Retrocausality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Retrocausality

Retrocausality, or backwards causation, is a concept of cause and effect in which an effect precedes its cause in time and so a later event affects an earlier one. In quantum physics, the distinction between cause and effect is not made at the most fundamental level and so time-symmetric systems can be viewed as causal or retrocausal. Philosophical considerations of time travel often address the same issues as retrocausality, as do treatments of the subject in fiction, but the two phenomena are distinct.

Philosophy

See also: Principle of sufficient reason § Proposed proofs of universal validity

Philosophical efforts to understand causality extend back at least to Aristotle's discussions of the four causes. It was long considered that an effect preceding its cause is an inherent self-contradiction because, as 18th century philosopher David Hume discussed, when examining two related events, the cause is by definition the one that precedes the effect.

The idea of retrocausality is also found in Indian philosophy. It was defended by at least two Indian Buddhist philosophers, Prajñākaragupta (ca. 8th–9th century) and Jitāri (ca. 940–1000), the latter wrote a specific treatise on the topic, the Treatise on Future Cause (Bhāvikāraṇavāda). The idea is also found in some Chinese Buddhist philosophers, like Fazang.

In the 1950s, Michael Dummett wrote in opposition to such definitions, stating that there was no philosophical objection to effects preceding their causes. This argument was rebutted by fellow philosopher Antony Flew and, later, by Max Black. Black's "bilking argument" held that retrocausality is impossible because the observer of an effect could act to prevent its future cause from ever occurring. A more complex discussion of how free will relates to the issues Black raised is summarized by Newcomb's paradox. Essentialist philosophers have proposed other theories, such as the existence of "genuine causal powers in nature" or by raising concerns about the role of induction in theories of causality.^{[9][page needed][10][page needed]}

Physics

Most physical theories are time symmetric: microscopic models like Newton's laws or electromagnetism have no inherent direction of time. The "arrow of time" that distinguishes cause and effect must have another origin. To reduce confusion, physicists distinguish strong (macroscopic) from weak (microscopic) causality.

Macroscopic causality

The imaginary ability to affect the past is sometimes taken to suggest that causes could be negated by their own effects, creating a logical contradiction such as the grandfather paradox. This contradiction is not necessarily inherent to retrocausality or time travel; by limiting the initial conditions of time travel with consistency constraints, such paradoxes and others are avoided.

Aspects of modern physics, such as the hypothetical tachyon particle and certain time-independent aspects of quantum mechanics, may allow particles or information to travel backward in time. Logical objections to macroscopic time travel may not necessarily prevent retrocausality at other scales of interaction. Even if such effects are possible, however, they may not be capable of producing effects different from those that would have resulted from normal causal relationships.

Physicist John G. Cramer has explored various proposed methods for nonlocal or retrocausal quantum communication and found them all flawed and, consistent with the no-communication theorem, unable to transmit nonlocal signals.

Relativity

"In relativity, time and space are intertwined in the fabric of space-time, so time can contract and stretch under the influence of gravity." Closed timelike curves (CTCs), sometimes referred to as time loops, in which the world line of an object returns to its origin, arise from some exact solutions to the Einstein field equation. However, the chronology protection conjecture of Stephen Hawking suggests that any such closed timelike curve would be destroyed before it could be used. Although CTCs do not appear to exist under normal conditions, extreme environments of spacetime, such as a traversable wormhole or the region near certain cosmic strings, may allow their brief formation, implying a theoretical possibility of retrocausality. The exotic matter or topological defects required for the creation of those environments have not been observed.

Microscopic causality

Most physical models are time symmetric; some use retrocausality at the microscopic level.

Electromagnetism

Wheeler–Feynman absorber theory, proposed by John Archibald Wheeler and Richard Feynman, uses retrocausality and a temporal form of destructive interference to explain the absence of a type of converging concentric wave suggested by certain solutions to Maxwell's equations. These advanced waves have nothing to do with cause and effect: they are simply a different mathematical way to describe normal waves. The reason they were proposed is that a charged particle would not have to act on itself, which, in normal classical electromagnetism, leads to an infinite self-force.

Quantum physics

Time runs left to right in this Feynman diagram of electron–positron annihilation. When interpreted to include retrocausality, the electron (marked e⁻) was not destroyed, instead becoming the positron (e⁺) and moving backward in time.

