Wormholes and Entanglement
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Wormholes and Entanglement Juan Maldacena Institute for Advanced Study We will talk about some curious spacetime configurations. Outline • Wormholes • Science fiction wormholes. • Simple wormholes in general relativity • Traversable wormhole solutions. • A wormhole you could travel through? • Black holes and quantum systems • Wormholes and entanglement. General relativity • Einstein realized that gravity can be described by allowing spacetime to be curved. Curvature = energy density • Einstein equations: <latexit sha1_base64="HMjy1/kc+/aGtRITRnvKTJvBRAA=">AAACKXicbVBNS8NAFNz4WetX1aOXh0XwYklEtBeh6EFPomJboQlhs920SzebsLsRSsjf8eJf8aKgqFf/iNtapLYOLAwz83j7Jkg4U9q2P6yZ2bn5hcXCUnF5ZXVtvbSx2VBxKgmtk5jH8i7AinImaF0zzeldIimOAk6bQe9s4DfvqVQsFre6n1Avwh3BQkawNpJfqt34mRul4Io0h33IHDc2cTiAHDpjzg2cQBXchMG5fwm3vw7kfqlsV+whYJo4I1JGI1z5pRe3HZM0okITjpVqOXaivQxLzQinedFNFU0w6eEObRkqcESVlw0vzWHXKG0IY2me0DBUxycyHCnVjwKTjLDuqklvIP7ntVIdVr2MiSTVVJCfRWHKQccwqA3aTFKied8QTCQzfwXSxRITbcotmhKcyZOnSeOg4hxVnOvDcu10VEcBbaMdtIccdIxq6AJdoToi6AE9oVf0Zj1az9a79fkTnbFGM1voD6yvbzzEpCo=</latexit> 1 R g R =8⇡G T µ⌫ − 2 µ⌫ N µ⌫ • Light rays are typically focused as they propagate on a curved spacetime. General relativity and spacetime geometry • General relativity is based on the idea that spacetime is curved. • In many cases typically considered the deformation of spacetime is small and its topology is the same as that of Minkowski space. • Here we will analyze spacetimes with non-trivial spacetime topologies. • We will first see what is allowed and what is forbidden. • Traversable and non-traversable wormholes. • Relation to black holes and entanglement. Wormhole = spacetime tunnel Wheeler 1966 Science fiction wormholes (used to travel in time or faster than light) d Science fiction wormhole: travel time is shorter than d. à would allow faster than light travel, or travel to the past. (speed of light =1) (Travel time as seen by somebody that remains outside) Science fiction wormholes are problematic • They would violate the principle that signals cannot propagate faster than light in the ambient space. • We would have a violation of the principle on which special relativity rests. • This is a basic principle of our current physics framework!. • It is one principle that lead to general relativity itself. • Combined with special relativity, it can lead to travel into the past, with the associated paradoxes. • But they are certainly possible as mathematical geometries… <latexit sha1_base64="Acf2ACnnftW7Gh7lh2BvIzPem2k=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBCEQNiVgHoLevEYIS9IljA76U2GzD4yMxsIS77DiwdFvPox3vwbJ8keNLGgoajqprvLiwVX2ra/rY3Nre2d3dxefv/g8Oi4cHLaVFEiGTZYJCLZ9qhCwUNsaK4FtmOJNPAEtrzRw9xvTVAqHoV1PY3RDegg5D5nVBvJrffSUmlGugMcE7tXKNplewGyTpyMFCFDrVf46vYjlgQYaiaoUh3HjrWbUqk5EzjLdxOFMWUjOsCOoSENULnp4ugZuTRKn/iRNBVqslB/T6Q0UGoaeKYzoHqoVr25+J/XSbR/66Y8jBONIVsu8hNBdETmCZA+l8i0mBpCmeTmVsKGVFKmTU55E4Kz+vI6aV6XnUr57qlSrN5nceTgHC7gChy4gSo8Qg0awGAMz/AKb9bEerHerY9l64aVzZzBH1ifPyh/kRI=</latexit> Einstein’s equations to the rescue • The geometry should obey Einstein equations. Curvature = energy density • Could we engineer the appropriate matter distribution? • But energy is positive (classically) ( T ++ 0 ) ≥ • This results in the condition that gravity always focuses light rays. Side comment: Focusing of light and black holes • The fact that matter focuses light is central to one of Roger Penrose’s most famous results, which is that black holes can form and that they contain a singularity. Traversable wormholes need defocusing They focus as they go into one mouth They defocus as they go out. - - - à There must be negative null energy in the wormhole • This seems a happy resolution. Einstein’s equations do not allow science fiction wormholes! • Einstein theory is more than geometry. The geometry is constrained by the equations. And these constraints depend on the properties of matter. • It is a delicate dance between geometry and matter. But in quantum mechanics the energy can be negative! Casimir forces: T++ < 0 in some regions. Can this allow wormholes ? <latexit sha1_base64="ocZMu1Vd08UfNEZp2Pdh/p7fPHU=">AAACE3icbVDLSgMxFM3UV62vqks3wSKIxTIjBXVXdOOyQl/Qx5BJM21oJjMmd8Qy9B/c+CtuXCji1o07/8b0sdDWA+EezrmXm3u8SHANtv1tpZaWV1bX0uuZjc2t7Z3s7l5Nh7GirEpDEaqGRzQTXLIqcBCsESlGAk+wuje4Hvv1e6Y0D2UFhhFrB6Qnuc8pASO52ZMWl+Amp6b4MBx1phV3Hzp5XHGTfH6EcavH7rCN3WzOLtgT4EXizEgOzVB2s1+tbkjjgEmggmjddOwI2glRwKlgo0wr1iwidEB6rGmoJAHT7WRy0wgfGaWL/VCZJwFP1N8TCQm0Hgae6QwI9PW8Nxb/85ox+BfthMsoBibpdJEfCwwhHgeEu1wxCsJkwAlV3PwV0z5RhIKJMWNCcOZPXiS1s4JTLFzeFnOlq1kcaXSADtExctA5KqEbVEZVRNEjekav6M16sl6sd+tj2pqyZjP76A+szx/li5zf</latexit> <latexit