Traversable wormholes in general relativity explore the theoretical framework for faster-than-light travel. The paper discusses how a spacecraft could warp spacetime without the need for wormholes, making it a pivotal work in theoretical physics. It delves into the implications of such travel on our understanding of the universe and the laws of physics. This research is essential for physicists and students interested in advanced concepts of general relativity and cosmology.

Key Points

  • Explores the concept of traversable wormholes within general relativity.
  • Discusses faster-than-light travel through spacetime warping.
  • Analyzes the implications of theoretical physics on real-world applications.
  • Essential reading for students and researchers in theoretical physics.
Noobmaster 69
Author:R. A. Konoplya, A. Zhidenko
8 pages
Language:English
Type:Research Paper
Arts and Humanities
Noobmaster 69
Author:R. A. Konoplya, A. Zhidenko
8 pages
Language:English
Type:Research Paper
Arts and Humanities
107
/ 8
arXiv:2106.05034v4 [gr-qc] 4 Mar 2022
Traversable wormholes in General Relativity
R. A. Konoplya
1, ∗
and A. Zhidenko
1, 2, †
1
Research Centre for Theoretical Physics and Astrophysics,
Institute of Physics, Silesian University in Opava,
Bezručovo náměstí 13, CZ-74601 Opava, Czech Republic
2
Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC (UFABC),
Rua Abolição, CEP: 09210-180, Santo André, SP, Brazil
In [1] asymptotically flat traversable wormhole solutions were obtained in the Einstein-Dirac-
Maxwell theory without using phantom matter. The normalizable numerical solutions found therein
require a peculiar behavior at the throat: the mirror symmetry relatively the throat leads to the
nonsmoothn ess of gravitational and matter fields. In particular, one must postulate changing of the
sign of the fermionic charge density at the throat, requiring coexistence of particle and antiparti-
cles without annihilation and posing a membrane of matter at the throat with specific properties.
Apparently this k ind of configuration could not exist in nature. We show that there are worm-
hole solutions, which are asymmetric relative t he th roat and endowed by smooth gravitational and
matter fields, thereby being free from all the above problems. This indicates that such wormhole
configurations could also b e supported in a realistic scenario.
PACS numbers: 04.20.-q,04.25.dg
Introduction. Wo rmholes are hypothetical objects
connecting disparate points of space time or even different
universe s [2]. Wormholes have never been observed and
even their ex istence and formation scenarios are highly
disputable questions. Nevertheless, the chance to have
a traversable wormhole or construct it in a laboratory
in the distant future pays off the efforts of theoreticians,
attracting a lo t of attention recent years. Existence of
humanly traversable wormholes requires gravitational re-
pulsion, which usually could be supported by matter
with negative kinetic terms, restraining the throat from
shrinking. Examples of wormholes without adding such
phantom matter come at the price of modifications of
the gravitational theory [3–11]. Frequently, wormholes
in such theories are unstable agains t linear perturbations
[12, 13]. Miniature s e lf-supported wormholes could possi-
bly exis t due to vacuum polarization in their v ic inity [14].
Cylindrical wormhole solutions found in [15] are noncom-
pact and glued with the asymptotically flat spacetime.
Therefore, the crucial question is whether asymptoti-
cally flat traversable wormholes could ex ist as compact
objects within the Einstein gravity without adding phan-
tom matter. In this case normal matter fields must any-
way violate the null energy conditions [16, 17]. Until the
recent work [1], no solutions of Einstein equations with
