacceptable: not the correct asymptotic behavior CTC’s are possible matching problems at the boundary

However: cosmic string solution in GR : could be physically acceptable .

Now: spinning cosmic strings: Some additional fields are necessary to compensate the energy failure close to the core. THEN: How do we solve the CTC problem and matching problem??

By Conformal invariant model? Bondi-Marder spacetime as “unphysical” metric

Remember: Bondi-Marder spacetime [needed because 푇푡푡 + 푇푟푟 ≠ 0 for CS ] 푑푠2 = 푒−2휓 푒2훾 푑푟2 − 푑푡2 + 푟2푑휑2 + 푒2휓+2휇푑푧2 = 휔 2 −푑푡2 + 푑푟2 + 푒2휏푑푧2 + 푟2푒−2훾푑휑2 ퟐ So 품 흁흂 = 흎 품 흁흂 ↑ ↑ Ricci-flat un-physical metric from 5D

휔 is a conformal factor. We consider first the exterior vacuum situation: ퟐ (흎 ) Einstein equation: 흎 푮흁흂 = 푻흁흂 ퟏ 흎 ̂ - equation: 휵 흁흏 흎 − 흎 푹 = ퟎ 흁 ퟔ ퟏ Check: 푻풓 푮 − 푻(흎) = 0 흁흂 흎 ퟐ 흁흂 One can solve equation for 휔 : ퟏ ퟐ ퟐ ퟏ ퟐ 흎 = 퓑풆ퟐ𝝇ퟏ 풓 +풕 −ퟐ흊풓 +𝝇ퟐ풕+풓

4 constants . Generation of curvature from Ricci flat spacetimes. [Slagter, Phys. Dark Univ.,2019] Numerical solution ω

Quantum amplitudes are obtained by

퐷휔 푥 … . .

No problem here. Spinning U(1) gauged cosmic strings Let us consider now the 4D stationary axially symmetric spacetime with rotation: [for the moment no t-dependency]

2 −2푓 푟 2 2푓(푟) 2 2 2훾 푟 2 2 푑푠 = −푒 푑푡 − 퐽 푟 푑휑 + 푒 푙(푟) 푑휑 + 푒 (푑푟 + 푑푧 ) rewritten as ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ흁 풓 ퟐ ퟐ 풅풔 = 흎(풓) − 풅풕 − 푱 풓 풅흋 + 풃 풓 풅흋 + 풆 (풅풓 + 풅풛 )

dilaton ↓ decoupled from 품 흁흂

Some results: 1. obtainable from Weyl form by: 풕 → 풊풛, 풛 → 풊풕, 푱 → 풊푱 2. interesting relation with (2+1) dim gravity [cosmon’s; ‘tHooft ,2000] 3. Gott-spacetime: no CTC’s [parallel and opposite moving pair] 4. for constant J: ► conical exterior spacetime [angle-deficit] ►if one transform: 푡 → 푡 − 퐽휑: results in local Minkowski but then t jumps by 8휋퐺퐽 [ helical time] QM-solution? Quantized angular momentum→ also t ! 5. What happed at the boundary 푟푐 of the string? r=0: J = 0 and b→r r = 풓풄: J = constant and 푏 = 퐵(푟 + 푟푐) Then: problems at the boundary for 푱풓 and WEC violated!!

Spinning U(1) gauged cosmic strings in CI gravity

No choice yet for 푉 휔, Φ . From tracelessness and Bianchi: 2 푑푉 푑푉 1 푑푉 푑푉 푉 = Φ ∗ + 휔 푉′ = Φ ∗′ + 휔 ′ 3 푑Φ ∗ 푑휔 6 푑Φ ∗ 푑휔

