Null energy condition

New paper! A bit too technical to discuss here. But for Sci-Fi fans, the following tidbit: wormholes and time machines generally require violation of the null energy condition. Our paper shows, among other things, that none of the particles or fields of the type generally considered in particle physics will allow construction of a (stable) wormhole or time machine.

For those familiar with the Einstein equations, they are of the form:

{Curvature tensor R, fn of spacetime geometry} = {matter energy-momentum tensor T}

The set of possible spacetime geometries (could there be wormholes? loops in time?) satisfying the Einstein equations is unconstrained without some information about the right hand side of the equation. Traditionally in general relativity, one assumes that the matter fields satisfy certain energy conditions, for example the null energy condition, which requires that the tensor T, when contracted with any null (lightlike) four-vector n, is positive: Tnn > 0. We show that when this condition is violated, the system is necessarily unstable to small perturbations.


Instabilities and the null energy condition
Authors: Roman V. Buniy, Stephen D.H. Hsu

We show that violation of the null energy condition implies instability in a broad class of models, including all gauge theories with scalar and fermionic matter as well as any perfect fluid. When applied to the dark energy, our results imply that $w = p / \rho$ is unlikely to be less than -1.

http://arxiv.org/abs/hep-th/0502203

Stability and null energy condition

My collaborator, UO postdoc Roman Buniy, is off to sunny Miami to attend a particle theory conference. His talk is on stability of quantum field theories and the null energy condition.

Recently, cosmologists have discovered that 80% of the energy in the universe is in a very unusual form called "dark energy" (not to be confused with "dark matter," which also exists but is clumped around galaxies, rather than diffuse, and doesn't seem as weird). Dark energy causes the expansion of the universe to accelerate, due to an equation of state with negative pressure. How negative can the pressure be? There is some theoretical prejudice that it cannot be more negative than minus its energy density (in natural units where Planck's constant and the speed of light are unity), even though the data allow for and perhaps even suggest this possibility.

What do I mean by "theoretical prejudice"? Well, I mean many theorists would be shocked if things turned out otherwise. If the pressure is too negative something called the "null energy condition" (NEC) used in general relativity is violated. The NEC says that the contraction of any null, or light-like, four vectors with the stress energy tensor must be non-negative. Assuming the NEC, one can prove a number of pleasing properties of solutions to the Einstein equations. It is believed to be satisfied by any reasonable types of matter.

In our paper we show that, in a broad class of models including any constructed out of interacting scalar and gauge fields, or any model describing a perfect fluid, if the NEC is violated by some configuration of the fields, then that configuration is unstable (i.e., will fall apart). This makes it very unlikely that the dark energy violates the NEC, since it seems to have been stable over billions of years.

More null energy condition

hep-th/0606091

This is an extension of an earlier paper hep-th/0502203 (see blog post here). We show that in a broad class of models violation of the null energy condition indicates instability. The results are relevant to general relativity and to cosmology.

A paper by Dubovsky et al. (Ref. [6] in the new paper) exhibited a loophole in our earlier results involving superluminal (acausal) excitations. We traced this to an equation in our analysis (Eq. (47) in the new paper) that needs to be solved to find the unstable mode. In models with superluminal excitations this equation is solvable in some frames but not others (this is a sickness of acausality). Correspondingly, in some frames the Hamiltonian has no negative mode. In the appendix of the new paper we show that requiring causality is enough to restore the link between NEC violation and instability.

Coincidentally, the paper posted just before ours, hep-th/0606090 (Creminelli et al.) discusses some effective theories that can violate the NEC and be stable. Our results don't cover these models as they have four-derivative terms in the Lagrangian. We restrict ourselves to models which have classical equations of motion that are second order PDEs. I don't know what to make of third and higher order PDEs -- who knows what initial conditions are well-posed in those models.

However, the Creminelli et al. paper and the Dubovsky et al. paper are examples of how the creativity of model builders tends to evade the results of theorem-provers. No go theorems are only as good as their assumptions!


The null energy condition and instability

Authors: Roman V. Buniy, Stephen D. H. Hsu, Brian M. Murray

We extend previous work showing that violation of the null energy condition implies instability in a broad class of models, including gauge theories with scalar and fermionic matter as well as any perfect fluid. Simple examples are given to illustrate these results. The role of causality in our results is discussed. Finally, we extend the fluid results to more general systems in thermal equilibrium. When applied to the dark energy, our results imply that w is unlikely to be less than -1.

