On Spacetimes with Given Kinematical Invariants: Construction and Examples

0.22 MB·English

by  M. Plaue

ON SPACETIMES WITH GIVEN KINEMATICAL
INVARIANTS: CONSTRUCTION AND EXAMPLES
M. PLAUE, M. SCHERFNER, AND L. A. M. DE SOUSA JR.
Abstract. We present a useful method for the construction of cosmologi-
cal models by solving the differential equations arising from calculating the
kinematical invariants (shear, rotation, expansion and acceleration) of an ob-
server field in proper time description. As an application of our method we
8 present two generalizations of the G¨odel spacetime that follow naturally from
0
our approach.
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2

n
a
J 1. Introduction

4
2 The construction of viable cosmological and astrophysical models often requires

] particular restrictions on the kinematic properties of those models. For example,
c
q parallax-free models must necessarily be shear-free [8].
-
r Here the cosmological observer field is analyzed in proper time description to
g
[ obtain expressions for the metric that depend explicitly on the kinematical in-

2 variants. This approach leads to a useful toolkit for constructing cosmological
v
models with given kinematical properties. In addition, the analysis given here
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6
leads to a deeper understanding of the kinematical invariants and the relations
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3 between them. Of course, restrictions regarding the kinematical quantities also
.
1 give rise to a limited range of possible matter models – it is often not possible
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8 to give energy-momentum tensors representing simple matter like a perfect fluid
0
with a particular combination of rotation, shear, expansion and acceleration. Re-
:
v
call for example the well-known shear-free fluid conjecture which has been proven
i
X
for a number of special cases (cf. [7]).
r
a
As an application of our method, we will construct a spacetime that general-
izes the well-known G¨odel metric [1] which represents a dust model with negative
cosmologicalconstant thathasnon-vanishingrotation,butvanishingshear, accel-
eration and expansion. It can be shown that this model contains closed timelike
curves. Following [3], p. 170 those properties suggest that the G¨odel spacetime
is not very physical. In 1952, G¨odel [2] made some remarks about the necessity
of constructing a solution with non-vanishing expansion, but without giving an
1
explicit metric. In this paper, we will construct one generalization of the G¨odel
spacetime for which the acceleration vanishes as well as another model that is
parallax-free.
Theauthorswish tothankProfessor YuriObukhov formanyvaluablesuggestions
while preparing the paper.
2. Preliminaries
We consider an (N +1)-dimensional smooth Lorentzian manifold (M,g) of sig-
nature (+, ,..., ). Given coordinates (x0,...,xN), by ∂ we denote the partial
l
− −
derivative with respect to the lth coordinate and stands for the covariant de-
j
∇
rivative indirection of the jth coordinatevector. Latinindices will takethe values
0,...,N, whereas greek indices will range from 1 to N. In the following we fix a
timelike unit vector field Xi, the cosmological observer field. The usual decom-
position of the covariant derivative of X into irreducible parts (see for example
i
[4]) reads
1
(1) X = ω +σ + ΘP X˙ X
k i ik ik ik i k
∇ N −
with the antisymmetric part ω (rotation), the symmetric traceless part σ
ik ik
(shear) and the trace Θ (expansion) itself. In detail, the parts read
(2) ω = X X˙ X ,
ik [i k] [i k]
∇ −
1
(3) σ = X X˙ X ΘP ,
ik ∇(i k) − (i k) − N ik
and
(4) Θ = Xa.
a
∇
Here the brackets (parentheses) denote the antisymmetric (symmetric) parts and
(5) P = g X X
ik ik i k
−
is the projection tensor on the N-dimensional subspace perpendicular to Xi. The
acceleration is given by
(6) X˙ = Xa X .
i a i
∇
In addition we have
(7) X˙ Xa = 0, σ Xa = ω Xa = P Xa = 0.
a ai ai ai
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3. Proper time description of the observer field
We adopt comoving coordinates with respect to the observer field such that Xi =
δi; this is also known as the proper time description. From Xi being a timelike
0
unit vector field we infer 1 = X Xa = g XaXb = g . The first coordinate
