black holes in an expanding universe: the mcvittie metric · 2013. 9. 25. · d. c. guariento vii...
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D. C. Guariento VII Aegean Summer School – 1
Black holes in an expanding universe:
The McVittie metric
Daniel C. Guariento
Universidade de São Paulo – Brazil
Perimeter Institute for Theoretical Physics – Canada
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Motivation
Introduction
Overview
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
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Motivation
Introduction
Overview
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
Two competing effects:
Gravitationally bound objects;
Expanding universe.
![Page 4: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/4.jpg)
Motivation
Introduction
Overview
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
Two competing effects:
Gravitationally bound objects;
Expanding universe.
Coupling between local effects and cosmological evolution
Causal structure
Accretion through Einstein equations;
Generalizations with other types of matter;
Consistency and stability analyses;
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Motivation
Introduction
Overview
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
Two competing effects:
Gravitationally bound objects;
Expanding universe.
Coupling between local effects and cosmological evolution
Causal structure
Accretion through Einstein equations;
Generalizations with other types of matter;
Consistency and stability analyses;
Experimental tests:
Structure formation;
Observations of galaxy clusters.
![Page 6: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/6.jpg)
Overview
Introduction
Overview
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 3
The McVittie metric
Properties
Apparent horizons
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with cosmological history
Example: Black hole in ΛCDM
Conclusions
![Page 7: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/7.jpg)
The McVittie metric
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 4
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −
(
1− m
2a(t)r
)2
(
1 + m
2a(t)r
)2dt2+a2(t)
(
1 +m
2a(t)r
)4(
dr2 + r2dΩ2)
![Page 8: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/8.jpg)
The McVittie metric
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 4
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −
(
1− m
2a(t)r
)2
(
1 + m
2a(t)r
)2dt2+a2(t)
(
1 +m
2a(t)r
)4(
dr2 + r2dΩ2)
a(t) constant: Schwarzschild metric
m = 0: FLRW metric
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The McVittie metric
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 4
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −
(
1− m
2a(t)r
)2
(
1 + m
2a(t)r
)2dt2+a2(t)
(
1 +m
2a(t)r
)4(
dr2 + r2dΩ2)
a(t) constant: Schwarzschild metric
m = 0: FLRW metric
Unique solution that satisfies
Spherical symmetry
Perfect fluid
Shear-free
Asymptotically FLRW
Singularity at the center
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Metric features
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 5
Past spacelike singularity at a = m
2r Event horizons only defined if H ≡ a
aconstant as t → ∞
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Metric features
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 5
Past spacelike singularity at a = m
2r Event horizons only defined if H ≡ a
aconstant as t → ∞
Fluid has homogeneous density
ρ(t) =3
8πH2
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Metric features
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 5
Past spacelike singularity at a = m
2r Event horizons only defined if H ≡ a
aconstant as t → ∞
Fluid has homogeneous density
ρ(t) =3
8πH2
Expansion is homogeneous (Hubble flow) and shear-free
Mean extrinsic curvature is constant on comoving foliation
Kαα = 3H
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Metric features
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 5
Past spacelike singularity at a = m
2r Event horizons only defined if H ≡ a
aconstant as t → ∞
Fluid has homogeneous density
ρ(t) =3
8πH2
Expansion is homogeneous (Hubble flow) and shear-free
Mean extrinsic curvature is constant on comoving foliation
Kαα = 3H
Pressure is inhomogeneous
p(r, t) =1
8π
[
H2−5m+ 2ar
m− 2ar+
2a
a
m+ 2ar
m− 2ar
]
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Areal radius coordinates
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 6
Causal structure is more easily seen on non-comoving coordinates
Areal radius
r = a(
1 +m
2r
)2r
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Areal radius coordinates
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 6
Causal structure is more easily seen on non-comoving coordinates
Areal radius
r = a(
1 +m
2r
)2r
Two branches:
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Areal radius coordinates
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 6
Causal structure is more easily seen on non-comoving coordinates
Areal radius
r = a(
1 +m
2r
)2r
Two branches:
0 < a < r
2m (not used)a
2m < r < ∞ =⇒ 2m < r < ∞
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Areal radius coordinates
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 6
Causal structure is more easily seen on non-comoving coordinates
Areal radius
r = a(
1 +m
2r
)2r
Two branches:
0 < a < r
2m (not used)a
2m < r < ∞ =⇒ 2m < r < ∞
McVittie in new (canonical) coordinates
[Kaloper, Kleban, Martin, PRD 81,104044 (2010)]
ds2 = −R2dt2 +
[
dr
R−Hrdt
]2
+ r2dΩ2
where
(
R =√
1− 2mr
)
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Apparent horizons
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 7
Apparent horizons: zero expansion of null radial geodesics
(
dr
dt
)
±= R (rH ±R) = 0