For the exterior we obtain 푏′ 휔 1 2 1 푏′ 휔 ′ 퐽′′ = 퐽′ − 2 푏′′ = 퐽′2 − 푏′휔 ′ 휇′′ = 퐽′2 − 휇′ + 2 푏 휔 푏 휔 2푏2 푏 휔 ′2 ′ ′ ′′ 3휔 ′2 휔 1 ′ 휔 푏 ′ ”spin-mass rel” 휔 = − 퐽 + + 휇 + 2휔 ↓ 8푏2 2휔 2 푏 풃 푱(풓) = const. ퟐ 풅풓 흎 ′ with exact solution: 풄ퟑ 흁 풓 = 풄ퟏ풓 + 풄ퟐ − 풍풐품 ( 풄ퟒ풓 + 풄ퟓ) 풃 풓 = 흎 풓 = ퟐ풄ퟒ풓 + ퟐ풄ퟓ ퟐ풄ퟒ풓+ퟐ풄ퟓ 풄ퟑ 푱 풓 = 풄ퟔ ± ퟐ풄ퟒ풓 + ퟐ풄ퟓ

► J has correct asymptotic form! ► 𝑔휇휈 Ricci flat! [while 𝑔 휇휈 푛표푡 ] 푐 −푐 푐 ► CTC for 푟 = 3 5 6 which can be pushed to ±∞ or 0. 푐4푐6 Numerical verification The interior solution

ퟏ For the gauge field we can take: 푨 = 푷 풓 , ퟎ, ퟎ, (푷 풓 − 풏) 흁 ퟎ 풆

The field equation contain now terms like

푏 푃′ 푒퐽푃′ + 푃′ 퐽′′ = 퐽′ 휕 푙표𝑔 − 2 0 0 + ⋯ 푟 휂2푋2 + 휔 2 푒 휂2푋2 + 휔 2

The “spin-mass” relation becomes in case of global strings (P=푃0 =0) 푏 퐽 = 푐표푛푠푡 푑푟 휂2푋2 + 휔 2

Energy momentum: ′ ′ ′ 3 2 휇 푏 푏 푇 = − 퐽′ + + (휇′ + )휕 log (휂2푋2 + 휔 2) 푡푡 4푏2 푏 푏 푟

This can be made positive due to the additional matter!

Scalar curvature: 푘′푏′ 푏′ 5 휂2푋′2 + 휔′2 푅 = + 푘′ 휕 휂2푋2 + 휔2 + 푏 푏 푟 2 휂2푋2 + 휔2

regular everywhere Numerical solution Local observer

Local orthonormal frame: Θ 푡 = 푑푡 − 퐽푑휑 Θ푟 = 푒휇푑푟 Θ 푧 = 푒휇푑푧 Θ 휑 = 푏푑휑 1 Timelike 4-velocity: 푈 = 1,0, 훼, 훽 휈 휀 Local energy density measured by the observer moving at constant 푟 = 푟푐

훽2 + 훼2 푏′ + 훽퐽′ 2훼2 − 휀2 휀2퐺휇 휈 푈 푈 = 휕 푙표𝑔 휂2푋2 + 휔 2 + 퐽′2 휇 휈 푏 푟 4푏2

Can be made positive for suitable physically acceptable behavior of 푏′, 퐽′, 푋′, 휔′ and 휀2 < 2훼2 ( for sufficiently high velocity) ======Main conclusion:

It seems that there are no obstructions for a physically acceptable solution for a spinning cosmic string in conformal gravity.

No CTC’s No violation of WEC Interior: regular and easily matched on exterior

Thank you! Extra background

3-th TMF-conf Torino 2019 Some history of QFT Calculations in QFT: ■ In perturbation theory the effect of interactions is expressed in a powerserie of the coupling constant ( <<1 !) ■ Regularization scheme necessary in order to deal with divergent integrals over internal 4- momenta. ■ Introduce cut-off energy/mass scale Λ and stop integration there. [however, invisible in physical constants and partcle data tables] So renormalization comes in ■ Covariant theory of gravitation cannot be renormalized [in powercounting sense ] Non-renormalizable interactions is suppressed at low energy, but grows with energy. At energies much smaller than this “catastrophe-scale”, we have an effective field theory.

Standard model is too an effective field theory. ■ In curved background: geometry of spacetime remaims in first instance non-dynamical! However: in GRT it is.

String theory solution? ퟐ 휶휷 ■ Nambu-Goto action (Polyakov) 푨 = −푻 풅 𝝈 −품품 풉 ∗ 휼휶휷 Some history of QFT

ퟐ New gauge symmetry: 품휶휷 → 휴(𝝈) 품휶휷 [ Ω smooth function on the worldsheet]

훼 After quantization: 푇훼 depends on Ω, unless a crucial number in 2d-CFT 훼 (central charge) is zero! [in conformal gravity 푇훼 = 0 ] The Fadeev-Popov ghost field ( needed for quantisation) contribute a central charge of -26, which can be canceled by 26-dimensional background.

