Black holes, wormholes, and singularities
beyond
Riemannian geometry
Gonzalo J. Olmo
Dept. Física Teórica and IFIC - UV & CSIC (Valencia,Spain)
in collaboration with
D. Rubiera-Garcia, A. Sanchez-Puente, , F.S.N. Lobo, D. Bazeia, L. Losano, . . .
Gonzalo J. Olmo
Schwarzschild solution
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
One hundred years ago Schwarzschild found his famous solution:
2
1
2
2
2
dt
+
ds2 = − 1 − 2M
2M dr + r dΩ
r
(1− r )
◆
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
◆
◆
◆
It describes the exterior gravitational field of a spherical object of mass M.
The observational success of this solution is impressive.
Planetary motion, light bending, . . . verified using its linearized form.
This geometric theory was capturing fundamental aspects of Nature.
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 2/29
Schwarzschild solution
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
One hundred years ago Schwarzschild found his famous solution:
2
1
2
2
2
dt
+
ds2 = − 1 − 2M
2M dr + r dΩ
r
(1− r )
◆
● So . . . what is a singularity?
● Living with holes
◆
On the geometry of space-time
◆
Palatini Gravity
◆
It describes the exterior gravitational field of a spherical object of mass M.
The observational success of this solution is impressive.
Planetary motion, light bending, . . . verified using its linearized form.
This geometric theory was capturing fundamental aspects of Nature.
Graphene Wormhole
■
Conclusions
But we had to wait until the 1960’s to fully understand its complexity:
The End
◆
◆
Gonzalo J. Olmo
The exterior solution could be
extended below r = 2M until r → 0.
It describes a Black Hole, with an
event horizon and a singular region
inside.
Lisbon, 12-15 Sept. 2016 - p. 2/29
Schwarzschild solution
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
One hundred years ago Schwarzschild found his famous solution:
2
1
2
2
2
dt
+
ds2 = − 1 − 2M
2M dr + r dΩ
r
(1− r )
◆
● So . . . what is a singularity?
● Living with holes
◆
On the geometry of space-time
◆
Palatini Gravity
◆
It describes the exterior gravitational field of a spherical object of mass M.
The observational success of this solution is impressive.
Planetary motion, light bending, . . . verified using its linearized form.
This geometric theory was capturing fundamental aspects of Nature.
Graphene Wormhole
■
Conclusions
But we had to wait until the 1960’s to fully understand its complexity:
The End
◆
◆
■
Gonzalo J. Olmo
The exterior solution could be
extended below r = 2M until r → 0.
It describes a Black Hole, with an
event horizon and a singular region
inside.
Exact solutions of this type are very important for the observational verification
of the theory. But their physical interpretation is not fully satisfactory.
Lisbon, 12-15 Sept. 2016 - p. 2/29
Troubles and implications
■
The Schwarzschild solution describes a vacuum space-time: Rµν = 0
◆
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
◆
◆
Palatini Gravity
Graphene Wormhole
◆
The equations are not solved at the
location of the sources.
gµν is not known at r = 0 .
Point particles DO NOT generate
this solution.
What sources this field?
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 3/29
Troubles and implications
■
The Schwarzschild solution describes a vacuum space-time: Rµν = 0
◆
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
◆
● Living with holes
◆
On the geometry of space-time
Palatini Gravity
◆
Graphene Wormhole
The equations are not solved at the
location of the sources.
gµν is not known at r = 0 .
Point particles DO NOT generate
this solution.
What sources this field?
Conclusions
■
The End
Gonzalo J. Olmo
Gravitational collapse models show that (parts of) I and II are physical.
◆ The energy density grows unboundedly at r = 0 as the collapse proceeds.
◆ Curvature scalars diverge as r → 0 in the exact solution.
◆ Infinities. Divergences. Singularities. Something weird is going on there!