Only ingoing geodesics have a solution
1− 2m
r−Hr2 = 0
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Apparent horizons
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 7
Apparent horizons: zero expansion of null radial geodesics
(
dr
dt
)
±= R (rH ±R) = 0
Only ingoing geodesics have a solution
1− 2m
r−Hr2 = 0
Real positive solutions only exist if 13√3m
> H > 0
r+ Outer (cosmological) horizon
r− Inner horizon
If H(t) → H0 for t → ∞ apparent horizons become
Schwarszchild–de Sitter event horizons (r− → r∞)
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Light cones and apparent horizons
Introduction
The McVittie metric
Properties
Apparent horizons
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 8
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Asymptotic behavior
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 9
Inner horizon is an anti-trapping surface for finite coordinate times
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Asymptotic behavior
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 9
Inner horizon is an anti-trapping surface for finite coordinate times
Singular surface r∗ = 2m lies in the past of all events (McVittie big
bang)
d
dt(r − r∗) = RrH +O
(
R2)
> 0
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Asymptotic behavior
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 9
Inner horizon is an anti-trapping surface for finite coordinate times
Singular surface r∗ = 2m lies in the past of all events (McVittie big
bang)
d
dt(r − r∗) = RrH +O
(
R2)
> 0
If H0 > 0 the inner horizon (r∞ is traversable in finite proper time
Black hole at Shwarzschild–de Sitter limit
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Asymptotic behavior
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 9
Inner horizon is an anti-trapping surface for finite coordinate times
Singular surface r∗ = 2m lies in the past of all events (McVittie big
bang)
d
dt(r − r∗) = RrH +O
(
R2)
> 0
If H0 > 0 the inner horizon (r∞ is traversable in finite proper time
Black hole at Shwarzschild–de Sitter limit
If H0 = 0, some curvature scalars become singular at r∞
Possibly no BH interpretation at Schwarzschild limit
[Kaloper, Kleban, Martin, PRD 81,104044 (2010)][Lake, Abdelqader, PRD 84,044045 (2011)]
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Penrose diagrams
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 10
Geodesic completion with Schwarzschild–de Sitter
i+
b +i0
+
r = 0
r = 2m
( -)
r+r-
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Penrose diagrams
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 10
Geodesic completion with Schwarzschild–de Sitter
<
∞
>
∞
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Penrose diagrams
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 10
Geodesic completion with Schwarzschild–de Sitter
i+
b +i0
+
-
+
-
r = 0
r = 0
…
r = 2m
( -)
r+r-
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Convergence to H0
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 11
Causal structure depends on cosmological history
Horizon behavior at t → ∞ depends on the set Φ(H−)
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Convergence to H0
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 11
Causal structure depends on cosmological history
Horizon behavior at t → ∞ depends on the set Φ(H−)
Φ non-compact
All causal curves departing r⋆ cross r− before t → ∞ Spacetime connects to the inner region of
Schwarzschild–de Sitter
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Convergence to H0
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 11
Causal structure depends on cosmological history
Horizon behavior at t → ∞ depends on the set Φ(H−)
Φ non-compact
All causal curves departing r⋆ cross r− before t → ∞ Spacetime connects to the inner region of
Schwarzschild–de Sitter
Φ compact
Causal curves may depart r⋆ and never reach r− Spacetime connects to both inner and middle regions of
Schwarzschild–de Sitter
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McVittie with compact Φ
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 12
i+
-i0 +i0
+
+
-
r = 0
…
r = 2m
te
b
( -)
r+r-
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Cosmological history
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 13
We can determine the fate of null geodesics via the intermediate value
theorem [da Silva, Fontanini, DCG, PRD 87,064030 (2013)]
Find known curves that bind the image of ingoing geodesics from
above and below
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Cosmological history
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 13
We can determine the fate of null geodesics via the intermediate value
theorem [da Silva, Fontanini, DCG, PRD 87,064030 (2013)]
Find known curves that bind the image of ingoing geodesics from
above and below
A = R(r∞) + r∞R′(r∞), B = R(r∞) (R′(r∞)−H0),∆H = H −H0, ti > 0
F+(ti, t) =
∫
t
ti
e(B−δ)ue−A
∫u
ti∆H(s)ds
∆H(u)du
F−(ti, t) =
∫
t
ti
e(B+δ)ue−A
∫u
ti∆H(s)ds
∆H(u)du
If F+ diverges for some δ > 0, then Φ(H−) is unbounded
If F− converges for some δ > 0, then Φ(H−) is bounded
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Cosmological history
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 13
We can determine the fate of null geodesics via the intermediate value
theorem [da Silva, Fontanini, DCG, PRD 87,064030 (2013)]
Find known curves that bind the image of ingoing geodesics from
above and below
A = R(r∞) + r∞R′(r∞), B = R(r∞) (R′(r∞)−H0),∆H = H −H0, ti > 0
F+(ti, t) =
∫
t
ti
e(B−δ)ue−A
∫u
ti∆H(s)ds
∆H(u)du
F−(ti, t) =
∫
t
ti
e(B+δ)ue−A
∫u
ti∆H(s)ds
∆H(u)du
If F+ diverges for some δ > 0, then Φ(H−) is unbounded
If F− converges for some δ > 0, then Φ(H−) is bounded
Causal structure of McVittie depends on how fast H → H0.