Ernst Stueckelberg, and later Richard Feynman, proposed an interpretation of the positron as an electron moving backward in time, reinterpreting the negative-energy solutions of the Dirac equation. This is called the Stückelberg interpretation. Electrons moving backward in time would have a positive electric charge according to this interpretation. This time-reversal of anti-particles is required in modern quantum field theory, and is for example a component of how nucleons in atoms are held together with the nuclear force, via exchange of virtual mesons such as the pion. A meson is made up by an equal number of normal quarks and anti-quarks, and is thus simultaneously both emitted and absorbed.

Wheeler invoked this time-reversal concept to explain the identical properties shared by all electrons, suggesting that "they are all the same electron" with a complex, self-intersecting world line. Yoichiro Nambu later applied it to all production and annihilation of particle-antiparticle pairs, stating that "the eventual creation and annihilation of pairs that may occur now and then is no creation or annihilation, but only a change of direction of moving particles, from past to future, or from future to past." The backwards-in-time point of view is nowadays accepted as completely equivalent to other pictures, but it has nothing to do with the macroscopic terms "cause" and "effect", which do not appear in a microscopic physical description.

Retrocausality is associated with the Double Inferential state-Vector Formalism (DIVF), later known as the two-state vector formalism (TSVF) in quantum mechanics, where the present is characterised by quantum states of the past and the future taken in combination.

Retrocausality is sometimes associated with nonlocal correlations that generically arise from quantum entanglement, including for example the delayed choice quantum eraser. However accounts of quantum entanglement can be given which do not involve retrocausality. They treat the experiments demonstrating these correlations as being described from different reference frames that disagree on which measurement is a "cause" versus an "effect", as necessary to be consistent with special relativity. That is to say, the choice of which event is the cause and which the effect is not absolute but is relative to the observer. The description of such nonlocal quantum entanglements can be described in a way that is free of retrocausality if the states of the system are considered.

Tachyon visualization: since a tachyon moves faster than the speed of light, we can not see it approaching. After a tachyon has passed nearby, we would be able to see two images of it, appearing and departing in opposite directions. The black line is the shock wave of Cherenkov radiation, shown only in one moment of time.

Tachyons

Hypothetical superluminal particles called tachyons have a spacelike trajectory, and thus can appear to move backward in time, according to an observer in a conventional reference frame. Despite frequent depiction in science fiction as a method to send messages back in time, hypothetical tachyons do not interact with normal tardyonic matter in a way that would violate standard causality. Specifically, the Feinberg reinterpretation principle means that ordinary matter cannot be used to make a tachyon detector capable of receiving information.

Parapsychology

Retrocausality is claimed to occur in some psychic phenomena such as precognition. J. W. Dunne's 1927 book An Experiment with Time studied precognitive dreams and has become a definitive classic. Parapsychologist J. B. Rhine and colleagues made intensive investigations during the mid-twentieth century. His successor Helmut Schmidt presented quantum mechanical justifications for retrocausality, eventually claiming that experiments had demonstrated the ability to manipulate radioactive decay through retrocausal psychokinesis. Such results and their underlying theories have been rejected by the mainstream scientific community and are widely accepted as pseudoscience, although they continue to have some support from fringe science sources.

Efforts to associate retrocausality with prayer healing have been similarly rejected.

From 1994, psychologist Daryl J. Bem has argued for precognition. He subsequently showed experimental subjects two sets of curtains and instructed them to guess which one had a picture behind it, but did not display the picture behind the curtain until after the subject made their guess. Some results showed a higher margin of success (p. 17) for a subset of erotic images, with subjects who identified as "stimulus-seeking" in the pre-screening questionnaire scoring even higher. However, like his predecessors, his methodology has been strongly criticised and his results discounted.

Wormhole

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Wormhole

A wormhole visualized as a two-dimensional surface. Route (a) is the shortest path through normal space between points 1 and 2; route (b) is a shorter path through a wormhole.

A wormhole is a hypothetical structure that connects disparate points in spacetime. It can be visualized as a tunnel with two ends at separate points in spacetime (i.e., different locations, different points in time, or both). Wormholes are based on a special solution of the Einstein field equations. Wormholes are consistent with the general theory of relativity, but whether they actually exist is unknown. Many physicists postulate that wormholes are merely projections of a fourth spatial dimension, analogous to how a two-dimensional (2D) being could experience only part of a three-dimensional (3D) object.