Lisbon, 12-15 Sept. 2016 - p. 3/29
Troubles and implications
■
The Schwarzschild solution describes a vacuum space-time: Rµν = 0
◆
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
◆
● Living with holes
◆
On the geometry of space-time
Palatini Gravity
◆
Graphene Wormhole
The equations are not solved at the
location of the sources.
gµν is not known at r = 0 .
Point particles DO NOT generate
this solution.
What sources this field?
Conclusions
The End
Gonzalo J. Olmo
■
Gravitational collapse models show that (parts of) I and II are physical.
◆ The energy density grows unboundedly at r = 0 as the collapse proceeds.
◆ Curvature scalars diverge as r → 0 in the exact solution.
◆ Infinities. Divergences. Singularities. Something weird is going on there!
■
The simplicity of the Schwarzschild solution is complex enough to hide
fundamental basic limitations of the theory it comes from:
◆ The theory seems unable to determine the geometry in some regions.
◆ Without a well defined background geometry, physical laws as we know
them cannot be formulated.
Lisbon, 12-15 Sept. 2016 - p. 3/29
Troubles and implications
■
The Schwarzschild solution describes a vacuum space-time: Rµν = 0
◆
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
◆
● Living with holes
◆
On the geometry of space-time
Palatini Gravity
◆
Graphene Wormhole
The equations are not solved at the
location of the sources.
gµν is not known at r = 0 .
Point particles DO NOT generate
this solution.
What sources this field?
Conclusions
The End
■
Gravitational collapse models show that (parts of) I and II are physical.
◆ The energy density grows unboundedly at r = 0 as the collapse proceeds.
◆ Curvature scalars diverge as r → 0 in the exact solution.
◆ Infinities. Divergences. Singularities. Something weird is going on there!
■
The simplicity of the Schwarzschild solution is complex enough to hide
fundamental basic limitations of the theory it comes from:
Despite its enormous observational success, predictability and determinism are seriously threatened in this theory.
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 3/29
What is a singularity?
■
We need to improve our understanding of the theory to overcome the
difficulties raised by singularities.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 4/29
What is a singularity?
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
■
We need to improve our understanding of the theory to overcome the
difficulties raised by singularities.
■
In gravitation theories the notion of singularity differs from that in Minkowski:
◆ In Minkowski if a certain field attains infinite values at a given space-time
point, then we say that the field has a singularity at that point.
◆ In the Schwarzschild solution the field equations are not solved at r = 0:
strictly speaking that is not a region of the space-time!!!
◆ The idea of singularity as a place is, therefore, not appropriate.
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 4/29
What is a singularity?
● Schwarzschild solution
■
We need to improve our understanding of the theory to overcome the
difficulties raised by singularities.
■
In gravitation theories the notion of singularity differs from that in Minkowski:
◆ In Minkowski if a certain field attains infinite values at a given space-time
point, then we say that the field has a singularity at that point.
◆ In the Schwarzschild solution the field equations are not solved at r = 0:
strictly speaking that is not a region of the space-time!!!
◆ The idea of singularity as a place is, therefore, not appropriate.
■
Regions where curvature invariants blow up as we approach them?
◆ There are solutions with vanishing curvature scalars but divergent Riemann.
◆ In a cone, Rα βµν = 0 everywhere but the metric is not defined at the vertex.
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 4/29
What is a singularity?
● Schwarzschild solution
■
We need to improve our understanding of the theory to overcome the
difficulties raised by singularities.
■
In gravitation theories the notion of singularity differs from that in Minkowski:
◆ In Minkowski if a certain field attains infinite values at a given space-time
point, then we say that the field has a singularity at that point.
◆ In the Schwarzschild solution the field equations are not solved at r = 0:
strictly speaking that is not a region of the space-time!!!
◆ The idea of singularity as a place is, therefore, not appropriate.
■
Regions where curvature invariants blow up as we approach them?
◆ There are solutions with vanishing curvature scalars but divergent Riemann.
◆ In a cone, Rα βµν = 0 everywhere but the metric is not defined at the vertex.
