Hideki Maeda presented a simple traversable wormhole solution which violate energy conditions only at the planck scale
Wormhole is a configuration of spacetimes containing distinct non-timelike infinities. In particular, a wormhole that contains a casual curve connecting such infinities are referred to as a traversable wormhole. Several examples of traversable wormholes have been already discussed by us in our articles. Now, Hideki Maeda presented a simple static spacetime which describes a spherically symmetric traversable worm-hole characterized by a length parameter, l and reduces to Minkowski in the limit l → 0. His findings recently appeared in Arxiv.
He showed that, this wormhole connects two distinct asymptotically flat regions with vanishing ADM mass and the areal radius of its throat is exactly l. Additionally, all the standard energy conditions i.e. null-energy condition, weak energy condition, dominant energy condition and standard energy condition are violated outside the proper radial distance approximately 1.60 l from the wormhole throat.
Finally, he computed the total amount of negative energy required to support this wormhole and found that, if l is identified as the Planck length lp (≃ 1.616 × 10¯35 m), the total amount of the negative energy supporting this wormhole is only E ≃ −2.65 mpc², which is the rest mass energy of about – 5.77 × 10¯5 g. For a “humanly traversable” wormhole with l = 1m, he obtained mass of −3.57 × 1027 kg, which is about –1.88 times as large as Jupiter’s mass.
“Ofcourse, an important task is to identify a theory of gravity which admits the static spacetime as a solution. Once such theory is identified, a subsequent task is to study the stability. Since the region of the NEC violation is tiny, our wormhole could be a dynamically stable configuration in the semiclassical regime. These important tasks are left for future investigations.”
— he concluded.
Reference: Hideki Maeda, “A simple traversable wormhole violating energy conditions only at the Planck scale”, Arxiv, pp. 1-4, 2021. arXiv:2107.07052
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Just search on google, “what are the types of wormholes?”, multiple results will appear on your screen like Morris Throne, Ellis wormhole, Schwarzschild wormhole etc. But, all these, are not at all same. Some are constructed in an artificial manner while, some are exact solutions of general relativity. Belonging to latter class is the wormhole called “Ellis-Bronnikov wormhole (EBWH)”. These wormholes are built by energy-violating Einstein minimally coupled scalar field. Now, Rosaliya Yusupova and colleagues carried out study of accretion onto Ellis-Bronnikov wormhole.
There is an important caveat relating to the stability of EBWH. EBWHs are unstable to perturbations. This means that they should not be existing in the universe today—they should have died almost as soon as they were born. In that case, what is the use of studying accretion to such an object? However, some time ago, the question of (in)stability was analyzed from a completely different standpoint based not on challenging the mechanism of instability but on exploiting the fact that, in general relativity, observations are dependent on the location of the observer. It was then shown that while some observers observe instability of EBWH, there is a nonzero probability that some other observers could observe its stability from different locations. Thus, their accretion scenario is relevant only to the latter types of observers.
They first identified a novel feature of EBWH, namely that it reduces to the Schwarzschild black hole under the combination of a complex wick rotation and an inversion.
Then, they analyzed the profiles of fluid radial velocity, density and the rate of mass variation of the EBWH due to accretion and compared the profiles with those of the Schwarzschild black hole (SBH). Their results are described below:
The EBWH velocity profiles remain lower than that of SBH at all radii, showing that the phantom matter also accretes to EBWH. The same behavior also emerges for massless EBWH. As one moves away from the central object, all profiles tend to merge close to one another.
The density of fluid accreting to EBWH is less than that accreting to SBH. In the case of massive EBWH, density increases in the vicinity of the throat (and horizon is case of SBH). While the density becomes minimums , near the throat of massless EBWH.
The accretion of phantom energy decreases the mass of SBH, but increases the mass of EBWH.
Accretion to massless EBWH shares the same patterns as those of the massive EBWH; hence, there is no way to distinguish massive and massless objects by means of accretion flow.
For quintessence, dust and stiff matter, it has been shown that the massless EBWH has the highest velocity of accreting fluid, and the SBH profile shows lowest velocity profiles. While, highest density and high values of accreting matter achieved near the central source.