sha1_base64="pdNtgpJ2nWcdMWNw9luIS5GHftY=">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRZBEMquFNRb0YvHCm5baNeSTbNtaDZZkqxYlv4GLx4U8eoP8ua/MW33oK0PBh7vzTAzL0w408Z1v53Cyura+kZxs7S1vbO7V94/aGqZKkJ9IrlU7RBrypmgvmGG03aiKI5DTlvh6Gbqtx6p0kyKezNOaBDjgWARI9hYyX96OEOoV664VXcGtEy8nFQgR6NX/ur2JUljKgzhWOuO5yYmyLAyjHA6KXVTTRNMRnhAO5YKHFMdZLNjJ+jEKn0USWVLGDRTf09kONZ6HIe2M8ZmqBe9qfif10lNdBlkTCSpoYLMF0UpR0ai6eeozxQlho8twUQxeysiQ6wwMTafkg3BW3x5mTTPq16tenVXq9Sv8ziKcATHcAoeXEAdbqEBPhBg8Ayv8OYI58V5dz7mrQUnnzmEP3A+fwCx5435</latexit> There is an integrated null energy condition • Achronal, average null energy condition: x+ 1 dx+T 0 ++ ≥ Z1 • If you have a complete fastest light ray in a given geometry, then the integral of T++ should be positive (or zero). • You can have negative T++ in some regions, but the total integral should be positive. This is a conjecture in general. In flat space it was proven for general interacting theories. Faulkner, Leigh, Parrikar, Wang; Hartman, Kudu, Tajdini Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, Wooglar Science fiction wormholes are forbidden • If the wormhole allows you to go faster than light in the ambient spacetime. Then the light ray going through the wormhole is a “fastest” (achronal) light ray. • Total focusing is equal to the integral of the null energy. • So they cannot defocus…. They focus as they go into one mouth They defocus as they go out. - - - Conclusion • Basic principles of special relativity and quantum mechanics imply positive energy. • Together with general relativity, these forbid traversable wormholes violating the faster than light travel ban. (not a rigorous theorem yet) • These are very basic principles, not the specific properties to the particles know. Dark matter is also expected to obey these properties. Comment • An interesting link between causality and a certain positivity property of property of quantum matter. • Gravity and quantum mechanics are good partners. • The beast saving the beauty. • ``Exotic matter’’ à matter outside the framework of Quantum Field Theory. But the laws of physics allow non-traversable wormholes These are time dependent wormholes. In fact, these arise in the simplest solution of general relativity: the Schwarzschild solution. Full Schwarzschild solution Eddington, Lemaitre, Einstein, singularity Rosen, Finkelstein, Kruskal ER Left Right Einstein-Rosen bridge exterior exterior R3 R3 Vacuum solution. No exotic matter. Figure 2: Maximally extended SchwarzschildTwo exteriors, sharing spacetime. the interior. There are two asymptotic regions. The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions. not in causal contact and information cannot be transmitted across the bridge. This can easily seen from the Penrose diagram, and is consistent with the fact that entanglement does not imply non-local signal propagation. (a) (b) Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neck connecting two asymptotically flat regions. (b) Here we have two distant entangled black holes in the same space. The horizons are identified as indicated. This is not an exact solution of the equations but an approximate solution where we can ignore the small force between the black holes. All of this is well known, but what may be less familiar is a third interpretation of the eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we can consider two very distant black holes in the same space. If the black holes were not entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow created at t =0intheentangledstate(2.1),thenthebridgebetweenthemrepresentsthe entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not 7 A collapsing bridge Eddington, Lemaitre, Einstein, singularity Rosen, Finkelstein, Kruskal ER Left Right Einstein-Rosen bridge exterior exterior R3 R3 Vacuum solution. No exotic matter. Figure 2: Maximally extended SchwarzschildTwo exteriors, sharing spacetime. the interior. There are two asymptotic regions. The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions. not in causal contact and information cannot be transmitted across the bridge. This can easily seen from the Penrose diagram, and is consistent with the fact that entanglement does not imply non-local signal propagation. (a) (b) Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neck connecting two asymptotically flat regions. (b) Here we have two distant entangled black holes in the same space. The horizons are identified as indicated. This is not an exact solution of the equations but an approximate solution where we can ignore the small force between the black holes. All of this is well known, but what may be less familiar is a third interpretation of the eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we can consider two