Weyl metrics and wormholes Mikhail S. Volkov LMPT, University of - PowerPoint PPT Presentation

Weyl metrics and wormholes Mikhail S. Volkov LMPT, University of Tours, FRANCE Kyoto, YITP Workshop on Gravity and Cosmology, 14-th February 2018 G.W.Gibbons and M.S.V. Phys.Lett. B760 (2016) 324 JCAP 1705 (2017) 039 Phys.Rev.D96 (2017) 024053 Mikhail S. Volkov Weyl metrics and wormholes

Introduction I. Gravitating scalar field II. Vacuum wormholes III. Zero mass limit of Kerr spacetime is a wormhole IV. Wormholes in massive bigravity

Wormholes – spacetime bridges Wormholes interpolate between different universes or between different parts of the same universe. Could supposedly be used for interstellar and time travels.

Some history /Flamm, 1916/ – The spatial part of the Schwarzschild geometry contains a throat dr 2 dl 2 = 1 − 2 M / r + r 2 d Ω 2 = dr 2 + r 2 d Ω 2 + dZ 2 dr 2 r = r ( Z ) ≡ 2 M + Z 2 dZ 2 = where ⇒ 8 M . r / (2 M ) − 1 Flamm assumed Z > 0. Einstein-Rosen considered Z ∈ ( −∞ , + ∞ ) Z a c 0 b 0 10 r

Some history /Einstein-Rosen, 1935/ – Schwarzschild black hole has two exterior regions connected by a bridge. The ER bridge is spacelike and cannot be traversed by classical objects. /Maldacena-Susskind, 2013/ – the ER bridge may connect quantum particles to produce quantum entanglement and the Einstein-Pololsky-Rosen (EPR) effect, hence ER=EPR.

Some history /Wheeler, 1957/ wormholes may provide geometric models of elementary particles – handles of space trapping inside an electric flux. /Misner, 1960/ Wormholes can describe initial data for the Einstein equations. The time evolution of these data corresponds to the black hole collisions of the type observed in the GW events like GW150914. /Morris, Thorn, Yurtsever, 1988/ wormholes traversable by classical object may be supported by vacuum polarisation.

Can wormholes be solutions of Einstein equations ? ds 2 = − Q 2 ( r ) dt 2 + dr 2 + R 2 ( r )( d ϑ 2 + sin 2 ϑ d ϕ 2 ) , 3.0 R(r) 2.5 2.0 1.5 Q(r) 1.0 0.5 � 3 � 2 � 1 1 2 3 G µν = T µν ⇒ energy ρ = − T 0 0 and pressure p = T r r fulfill ρ + p = − 2 R ′′ p = − 1 R < 0 , R 2 < 0 . ⇒ the Null Energy Condition (NEC) must be violated. / T µν v µ v ν = R µν v µ v ν ≥ 0 for any null v µ /

NEC violation The general case without symmetry ⇒ topological censorship: compact two-surface of minimal area can exit if only NEC is violated /Friedman, Schleich, Witt, 1993/ ⇒ traversable wormholes are possible if only energy is negative. This may be, for example, due to vacuum polarization exotic matter: phantom fields, etc. Wormholes may exist in alternative gravity models: Gauss-Bonnet brainworld, etc. theories with non-minimally coupled fields (Horndeski) massive (bi)gravity

Best known example – phantom-supported wormhole L = R +2( ∂ψ ) 2 Bronnikov-Ellis wormhole: � r ds 2 = − dt 2 + dr 2 + ( r 2 + a 2 )( d ϑ 2 + sin 2 ϑ d ϕ ) 2 , ψ = arctan � ; a r ∈ ( −∞ , ∞ ) Figure: Isometric embedding of the equatorial section of the BE wormhole to the 3-dimensional Euclidean space

Phantom wormholes from the Kaluza-Klein viewpoint ds 2 = − dt 2 + dl 2 where dl 2 = γ ik dx i dx k fulfills (3) R ik ( γ ) = − 2 ∂ i ψ∂ k ψ, ∆ ψ = 0 ( ∗ ) The simplest solution is the Bronnikov-Ellis wormhole, more general solutions – superposition of wormholes. Eqs.( ∗ ) coincide with the vacuum Einstein equations for 5-metric ds 2 5 = cos(2 ψ )[ − dx 2 0 + dx 2 4 ] + 2 sin(2 ψ ) dx 0 dx 4 + dl 2 ⇒ wormholes can be interpreted as 5-geometries without invoking phantom fields /Clement/.