Non-phantom accretion (quintessence, dust, stiff matter) increases the mass of SBH but decreases the mass of EBWH.
“The above contrasting behavior of accretion could be the physical signatures of the distinct topologies of the accreting central objects”
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Crispino and colleagues studied the absorption of massless scalar waves in a geometry that interpolates between the Schwarzschild solution and a wormhole that belongs to the Morris-Thorne class of solutions.
Their results showed that black holes and wormholes present distinctive absorption spectra.
They concluded that, wormhole results are characterized by the existence of quasibound states which generate Breit-Wigner-like resonances in the absorption spectrum.
Crispino and colleagues investigated the propagation of massless scalar waves in the geometry proposed by Simpson and Visser. The line element of this geometry depends on a parameter. Depending on the values of a parameter, this geometry describes a Schwarzschild black hole (BH), a regular BH, or a wormhole spacetime belonging to the Morris-Thorne class. Their study recently appeared in the Journal Physical Review D.
Within General Relativity (GR) , BHs are solutions of the Einstein’s field equations that posses an event horizon. The first exact solution of Einstein’s equation is known as Schwarzschild geometry, which describes a spherically symmetric, electrically uncharged and non-rotating BH. The spherically symmetric, electrically charged and non-rotating BH geometry is known as the Reissner-Nordström solution. The first exact uncharged rotating BH solution was obtained by Kerr, while the charged rotating BH was presented in Newman and colleagues. Such standard BH solutions of GR are cursed with singularities, where geometrical quantities diverge and physics predictability breaks down. Later, regular BHs, i.e. non-singular BH solutions, were proposed as an alternative to avoid the singularity problem. While, wormholes are solutions that connect two asymptotically flat regions by a throat.
Absorption of matter and fields is of great interest in GR, for instance, in explaining the role of accretion by BHs in active galactic nuclei. There have been many studies on the absorption of scalar waves by black holes. But, in comparison to BHs, few results for the absorption of scalar waves by wormhole spacetimes are available.
Thus, Crispino and colleagues now carried out the study on the scalar absorption for Schwarzschild and regular BHs, as well as for Morris-Thorne wormholes, considering the simpson-Visser line element. They used the partial wave approach to compute the scalar absorption cross section in this geometry.
“We found that the absorption spectrum of this geometry presents interesting features. For instance, the wormhole solution can show imprints of quasibound states around the throat, leading to narrow peaks (or Breit-Wigner-like resonances) in the absorption spectrum.”
They found that the BH and the wormhole configurations can be quite distinctive concerning the absorption of scalar waves. The distinction is due to the presence of trapped modes around the wormhole’s throat (as shown in table 1) and to the different values of the total absorption cross section in the low- and high-frequency limits (as shown in fig 1 below). The absorption cross section of the wormhole branch of interpolation can present narrow resonant peaks due to a potential well at the throat of the wormhole (refer fig 4 and 6 below). Moreover, these peaks become broader as they increase the parameter ‘a’, due to the decreasing of the depth of the potential well around the wormhole throat.
“Due to the shape of the potential of the wormhole case, the results are, in general quite different from the BH ones. Such differences arises due to the presence of potential well. This potential well allows quasibound states to exist around r = 0. These quasibound states are similar to the trapped modes and are associated to stable null geodesics at the wormhole throat in the eikonal limit. These trapped modes have complex frequency and the imaginary part is usually small i.e. they are long-lived modes. “
— told Crispino, one of the author of the study
They also reported similar resonance effects in the absorption cross section for extreme/exotic compact objects (ECOs) and for BH remnants, where the partial transmission amplitudes also present Breit-Wigner-type resonances, analogously to the phenomenon present in nuclear scattering theory.
Reference: Haroldo C. D. Lima Junior, Carolina L. Benone, and Luís C. B. Crispino, “Scalar absorption: Black holes versus wormholes”, Phys. Rev. D 101, 124009 – Published 5 June 2020. DOI: https://doi.org/10.1103/PhysRevD.101.124009
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The rotating BTZ black hole is a solution of Einstein gravity in 2+1 dimensions with a negative cosmological constant Λ = – 1 l². The solution can be constructed from a quotient of global AdS3. Now, Xiao and colleagues considered a rotating eternal black hole in three dimensions (rotating BTZ) and studied the traversable wormhole produced by including a double trace deformation at the boundary.