In 1995, Matt Visser suggested there may be many wormholes in the universe if cosmic strings with negative mass were generated in the early universe. Some physicists, such as Kip Thorne, have suggested how to create wormholes artificially.

Terminology

In 1928, German mathematician, philosopher and theoretical physicist Hermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis of electromagnetic field energy;however, he did not use the term "wormhole" (he spoke of "one-dimensional tubes" instead).

The term "worm-holes" appears in British astronomer Arthur Eddington's 1928 popular exposition The Nature of the Physical World, in which he uses it metaphorically for "worm-holes" (material particles) crossing the "grain" of spacetime, rather than for a spacetime shortcut.

American theoretical physicist John Archibald Wheeler (inspired by Weyl's work) used the term "wormhole". In a 1957 paper that he wrote with Charles W. Misner, they write:

This analysis forces one to consider situations ... where there is a net flux of lines of force, through what topologists would call "a handle" of the multiply-connected space, and what physicists might perhaps be excused for more vividly terming a "wormhole".

— Charles Misner and John Wheeler in Annals of Physics

Modern definitions

Wormholes have been defined both geometrically and topologically. From a topological point of view, an intra-universe wormhole (a wormhole between two points in the same universe) is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes (1996).

If a Minkowski spacetime contains a compact region ${\displaystyle \mathrm{\Omega }}$, and if the topology of ${\displaystyle \mathrm{\Omega }}$ is of the form ${\displaystyle \mathrm{\Omega }\sim S\times \mathrm{\Sigma }}$, where ${\displaystyle \mathrm{\Sigma }}$ is a three-manifold of the nontrivial topology, whose boundary has the topology of the form ${\displaystyle \delta \mathrm{\Sigma }\sim S^2}$, and if, furthermore, the hypersurfaces ${\displaystyle \mathrm{\Sigma }}$ are all spacelike, then the region ${\displaystyle \mathrm{\Omega }}$ contains a quasi-permanent intrauniverse wormhole.

Geometrically, wormholes can be described as regions of spacetime that constrain the incremental deformation of closed surfaces. For example, in Enrico Rodrigo's The Physics of Stargates, a wormhole is defined informally as:

a region of spacetime containing a "world tube" (the time evolution of a closed surface) that cannot be continuously deformed (shrunk) to a world line (the time evolution of a point or observer).

Development

"Embedding diagram" of a Schwarzschild wormhole

Schwarzschild wormholes

The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that it would collapse too quickly for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes, were thought to be possible only if exotic matter with negative energy density could be used to stabilize them. Later, physicists reported that microscopic traversable wormholes may be possible and not require any exotic matter, instead requiring only electrically charged fermionic matter with small enough mass that it cannot collapse into a charged black hole. While such wormholes, if possible, may be limited to transfers of information, humanly traversable wormholes may exist if reality can broadly be described by the Randall–Sundrum model 2, a brane-based theory consistent with string theory.

Einstein–Rosen bridges

Einstein–Rosen bridges (or ER bridges), named after Albert Einstein and Nathan Rosen, are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, and that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": it should be possible to continue this path arbitrarily far into the particle's future or past for any possible trajectory of a free-falling particle (following a geodesic in the spacetime).

In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see the light that fell in from the other one), and likewise particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal–Szekeres coordinates.

In this spacetime, it is possible to come up with coordinate systems such that if a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') is picked and an "embedding diagram" drawn depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge". The Schwarzschild metric describes an idealized black hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from a collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's geography, it removes the part of the diagram corresponding to the white hole interior region, along with the part of the diagram corresponding to the other universe.

The Einstein–Rosen bridge was discovered by Ludwig Flamm in 1916, a few months after Schwarzschild published his solution, and was rediscovered by Albert Einstein and his colleague Nathan Rosen, who published their result in 1935. In 1962, John Archibald Wheeler and Robert W. Fuller published a paper showing that this type of wormhole is unstable if it connects two parts of the same universe, and that it will pinch off too quickly for light (or any particle moving slower than light) that falls in from one exterior region to make it to the other exterior region.

According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular Schwarzschild black hole. In the Einstein–Cartan–Sciama–Kibble theory of gravity, however, it forms a regular Einstein–Rosen bridge. This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamic variable. Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum (spin) of matter. The minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction that is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity (e.g. a black hole). Instead, the collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge.