usual matter fields were known to provide existence of
such wor mholes. Wormhole solutions in Einstein gravity
with added Maxwell and two Dirac fields with the usual
coupling between them were found in [1]. Two kinds of
wormhole s olutions were represented there: The first one
is an analytical solution, describing sy mmetric relative
the throat wormhole supported by massless and neutral
fermions, which, themselves, are nonsymmetric relative
∗
roman.konoply[email protected]
†
olexandr.zh[email protected]
the throat. However, the fermions do not decay at in-
finity and are, therefore, non-normalizable
1
. The o ther,
normalizable, solution was obtained numerically and cor-
responds to symmetric configuration of both the metric
tensor and matter fields. The solution was obtained by
integrating the field equations between the throat and
infinity, and requiring the mirror symmetry, what led to
other “exotic” properties (see [19] for details):
1. The throat becomes a special point where a massive
shell of some matter must be posed.
2. The infinitely thin shell separ ates the fermion par-
ticles and antiparticles which, therefore, must meet
at the throat without annihilation.
3. The metric tensor and matter fields are not con-
tinuously differentiable at the throat (although the
metric and Riemann tensors are continuous).
Thus, the consistent quantum description of such classi-
cal configuration is evidently impossible.
In this context, we are interested to know whether
traversable wormholes can exist in a more r e alistic sit-
uation, i. e., without the above exotic fac tors, such as
the mass shell on the throat or coexistence of particles
and antiparticles without annihilation. Here we show
that there are nonsymmetric, relative the throat, contin-
uously differentiable solutions that describe asymptoti-
cally flat, traversable wormholes supported by nor maliz-
able and smooth matter fields. Thus, our solutions are
free from all of the above disadvantages of [1].
1
Note that traversable wormholes in the four-dimensional anti-de
Sitter spacetime can be supported by massless fermions, which
are localized near the throat [18].
2
Basic equations. We consider the action [1]
S =
1
4π
Z
√
−g
1
4
R + L
M
+ L
1
+ L
2
d
4
x, (1)
where the Lagrangians fo r the Maxwell and two Dirac
fields with mass µ are defined as follows:
L
M
= −
1
4
F
µν
F
µν
, F
µν
≡ ∂
µ
A
ν
− ∂
ν
A
µ
; (2)
L
1
=
i
2
¯
Ψ
1
γ
µ
D
µ
Ψ
1
−
i
2
(
¯
Ψ
1
γ
µ
D
µ
Ψ
1
)
∗
− iµ
¯
Ψ
1
Ψ
1
; (3)
L
2
=
i
2
¯
Ψ
2
γ
µ
D
µ
Ψ
2
−
i
2
(
¯
Ψ
2
γ
µ
D
µ
Ψ
2
)
∗
− iµ
¯
Ψ
2
Ψ
2
. (4)
A spherically symmetric configuration is given by the fol-
lowing line element and four-potential:
ds
2
= −N (x)
2
dt
2
+
r
′
(x)
2
B(x)
2
dx
2
+ r(x)
2
(dθ
2
+ sin
2
θdϕ
2
),
A
µ
dx
µ
= V (x)dt. (5a)
We employ the fo llowing ansatz for the spinors (cf. [20]):
Ψ
1
= e
−iωt+iϕ/2
φ(x) cos
θ
2
iκφ
∗
(x) sin
θ
2
−iφ
∗
(x) cos
θ
2
−κφ(x) sin
θ
2
, (5b)
Ψ
2
= e
−iωt−iϕ/2
iφ(x) sin
θ
2
κφ
∗
(x) cos
θ
2
φ
∗
(x) sin
θ
2
iκφ(x) cos
θ
2
, (5c)
with κ = ±1 and
φ(x) = e
iπ/4
F (x) − e
−iπ/4
G(x), (5d)
where F (x) and G(x) are real. The nonzero component
of the current is
j
0
=
4|φ(x)|
2
N(x)
=
4(F (x)
2
+ G(x))
2
N(x)
. (6)
Varying the action (1) and substituting (5) in the field
equations, we obtain a set of the ordinary differential
equations for functions N(x), B(x), V (x), G(x) and F (x)
(see Supplemental Material for details).
Wormholes with Z
2
symmetry. Junction condi-
tions at the throat lead to the choice of the opposite
signs for κ o n different sides of the throat and changing
the sign o f one of the fermio n functions, G(x) or F (x),
at the throat, which corresponds to the transfor mation
φ
+
→ ±iφ
∗
−
[1]. Therefore, it is convenient to associate
the two sides of space relative the throat with the oppo-