“We explored the effects of rotation in the size of a traversable wormhole obtained via a double trace boundary deformation.”
— told Xiao, third author of the study.
For simplicity, they first considered a constant boundary coupling. They showed that, at fixed temperature, the size of the wormhole opening increases with the angular momentum. They also established a bound on information that can be transferred and showed that in a rotating background more information can be sent through the wormhole as compared to the non-rotating scenario. However, for the type of interaction considered, the wormhole closes as the temperature approaches the extremal limit.
Finally, they briefly considered a scenario where the boundary coupling is not spatially homogenous (i.e. it is non-homogenous) and showed how this is reflected in the wormhole opening. They found that the wormhole opening is peaked near x1 (as shown in fig 1), but due to the presence of rotation the maximum value of the opening is slightly shifted to the right (i.e. y – x1).
Miranda and colleagues in their paper, studied a Kerr-like wormhole with phantom matter as source. It has three parameters: mass, angular momentum and scalar field charge. This wormhole has a naked ring singularity (very similar to kerr black hole), otherwise it is regular everywhere (except on ring) (please refer note below if you don’t know about ring singularity). It has also a gauge parameter which determine the radius of the throat and a naked ring singularity, exactly the same one as the Kerr solution.
The mean feature of this wormhole is that throat is found outside of the ring singularity, the mouth of the throat lie on a sphere of the same radius as the ring singularity around the wormhole, this avoids any observer to see or to reach the singularity, it behaves like an anti-horizon.
They analyzed the geodesics of the wormhole and showed that the null polar geodesics/light-rays, that means, geodesics going through the polar line are regular, an observer can go through the throat if the observer trajectory remains on the polar geodesics and contains an energy bigger than 1/2f, for any values of the free parameters. On this trajectory the tidal forces are very small, therefore this wormhole is traversable. For any other angle the mouth of the wormhole lie on the sphere, but close to the equator, the effect of the wormhole is to repel the test particles. On the equator, the repulsion is infinity and nothing can reach the singularity, even the light is repealed by the wormhole.
“We analyse the geodesics of the wormhole and find that an observer can go through the geodesics without troubles, but the equator presents an infinity potential barrier which avoids to reach the throat.”
— told Miranda, first author of the study
Thus, the sphere of the same radius as the ring singularity has an effect contrary to the horizon of a black hole, namely, an observer can reach the sphere, goes thought the throat, but this sphere avoids the traveller to observe or to reach the singularity. Of course, the traveller can come back to its original world without much troubles.
“From an analysis of the Riemann tensor we obtain that the tidal forces permits the wormhole to be traversable for an observer like a human being.”
— told Miranda, first author of the study
Note: Ring singularity: A ring singularity is the gravitational singularity of a rotating kerr black hole/ wormhole shaped like a ring (as shown in fig below).
Rahaman and colleagues, in their work taking cosmic fluid as source, have provided a new class of wormhole solutions under the framework of general relativity. Here, this matter source would supply fuel to construct the exact wormhole spacetime. Their matter sources consists of two non-interacting fluids, as follows: the first one is real matter in the form of perfect fluid and the second one is anisotropic dark energy which is phantom energy type. The mining of this second ingredient can be done from cosmic fluid that is responsible for acceleration of the Universe.
1.1.Energy Conditions
By checking the material compositions comprising the wormhole, whether it will satisfy or not the null energy condition (NEC), weak energy condition (WEC) and strong energy condition (SEC) simultaneously at all points outside the source, they found that the null energy condition (NEC), weak energy condition (WEC) are violated, however, the strong energy condition (SEC) is satisfied marginally. Hence, in their models, the null energy condition (NEC) is violated to hold a wormhole open.