Through the wormhole

Tomorrow a team from the Science Channel show Through the Wormhole (Morgan Freeman is the narrator) is flying up to Berkeley to interview me for an upcoming episode. They found me because of a paper I wrote with my former postdoc Roman Buniy (see below). Roman is a brilliant guy who twice placed first in the Ukrainian physics Olympiad. He made the beautiful figures below.

Doing this kind of show isn't exactly a win-win: I would guess the volume of crackpot mail I receive could go up by an order of magnitude :-(

Semi-classical wormholes and time machines are unstable

hep-th/0504003

Abstract: We show that Lorentzian (traversable) wormholes and time machines with semi-classical spacetimes are unstable due to their violation of the null energy condition (NEC). Semi-classicality of the energy-momentum tensor in a given quantum state (required for semi-classicality of the spacetime) implies localization of its wavefunction in phase space, leading to evolution according to the classical equations of motion. Previous results related to violation of the NEC then require that the configuration is unstable to small perturbations.

Here are some slides on the subject. Click for larger version. For more background on why it is difficult to construct effective field theories which lead to (stable) violation of the null energy condition (NEC), see Phys. Rev. D 74, 063518 (2006).

New paper

This paper follows up on our earlier work on the null energy condition (NEC). Imagine you've built a device which "warps" spacetime enough to create a wormhole or time machine. General theorems show that somewhere the matter in your device has to violate the NEC. In our earlier work we showed that any classical system which does so is unstable to small perturbations. In this paper, we show that semi-classicality of the device spacetime is a strong enough condition to require semi-classicality of the matter fields from which it is constructed. In other words, a device which warps space in a deterministic (nearly classical) manner is subject to the earlier results and is unstable.

We leave open the possibility of intrinsically quantum (or "fuzzy" devices) whose spacetime is strongly fluctuating. However, these might not be the most safe or realiable means of transportation! A wormhole or time machine cannot be both predictable and stable.


Semi-classical wormholes and time machines are unstable
Authors: Roman V. Buniy, Stephen D.H. Hsu

We show that Lorentzian (traversable) wormholes and time machines with semi-classical spacetimes are unstable due to their violation of the null energy condition (NEC). Semi-classicality of the energy-momentum tensor in a given quantum state (required for semi-classicality of the spacetime) implies localization of its wavefunction in phase space, leading to evolution according to the classical equations of motion. Previous results related to violation of the NEC then require that the configuration is unstable to small perturbations.

http://arxiv.org/abs/hep-th/0504003

BBC on wormholes

We get a mention in this nice BBC article. Here is the latest version (PDF) of the talk Roman Buniy will give on Tuesday at a conference at Vanderbilt.

Wormhole 'no use' for time travel
By Paul Rincon
BBC News science reporter

Artist's impression of a wormhole Image: SPL
Wormholes contort the fabric of the Universe
For budding time travellers, the future (or should that be the past?) is starting to look bleak.

Hypothetical tunnels called wormholes once looked like the best bet for constructing a real time machine.

These cosmic shortcuts, which link one point in the Universe to another, are favoured by science fiction writers as a means both of explaining time travel and of circumventing the limitations imposed by the speed of light.

The concept of wormholes will be familiar to anyone who has watched the TV programmes Farscape, Stargate SG1 and Star Trek: Deep Space Nine.

The opening sequence of the BBC's new Doctor Who series shows the Tardis hurtling through a "vortex" that suspiciously resembles a wormhole - although the Doctor's preferred method of travel is not explained in detail.

But the idea of building these so-called traversable wormholes is looking increasingly shaky, according to two new scientific analyses.

Remote connection

A common analogy used to visualise these phenomena involves marking two holes at opposite ends of a sheet of paper, to represent distant points in the Universe. One can then bend the paper over so that the two remote points are positioned on top of each other.

[The wormholes] you would like to build - the predictable ones where you can say Mr Spock will land in New York at 2pm on this day - those look like they will fall apart
Stephen Hsu, University of Oregon
If it were possible to contort space-time in this way, a person might step through a wormhole and emerge at a remote time or distant location.

The person would pass through a region of the wormhole called the throat, which flares out on either side.

According to one idea, a wormhole could be kept open by filling its throat, or the region around it, with an ingredient called exotic matter.