site signs of κ = ±1.
The analytic solution given in [1] suggests the
appropria te choice of the compact coordinate,
x = κ
p
1 − r
0
/r, so that the two signs of x describe the
wormhole on both sides of the throat located at x = 0
(r = r
0
). Without loss of generality, we take r
0
= 1 and
measure all the dimensional quantities in units of the
wormhole radius.
In order to obtain a symmetric wormhole, we solve the
field equations for x > 0 (κ = 1) and employ the above
junction condition to produce a symmetric solution. One
can check that the equations are automatically satisfied
for x < 0 (κ = −1) if
N
−
(x) = −N
+
(−x),
B
−
(x) = −B
+
(−x),
V
−
(x) = −V
+
(−x),
F
−
(x) = −F
+
(−x),
G
−
(x) = G
+
(−x),
ω
−
= −ω
+
.
(7)
Note that the mirror symmetry r equires changing of
the sign of frequency ω and lapse function N(x) at the
throat. The latter implies that the charge density (6) also
changes its sign, i. e., particles and antiparticles meet at
the throat. Since the Maxwell potential is an odd func-
tion of x, the electric strength V
′
(x) is even, having the
extreme value at the throat. Therefore, we require that
V
′′
(0) = 0. (8)
With the additional condition (8), for the fixed field
charge q and mass µ > 0, all the series coefficients for
the functions N(x), B(x), V (x), F (x) and G(x) ca n be
calculated in terms of the following four parameters (see
the Appendix)
n
i
≡ N(0), b
i
≡ B
′
(0),
f
i
≡ F (0)
√
r
0
, g
i
≡ G(0)
√
r
0
.
(9)
We use the series expansions to calculate values of the
functions near the throat and use them as initial values
for numerical integration. In order to solve the system of
six first-order differential equations (31 ) it is sufficient
to use the sta ndard Mathematica® Livermore Solver
with the quadruple-prec ision floating-point arithmetics.
We have checked that, within the numerical tolerance of
10
−6
, the explicit Runge-Kutta method with increase d
floating-point precision yields the same results, including
the asymptotic behavior of the functions in a vicinity of
the singular points x = ±1. Therefo re, we are c onvinced
that all six decimal cases in the presented numerical data
are accurate.
Considering fix e d n
i
, b
i
, and f
i
, and va rying g
i
, we find
that the fermion fields F(x) and G(x) diverge as
lim
x→1
F (x) = ±∞, lim
x→1
G(x) = ∓∞,
changing the sign at ce rtain values of g
i
. Thus, one can
use the shooting method to find the value g
i
, s uch that
F (x) and G(x) vanish as x → 1. The convergent solu-
tion is such that B(1) = 1 and N(1) = σ, so that the
asymptotic observer time is τ = σt. The values of the
asymptotic mass M, charge Q, and post-Newtonian pa-
rameter γ c an be read o ff from the asymptotic b e havior
3
of the functions
N
+
(x) = σ
1 −
2M
r
0
(1 − x) + O(1 − x)
2
,
B
+
(x) = 1 − γ
2M
r
0
(1 − x) + O(1 − x)
2
,
V
′
+
(x) = σ
2Q
r
0
+ O(1 − x)
.
(10)
It follows that variation of the parameter n
i
scales σ,
and we fix n
i
in such a way that σ = 1 (t = τ). Fol-
lowing [1], we choose f
i
= 0. Then b
i
parametrizes a
family of wormholes with Q > M, approaching the ex-
tremally charged Reissner-Nordström black hole in the
limit b
i
→ 0. Large r b
i
corresponds to smaller values
of Q/r
0
, M/r
0
, and M/Q. We notice that this family
of wormholes differs from the one r e ported in [1], where
the condition γ = 1 was imposed instead of (8). The
condition (8) is more relevant, since the Maxwell equa-
tion (21) has no discontinuity at the throat because of
changing the sign of V
′′
(0).
We conclude tha t the junction conditions for such sym-
metric wormholes lead to nonsmooth g e ometries and con-
figuration of matter fields. Although the metric tensor
and matter fields are continuous, their higher derivatives
have discontinuity at the throat. The geo metries consid-