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
The general characterization of singularities by the “blowing up” of curvature is unsatisfactory. The characterization of singularities by a detailed
enumeration of the possible other types of pathological behavior of the
space-time metric also appears to be a hopeless task because of the infinite variety of possible pathological behavior. [R.M.Wald (1984), p.214]
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 4/29
So . . . what is a singularity?
■
Singularities are neither places nor things that explode. So, how do we
characterize them?
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 5/29
So . . . what is a singularity?
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
■
Singularities are neither places nor things that explode. So, how do we
characterize them?
■
The presence of singularities should be detectable by the fact that certain curves
(geodesics) will have finite affine length.
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
◆
◆
Gonzalo J. Olmo
We say that a space-time is singular when it is geodesically incomplete.
The Schwarzschild and Reissner-Nordström space-times are singular.
Lisbon, 12-15 Sept. 2016 - p. 5/29
So . . . what is a singularity?
● Schwarzschild solution
■
Singularities are neither places nor things that explode. So, how do we
characterize them?
■
The presence of singularities should be detectable by the fact that certain curves
(geodesics) will have finite affine length.
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
◆
◆
■
Gonzalo J. Olmo
We say that a space-time is singular when it is geodesically incomplete.
The Schwarzschild and Reissner-Nordström space-times are singular.
The singularity theorems determine conditions for geodesic incompleteness.
◆ They occur in GR under very general assumptions.
◆ They do not provide any information about the behavior of curvature tensors
or the metric.
Lisbon, 12-15 Sept. 2016 - p. 5/29
Living with holes
■
The singularity theorems confirm that singularities, understood as “holes” in
the fabric of space-time, are endemic in General Relativity.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 6/29
Living with holes
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
■
The singularity theorems confirm that singularities, understood as “holes” in
the fabric of space-time, are endemic in General Relativity.
■
Interestingly, many of the problems in singular space-times can be illustrated
with Minkowskian examples:
● Living with holes
On the geometry of space-time
◆
Palatini Gravity
Graphene Wormhole
◆
Conclusions
The End
Gonzalo J. Olmo
This implies the possibility of creating and
destroying particles and radiation.
Predictability and determinism are seriously
compromised.
Lisbon, 12-15 Sept. 2016 - p. 6/29
Living with holes
● Schwarzschild solution
■
The singularity theorems confirm that singularities, understood as “holes” in
the fabric of space-time, are endemic in General Relativity.
■
Interestingly, many of the problems in singular space-times can be illustrated
with Minkowskian examples:
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
◆
Palatini Gravity
Graphene Wormhole
◆
Conclusions
The End
■
Gonzalo J. Olmo
This implies the possibility of creating and
destroying particles and radiation.
Predictability and determinism are seriously
compromised.
Numerous attempts to build nonsingular theories of gravity have focused on
keeping curvature invariants and the matter fields under control:
◆ Theories with bounded scalars and energy densities (NEDs, scalar fields),
surgically joined space-times, . . .
Lisbon, 12-15 Sept. 2016 - p. 6/29
Living with holes
● Schwarzschild solution
■
The singularity theorems confirm that singularities, understood as “holes” in
the fabric of space-time, are endemic in General Relativity.
■
Interestingly, many of the problems in singular space-times can be illustrated
with Minkowskian examples:
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
◆
Palatini Gravity
Graphene Wormhole
◆
Conclusions
The End
Gonzalo J. Olmo
This implies the possibility of creating and
destroying particles and radiation.
Predictability and determinism are seriously
compromised.
■
Numerous attempts to build nonsingular theories of gravity have focused on
keeping curvature invariants and the matter fields under control:
◆ Theories with bounded scalars and energy densities (NEDs, scalar fields),
surgically joined space-times, . . .
■
Infinities in the matter sector are a technical difficulty.
The fundamental problem with singularities is the presence of holes:
◆ Is there a physically sensible way to deal with holes in geometry?