This is strange stuff indeed, and explaining it requires scientists to look beyond the laws of classical physics to the world of quantum mechanics.

Exotic matter is repelled, rather than attracted, by gravity and is said to have negative energy - meaning it has even less than empty space.

Law breaker

But according to a new study by Stephen Hsu and Roman Buniy, of the University of Oregon, US, this method of building a traversable wormhole may be fatally flawed. In a paper published on the arXiv pre-print server, the authors looked at a kind of wormhole in which the space-time "tube" shows only weak deviations from the laws of classical physics.

These "semi-classical" wormholes are the most desirable type for time travel because they potentially allow travellers to predict where and when they would emerge.

The Tardis (BBC)
The concept is a favourite of science fiction writers
Wormholes entirely governed by the laws of quantum mechanics, on the other hand, would likely transport their payloads to an undesired time and place.

Calculations by the Oregon researchers show a wormhole that combines exotic matter with semi-classical space-time would be fundamentally unstable.

This result relies in part on a previous paper in which Hsu and Buniy argued that systems which violate a physical principle known as the null energy condition become unstable.

"We aren't saying you can't build a wormhole. But the ones you would like to build - the predictable ones where you can say Mr Spock will land in New York at 2pm on this day - those look like they will fall apart," Dr Hsu said.

Tight squeeze

A separate study by Chris Fewster, of the University of York, UK, and Thomas Roman, of Central Connecticut State University, US, takes a different approach to tackling the question of wormholes.

Amongst other things, their analysis deals with the proposal that wormhole throats could be kept open using arbitrarily small amounts of exotic matter.

Fewster and Roman calculated that, even if it were possible to build such a wormhole, its throat would probably be too small for time travel.

It might - in theory - be possible to carefully fine-tune the geometry of the wormhole so that the wormhole throat became big enough for a person to fit through, says Fewster.

But building a wormhole with a throat radius big enough to just fit a proton would require fine-tuning to within one part in 10 to the power of 30. A human-sized wormhole would require fine-tuning to within one part in 10 to the power of 60.

"Frankly no engineer is going to be able to do that," said the York researcher.

The authors are currently preparing a manuscript for publication.

Supporting view

However, there is still support for the idea of traversable wormholes in the scientific community. One physicist told BBC News they could see problems with Hsu's and Buniy's conclusions.

"Violations of the null energy condition are known to occur in a number of situations. And their argument would prohibit any violation of it," they commented.

"If that's true, then don't worry about Hawking radiation from a black hole; the entire black hole vacuum becomes unstable."

The underlying physics was not in doubt, the researcher argued. The real challenge was in explaining how to engineer wormholes big enough to be of practical use.

Cambridge astrophysicist Stephen Hawking is amongst those researchers who have pondered the question of wormholes.

In the 1980s, he argued that something fundamental in the laws of physics would prevent wormholes being used for time travel. This idea forms the basis of Hawking's Chronology Protection Conjecture.

Macroscopic Superposition States in Isolated Quantum Systems

Happy Thanksgiving! :-)

Macroscopic Superposition States in Isolated Quantum Systems 

https://arxiv.org/abs/2011.11661   

Roman V. Buniy and Stephen D.H. Hsu 

For any choice of initial state and weak assumptions about the Hamiltonian, large isolated quantum systems undergoing Schrodinger evolution spend most of their time in macroscopic superposition states. The result follows from von Neumann's 1929 Quantum Ergodic Theorem. As a specific example, we consider a box containing a solid ball and some gas molecules. Regardless of the initial state, the system will evolve into a quantum superposition of states with the ball in macroscopically different positions. Thus, despite their seeming fragility, macroscopic superposition states are ubiquitous consequences of quantum evolution. We discuss the connection to many worlds quantum mechanics.

It may come as a surprise to many physicists that Schrodinger evolution in large isolated quantum systems leads generically to macroscopic superposition states. For example, in the familiar Brownian motion setup of a ball interacting with a gas of particles, after sufficient time the system evolves into a superposition state with the ball in macroscopically different locations. We use von Neumann's 1929 Quantum Ergodic Theorem as a tool to deduce this dynamical result. 

The natural state of a complex quantum system is a superposition ("Schrodinger cat state"!), absent mysterious wavefunction collapse, which has yet to be fully defined either in logical terms or explicit dynamics. Indeed wavefunction collapse may not be necessary to explain the phenomenology of quantum mechanics. This is the underappreciated meaning of work on decoherence dating back to Zeh and Everett. See talk slides linked here, or the introduction of this paper.