1.2.Equilibrium condition
By obtaining modified TOV equation they showed that, equilibrium stage for the wormhole can be achieved due to the combined effect of 3 forces: gravitational force, hydrostatic force, plus another force due to the anisotropic nature (Fa) of the matter comprising the wormhole.
1.3.Effective gravitational mass
They also calculated effective mass of the wormhole, and found that it was up to radius 8 km from the throat (assuming the throat radius r0 = 4 km) is obtained as Meff = 0.828 km = 0.561 M (where 1 Solar Mass = 1.475 km). They noted from the effective gravitational mass equation that though wormholes are supported by the exotic matter, but the effective mass is positive. This implies that for an observer sitting at large distance could not distinguish the gravitational nature between wormhole and a compact mass M.
1.4. Total gravitational energy
It is known that total gravitational energy of a localized real matter obeying all energy conditions is negative. Naturally, authors would like to know how the gravitational energy behaves for the matters that supply fuel of their wormhole structure.
They found that the total gravitational energy from the throat r0 = 4 to the embedded radial space 1.5r0 = 7 (i.e. a = 1.5) as Eg = 1.9397, which indicates that Eg > 0, in other words, there is a repulsion around the throat. This result is very much expected for constructing a physically valid wormhole.
1.5.Traversability conditions
If the tidal gravitational forces felt by a traveler be reasonably small, then travel through wormhole is possible. It means that, acceleration felt by a traveler should less than the gravitational acceleration at earth surface. In their model, they found that the condition imposed by morris et al.
is automatically satisfied, the traveler feels a zero gravitational acceleration since ν = 0.
Based on the all these observations they concluded that their wormhole model with anisotropic dark energy and real matter is fascinating in several aspects and hence very promising one.
However, they observed in the present investigation that anisotropic dark energy with different energy density and radial pressure may also provide the exotic fuel in constructing the wormhole. So, interpretations within dark energy or other than dark energy is needed for exotic sector of the energy-momentum tensor which can be sought for in a future work.
In recent observations it is strongly believed that the universe is experiencing an accelerated expansion. The type Ia Supernovae and Cosmic Microwave Background (CMB) observations have shown the evidences to support cosmic acceleration. This acceleration is caused by some unknown matter which has the property that positive energy density and negative pressure satisfying ρ + 3p < 0 is dubbed as “dark energy” (DE). If ρ + p < 0, it is dubbed as “phantom energy”. The combined astrophysical observations suggests that universe is spatially flat and the dark energy occupies about 70% of the total energy of the universe, the contribution of dark matter is ∼ 26%, the baryon is 4% and negligible radiation. A cosmological property in which there is an infinite expansion in scale factor in a finite time termed as ‘Big Rip’. In the phantom cosmology, big rip is a kind of future singularity in which the energy density of phantom energy will become infinite in a finite time. To realize the Big Rip scenario the condition ρ + p < 0 alone is not sufficient. Distinct data on supernovas showed that the presence of phantom energy with – 1.2 < w < – 1 in the Universe is highly likely. In this case the cosmological phantom energy density grows at large times and disrupts finally all bounded objects up to subnuclear scale.
A wormhole is a feature of space that is essentially a “shortcut” from one point in the universe to another point in the universe, allowing travel between them that is faster than it would take light to make the journey through normal space. So the wormholes are tunnels in spacetime geometry that connect two or more regions of the same spacetime or two different spacetimes. Wormholes may be classified into two categories – Euclidean wormholes and Lorentzian wormholes. The Euclidean wormholes arise in Euclidean quantum gravity and the Lorentzian wormholes which are static spherically symmetric solutions of Einstein’s general relativistic field equations. In order to support such exotic wormhole geometries, the matter violating the energy conditions (null, weak and strong), but average null energy condition is satisfied in wormhole geometries. For small intervals of time, the weak energy condition (WEC) can be satisfied.
Ujjal Debnath and colleagues in their work studied effects of accretion of the dark energies onto Morris-Thorne wormhole using a dark-energy accretion model for wormholes which have been obtained by generalizing the Michel theory. They found that for quintessence like dark energy, the mass of the wormhole decreases and phantom like dark energy, the mass of wormhole increases.
