A spacetime with closed timelike geodesics everywhere

Abstract

In the present article we find a new class of solutions of Einstein's field equations. It describes stationary, cylindrically symmetric spacetimes with closed timelike geodesics everywhere outside the symmetry axis. These spacetimes contain a magnetic field parallel to the axis, a perfect fluid with constant density and pressure, and Lorentz invariant vacuum with energy density represented by a negative cosmological constant. Comment: 7 pages, 1 figure

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arXiv:1004.3235v1 [gr-qc] 19 Apr 2010
A spacetime with closed timelike geodesics everywhere
Øyvind Grøn∗and Steinar Johannesen∗
∗ Oslo University College, Department of Engineering, P.O.Box 4 St.Olavs Plass, N-0130 Oslo, Norway
Abstract In the present article we find a new class of solutions of Einstein’s field equations.
It describes stationary, cylindrically symmetric spacetimes with closed timelike geodesics every-
where outside the symmetry axis. These spacetimes contain a magnetic field parallel to the axis,
a perfect fluid with constant density and pressure, and Lorentz invariant vacuum with energy
density represented by a negative cosmological constant.
1. Introduction
Only three known solutions of Einstein’s field equations representing spacetimes in R4
with closed timelike geodesics have previously been found. It was demonstrated by Stead-
man [1] that such curves exist in the exterior van Stockum spacetime [2] representing
spacetime inside and outside a cylindrically symmetric rotating dust distribution. It has
also been shown by Bonnor and Steadman [3] that a spacetime with two spinning par-
ticles, each one with a magnetic moment and mass equal to their charge, permit special
cases in which there exist closed timelike geodesics. More recently Grøn and Johannesen
[4] found a class of solutions of Einstein’s field equations with closed timelike geodesics
representing spacetime outside a spinning cosmic string surrounded by a region of finite
radial extension with vacuum energy and a gas of non-spinning strings. In all of these
spacetimes there are closed timelike geodesics only at particular radii.
In the present article we present a solution of Einstein’s field equations representing
a cylindrically symmetric spacetime with closed timelike geodesics everywhere outside the
axis.
2. A spacetime with closed timelike geodesics everywhere
We shall here investigate a class of stationary cylindrically symmetric spacetimes described
by a line element of the form
ds2= −(dt − 2ωa(r)dφ)2+ b(r)2dφ2+ dr2+ dz2, (1)
where ω is a constant, using units so that c = 1. The coordinate time is shown on standard
clocks at rest in the coordinate system. The presence of the product term 4ωa(r)dtdφ in
the line element means that the coordinate clocks are not Einstein synchronized. Further-
more we assume that there is no gap in the coordinate time along a closed curve around
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the axis.
With this form of the line element, Weyssenhoff’s formula for the vorticity gives [5]
Ω =a′
bω ,(2)
where a′is the derivative of a with respect to r. We shall consider a space with constant
vorticity, Ω = ω. Hence b = a′.
In the present paper we shall search for solutions of Einstein’s field equations with
closed timelike geodesics at all radii r > 0.
Let us consider circular timelike curves in the plane z = constant with center on the
z-axis. For such curves to be closed in spacetime, the condition a′2− 4ω2a2< 0, must
be fullfilled [4]. Closed timelike geodesics of this type exist in a region where in addition
a′(a′′− 4ω2a) = 0. We have two cases:
1. a′= 0 which is not permitted because it implies that the determinant of the metric
tensor vanishes.
2. a′?= 0 and a′′− 4ω2a = 0 which gives that
a = C1cosh(2ωr) + C2sinh(2ωr) .(3)
From the conditions above it follows that the constants C1and C2must fullfill C1> C2≥
0, where we have assumed that a > 0, which involves no loss of physical generality. Since
these conditions may be fullfilled everywhere outside the z-axis, we can conclude that
there are closed timelike geodesics of the type described above in the entire spacetime
outside the z-axis.
Einstein’s field equations now give for the mixed components of the total energy
momentum tensor