Lisbon, 12-15 Sept. 2016 - p. 6/29
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 7/29
Effective geometries in Physics
■
Wave propagation on the continuum effective geometry of bilayer graphene.
■
A microstructure with a macroscopic continuum limit is found in condensed
matter systems such as graphene or Bravais crystals.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 8/29
Effective geometries in Physics
■
Wave propagation on the continuum effective geometry of bilayer graphene.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
◆
Gonzalo J. Olmo
Microscope image of a graphene layer with defects.
Lisbon, 12-15 Sept. 2016 - p. 8/29
Microscopic defects
● Schwarzschild solution
● Troubles and implications
■
A microstructure with a macroscopic continuum limit is found in condensed
matter systems such as graphene or Bravais crystals.
■
Crystalline structures may have different kinds of defects:
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
◆
◆
Gonzalo J. Olmo
In real crystals, the density of defects is generally non-zero.
There are interactions between different kinds of defects.
Lisbon, 12-15 Sept. 2016 - p. 9/29
Microscopic defects
● Schwarzschild solution
● Troubles and implications
■
A microstructure with a macroscopic continuum limit is found in condensed
matter systems such as graphene or Bravais crystals.
■
Crystalline structures may have different kinds of defects:
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
◆
◆
Gonzalo J. Olmo
Defects have dynamics.
Upon the action of forces or heat, defects can move and interact.
Lisbon, 12-15 Sept. 2016 - p. 9/29
The geometry of perfect crystals
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
The continuum limit of these structures is most naturally described in terms of
differential geometry:
◆ At each point we find 2 or 3 lattice vectors defining the microstructure.
◆ Moving along those vectors we jump from atom to atom.
◆ Distances can be measured by step counting along lattice vectors.
◆
ds2 = gi j dxi dx j , with gi j = δi j and Γabc = 0 in suitable coordinates.
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 10/29
The geometry of defected crystals
■
The step-counting procedure breaks down with point defects:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 11/29
The geometry of defected crystals
■
The step-counting procedure breaks down with point defects:
■
The continuum limit must be reconsidered with care:
◆ One should determine the density of defects, whose average separation
scale can be much larger than the interatomic separation.
◆ Knowledge of that density allows to determine the deformations of lengths,
areas, and volumes w.r.t. an idealized reference structure without defects:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
Phys
gµν
α
= Dµ α hAux
αν where Dµ depends on the density of defects.
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 11/29
The geometry of defected crystals
■
The step-counting procedure breaks down with point defects:
■
The continuum limit must be reconsidered with care:
◆ One should determine the density of defects, whose average separation
scale can be much larger than the interatomic separation.
◆ Knowledge of that density allows to determine the deformations of lengths,
areas, and volumes w.r.t. an idealized reference structure without defects:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Phys
gµν
Conclusions
The End
■
The idealized geometry has a well-defined parallel transport: ∇Γα hµν = 0 .
◆
Gonzalo J. Olmo
α
= Dµ α hAux
αν where Dµ depends on the density of defects.
But ∇Γµ gαβ 6= 0 ⇒ non-metricity tensor ∇Γµ gαβ = Qµαβ
Lisbon, 12-15 Sept. 2016 - p. 11/29
The geometry of defected crystals
■
The step-counting procedure breaks down with point defects:
■
The continuum limit must be reconsidered with care:
◆ One should determine the density of defects, whose average separation
scale can be much larger than the interatomic separation.
◆ Knowledge of that density allows to determine the deformations of lengths,
areas, and volumes w.r.t. an idealized reference structure without defects:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Phys
gµν
Conclusions
The End
■
The idealized geometry has a well-defined parallel transport: ∇Γα hµν = 0 .
◆
■
Gonzalo J. Olmo
α
= Dµ α hAux
αν where Dµ depends on the density of defects.