They have also assumed recently proposed two types of dark energy like variable modified Chaplygin gas (VMCG) and generalized cosmic Chaplygin gas (GCCG). They obtained the expression of wormhole mass in both cases and found that the mass of the wormhole is finite at late universe. Their dark energy fluids violate the strong energy condition (ρ + 3p < 0 in late epoch), but do not violate the weak energy condition (ρ + p > 0). So the models drive only quintessence scenario in late epoch, but do not generate the phantom epoch (in their choice). So wormhole mass decreases during evolution of the universe for these two dark energy models.
Since their considered dark energy candidates do not violate weak energy condition, so the dynamical mass of the wormhole are decaying by the accretion of their considered dark energies, though the pressures of the dark energies are outside the wormhole. From figures 1 and 2, they observed that the wormhole mass decreases as z increases for both VMCG and GCCG, which accrete onto the wormhole in our expanding universe.
Figs. 1 and 2 show the variations of wormhole mass M against redshift z for VMCG and GCCG models.
Next they have assumed 5 kinds of parametrizations (Models I-V) of well known dark energy models (some of them are Linear, CPL, JBP models). These models generate both quintessence and phantom scenarios for some restrictions of the parameters. So if these dark energies accrete onto the wormhole, then for quintessence stage, wormhole mass decreases upto a certain value (finite value) and then again increases to infinite value for phantom stage during whole evolution of the universe. They also showed these results graphically clearly. Figures 3-7 shows the mass of wormhole first decreases to finite value and then increases to infinite value.
Figs. 3-7 show the variations of wormhole mass M against redshift z for Models I-V respectively
In future work, it will be interesting to show the natures of mass for various types of wormhole models if different kinds of dark energies accrete upon wormhole in accelerating universe also.
Previously on “Can We Able to escape big rip? PART 1: Achronal Cosmic Future”, We saw that In 2004, Pedro F. González Diaz obtained that the accretion of dark energy leads to a gradual increase of the wormhole throat radius which eventually overtakes the superaccelerated expansion of the Universe and becomes infinite at a time in the future before the occurrence of the big rip singularity.
After that time, as it continues accreting dark energy, the wormhole becomes an Einstein-Rosen bridge which can, in principle, be used by the future advanced civilizations in their efforts to escape from the big rip. However, in this activity the future civilizations will have to face but another problem: an “extremely stringent information’s bound”.
Guys during 1980’s, Bekenstein proposed a information bound, which is also called as “Berkenstein Bound”, which shows that the total amount of information, which can be stored in region of radius R is I < Im = 2πRE/(hc log 2). Thus,
So, in his paper, he calculated the horizon radius and found that its in the order of Planck scale. Thus, he found the Berkenstein Bound results in,
What it actually mean? Well, it means that, the largest amount of information to be processed and therefore – sent through the Einstein-Rosen bridge is no way greater then 69 bits. Yeah, just 69 bits. As a comparison, the superior limit of amount of information encoded in a human being is about 1045 bits. This value is surely an excess, for any object existing in current universe is encoded far less than any quantum field theory’s constraints, but in any case the amount of 69 bits seems terribly small. For example, the amount of information in a typical book is I ∼ 107 bits, and 1015 bits stands for all books in the Library of Congress. 69 bits is, in fact, slightly higher then the total amount of information that can be coded in the proton, viz. 44 bits!
So, it means you wont be able to leave this universe. The only way to save yourself, is to prevent big rip. And for that, you will need quantum effects.
○ Alexander Shatskiy considered a technique of calculating deflection of the light passing through wormholes (from one universe to another). He found fundamental and characteristic features of electromagnetic radiation passing through the wormholes.
○ He showed that the distortion of the light rays that had passed through the WH throat is caused not only by redistributing of the star density, but also by changes in their apparent brightness.
○ He also implied that if angular resolution of the observer’s instrument in our Universe is high enough, he will be able to discover the changing star density in the throat J(h).
○ He also showed that the apparent brightness of the WH’s part inside its throat does not depend on impact parameter.