ered in [1] have additional discontinuity of V
′′
(0). Never-
theless, the Riemann tensor is continuous in both cases.
Smooth asymmetric wormholes. In order to avoid
the above discontinuities, instead of the matching (7) at
the throat, we substitute
F (x) →
˜
F (x) ≡ κF (x), B(x) →
˜
B(x) ≡ κB(x), (11)
in the Einstein-Dirac-Maxwell equations (31)
2
and search
for a solution in the complete region −1 < x < 1. The
equations for the tilted functions do not depend on κ.
Therefore, we fix κ = 1 in a uniform way
3
for the solu-
tion in the whole space −1 < x < 1. The obtained solu-
tions for N(x), B(x), V (x), F (x) a nd G(x) are smooth
functions everywhere, so that the densities of the elec-
tromagnetic and Dirac fields do not have disco ntinuities,
and the resulting co nfiguration does no t req uire posing
any shell of matter at the throat. The function B(x)
changes its sign at the throat (x = 0) where it crosses
the x-a xis. The latter condition follows from the conti-
nuity of the radial vielbein (see Appendix) [21].
Since we do not have Z
2
symmetry with respect to
the thr oat, we need to impose two asymptotic conditions
independently. For fixed values of n
i
and b
i
we find that,
depending on the choice of f
i
and g
i
, F (x) and G(x) can
have the following asymptotic behavior (see Fig. 3):
2
Equivalently, we can substitute G(x) →
˜
G(x) ≡ κG(x).
3
The physically equivalent solution for κ = −1 can be obtained by
changing the sign of the functions F (x) and B(x) and mirroring
the configuration by replacing x → −x.
1. lim
x→±1
G(x) = +∞, lim
x→±1
F (x) = ∓∞ (red);
2. lim
x→±1
G(x) = −∞, lim
x→±1
F (x) = ±∞ (blue);
3. lim
x→±1
G(x) = ±∞, lim
x→±1
F (x) = −∞ (magenta);
4. lim
x→±1
G(x) = ∓∞, lim
x→±1
F (x) = +∞ (green).
When F (x) and G(x) change the sign there ar e expo-
nentially dec aying solutions, which we find by shooting
f
i
and g
i
. The resulting metric is asymptotically flat on
both sides of the throat,
N(x) = σ
±
1 −
2M
±
r
0
(1 ∓ x) + O(1 ∓ x)
2
,
B(x) = ±
1 − γ
±
2M
±
r
0
(1 ∓ x) + O(1 ∓ x)
2
, (12)
V
′
(x) = σ
±
±
2Q
±
r
0
+ O(1 ∓ x)
.
Note tha t, due to asymmetry, σ
+
6= σ
−
, two station-
ary asymptotic o bservers on the opposite sides o f the
wormhole’s throat have relativistic time dilation (red-
shift). By scaling n
i
we can fix the c oordinate time ac-
cording to one of the observers, so that either σ
+
= 1 or
σ
−
= 1. In addition, unles s q = 0, we have M
+
6= M
−
and Q
+
6= Q
−
. However, for all the solutions we have
obtained, γ
+
= γ
−
≈ 1, at least within the numerical
accuracy.
Since the electric potential V (x) is now a smooth func-
tion everywhere, requirement V
′′
(0) = 0 (8) seems not
relevant for this case. Although we have obtained the
wormholes, which have both smooth asymmetric contin-
uation and a continuous symmetric one, the mo st phys-
ically relevant condition on the throat is, apparently
(cf. [17]),
N
′
(0) = 0, (13)
which leads to no gravitational force experienced by a
stationary observer at the throat (x = 0). The corre-
sp onding time delation is given by σ
0
= N (0) = n
i
.
Since the field equations (31) are invariant under
changing sign of the fermionic functions G(x) → −G(x)
and F (x) → −F (x), we can study only g
i
> 0 without
loss of generality. For a given b
i
there are various possi-
ble solutions, corresponding to different values f
i
and g
i
.
We compare two such solutions on Fig. 1. The largest
absolute values of g
i
and f
i
(producing asymptotically
flat solutions) with opposite signs (solution (1) in Fig. 1)
correspond to the fermion configuration with G(x) 6= 0
and F (x) 6= 0. The sec ond largest values of g
i
and f
i
lead to the fermion c onfiguration, for which both G(x)
and F (x) cross zero once near the throat. The closer
the solution to the origin, the more zeroth the fermionic
/ 8
End of Document
107