a ab 00
denotes the proper time of the cosmological observer; a fact which we will from
now on emphasize by denoting x0 with t.
In proper time description the components of the metric can be expressed by
components of the observer field – this is crucial to our analysis and we have:
(8) X = g Xa = g
i ai 0i
and
(9) P = g g g .
ik ik 0i 0k
−
For the covariant derivative one has
(10) X = ∂ X Γa X
∇k i k i − ik a
= ∂ X Γa g
k i − ik 0a
1
= ∂ g gabg (∂ g +∂ g ∂ g )
k 0i 0a k ib i kb b ik
− 2 −
1
= ∂ g δb(∂ g +∂ g ∂ g )
k 0i − 2 0 k ib i kb− b ik
1
= (∂ g ∂ g +∂ g ).
k 0i i 0k 0 ik
2 −
For the acceleration, one has in particular:
(11) X˙ = Xa X
i a i
∇
= X
0 i
∇
1
= ∂ g ∂ g
0 0i i 00
− 2
= ∂ g .
0 0i
3
In the same manner, we obtain similar expressions for the other kinematical
quantities in proper time description:
1
(12) Θ = gia∂ g
0 ia
2
1
= ∂ (logdet(g )),
0 ik
2
1
(13) ω = (g ∂ g g ∂ g ∂ g +∂ g ),
ik 0k 0 0i 0i 0 0k k 0i i 0k
2 − −
1 1
σ = (∂ g g ∂ g g ∂ g ) ΘP
ik 2 0 ik − 0k 0 0i − 0i 0 0k − N ik
1 1
(14) = ∂ P ΘP .
2 0 ik − N ik
Itcanalsobeeasilyseenfrom(7)thatthekinematicalquantitiesarecharacterized
by their spatial components:
(15) X˙ = 0, ω = σ = P = 0.
0 0i 0i 0i
4. Models with given kinematical invariants
Equation (14) can be solved for the projection tensor P = g g g , which
ik ik 0i 0k
−
leads to
(16) P (t,xγ) = (Σ (t,xγ)+P (0,xγ))S2(t,xγ).
ik ik ik
with
t
1
(17) S(t,xγ) = exp Θ(τ,xγ)dτ
N Z
 0 
and
t
σ (τ,xγ)
(18) Σ (t,xγ) = 2 ik dτ.
ik Z S2(τ,xγ)
0
Denoting derivation with respect to t by a dot, we have
S˙
(19) Θ = N
S
with S(0,xγ) = 1; S(t,xγ) is the so-called scale parameter.
We interpret this equation as an evolution equation for the spatial components
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of the metric for a given “start metric” g (0,xγ) and a given observer field
αβ
g (t,xγ), expansion Θ(t,xγ) and shear σ (t,xγ) that drive the evolution:
0α αβ
(20) g (t,xγ) = g (t,xγ)g (t,xγ)+
αβ 0α 0β
S2(t,xγ)(g (0,xγ) g (0,xγ)g (0,xγ))+
αβ 0α 0β
−
S2(t)Σ (t,xγ).
αβ
Remark: The initial value “t = 0” is not a specific coordinate value like e.g.
a singularity. In fact, the construction is not valid for singular start metrics.
One should regard g (0,xγ) as the spatial metric at some arbitrary time in the
αβ
history of the observer.
5. Some special cases
Whereas expansion, shear and the observer field itself enter the evolution equa-
tion directly, the rotation and acceleration impose additional constraints on the
observer field.
Models with vanishing acceleration. In the case of a vanishing acceleration,
we have ∂ g = 0, which implies g (t,xγ) = g (0,xγ), and (20) reduces to
0 0α 0α 0α
(21) g (t,xγ) = S2(t,xγ)g (0,xγ)+
αβ αβ
1 S2(t,xγ) g (0,xγ)g (0,xγ)+
0α 0β
−
(cid:0) (cid:1)
S2(t)Σ (t,xγ).
αβ
Irrotational models. The rotation is calculated from the g components of
0α
the metric alone. The equation for an observer field with vanishing rotation
can be solved by setting g = h(t)∂ φ(xγ). If the model also has vanishing
0α α
−
acceleration, h(t) is constant and the observer field is a gradient.
Parallax-free models. Spacetimeswithanobserver fieldparalleltoaconformal
vectorfieldareimportantsincesuchmodelsarepreciselytheparallax-freemodels.
As was shown in [8], this condition holds if and only if Xi is shear-free and the
exterior derivative of X˙ ΘX vanishes. The last condition can be met by
i − N i
assuming
Θ
(22) X˙ = X ∂ f
i i i
N −
for some function f.
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Integrating (22), and substituting into (20) with σ = 0, we have
αβ
(23) g (t,xγ) = S2(t,xγ)g (0,xγ)+
αβ αβ
S2(t,xγ)F (t,xγ)F (t,xγ)
α β
−
S2(t,xγ)(F (t,xγ)g (0,xγ)+F (t,xγ)g (0,xγ))
α 0β β 0α
with the functions
t
∂ f(τ,xγ)
(24) F (t,xγ) = α dτ.
α Z S(τ,xγ)
0
Toolkit. Figure 1 can be used as a toolkit to construct spacetimes with specific
kinematic properties. We have only listed shear-freemodels since anon-vanishing
shear can be easily included via the functions Σ . Also, we have notationally
αβ
suppressed the dependence on the spatial coordinates xγ.
6. Generalizations of the Go¨del spacetime
WeliketogeneralizetheG¨odelspacetime(R4,g˜)equippedwiththeusualobserver
field to a shear-free model with non-vanishing expansion. Written in canonical
(cartesian) coordinates (t,x,y,z), the G¨odel metric reads:
1 0 e√2ω0x 0
 