Although Schwarzschild wormholes are not traversable in both directions, their existence inspired Kip Thorne to imagine traversable wormholes created by holding the "throat" of a Schwarzschild wormhole open with exotic matter (material that has negative mass/energy).

Other non-traversable wormholes include Lorentzian wormholes (first proposed by John Archibald Wheeler in 1957), wormholes creating a spacetime foam in a general relativistic spacetime manifold depicted by a Lorentzian manifold, and Euclidean wormholes (named after Euclidean manifold, a structure of Riemannian manifold).

Traversable wormholes

The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary matter vacuum energy, but it has been shown theoretically that quantum field theory disallows states where energy can be arbitrarily negative for an arbitrary length of time. Some physicists, such as Stephen Hawking, Kip Thorne, and others, argued that such effects might make it possible to stabilize a traversable wormhole. The only known natural process that is theoretically predicted to form a wormhole in the context of general relativity and quantum mechanics was put forth by Juan Maldacena and Leonard Susskind in their ER = EPR conjecture. The quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale, and stable versions of such wormholes have been suggested as dark matter candidates. It has also been proposed that, if a tiny wormhole held open by a negative mass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.

Image of a simulated traversable wormhole that connects the square in front of the physical institutes of University of Tübingen with the sand dunes near Boulogne-sur-Mer in the north of France. The image is calculated with 4D raytracing in a Morris–Thorne wormhole metric, but the gravitational effects on the wavelength of light have not been simulated.

Lorentzian traversable wormholes would allow travel in both directions from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis and independently in a 1973 paper by K. A. Bronnikov. Ellis analyzed the topology and the geodesics of the Ellis drainhole, showing it to be geodesically complete, horizonless, singularity-free, and fully traversable in both directions. The drainhole is a solution manifold of Einstein's field equations for a vacuum spacetime, modified by inclusion of a scalar field minimally coupled to the Ricci tensor with antiorthodox polarity (negative instead of positive). (Ellis specifically rejected referring to the scalar field as 'exotic' because of the antiorthodox coupling, finding arguments for doing so unpersuasive.) The solution depends on two parameters: m, which fixes the strength of its gravitational field, and n, which determines the curvature of its spatial cross sections. When m is set equal to 0, the drainhole's gravitational field vanishes. What is left is the Ellis wormhole, a nongravitating, purely geometric, traversable wormhole.

Kip Thorne and his graduate student Mike Morris independently discovered in 1988 the Ellis wormhole and argued for its use as a tool for teaching general relativity. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is also known as a Morris–Thorne wormhole.

Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made where the traversing path does not pass through a region of exotic matter. In the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions that is sometimes studied in the context of brane cosmology), however, exotic matter is not needed in order for wormholes to exist—they can exist even with no matter. A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al., in which it was proposed that such wormholes could have been naturally created in the early universe.

Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. In 1988, Morris, Thorne and Yurtsever worked out how to convert a wormhole traversing space into one traversing time by accelerating one of its two mouths. According to general relativity, however, it would not be possible to use a wormhole to travel back to a time earlier than when the wormhole was first converted into a time "machine". Until this time it could not have been noticed or have been used.

Raychaudhuri's theorem and exotic matter

To see why exotic matter is required, consider an incoming light front traveling along geodesics, which then crosses the wormhole and re-expands on the other side. The expansion goes from negative to positive. As the wormhole neck is of finite size, we would not expect caustics to develop, at least within the vicinity of the neck. According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition. Quantum effects such as the Casimir effect cannot violate the averaged null energy condition in any neighborhood of space with zero curvature, but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime. Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition, violations have nevertheless been found, so it remains an open possibility that quantum effects might be used to support a wormhole.

Modified general relativity

In some hypotheses where general relativity is modified, it is possible to have a wormhole that does not collapse without having to resort to exotic matter. For example, this is possible with R² gravity, a form of f(R) gravity.

Faster-than-light travel

Further information: Faster-than-light

Wormhole travel as envisioned by Les Bossinas for NASA, c. 1998

The impossibility of faster-than-light relative speed applies only locally. Wormholes might allow effective superluminal (faster-than-light) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slower-than-light) speeds are used. If two points are connected by a wormhole whose length is shorter than the distance between them outside the wormhole, the time taken to traverse it could be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole. A light beam traveling through the same wormhole would still beat the traveler.