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FAQs

what are traversable wormholes in general relativity

Traversable wormholes in General Relativity are hypothetical structures that connect disparate points in spacetime, allowing for travel between them.

  • They are solutions to the Einstein field equations.
  • Wormholes require exotic matter to keep them stable and open.
  • They could theoretically allow for shortcuts through space and time.

how do traversable wormholes work in general relativity

Traversable wormholes work by creating a bridge between two separate regions of spacetime, theoretically allowing for travel through them.

  • They involve a throat connecting two mouths, which can be visualized as two openings in spacetime.
  • The geometry around a traversable wormhole must allow for negative energy density to prevent collapse.
  • They challenge our understanding of physics, particularly regarding causality and the nature of spacetime.

what is the significance of traversable wormholes in physics

The significance of traversable wormholes in physics lies in their potential implications for time travel and the structure of the universe.

  • They provide insights into the nature of gravity and quantum mechanics.
  • Exploring their properties can lead to a better understanding of the fundamental laws of physics.
  • They also raise philosophical questions about the nature of reality and time.

can traversable wormholes exist in our universe

Currently, traversable wormholes are purely theoretical constructs and have not been observed in our universe.

  • They require conditions and materials, such as negative energy, that have not been found in nature.
  • Research continues to explore their feasibility within the framework of General Relativity.
  • Some scientists believe they could exist under specific conditions, but this remains speculative.

what are the challenges of creating traversable wormholes

Creating traversable wormholes poses significant challenges, primarily due to the need for exotic matter.

  • Exotic matter is required to maintain the stability of the wormhole throat.
  • Current physics does not provide a known source of such matter.
  • Additionally, the energy requirements and potential instabilities present further obstacles.

what findings are presented in the paper on traversable wormholes

The paper on traversable wormholes presents findings that explore new solutions to the Einstein equations without the need for phantom matter.

  • It discusses asymmetric wormhole configurations that are free from exotic properties.
  • The solutions are shown to be smooth and continuous, avoiding the issues found in previous models.
  • This research indicates that realistic traversable wormholes could exist under certain conditions.

how do traversable wormholes relate to black holes

Traversable wormholes are often compared to black holes, as both involve extreme gravitational fields.

  • Unlike black holes, which are regions from which nothing can escape, traversable wormholes theoretically allow for travel between different points in spacetime.
  • Both concepts challenge our understanding of gravity and spacetime.
  • Research into wormholes can provide insights into the nature of black holes and vice versa.

what is the role of exotic matter in traversable wormholes

Exotic matter plays a crucial role in the theoretical construction of traversable wormholes.

  • It is required to create negative energy density, which helps stabilize the wormhole throat.
  • Without exotic matter, wormholes would collapse under gravitational forces.
  • Research continues to investigate the properties and potential sources of exotic matter.