0 1 0 0
(25) (g˜ ) = − .
ik e√2ω0x 0 1e2√2ω0x 0 
 2 
 0 0 0 1
 − 
In this coordinates, the observer field is already of the desired form, Xi = δi. We
0
take as a start metric
1 0 0
−
(26) (g (0,xγ)) =  0 1e2√2ω0x 0 ,
αβ
2
 0 0 1
 − 
and as the observer field we choose
(27) (g (t,xγ)) = 0 f(t)e√2ω0x 0 .
0α
(cid:16) (cid:17)
The time-dependent function f(t) accounts for a possible non-vanishing acceler-
ation.
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Spatial metric g (t) = g (t,xγ)
αβ αβ
General S2(t)g (0)+g (t)g (t) S2(t)g (0)g (0)
αβ 0α 0β 0α 0β
−
X˙ = 0 S2(t)g (0)+(1 S2(t))g (0)g (0)
i αβ 0α 0β
−
Θ = 0 g (0)+g (t)g (t) g (0)g (0)
αβ 0α 0α 0α 0α
−
Θ = 0,X˙ = 0 g (0)
i αβ
ω = 0 S2(t)g (0)+(h2(t) h2(0)S2(t))∂ φ∂ φ
ik αβ α β
−
Parallax-free S2(t)g (0)+S2(t)(F (t)F (t) F (t)g (0) F (t)g (0))
αβ α β α 0β β 0α
− −
Figure 1. Table of shear-free cosmological models
In order to evolve this metric in a shear-free manner we use equation (20) with
σ = 0. To simplify things, we assume that the scale parameter S depends only
αβ
on the cosmological time parameter t. In this way we arrive at
(28)
S2(t) 0 0
−
(g (t,xγ)) =  0 1 [S2(t)(1 2f2(0))+2f2(t)]e2√2ω0x 0 .
αβ
2 −
 0 0 S2(t)
 − 
We like to model the matter content of this spacetime by a fluid (not necessarily
perfect). Thus, the energy-momentum tensor takes the form
(29) T = pg +(ǫ+p)X X +2q X +π ,
ik ik i k (i k) ik
−
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with energy density ǫ, isotropic pressure p, heat flow q and anisotropic pressure
i
π .
ik
The heat flow and anisotropic pressure have to satisfy the conditions q Xa = 0,
a
π Xa = 0 and πa = 0. The first condition directly implies q = 0, whereas the
ai a 0
second condition fixes q by inspection of the Einstein equation R 1Rg
γ ik − 2 ik −
Λg = κT after solving for ǫ and p:
ik ik
√2ω f(t)
(30) q = 0 3f˙(t)S(t) S˙(t)f(t) ,
1 κ(2f2(0) 1)S3(t) (cid:16) − (cid:17)
−
2f(t)e√2ω0x
(31) q = S(t)S¨(t) S˙(t) ,
2 κ S2(t) (cid:16) − (cid:17)
(32) q = 0.
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Models with vanishing acceleration. First we dealwith the case of vanishing
acceleration and set f(t) 1. The evolved metric then reads
≡
1 0 e√2ω0x 0
 0 S2(t) 0 0 
(33) (g ) = −
ik e√2ω0x 0 1e2√2ω0x(2 S2(t)) 0 
 2 − 
 0 0 0 S2(t)
 − 
and the condition πa = 0 enforces
a
(34) 2ω2 1 S2(t) S¨(t)S(t)+2(S˙(t))2 = 0.
0 − −
(cid:0) (cid:1)
This equation can be reduced to the separable first order equation
(35) (S˙(t))2 = ω2 (α 1)S4(t)+2S2(t) 1
0 − −
(cid:0) (cid:1)
which implies
(36) S¨(t) = 2ω2S(t) (α 1)S2(t)+1 .
0 −
(cid:0) (cid:1)
Substituting (35) and (36) into the differential equation (34) one sees that it is
identically satisfied for any value of the constant α. Furthermore, the equation
canbesolved interms ofelliptic integrals (thesolutionisoneof theJacobielliptic