0 0 0 3
κTµ
ν= ω2
1 0 0 0
0 1 0 0
0 0 1 0


+ Λδµ
ν,(4)
where κ is Einstein’s gravitational constant, and Λ is the cosmological constant represent-
ing Lorentz invariant vacuum energy (LIVE).
3. The physical contents of this spacetime
In addition to LIVE the spacetime is filled with a perfect fluid and a cylindrically sym-
metric magnetic field along the z-axis.
The energy momentum tensor of the perfect fluid has components
Tµ
ν= (ρ + p)uµuν+ pδµ
ν,(5)
where δµ
contravariant components of the fluid’s 4-velocity are ut= 1, ui= 0. The corresponding
non-zero covariant components are ut= −1, uφ= 2ωa. The non-zero mixed components
of the energy momentum tensor are
νis the Kronecker symbol. The fluid is at rest in the coordinate system. Then the
Tt
t= −ρ , Tr
r= Tφ
φ= Tz
z= p , Tt
φ= 2ωa(ρ + p) .(6)
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The mixed components of the energy momentum tensor for the magnetic field are
Tµ
ν= FµαFαν−1
4δµ
νFαβFαβ. (7)
The non-vanishing covariant components of the field tensor are
Frφ= −Fφr= B , (8)
where B is the magnetic field strength. This gives the following non-vanishing mixed
components of the energy momentum tensor
Tt
t= −Tr
r= −Tφ
φ= Tz
z=B2
2a′2, Tt
φ= −2ωa
a′2B2. (9)
We then obtain the following independent field equations
ω2+ Λ = κ
?B2
2a′2− ρ
?
, (10)
ω2+ Λ = κ
?
−B2
2a′2+ p
?
, (11)
3ω2+ Λ = κ
?B2
2a′2+ p
?
. (12)
Combining equations (10) and (11) we obtain
B2= (ρ + p)a′2
(13)
in accordance with the field equation for Tt
equations. Subtracting equation (11) from (12) and assuming that the magnetic field
points in the positive z-direction, we get
φ, which hence follows from the other field
B =
?2
κωa′.(14)
The last two equations give
κ(ρ + p) = 2ω2,(15)
which is the equation of state of the perfect fluid. Substituting equation (14) into equation
(10) leads to
κρ = −Λ = −κρv, (16)
where ρv is the density of the vacuum energy. Equations (15) and (16) show that the
density and pressure of the perfect fluid are constant, and that the cosmological constant
must be negative in order that the density of the perfect fluid shall be positive. The total
energy density of the perfect fluid and the vacuum energy vanishes. The energy density
in the spacetime comes from the magnetic field.
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4. A simple special case
4.1. Kinematics
As a simple illustrating example we will consider a solution of the field equations with
C1= 1 and C2= 0 in equation (3), giving
a = cosh(2ωr) .(17)
The line element then takes the form
ds2= −dt2+ 4ω cosh(2ωr)dφdt− 4ω2dφ2+ dr2+ dz2, (18)
In spite of the minus sign in front of dφ2the signature is correct. This is seen from the form
(1) of the line element corresponding to the orthogonal basis (et,eφ+ 2ωa(r)et,er,ez),
since et· et< 0, (eφ+ 2ωa(r)et) · (eφ+ 2ωa(r)et) = b(r)2> 0, er· er> 0 and ez· ez> 0.
The 3-space defined by the simultaneity t = constant is given by the spatial line
element
dl2= −4ω2dφ2+ dr2+ dz2.(19)
This shows that the vector eφ, which is a tangent vector in this space, has eφ· eφ =
−4ω2< 0. Hence this vector is timelike.
In the present spacetime the equation of circular null curves in the plane z = constant
with center on the z-axis is
4ω2?dφ
dt
?2
− 4ωcosh(2ωr)dφ
dt+ 1 = 0 .(20)
The physical velocities of light moving in opposite φ-directions are
v±=
?
|gφφ|
?dφ
dt
?
±= 2ω
?dφ
dt
?
±= e±2ωr. (21)
The speed of light is diffent from 1 since the coordinate clocks are not Einstein synchro-
nized. These expressions give the intersections of light cones with the plane in the tangent
space spanned by eφand etas shown in Figure 1. The reason for the direction of the light
cone is the following. We have considered light signals submitted in the positive and the
negative φ-direction. Their angular velocities are denoted by plus and minus in equation
(21). One signal has dφ > 0 and the other dφ < 0, but the angular velocities of both have
the same sign. Hence the coordinate time interval dt > 0 for the first signal, and dt < 0
for the second one. Also |(dφ/dt)+| > |(dφ/dt)−|.
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-
6
?
?
?
?
?
??
?????
?
x
y
t
Figure 1. Light cones in the spacetime with line element (18) where the x-, y- and t-axes correspond
to the basis vectors er, eφand et.
This figure shows that the tangent vector to a curve in the φ direction with t =
constant is inside the light cone at each point. Hence such a curve is timelike.
The light signal travelling in the negative φ-direction actually travels backwards in
time, and hence can be used to warn people living in previous times about possible catas-
trophic events in their future. This also involves the possibility of causal paradoxes, and
may demand some sort of chronology protection [6].
Let us calculate how far backwards in time light can come by travelling one time
around the axis. Integrating equation (21) we find
∆t1= −4πωe−2ωr1
(22)
for light travelling in the negative φ-direction. Hence by travelling an arbitrary number
of times around the axis the light may arrive arbitrarily far backwards in time.
We now consider light moving in both the φ- and r-directions. Then the 4-velocity
identity for the light takes the form
−˙t2+ 4ω cosh(2ωr)˙φ˙t − 4ω2˙φ2+ ˙ r2= 0 ,(23)
where the dot denotes differentiation with respect to an invariant parameter. This equa-
tion shows that there exist null curves with˙t = 0 given by
4ω2˙φ2= ˙ r2.(24)
Integrating this equation, we obtain
r = r0± 2ωφ(25)
describing Archimedean spirals. Light signals moving along these curves travel infinitely
fast, i.e. they arrive at the same point of time t as they are emitted.
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