Wormhole Physics - In Classical and Quantum Theories of...

Wormhole Physics Quantum Wormhole Physics Summary

Wormhole PhysicsIn Classical and Quantum Theories of Gravity

S. Al Saleh A. Mahrousseh L.A. Al Asfar

Department of Physics and Astronomey ,King Saud University

The 100th Anniversary of General Relativity.The 30th of November, 2015


Outline

Wormhole PhysicsIntroductionWhat are Wormholes?Can Einstein Rosen Bridges Allow Spacetime Travel ?Traversable wormholesWormhole and Time Travel

Quantum Wormhole PhysicsPreliminariesMathematical and Physical Results


Introduction

• Einstein’s Theory of general relativity is one of the mostmagnificent achievements humanity had ever encountered. Itdemonstrates the marriage between matter and backgroundgeometry. Matter and spacetime had became one!

Gµν = κTµν (1)

• These equations (1) were published a 100 years from today[3]. Are known as the Einstein Field Equations.Although Theyare not the only equations for gravity. But they are the onlytested ones.[10].


Introduction

Many solutions to (1) were found, a lot of them were veryinteresting and related to observations. But one solution was sostrange that Einstein himself was very skeptical about. Thissolution is called the Schwazchild solution It predicted theexistence of Black holes a very dense object having an enormousgravity even spacetime will not make sense because of it! Later, in1935 Einstein and Rosen [4] studied the extension of Schwarzchildsolution to predict an even stranger object, knows asEinstein-Rosen Bridge. Or what is known as a Wormhole.


III

III

IV

i+

i−

i0

J +

u=∞ u

=0

u=

0

v=−∞

J −

r = 0

r = 0

Figure: Penrose diagram of the maximally- extended Schwarzchildsolution, demonstrating two universes with an Einstein-Rosen Bridge .


Outline




Definition of a Wormhole

• We can start with a mathematical definition of a wormhole,

Definition ( Einstein-Rosen Bridge)

A compact region of Minkowskian spacetime Ω is homeomorphicto R× Σ. Where Σ having non-trivial topology and boundary∂Σ ' S2. All hyper surfaces Σ are spacelike/ The region Ω iscalled a Wormhole / Einstein-Rosen Bridge .

• Despite the abstractness of this definition. It contain a lot ofinsight of the physics and geometry of a wormhole. Thisdefinition describes exactly what one pictures a wormhole to”look like”


The picture that approximate what the previous definitiondescribes:


• It is not required to have two separate spaces connected by anER bridge. We can imagine such topologies forming with asingle space. It is similar to folding a paper and making a”shortcut” route between two distant points.

• This will make our spacetime multiply connected!


Outline




Schwazchild Wormholes

• A paper by Wheeler and Fuller had shown that ER bridgesresulting from Schwazchild solutions are unstable. They willpinch off very quickly at the speed of light. and Not allowingany information to pass through the other side [6].

• In other words, the ”bridge” connecting the two black holeswill keep getting longer and longer at the speed of light. Suchthat no particle entering one side is able to cross the otherone.


Wormholes In Einstien-Cartan Theory

• General Relativity is not the only theory of gravitation.Einstein-Cartan theory is an alternative one. It is amodification of EFE’s by including how spin couples to thegeometry of spacetime via changing its torsion.

• In this theory, the gravitational singularity formed by acollapsing matter of the blackhole cannot form due to thecoupling of spin of the fermions. Instead, the collapsingmatter bounces back and forms an Einstein-Rosen Bridge aswell. Hence, every black hole in Einstein -Cartan theory is awormhole.[15]


Outline




Traversable wormholesA Traversable wormhole is a wormhole that allow matter andinformation to cross from one mouth to the other one.

Figure: Image of a simulated traversable wormhole that connects thesquare in front of the physical institutes of University of Tbingen with thesand dunes near Boulogne sur Mer in the north of France. Here thegravito-optical effects are ignored.


Examples of Traversable Wormholes

• Traversable wormholes seem not to exist naturally. As there isno known natural cosmological process that allows them toform (unlike black holes)

• However, semi-classical theories of gravity that predicts morethan 3+1 D of spacetime such as Gauss Bonnet gravity[12][7]predict the existence of trasversable wormholes with ordinarymatter or even without matter!

• Moreover, Brane cosmology theories allow traversablewormholes open by cosmic strings of negatives mass . [15]


Examples of Wormholes

• In General Relativity however, it is required to have wormholesmade from exotic matter to be stable and allowtransportation. Such wormholes are knows are Morris-Tornwormholes [18] [14]

• Wormholes do not only connect two distinct spacelike pointsin one universe. They could link distinct ”times” or spacetimepoints in multiverse. For example, if the spacetime is multiplyconnected at the quantum information inside a blackholecould ” leak out ” to other universes by going into awormhole.