But ∇Γµ gαβ 6= 0 ⇒ non-metricity tensor ∇Γµ gαβ = Qµαβ
Dislocations are the microscopic realization of torsion: Tβγα = Γαβγ − Γαγβ
Lisbon, 12-15 Sept. 2016 - p. 11/29
The geometry of defected crystals
■
The step-counting procedure breaks down with point defects:
■
The continuum limit must be reconsidered with care:
◆ One should determine the density of defects, whose average separation
scale can be much larger than the interatomic separation.
◆ Knowledge of that density allows to determine the deformations of lengths,
areas, and volumes w.r.t. an idealized reference structure without defects:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Phys
gµν
Conclusions
The End
■
The idealized geometry has a well-defined parallel transport: ∇Γα hµν = 0 .
◆
Gonzalo J. Olmo
α
= Dµ α hAux
αν where Dµ depends on the density of defects.
But ∇Γµ gαβ 6= 0 ⇒ non-metricity tensor ∇Γµ gαβ = Qµαβ
■
Dislocations are the microscopic realization of torsion: Tβγα = Γαβγ − Γαγβ
■
Independent gµν and Γαβγ are necessary to account for microscopic defects.
Lisbon, 12-15 Sept. 2016 - p. 11/29
Metric-affine geometry and gravitation
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
We have just seen that Riemannian geometry is not enough to deal with
microstructural defects in condensed matter systems.
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 12/29
Metric-affine geometry and gravitation
■
We have just seen that Riemannian geometry is not enough to deal with
microstructural defects in condensed matter systems.
■
Could this lack of versatility of Riemannian geometry be the reason for the
existence of black hole singularities?
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 12/29
Metric-affine geometry and gravitation
■
We have just seen that Riemannian geometry is not enough to deal with
microstructural defects in condensed matter systems.
■
Could this lack of versatility of Riemannian geometry be the reason for the
existence of black hole singularities?
■
What happens if gravitation is formulated in non-Riemannian spaces?
Who knows if space-time could have a microstructure.
Why rule out that possibility a priori???
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 12/29
Metric-affine geometry and gravitation
■
We have just seen that Riemannian geometry is not enough to deal with
microstructural defects in condensed matter systems.
■
Could this lack of versatility of Riemannian geometry be the reason for the
existence of black hole singularities?
■
What happens if gravitation is formulated in non-Riemannian spaces?
Who knows if space-time could have a microstructure.
Why rule out that possibility a priori???
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
■
Gonzalo J. Olmo
Whether the space-time geometry is Riemannian or otherwise should
be determined by experiments, not by convention. The metric-affine or
Palatini approach hinges on this point.
Lisbon, 12-15 Sept. 2016 - p. 12/29
Metric-affine geometry and gravitation
■
We have just seen that Riemannian geometry is not enough to deal with
microstructural defects in condensed matter systems.
■
Could this lack of versatility of Riemannian geometry be the reason for the
existence of black hole singularities?
■
What happens if gravitation is formulated in non-Riemannian spaces?
Who knows if space-time could have a microstructure.
Why rule out that possibility a priori???
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
● Effective geometries in Physics
● Microscopic defects
● The geometry of perfect crystals
● The geometry of defected
crystals
● Metric-affine geometry and
gravitation
Palatini Gravity
Graphene Wormhole
Conclusions
The End
■
■
Gonzalo J. Olmo
Whether the space-time geometry is Riemannian or otherwise should
be determined by experiments, not by convention. The metric-affine or
Palatini approach hinges on this point.
The phenomenology of Palatini gravity should thus be explored in detail.