4D wormholes without phantom field Write a phantom field solution in the Weyl form, ds 2 = − e 2 U dt 2 + e 2 U � e 2 k ( d ρ 2 + dz 2 ) + ρ 2 d ϕ 2 � , ψ = ψ A new solution of the same form is obtained by swapping U ↔ ψ, k → − k hence by setting U new = ψ, ψ new = U , k new = − k . For the BE wormhole U = 0, hence the new solution is vacuum, U new = ψ, ψ new = 0 , k new = − k but the topology with two asymptotic regions remains – wormhole. The negative energy is hidden in the singularity.

I. Gravitating scalar field

Ordinary vs. phantom scalar L = R − 2 ǫ ( ∂ Φ) 2 ǫ = +1: ordinary scalar Φ = φ ǫ = − 1: phantom Φ = ψ .

Static system ds 2 = − e 2 U dt 2 + e − 2 U γ ik dx i dx k , the field equations are 1 (3) = ∂ i U ∂ k U + ǫ ∂ i Φ ∂ k Φ , R ik 2 ∆ U = 0 , ∆Φ = 0 . Rotational symmetry for real scalar, Φ ≡ φ , ǫ = +1 : → U cos α + φ sin α , U φ → φ cos α − U sin α , γ ik → γ ik , Boost symmetry for phantom, Φ ≡ ψ ǫ = − 1 : U → U cosh α + ψ sinh α , ψ → ψ cosh α + U sinh α , γ ik → γ ik .

Solutions from Schwarzschild − r − m r + m dt 2 + r + m r − m dr 2 + ( r + m ) 2 d Ω 2 , ds 2 = Φ = 0 . Rotations with cos α = 1 / s give Fisher-Janis-Robinson-Winicour solutions for ordinary scalar � r + m � 1 / s � 1 / s � � r − m dt 2 + dx 2 + ( r 2 − m 2 ) d Ω 2 � ds 2 = − , r + m x − m √ s 2 − 1 � r − m � φ = ± ln , | s | ≥ 1 , 2 s r + m Boosts with cosh α = 1 / s give solutions for phantom � 1 / s � 1 / s � � r − m � r + m dt 2 + dx 2 + ( r 2 − m 2 ) d Ω 2 � ds 2 = − , r + m r − m √ 1 − s 2 � r − m � ψ = ± ln | s | ≤ 1 . 2 s r + m

Wormholes Upon analytic continuation m → i µ, s → − is . one obtains − e 2Ψ / s dt 2 + e − 2Ψ / s [ dx 2 + ( x 2 + a 2 ) d Ω 2 ] , ds 2 = √ s 2 + 1 ψ = ± Ψ , Ψ = arctan ( r / a ) . s Taking s → ∞ gives ultrastatic wormhole of Bronnikov-Ellis. − dt 2 + dx 2 + ( x 2 + a 2 ) d Ω 2 , ds 2 = ψ = Ψ .

Axial symmetry L = R − 2 ǫ ( ∂ Φ) 2 ǫ = +1: ordinary scalar Φ ≡ φ ǫ = − 1: phantom Φ ≡ ψ Weyl parametrization ds 2 = − e 2 U dt 2 + e − 2 U � d ρ 2 + dz 2 � e 2 k � + ρ 2 d ϕ 2 � where U , k , Φ depend on ρ, z .