Alexander Shatskiy considered a technique of calculating deflection of the light passing through wormholes (from one universe to another). He found fundamental and characteristic features of electromagnetic radiation passing through the wormholes. Making use of this, he proposed new methods of observing distinctive differences between wormholes and other objects as well as methods of determining characteristic parameters for different wormhole models.
By modifying Einstein equations, he first obtained the explicit analytical form of the solution to the first order in the small correction δ ( being defined by the equation-of-state of the matter in the WH) and then transcendental equation yielding the throat radius.
In order to know, what causes the distortion of light in wormhole, he considered that the other universe contain N stars with equal luminosities and supposed N >>> 1.
Then he assumed that, all the stars are homogeneously distributed over the celestial sphere in the other universe. An observer in our Universe who is looking at the stars in the other universe through the WH throat sees them inhomogeneously distributed over the throat. This is because of the fact that the WH throat refracts and distorts the light of these stars. The distortion will obviously be spherically symmetric with the throat center being the symmetry center.
Later, he assumed that the observer look only at the fraction of the stars seen in the thin ring with the center coinciding with the throat center, the ring radius being h and its width – dh. Hence, the observer surveys the solid angle dΩ of the other universe and, moreover, dΩ = 2π|sin θ| dθ. Here, θ(h) is the deflection angle of light rays passing through the WH throat measured relative to the rectilinear propagation (By convention, the rectilinear propagation means the trajectory passing through the center of the WH throat) . Since the total solid angle equals 4π, the observer can see dN = N dΩ/(4π) stars in the ring (Since the light deflection angle θ can exceed π, the total solid angle turns out to be more than 4π. This change, however, reduces to another constant (instead of 4π) and does not affect the final result.). Furthermore, the apparent density of the stars (per unit area of the ring dS = 2πh dh) is J = dN/dS. He, therefore, obtain:
Equation 1
The dependance θ(h),
Equation 2
where, the notations η ≡ 1/x and h˜ ≡ h/q were used. This yields
Equation 3
Taking advantage of all these formulae, they found the expression for the apparent density of the stars J(h) in the wormhole.
He noted that, 2nd formula/equation also gives the maximum possible impact parameter h = hmax which still allows the observer to see the stars of the other universe. This parameter corresponds to a zero of the second factor in the radicand in (2). Namely, hmax is equal to the least possible value of the function e^–φ / η. Having conducted trivial inquiry, he obtained:
as δ → 0.
What all these equations actually showed that the distortion of the light rays that had passed through the WH throat is caused not only by redistributing of the star density, but also by changes in their apparent brightness. Namely, as the impact parameter h increases the stellar brightness changes. This is because of the fact that as the radius h of the ring, through which the star light passes, increases, an element of the solid angle where this light scatters changes as well. The respective change in the stellar brightness is proportional to the quantity κ = dS/dΩ. Therefore, the total brightness of all the stars seen on unit area of the above-mentioned ring is dN · κ/dS.
Thus, he obtained that as N → ∞ the apparent brightness of the WH’s part inside its throat does not depend on impact parameter and, regardless of which WH model he used, the WH looks like a homogeneous spot in every wavelength range.
In spite the result obtained stating that the light distribution in the WH throat is homogeneous for each WH model, he also noted that in the real universe the number of visible stars is finite, though big. This implies that if angular resolution of the observer’s instrument in our Universe is high enough, he will be able to discover the changing star density in the throat J(h).
The left panel of Fig. 1 shows this plot for δ = 0.001. Sharp minima on the plot correspond to zeros of the sine in equation 1. This is because at sufficiently large impact parameters the light rays are deflected by large angles (θ > π) so that in the vicinities of the points θ = πn abrupt declines in distribution arise. But near these declines the observed stellar brightness tends to infinity (lensing), which ultimately provides the (on average) uniform light flow over the WH throat (see the right panel of Fig. 1).
Positions of the declines depend on the value of δ. Hence, registering them makes it possible to determine the equation-of-state parameters of the WH matter and features of the WH model (which is highly analogous to processing the light spectra).
Reference: Alexander Shatskiy, “Image of another universe being observed through a wormhole throat”, Astronomical Journal, pp. 1-6, 2012. https://arxiv.org/abs/0809.0362
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