Lisbon, 12-15 Sept. 2016 - p. 12/29
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
Palatini gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 13/29
Metric-affine gravity
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
In the metric-affine (or Palatini) formalism, one assumes that gµν and Γαβγ are
R n √
independent entities: S = d x −gL[gµν , Γαβγ ] + Smatter [gµν , ψm ]
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 14/29
Metric-affine gravity
■
In the metric-affine (or Palatini) formalism, one assumes that gµν and Γαβγ are
R n √
independent entities: S = d x −gL[gµν , Γαβγ ] + Smatter [gµν , ψm ]
■
Field equations in Palatini approach:
R n √
√
δL
α
δS = d x −g δgδLµν − L2 gµν δgµν + −g δΓ
α δΓβγ + δSmatter
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
βγ
δgµν ⇒
δL
δgµν
− L2 gµν = 8πGTµν
δΓαβγ ⇒
δL
δΓαβγ
=0
(assuming no coupling of Γ to the matter)
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 14/29
Metric-affine gravity
■
In the metric-affine (or Palatini) formalism, one assumes that gµν and Γαβγ are
R n √
independent entities: S = d x −gL[gµν , Γαβγ ] + Smatter [gµν , ψm ]
■
Field equations in Palatini approach:
R n √
√
δL
α
δS = d x −g δgδLµν − L2 gµν δgµν + −g δΓ
α δΓβγ + δSmatter
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
βγ
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
■
δgµν ⇒
δL
δgµν
− L2 gµν = 8πGTµν
δΓαβγ ⇒
δL
δΓαβγ
=0
Metric approach:
The relation δΓαβγ =
Graphene Wormhole
Conclusions
(assuming no coupling of Γ to the matter)
δL
δΓαβγ
δΓα
βγ
=
n
gαµ δΓδLα
λν
gαρ
2
−
∇β δgργ + ∇γ δgρβ − ∇ρ δgβγ
gαλ δL
2 δΓα
µν
The End
δgµν ⇒
Gonzalo J. Olmo
δL
δgµν
o
implies
∇λ δgµν and leads to
δL
αλ δL
− L2 gµν + ∇λ gγν δΓ
µ − gβµ gγν g
δΓα
λγ
βγ
= 8πGTµν
Lisbon, 12-15 Sept. 2016 - p. 14/29
Metric-affine gravity
■
In the metric-affine (or Palatini) formalism, one assumes that gµν and Γαβγ are
R n √
independent entities: S = d x −gL[gµν , Γαβγ ] + Smatter [gµν , ψm ]
■
Field equations in Palatini approach:
R n √
√
δL
α
δS = d x −g δgδLµν − L2 gµν δgµν + −g δΓ
α δΓβγ + δSmatter
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
βγ
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
■
● BH structure in f(R)
δgµν ⇒
δL
δgµν
− L2 gµν = 8πGTµν
δΓαβγ ⇒
δL
δΓαβγ
=0
Metric approach:
The relation δΓαβγ =
● Geodesics in f(R)
Graphene Wormhole
δL
δΓαβγ
δΓα
βγ
Conclusions
(assuming no coupling of Γ to the matter)
=
n
gαµ δΓδLα
λν
gαρ
2
−
∇β δgργ + ∇γ δgρβ − ∇ρ δgβγ
gαλ δL
2 δΓα
µν
The End
δgµν ⇒
■
Gonzalo J. Olmo
δL
δgµν
o
implies
∇λ δgµν and leads to
δL
αλ δL
− L2 gµν + ∇λ gγν δΓ
µ − gβµ gγν g
δΓα
λγ
βγ
= 8πGTµν
Metric and Palatini variations generally lead to different field equations.
Lisbon, 12-15 Sept. 2016 - p. 14/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
◆
1 R
κ2 ε
d4x
p
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 15/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
1 R
κ2 ε
d4x
p
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
Field equations (without torsion and with qµν ≡ gµν + εRµν (Γ) ):
√
|q|
◆ gµν ⇒ √ qµν − λgµν = −κ2 εT µν
|g|
◆ Γα
µν
⇒
∇Γα
√ µν qq
=0
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 15/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
1 R
κ2 ε
● Why a wormhole?