Time travel

Main article: Time travel

If traversable wormholes exist, they might allow time travel. A proposed time-travel machine using a traversable wormhole might hypothetically work in the following way: One end of the wormhole is accelerated to some significant fraction of the speed of light, perhaps with some advanced propulsion system, and then brought back to the point of origin. Alternatively, another way is to take one entrance of the wormhole and move it to within the gravitational field of an object that has higher gravity than the other entrance, and then return it to a position near the other entrance. For both these methods, time dilation causes the end of the wormhole that has been moved to have aged less, or become "younger", than the stationary end as seen by an external observer; time connects differently through the wormhole than outside it, however, so that synchronized clocks at either end of the wormhole will always remain synchronized as seen by an observer passing through the wormhole, no matter how the two ends move around. This means that an observer entering the "younger" end would exit the "older" end at a time when it was the same age as the "younger" end, effectively going back in time as seen by an observer from the outside. One significant limitation of such a time machine is that it is only possible to go as far back in time as the initial creation of the machine; it is more of a path through time rather than it is a device that itself moves through time, and it would not allow the technology itself to be moved backward in time.

According to current theories on the nature of wormholes, construction of a traversable wormhole would require the existence of a substance with negative energy, often referred to as "exotic matter". More technically, the wormhole spacetime requires a distribution of energy that violates various energy conditions, such as the null energy condition along with the weak, strong, and dominant energy conditions. It is known that quantum effects can lead to small measurable violations of the null energy condition and many physicists believe that the required negative energy may actually be possible due to the Casimir effect in quantum physics. Although early calculations suggested a very large amount of negative energy would be required, later calculations showed that the amount of negative energy can be made arbitrarily small.

In 1993, Matt Visser argued that the two mouths of a wormhole with such an induced clock difference could not be brought together without inducing quantum field and gravitational effects that would either make the wormhole collapse or the two mouths repel each other, or otherwise prevent information from passing through the wormhole. Because of this, the two mouths could not be brought close enough for causality violation to take place. In a 1997 paper, however, Visser hypothesized that a complex "Roman ring" (named after Tom Roman) configuration of an N number of wormholes arranged in a symmetric polygon could still act as a time machine, although he concludes that this is more likely a flaw in classical quantum gravity theory rather than proof that causality violation is possible.

Interuniversal travel

See also: Multiverse

A possible resolution to the paradoxes resulting from wormhole-enabled time travel rests on the many-worlds interpretation of quantum mechanics.

In 1991 David Deutsch showed that quantum theory is fully consistent (in the sense that the so-called density matrix can be made free of discontinuities) in spacetimes with closed timelike curves. Later, it was shown that such a model of closed timelike curves can have internal inconsistencies as it will lead to strange phenomena like distinguishing non-orthogonal quantum states and distinguishing proper and improper mixture. Accordingly, the destructive positive feedback loop of virtual particles circulating through a wormhole time machine, a result indicated by semi-classical calculations, is averted. A particle returning from the future does not return to its universe of origination but to a parallel universe. This suggests that a wormhole time machine with an exceedingly short time jump is a theoretical bridge between contemporaneous parallel universes.

Because a wormhole time-machine introduces a type of nonlinearity into quantum theory, this sort of communication between parallel universes is consistent with Joseph Polchinski's proposal of an Everett phone (named after Hugh Everett) in Steven Weinberg's formulation of nonlinear quantum mechanics.

The possibility of communication between parallel universes has been dubbed interuniversal travel.

Wormholes can also be depicted in a Penrose diagram of a Schwarzschild black hole. In the Penrose diagram, an object traveling faster than light will cross the black hole and will emerge from another end into a different space, time or universe. This will be an inter-universal wormhole.

Metrics

Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of a (traversable) wormhole metric is the following:

${\displaystyle d{s}^2=-c^{2}d{t}^2+d{\ell }^2+(k^2+{\ell }^2)(d{\theta }^2+{\sin }^2\theta d{\phi }^2),}$

first presented by Ellis (see Ellis wormhole) as a special case of the Ellis drainhole.