Outline




The Concept of Causality

Closed timelike curves:In a future-directed Lorentzian spacetime. A closed timelike curve( CTC) is a world line of a particle that allows it to ”return” tothe starting timelike point. In other words, the future of thatparticle lies in its past !


The Concept of Causality

Causality

Causality is a relation between two events. The first is called the”cause” while the second is called the ”effect”. Causality requiresthe cause proceeds the effect. Also due to relativity, they aretimelike or null like separated ( the effect lies in the light cone ofthe other). CTC’s seems to classically violate causality.

Causal relation between events (points on the spacetime manifold)are equivalence relations. They are mathematical rules for thepoints in the spacetime allows preservation of causality ( which israther a logical entity.)The set of all the causal relations on the spacetime manifold iscalled Causal Structure


Wormholes as Time Machines• Let’s have a traversable wormhole with two mouths , A and

B. If B is accelerated with respect to A, or put near anenormous gravity. The latter will be in different ”time” thanthe first. Hence, A and B are not only spatially separated, butalso temporally.


Wormholes as Time Machines

• If an observer jumps in A and comes out in B. They will betravelling back in time. This might not affect causality if Aand B are speacelike separated such as no immediate causalconnection between them is possible.

• One might ask however about the mass/energy and chargeconservation in the universe at each ”time”. A travellerthrough such wormhole might seem to violate thoseconservation laws. Nevertheless, calculation [17][5] suggeststhat the wormhole pays for the energy/charge/ angularmomentum of the traveller. This known from the no-hairtheorem .


Outline




Preliminaries: Quantum Entanglment

Pure statesA quantum particle is described by a state ray, or a ket |ψ〉 . If onehas more than one particle. Their overall state is described by atensor ( direct) product. |ψ〉 ⊗ |φ〉. Measurement of one particledoes not affect or give information about the other. Thisconfiguration for states is knows as pure states.



Entangeled states

However, there is another extreme configuration for quantumstates description of two particle system. Which is Bell Pair , thetwo particles are described by one quantum state:

|Ψ〉 =1√2

(|A, 1〉|B, 0〉+ |A.0〉|B, 1〉) (2)

Where A and B are the particles that take either state |0〉 or |1〉. (they are abstract information states/ bits).



Theorem (Monogamy of Entanglement)

Let A be entangled to B, A ∼ B . This relation implies that ifB ∼ C ⇒ A = C . A particle cannot be maximally entangled withmore than one independent system.

This is an important result that we will be needing in the nextdiscussion. It shall unfold a deep connection between quantummechanics and wormholes.


Preliminaries: Quantum vacuum

• The vacuum in quantum field theory is nothing but empty ! Itis composed of never-resting particles that pop-out fromnothing and disappear in short time. These particles are calledvirtual particles or quantum fluctuations

• One cannot detect those particles directly, unless a specificexperiment is conducted to affect or modify the vacuum sothat these particle effects appear e.g. Casimir effect.

• The quantum vacuum satisfies a special symmetry , calledconformal symmetry, one of the results of this symmetry thateach ”patch” of the vacuum is entangled with an opposite oneto it. When a boundary surface is taken into consideration.


Figure: A schematic showing a result of conformal symmetry of thequantum vacuum. Every ”imaginary”patch is entangled with the oneopposite to it of the same size. Such that measurement for virtualparticle in one implies knowledge of the other.


Outline




Chronology Protection Conjecture

• Time-machine solutions and others (like Godel solution toEFE’s or Tippler’s Cylinder) seems to violate the causalstructure, but not the principles of relativity.

• The causal structure is not rigorously proven, and seems to bethreatened by the above solutions

• Thus, S. Hawking had proposed a conjecture to savechronology of the universe by the quantum effects that areignored in classical GR.[9]


When quantum correction to GR are taken into an account, closedtimelike cures will not be stable. As quantum field vacuumfluctuations will be amplified and blow up to infinity [19] .Nevertheless, some semi-classical calculations had shown thatsome CTC’s do not do that [11]. It remains for a complete theoryof quantum gravity to solve this puzzle. ( String theory forexample seems to support chronology protection [2].)

Figure: A famous painting by Salvador dali La persistencia de la memoria.Often linked to the arrow of time and our modern view of it


The figure below shows blue shift of normal modes fluctuations byMinser spacetime that contains a CTC. This illustrates thequantum effects contribution to chronology protection.


Quantum Entanglment and Wormholes ER= EPR

• Quantum entanglement is not only for microsystems. One cancreate macrosystems entangled by constructing them out ofentangled microsystems [16]

• It was shown that Blackholes decay via Hawking-Unruhradiation. And that radiation is Entangled with the blackholematter [8]

• Hence, if we created two blackholes out of a collapsingentangled matter. The blackholes will seem to violate one ofthe well-tested principles Unitarity, General Covariance orQuantum Field theory.