We also derive some new (sharper) concentration of measure bounds that can be applied to small systems (e.g., fewer than 10 qubits). 

Related posts:

Classics on the arxiv: von Neumann and the foundations of quantum statistical mechanics 

Pure State Quantum Thermalization: from von Neumann to the Lab 

Entanglement and fast thermalization in heavy ion collisions 

Fun fact: Professor Buniy was a postdoc in my group at Oregon. Before coming to the US for graduate school in theoretical physics he was among the last group of young men to serve in the Soviet Army (Strategic Missile Forces IIRC!)

I suppose he has a document like this one:

Here he is in 2011, working on the null energy condition and instabilities in quantum field theories: 

 

Trackback and arXiv

arXiv.org, the physics research archive, now allows trackback links to appear on article abstract pages. I often give talks on my research, and I thought it would be nice to use my blog and the trackback functionality to make it easy for readers of the papers to find the PDFs of the corresponding talks. The slides of a talk are sometimes easier to follow than the paper itself!

THERMAL GRAVITY, BLACK HOLES AND COSMOLOGICAL ENTROPY, B. Murray and S. Hsu, hep-th/0512033: Slides of Talk given as ITS seminar by B. Murray.

ENTANGLEMENT ENTROPY, BLACK HOLES AND HOLOGRAPHY, R. Buniy and S. Hsu hep-th/0510021: Slides of Talk given as ITS seminar by S. Hsu.

INSTABILITIES AND THE NULL ENERGY CONDITION, R. Buniy and S. Hsu, hep-th/0502203: Slides of Talk given at Johns Hopkins by S. Hsu.

Let's see if arXiv accepts these trackbacks... Oops, Blogger doesn't support trackback, and arXiv doesn't support manual trackback pings, so it doesn't work right now.

JHU Talk

I'm going to be at Johns Hopkins next week, giving a talk on the null energy condition and instabilities. The slides (big PDF file, subject to change, no fair peeking if you are a JHU theorist) can be found here.

Curved space and monsters

New paper!

http://arxiv.org/abs/0706.3239

A simple question: how many different black holes can there be with mass M? Conventional wisdom: of order exp(A), where A is the surface area of the hole and scales as M^2.

Using curved space, we construct objects of ADM mass M with far more than exp(A) microstates. These objects have pathological properties, but, as far as we can tell, can be produced via quantum tunneling from ordinary (non-pathological) initial data. Our results suggest that the relation between black hole entropy and the number of microstates of the hole is more subtle than perhaps previously appreciated.

Update! Rafael Sorkin was kind enough to inform us of his earlier related work with Wald and Zhang. We've added the following end-note to the paper.

Note added: After this work was completed we were informed of related results obtained by Sorkin, Wald and Zhang [25]. Those authors investigated monster-like objects as well as local extrema of the entropy S subject to an energy constraint, which correspond to static configurations and obey $A^{3/4}$ scaling. For example, in the case of a photon gas the local extrema satisfy the Tolman--Oppenheimer--Volkoff equation of hydrostatic equilibrium. In considering monster configurations, Sorkin et al. show that requiring a configuration to be no closer than a thermal wavelength $\lambda \sim \rho^{-1/4}$ from its Schwarzschild radius imposes the bound $S < A$. While this may be a reasonable criterion that must be satisfied for the assembly of an initial configuration, it does not seem to apply to states reached by quantum tunneling. From a global perspective configurations with $S > A^{3/4}$ are already black holes in the sense that the future of parts of the object does not include future null infinity.




Black hole entropy, curved space and monsters

Stephen D.H. Hsu, David Reeb

(Submitted on 21 Jun 2007)

We investigate the microscopic origin of black hole entropy, in particular the gap between the maximum entropy of ordinary matter and that of black holes. Using curved space, we construct configurations with entropy greater than their area in Planck units. These configurations have pathological properties and we refer to them as monsters. When monsters are excluded we recover the entropy bound on ordinary matter $S < A^{3/4}$. This bound implies that essentially all of the microstates of a semiclassical black hole are associated with the growth of a slightly smaller black hole which absorbs some additional energy. Our results suggest that the area entropy of black holes is the logarithm of the number of distinct ways in which one can form the black hole from ordinary matter and smaller black holes, but only after the exclusion of monster states.