One type of non-traversable wormhole metric is the Schwarzschild solution (see the first diagram):

${\displaystyle d{s}^2=-c^2\left(1-\frac{2GM}{r{c}^2}\right)d{t}^2+\frac{d{r}^2}{1-\frac{2GM}{r{c}^2}}+r^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2).}$

The original Einstein–Rosen bridge was described in an article published in July 1935.

For the Schwarzschild spherically symmetric static solution

${\displaystyle d{s}^2=-\frac{1}{1-\frac{2m}{r}}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2)+\left(1-\frac{2m}{r}\right)d{t}^2,}$

where ${\displaystyle ds}$ is the proper time and ${\displaystyle c=1}$.

If one replaces ${\displaystyle r}$ with ${\displaystyle u}$ according to ${\displaystyle u^2=r-2m}$

${\displaystyle d{s}^2=-4(u^2+2m)d{u}^2-(u^2+2m{)}^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2)+\frac{u^2}{u^2+2m}d{t}^2}$

The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to ${\displaystyle u>0}$ and ${\displaystyle u<0}$, which are joined by a hyperplane ${\displaystyle r=2m}$ or ${\displaystyle u=0}$ in which ${\displaystyle g}$ vanishes. We call such a connection between the two sheets a "bridge".

— A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"

For the combined field, gravity and electricity, Einstein and Rosen derived the following Schwarzschild static spherically symmetric solution

${\displaystyle {\phi }_1={\phi }_2={\phi }_3=0,{\phi }_4=\frac{\epsilon }{4},}$

${\displaystyle d{s}^2=-\frac{1}{\left(1-\frac{2m}{r}-\frac{{\epsilon }^2}{2r^2}\right)}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2)+\left(1-\frac{2m}{r}-\frac{{\epsilon }^2}{2r^2}\right)d{t}^2,}$

where ${\displaystyle \epsilon }$ is the electric charge.

The field equations without denominators in the case when ${\displaystyle m=0}$ can be written

${\displaystyle {\phi }_{\mu \nu }={\phi }_{\mu ,\nu }-{\phi }_{\nu ,\mu }}$

${\displaystyle g^2{\phi }_{\mu \nu ;\sigma }{g}^{\nu \sigma }=0}$

${\displaystyle g^2(R_{ik}+{\phi }_{i\alpha }{\phi }_k^{\alpha }-\frac{1}{4}{g}_{ik}{\phi }_{\alpha \beta }{\phi }^{\alpha \beta })=0}$

In order to eliminate singularities, if one replaces ${\displaystyle r}$ by ${\displaystyle u}$ according to the equation:

${\displaystyle u^2=r^2-\frac{{\epsilon }^2}{2}}$

and with ${\displaystyle m=0}$ one obtains

${\displaystyle {\phi }_1={\phi }_2={\phi }_3=0}$ and ${\displaystyle {\phi }_4=\frac{\epsilon }{{\left(u^2+\frac{{\epsilon }^2}{2}\right)}^{1/2}}}$

${\displaystyle d{s}^2=-d{u}^2-\left(u^2+\frac{{\epsilon }^2}{2}\right)(d{\theta }^2+{\sin }^2\theta d{\phi }^2)+\left(\frac{2u^2}{2u^2+{\epsilon }^2}\right)d{t}^2}$

The solution is free from singularities for all finite points in the space of the two sheets

— A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"

In fiction

Main article: Wormholes in fiction

Wormholes are a common element in science fiction because they allow interstellar, intergalactic, and sometimes even interuniversal travel within human lifetime scales. In fiction, wormholes have also served as a method for time travel.

Warp portals and higher-dimensional shortcuts

In theoretical physics and science fiction, the concept of a warp or warp portal is frequently used to describe shortcuts through space that are only possible by accessing a higher spatial dimension. Just as a two-dimensional space would require a third spatial dimension to bend or fold its plane in order to bring two distant points together, a three-dimensional space must be embedded within a fourth spatial dimension to allow similar manipulation. This analogy underlies many depictions of warp portals, which function by bending or folding three-dimensional space through the fourth spatial axis, allowing distant regions to become adjacent.

The existence of such a mechanism would imply that the universe possesses, or is embedded within, a four-dimensional spatial framework, even if that dimension is not directly observable. The geometry of these constructs is often modeled using solutions to Einstein's field equations, such as wormholes, and the Alcubierre warp bubble, both of which rely on higher-dimensional curvature.