Field equations ∂ 2 U ∂ρ + ∂ 2 U 1 ∂ U + ∂ z 2 = 0 , ∂ρ 2 ρ ∂ 2 Φ ∂ρ + ∂ 2 Φ 1 ∂ Φ + ∂ z 2 = 0 , ∂ρ 2 ρ �� ∂ U � 2 � 2 � 2 � 2 � � ∂ U � ∂ Φ � ∂ Φ ∂ k = ρ − + ǫ − ǫ , ∂ρ ∂ρ ∂ z ∂ρ ∂ z ∂ k � ∂ U ∂ U ∂ z + ǫ ∂ Φ ∂ Φ � = 2 ρ . ∂ z ∂ρ ∂ρ ∂ z

Target space symmetries preserve spherical symmetry: � cos α � U � � � U � sin α rotations → , k → k φ − sin α cos α φ � U � � cosh α � � U � sinh α boosts → , k → k ψ sinh α cosh α ψ interchange BE wormhole and ring wormhole: swap U ↔ ψ, k → − k do not intermix scalar field and gravity amplitudes: k → λ 2 k , scaling U → λ U , Φ → λ Φ tachyon: U → ln ρ − U , k → k − 2 U + ln ρ, Φ → Φ Acting with this on vacuum metrics yields new solutions.

Simplest vacuum Weyl metrics

One rod – Schwarzschild ds 2 = − e 2 U dt 2 + e − 2 U � d ρ 2 + dz 2 � e 2 k � + ρ 2 d ϕ 2 � with � m � R − m � 1 = − 1 d ζ U ( ρ, z ) = 2 ln ρ 2 + ( z − ζ ) 2 � R + m 2 − m � R 2 − m 2 � 1 k ( ρ, z ) = 2 ln R + R − where R = 1 � ρ 2 + ( z ± m ) 2 . 2( R + + R − ) , R ± = U is the Newtonian potential of a massive rod of length 2 m along the z-axis with mass density 1/2.

Two rods along z-axis U = U 1 + U 2 , k = k 1 + k 2 + k 12 , where (with a = 1 , 2) � ( R a ) 2 − ( m a ) 2 � R a − m a � � 1 k a = 1 U a = 2 ln , 2 ln , R a + m a R a + R a − � ( R 1+ R 2 − + z 1+ z 2 − + ρ 2 )( R 1 − R 2+ + z 1 − z 2+ + ρ 2 ) 1 � k 12 = 2 ln , ( R 1+ R 2+ + z 1+ z 2+ + ρ 2 )( R 1 − R 2 − + z 1 − z 2 − + ρ 2 ) with R a = 1 � ρ 2 + ( z a ± ) 2 , z a ± = z − z a ± m a , R a ± = 2( R a + + R a − ) k � = 0 on the part of symmetry axis between the rods – strut /Israel and Khan 1964/ Similarly for many rods.

Point masses k = − m 2 ρ 2 U = − m R , 2 R 4 , ρ 2 + z 2 . For two masses m ± at z = ± m one has � with R = − m + − m − U = , R + R − � ρ 2 + z 2 − m 2 − m 2 2( R + ) 4 − m 2 + ρ 2 − ρ 2 � 2( R − ) 4 + m + m − k = − 1 , 2 m 2 R + R − ρ 2 + ( z ± m ) 2 /Chazy, Curzon 1924/ . � with R ± =

Summary of part I Applying the target space dualities to the vacuum Weyl metric gives all known and also many new static solutions. For example, the Fisher-Janis-Robinson-Winicour solutions and their generalizations to axially symmetric case, � λ/ s � − λ/ s � x − m � x − m dt 2 + ds 2 dl 2 , = − x + m x + m � 1 − λ 2 � � r 2 − m 2 cos 2 ϑ dr 2 + ( r 2 − m 2 ) d ϑ 2 � dl 2 = x 2 − m 2 √ s 2 − 1 λ � x − m � ( x 2 − m 2 ) sin 2 ϑ d ϕ 2 , + φ = 2 ln , s x + m BE wormholes and thei axially symmetric generalizations; many other solutions

II. Vacuum wormholes

Starting point Take the Schwarzschild metric ds 2 = − e 2 U dt 2 + e − 2 U � d ρ 2 + dz 2 � e 2 k � + ρ 2 d ϕ 2 � � R 2 − m 2 1 � R − m � k ( ρ, z ) = 1 � U ( ρ, z ) = 2 ln , 2 ln R + m R + R − R = 1 � ρ 2 + ( z ± m ) 2 . 2( R + + R − ) , R ± = and apply the scaling to get prolate vacuum metrics /Zipoy-Voorhees / k → λ 2 k U → λ U , Next step is the analytic continuation of parameters m → ia , λ → i σ