functions).
However, evaluating (35) at t = 0 one sees that 9αω2 = 9S˙2(0) = Θ2(0) and we
0
must have α 0 to obtain real valued solutions. For α = 0 = Θ(0) one has
≥
S(t) 1 and we recover the G¨odel metric. The phase portrait of the differential
≡
equationfordifferent valuesofΘ(0)isshowninfigure2: If0 < Θ(0) < 3 ω , the
0
| | | |
solutionsareperiodic,whereasfor3 ω Θ(0) thespacetimeexpands/collapses
0
| | ≤ | |
8
indefinitely. The metric does not become singular at any point since S(t) does
not vanish for any initial value.
Furthermore, the model exhibits an interesting causal structure that we will dis-
cuss in the following section.
1.5
1.0
0.5
dS/dt 0.0
−0.5
−1.0
−1.5
0.75 1.0 1.25 1.5 1.75 2.0 2.25
S
Figure 2. Phase portrait for the scale parameter S(t) (ω = 1)
0
Parallax-free models. Wewillnowchoosef(t)such thatthemodelisparallax-
free. Calculating the exterior derivative of X˙ ΘX , we find that the observer
i − N i
field of our already shear-free model (28) is parallel to a conformal vector field if
and only if the following set of differential equations is satisfied:
(37) f¨(t)S(t) S˙(t)f˙(t) = 0,
−
(38) f˙(t)S(t) S˙(t)f(t) = 0.
−
Together with the condition of a traceless anisotropic pressure we finally have
S(t) = e13Θt = f(t) with the constant expansion Θ.
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The resulting metric takes the form
1 0 e√2ω0x+13Θt 0
 0 e23Θt 0 0 
(39) (g ) = − ,
ik e√2ω0x+31Θt 0 1e2√2ω0x+23Θt 0 
 2 
 0 0 0 e32Θt
 − 
which is precisely the G¨odel metric for Θ = 0.
We compute some scalar invariants of this solution: Rotationscalar andthe norm
of the acceleration are given by
(40) ωabωab = 2ω02e−23Θt,
2
(41) X˙ X˙a = Θ2.
a
−9
The heat flow and anisotropic pressure are non-zero:
(42) qaqa = πabπab = 8ω02Θ2e−23Θt.
− 9 κ2
Finally, the energy density and pressure are given by:
(43) ǫ = 31κ (cid:16)3ω02e−23Θt −Θ2 −3Λ(cid:17),
(44) p = 31κ (cid:16)3ω02e−23Θt +Θ2 +3Λ(cid:17).
A particular interesting case is Λ = Θ2, where we have a stiff matter model
− 3
(ǫ = p = ωκ02e−23Θt) with non-vanishing anisotropic pressure and heat flow that
is asymptotically a lambda vacuum for large cosmological times. In general,
one must have Λ Θ2 to ensure that the cosmological observer measures an
≤ − 3
everywhere non-negative energy density T XaXb = ǫ.
ab
7. Closed causal curves
Any function on a smooth closed curve xi(s) is periodic with respect to the curve
parameter s. If the curve is contained in a chart of a Lorentzian manifold, this
holds in particular for the coordinate function x0(s) and there exist maximal and
minimal points of x0(s) for which dx0 = 0 holds. Let xi(s) be such a closed curve
ds
that is also causal, i.e.
dxidxk
(45) g 0
ik
ds ds ≥
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