To see this, recall that at the boundary of the blackhole - the eventhorizon- the vacuum is assumed to behave well, and satisfy theconformal symmetry discussed earlier. Therefore, and since the twoblackholes are entangled ( or similarly for hawking radiation) thiswill go against the monogamy of entanglement.



The previous paradox is knows as the Firewall Paradox or theAMPS-from the paper authors’ names: Ahmed Almheiri, DonaldMarolf, Joseph Polchinski, and James Sully- paradox . It is aserious threat to quantum feild theory and/or general relativity. [1].



One of the most elegant and rather captivating solutions to theAMPS paradox is suggested by Prof L. Susskind. This solution isgiven a short name ER= EPR [13].In his paper, Susskind shows that and Einstein-Rosen Bridge isformed between two entangled blackholes. This resolves theparadox by assuming the points A and C shown previously to beone point . He even go further to generalise this result and give ageometric meaning to entanglement as Plank-scale ER bridgesbetween particles.


Summary

• The general theory of relativity had changed how we viewspace and time Perhaps, wormholes could allow interstellar oreven intergalactic travel one day! .

• Paradoxes push theoretical physics forward, it makes scientiststhink and discover solution that advances our understandingof nature. Such as the chronology protection or ER=EPR.

• Outlook• We need a complete theory of quantum gravity to be able to

fully understand spacetime.• Plank-sized wormholes could be what makes up the spacetime

at the micro-scale and hold it together. In our current work,we are trying to study the implications of this conjecture suchas Bekenstein-Hawking entropy counting assumingmultiply-connected spacetime.


Thank You !

Appendix

Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and JamesSully.Black holes: complementarity or firewalls?Journal of High Energy Physics, 2013(2):1–20, 2013.

Marco M Caldarelli, Dietmar Klemm, and Pedro J Silva.Chronology protection in anti-de sitter.Classical and Quantum Gravity, 22(17):3461, 2005.

Albert Einstein.Die grundlage der allgemeinen relativitatstheorie.Annalen der Physik, 354(7):769–822, 1916.

Albert Einstein and Nathan Rosen.The particle problem in the general theory of relativity.Physical Review, 48(1):73, 1935.

Allen Everett and Thomas Roman.

Appendix

Time travel and warp drives: a scientific guide to shortcutsthrough time and space.University of Chicago Press, 2012.

Robert W Fuller and John A Wheeler.Causality and multiply connected space-time.Physical Review, 128(2):919, 1962.

Elias Gravanis and Steven Willison.mass without mass from thin shells in gauss-bonnet gravity.Physical Review D, 75(8):084025, 2007.

Stephen Hawking, Juan Maldacena, and Andrew Strominger.Desitter entropy, quantum entanglement and ads/cft.Journal of High Energy Physics, 2001(05):001, 2001.

Stephen W Hawking.Chronology protection conjecture.Physical Review D, 46(2):603, 1992.

Appendix

Michael Kramer.Tests of general relativity.In 25TH TEXAS SYMPOSIUM ON RELATIVISTICASTROPHYSICS (TEXAS 2010), volume 1381, pages 84–97.AIP Publishing, 2011.

Li-Xin Li.Must time machines be unstable against vacuum fluctuations?Classical and Quantum Gravity, 13(9):2563, 1996.

David Lovelock.The einstein tensor and its generalizations.Journal of Mathematical Physics, 12(3):498–501, 1971.

Juan Maldacena and Leonard Susskind.Cool horizons for entangled black holes.Fortschritte der Physik, 61(9):781–811, 2013.

Michael S Morris, Kip S Thorne, and Ulvi Yurtsever.

Appendix

Wormholes, time machines, and the weak energy condition.Physical Review Letters, 61(13):1446, 1988.

Nikodem J Pop lawski.Cosmology with torsion: An alternative to cosmic inflation.Physics Letters B, 694(3):181–185, 2010.

Nicolo Spagnolo, Chiara Vitelli, Fabio Sciarrino, and FrancescoDe Martini.Entanglement criteria for microscopic-macroscopic systems.Physical Review A, 82(5):052101, 2010.

Leonard Susskind.Wormholes and time travel? not likely.arXiv preprint gr-qc/0503097, 2005.

Matt Visser.Traversable wormholes: Some simple examples.Physical Review D, 39(10):3182, 1989.

Appendix

Matt Visser.The quantum physics of chronology protection.The future of theoretical physics and cosmology: celebratingStephen Hawkings 60th birthday, pages 161–175, 2003.

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