Pis’ma v ZhETF, vol. 111, iss. 6, pp. 343 - 344
© 2020
March 25
Spatial Kasner solution and an infinite slab with constant energy
density
A. Yu. Kamenshchik+∗1), T. Vardanyan×◦
+Dipartimento di Fisica e Astronomia, Università di Bologna and Istituto Nazionale di Fisica Nucleare,
via Irnerio 46, 40126 Bologna, Italy
∗L. D. Landau Institute for Theoretical Physics Russian Academy of Sciences, 117940 Moscow, Russia
×Dipartimento di Fisica e Chimica, Università di L’Aquila, 67100 Coppito, L’Aquila, Italy
◦Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy
Submitted 27 February 2020
Resubmitted 27 February 2020
Accepted 27
February 2020
DOI: 10.31857/S0370274X20060028
The Kasner solution [1] of the Einstein equations
coordinates. To avoid it Schwarzschild also invented an
for an empty Universe having the spatial geometry of
internal solution [4] generated by a ball with constant
Bianchi-I type is usually presented in the “cosmological
energy density and isotropic pressure. At the boundary
form”:
of the ball the pressure disappears and the external and
internal solutions are matched. Then there is no singu-
ds2 = dt2 - a20t2p1 dx2 - b20t2p2 dy2 - c20t2p3 dz2.
(1)
larity in the center of the ball.
The solutions of the Einstein equations in the pres-
In the original paper by Kasner [1] the positive defi-
ence of an infinite plane or an infinite slab of a finite
nite metric with the dependence on one coordinate was
thickness with the metric
considered. Introducing the normal spacetime signature,
one can recover not only the cosmological metric (1), but
ds2 = a2(x)dt2 - dx2 - b2(x)dy2 - c2(x)dz2,
(5)
also a stationary metric that depends on one spatial co-
were also discussed in literature [5]. When b(x) = c(x),
ordinate:
these solutions are matched with special cases of the
ds2 = a20(x - x0)2p1 dt2 - dx2 - b20(x - x0)2p2 dy2 -
Kasner metric (2) such as the Rindler solution [6] with
p1 = 1, p2 = p3 = 0 and the Weyl-Levi-Civita solution
- c20(x - x0)2p3dz2.
(2)
[7, 8] with p1 = -13 , p2 = p3 =23
In our paper [9] we found an explicit form of two
The metric (2) has a singularity at the hypersurface
x = x0, where the value x0 is arbitrary. The Kasner
exact solutions in the spacetime with an infinite slab
of thickness 2L. In both cases pressure vanishes at the
indices p1, p2 and p3 satisfy the relations
boundaries of the slab. Outside the slab these solutions
p1 + p2 + p3 = p21 + p22 + p23 = 1.
(3)
are matched with the Rindler spacetime and with the
Weyl-Levi-Civita spacetime. Here we describe general
A convenient parametrization of the Kasner indices was
properties of the solutions of the Einstein equations
presented in paper [2]:
when there is an isotropy in yz-plane, i.e., b(x) = c(x),
and explicitly construct a particular exact solution that
u
1+u
p1 = -
, p2 =
,
differs from two solutions found in paper [9]. Besides,
1+u+u2
1+u+u2
we discuss solutions with b(x) = c(x), that are matched
u(1 + u)
p3 =
(4)
in the empty part of the space with the general Kas-
1+u+u2
ner solutions and not with its particular cases where
p2 = p3. We are not able to write down an explicit solu-
It is interesting to compare the Kasner solution (2) with
the external spherically symmetric Schwarzschild solu-
tion of this kind, however, analyzing the corresponding
differential equations we can show that such solutions
tion [3]. This solution has a singularity in the center of
do exist. Moreover, we prove that one of these empty
1)e-mail: kamenshchik@bo.infn.it
half-spaces should have Kasner singularity.
Письма в ЖЭТФ том 111 вып. 5 - 6
2020
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