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Example: ΛCDM
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 14
Source: dark matter and cosmological constant
Einstein’s equations result in the same H as in FLRW
H(t) = H0 coth
(
3
2H0t
)
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Example: ΛCDM
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 14
Source: dark matter and cosmological constant
Einstein’s equations result in the same H as in FLRW
H(t) = H0 coth
(
3
2H0t
)
Convergence depends on the sign of B − 3H0
(a) B − 3H0 < 0 =⇒ F− converges
(b) B − 3H0 > 0 =⇒ F+ diverges
(c) B − 3H0 = 0 =⇒ inconclusive
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Example: ΛCDM
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 15
Convergence depends on the sign of η
η ≡ B
3H0− 1 =
R(r∞)
3
[
R′(r∞)
H0− 1
]
− 1
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Example: ΛCDM
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 15
Convergence depends on the sign of η
η ≡ B
3H0− 1 =
R(r∞)
3
[
R′(r∞)
H0− 1
]
− 1
Inner horizon on the limit H → H0
r∞ =2
H0
√3cos
[
π
3+
1
3arccos
(
3√3mH0
)
]
η depends only on the product mH0 ≡ λ
Non-extreme Schwarzschild–de Sitter at infinity
0 < λ <1
3√3
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Example: ΛCDM
Introduction
The McVittie metric
Causal structure
Asymptotic behavior
Penrose diagrams
Dependence with
cosmological history
Example: Black hole in
ΛCDM
Conclusions
D. C. Guariento VII Aegean Summer School – 16
-1
-0.5
0
0.5
1
1.5
0 0.0755 13 3
= m H0
< 0
F- converges (a)
> 0
F+ diverges (b) = 0 (c)
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Conclusions
Introduction
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 17
McVittie is a fully dynamical exact solution of Einstein equations
describing a black hole in an expanding universe
Non empty space
Incompressible fluid
Shear-free motion
![Page 41: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/41.jpg)
Conclusions
Introduction
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 17
McVittie is a fully dynamical exact solution of Einstein equations
describing a black hole in an expanding universe
Non empty space
Incompressible fluid
Shear-free motion
Rich causal structure requires careful analysis
Big Bang not on intuitive location
Apparent horizons depend on cosmological evolution
Local physics dependent on global structure
![Page 42: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/42.jpg)
Conclusions
Introduction
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 17
McVittie is a fully dynamical exact solution of Einstein equations
describing a black hole in an expanding universe
Non empty space
Incompressible fluid
Shear-free motion
Rich causal structure requires careful analysis
Big Bang not on intuitive location
Apparent horizons depend on cosmological evolution
Local physics dependent on global structure
Coupling to a scalar field opens possibilities for modified gravity
![Page 43: Black holes in an expanding universe: The McVittie metric · 2013. 9. 25. · D. C. Guariento VII Aegean Summer School – 13 We can determine the fate of null geodesics via the intermediate](https://reader035.vdocuments.us/reader035/viewer/2022071411/6107303d73d2d5656134a511/html5/thumbnails/43.jpg)
Conclusions
Introduction
The McVittie metric
Causal structure
Conclusions
D. C. Guariento VII Aegean Summer School – 17
McVittie is a fully dynamical exact solution of Einstein equations
describing a black hole in an expanding universe
Non empty space
Incompressible fluid
Shear-free motion
Rich causal structure requires careful analysis
Big Bang not on intuitive location
Apparent horizons depend on cosmological evolution
Local physics dependent on global structure
Coupling to a scalar field opens possibilities for modified gravity
A fluid with more degrees of freedom provides new mechanisms for
interaction between the black hole and the environment
(stay tuned for M. Fontanini’s talk)