Can we do better? New conformal field theory

Suppose: QFT is correct and GRT holds at least to the Planck scale

■ Advantages of CI: A. At high energy, the rest mass of partcles have negligible effects So no explicit mass scale. CI would solve this B. CI field theory renormalizable [ coupling constants are dimensionless] C. CI In curved spacetime: would solve the black hole complementarity through conformal transformations between infalling and stationary observers. D. Could be singular-free E. Success in CFT/ADS correspondence F. In standart model, symmetry methods also successful. G. CI put constraints on GRT . Very welcome! Related Issues ► Asymptopia: How to handle: “far from an isolated source?” αβ we have only locally: 훻αT = 0 푘 훼 훼훽 is there a Killing-vector 휇: then 훻훼퐽 = 훻훼 푇 푘훽 = 0 then integral conservation law. gravitational energy and mass? 2 ►Isotropic scaling trick: 𝑔휇휈 → 𝑔 휇휈 = 휔 𝑔휇휈 with ω → 0 far from the source. 1 [note: we shall see that Einstein equations yield: 퐺 = (… ), so small 휇휈 휔2 distance limit will cause problem, unless we add scalar field comparable 1 with “dilaton “ω : 퐺 = (… ) ] 휇휈 휔2+Φ2

ퟏ Example: Minkowski: 풅풔ퟐ = −풅풗풅풖 + (풗 − 풖)ퟐ 풅휽ퟐ + 풔풊풏ퟐ휽풅흋ퟐ ퟒ one needs information about behavior of fields at 푣 → ∞ ퟏ ퟏ then: 풅풔ퟐ = 풅풖풅푽 + (ퟏ − 풖푽)ퟐ 풅휽ퟐ + 풔풊풏ퟐ휽풅흋ퟐ and infinity : 푉 → 0 푽ퟐ ퟒ so singular! ퟐ ퟐ then: 품흁흂→ 품 흁흂 = 흎 휼흁흂 = 푽 휼흁흂 : smooth metric extended to V=0 and one can handle tensor analysis at infinity. ퟒ Even better: 품 = 휼 with 푇, 푅 = 푡푎푛−1푣 ± 푡푎푛−1푢 흁흂 (ퟏ+풗ퟐ)(ퟏ+풖ퟐ) 흁흂 풅풔ퟐ = −풅푻ퟐ + 풅푹ퟐ + 풔풊풏ퟐ푹 풅휽ퟐ + 풔풊풏ퟐ휽풅흋ퟐ ퟑ ퟒ ퟑ Static Einstein universe 푺 ⨂ℛ : conformal map (ℛ , 휼흁흂) → (푺 ⨂ℛ, 품 흁흂) Related Issues CI

► If spacetime is fundamental discrete: then continuum symmetries ( such as L.I.) are imperilled. To make it compatible: the price is locality. [ Dowker, 2012; ‘t Hooft, 2016] Can non-locality be tamed far enough to allow known local physics to emerge at large distances? ► The Causal Set approach to quantum gravity: atomic spacetime in which the fundamental degrees of freedom are discrete order relations. [’tHooft, Myrheim, Bombelli, Lee, Myer and Sorkin] ► The causal set approach claims that certain aspects of General Relativity and quantum theory will have direct counterparts in quantum gravity: 1. the spacetime causal order from General Relativity, 2. the path integral from quantum theory. Then: Is it possible to obtain our familiar physical laws described by PDE’s from discrete diff operators on causal sets? For example, discrete operators that approximate the scalar D’Alembertian in any spacetime dimension? Seems to be yes! ►ω is fixed when we specify our global spacetime and coordinate system, which is associated with the vacuum state. 푹 ퟔ [remember 푹 → − 휵흁휵 휴 ] If we not specify this state, then no specified ω. 휴ퟐ 휴ퟑ 흁 ‘t Hooft: “ In quantum field theory we work on a flat background. Then ω is unique On non-flat background: sizes and time stretches and become ambiguous”