476
M. C. Johnson et al.
1 Introduction
The Schwarzschild-de Sitter metric is a singular solution of the Einstein equations
with cosmological constant. The mass is a parameter that can be taken negative.
Here we show that it is possible to deform the metric and smooth out the singularity
with physically sensible energy-momentum, that which satisfies the dominant
energy condition everywhere. This means for any future directed time-like or lightlike vector u:
T
0ν u ν ≥ 0 and T
μν u ν T μα u
α
≥ 0.
(1)
The example of such a deformation was given in [1]. Then in [2] it was shown that
with an ideal fluid, there exist bubble like configurations with the exterior space-time
given exactly by the negative mass Schwarzschild-de Sitter space-time. The fluid
was not dynamical, no equation of state was specified. However, it was seen that an
ideal fluid, could in principle organize itself to correspond to localized regions of
negative mass.
In this work we show how to obtain dynamically stable, non-singular solutions
of negative mass which satisfy the dominant energy condition everywhere. They are
made up of a de Sitter space-time inside a thin wall separated from negative mass
Schwarzschild-de Sitter space-time on the outside. The Israel junction conditions
[3, 4] express the conservation of energy-momentum across the wall.
2 Thin Wall Bubbles
The stress-energy tensor of the wall will be taken as
S ˆ
a ˆ
b = diag. (σ, −ϑ, −ϑ).
(2)
The interior mass function is taken to be m − (r) which is not yet specified while the
exterior mass function is taken to be
m + (r) = −M +
Λr 3
6
(3)
that of a negative mass Schwarzschild-de Sitter space-time with mass −M and
cosmological constant Λ. The conservation of energy-momentum flux through the
wall gives the equations:
1 −
2m − (r)
r
+ ˙
r
2
1/2
−
1 +
2M
r
−
Λr 2
3
+ ˙
r
2
1/2
= 4πσ r
(4)
M. C. Johnson et al.
1 Introduction
The Schwarzschild-de Sitter metric is a singular solution of the Einstein equations
with cosmological constant. The mass is a parameter that can be taken negative.
Here we show that it is possible to deform the metric and smooth out the singularity
with physically sensible energy-momentum, that which satisfies the dominant
energy condition everywhere. This means for any future directed time-like or lightlike vector u:
T
0ν u ν ≥ 0 and T
μν u ν T μα u
α
≥ 0.
(1)
The example of such a deformation was given in [1]. Then in [2] it was shown that
with an ideal fluid, there exist bubble like configurations with the exterior space-time
given exactly by the negative mass Schwarzschild-de Sitter space-time. The fluid
was not dynamical, no equation of state was specified. However, it was seen that an
ideal fluid, could in principle organize itself to correspond to localized regions of
negative mass.
In this work we show how to obtain dynamically stable, non-singular solutions
of negative mass which satisfy the dominant energy condition everywhere. They are
made up of a de Sitter space-time inside a thin wall separated from negative mass
Schwarzschild-de Sitter space-time on the outside. The Israel junction conditions
[3, 4] express the conservation of energy-momentum across the wall.
2 Thin Wall Bubbles
The stress-energy tensor of the wall will be taken as
S ˆ
a ˆ
b = diag. (σ, −ϑ, −ϑ).
(2)
The interior mass function is taken to be m − (r) which is not yet specified while the
exterior mass function is taken to be
m + (r) = −M +
Λr 3
6
(3)
that of a negative mass Schwarzschild-de Sitter space-time with mass −M and
cosmological constant Λ. The conservation of energy-momentum flux through the
wall gives the equations:
1 −
2m − (r)
r
+ ˙
r
2
1/2
−
1 +
2M
r
−
Λr 2
3
+ ˙
r
2
1/2
= 4πσ r
(4)