● Geodesics in Born-Infeld
● BH structure in f(R)
● Geodesics in f(R)
In GR
√ µν qq
=0
⇒
∇Γα
∇Lα
√ µν = 0 ↔ ∇Lα gµν = 0 ↔ Lλµν =
gg
● Curvature
■
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
Field equations (without torsion and with qµν ≡ gµν + εRµν (Γ) ):
√
|q|
◆ gµν ⇒ √ qµν − λgµν = −κ2 εT µν
◆ Γα
µν
● Black Holes as Geons
p
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
|g|
● Wormhole structure
● Scattering of waves off WHs
d4x
gλρ
2
∂µ gρν + ∂ν gρµ − ∂ρ gµν
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 15/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
1 R
κ2 ε
● Why a wormhole?
● Geodesics in Born-Infeld
⇒
∇Lα
√ µν = 0 ↔ ∇Lα gµν = 0 ↔ Lλµν =
gg
■
In GR
■
In BI gravity: ∇Γα
● Geodesics in f(R)
Graphene Wormhole
Conclusions
√ µν qq
=0
∇Γα
● Curvature
● BH structure in f(R)
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
Field equations (without torsion and with qµν ≡ gµν + εRµν (Γ) ):
√
|q|
◆ gµν ⇒ √ qµν − λgµν = −κ2 εT µν
◆ Γα
µν
● Black Holes as Geons
p
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
|g|
● Wormhole structure
● Scattering of waves off WHs
d4x
√
qqµν = 0 ⇒ Γλµν =
qλρ
2
gλρ
2
∂µ gρν + ∂ν gρµ − ∂ρ gµν
∂µ qρν + ∂ν qρµ − ∂ρ qµν
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 15/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
1 R
κ2 ε
● Why a wormhole?
● Geodesics in Born-Infeld
⇒
∇Lα
√ µν = 0 ↔ ∇Lα gµν = 0 ↔ Lλµν =
gg
■
In GR
■
In BI gravity: ∇Γα
■
One finds gµν = |Σ|− 2 Σµ α qαν
● BH structure in f(R)
● Geodesics in f(R)
Conclusions
The End
Gonzalo J. Olmo
√ µν qq
=0
∇Γα
● Curvature
Graphene Wormhole
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
Field equations (without torsion and with qµν ≡ gµν + εRµν (Γ) ):
√
|q|
◆ gµν ⇒ √ qµν − λgµν = −κ2 εT µν
◆ Γα
µν
● Black Holes as Geons
p
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
|g|
● Wormhole structure
● Scattering of waves off WHs
d4x
√
qqµν = 0 ⇒ Γλµν =
1
qλρ
2
gλρ
2
∂µ gρν + ∂ν gρµ − ∂ρ gµν
∂µ qρν + ∂ν qρµ − ∂ρ qµν
with Σµ ν = λδµ ν − κ2 εTµ ν
Lisbon, 12-15 Sept. 2016 - p. 15/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
1 R
κ2 ε
● Why a wormhole?
● Geodesics in Born-Infeld
⇒
∇Lα
√ µν = 0 ↔ ∇Lα gµν = 0 ↔ Lλµν =
gg
In GR
■
In BI gravity: ∇Γα
■
One finds gµν = |Σ|− 2 Σµ α qαν
■
Non-metricity caused by the energy-density of the matter fields.
● Geodesics in f(R)
Conclusions
Gonzalo J. Olmo
gλρ
2
■
● BH structure in f(R)
The End
√ µν qq
=0
∇Γα
● Curvature
Graphene Wormhole
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
Field equations (without torsion and with qµν ≡ gµν + εRµν (Γ) ):
√
|q|
◆ gµν ⇒ √ qµν − λgµν = −κ2 εT µν
◆ Γα
µν
● Black Holes as Geons
p
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
|g|
● Wormhole structure
● Scattering of waves off WHs
d4x
√
qqµν = 0 ⇒ Γλµν =
1
qλρ
2
∂µ gρν + ∂ν gρµ − ∂ρ gµν
∂µ qρν + ∂ν qρµ − ∂ρ qµν
with Σµ ν = λδµ ν − κ2 εTµ ν
Lisbon, 12-15 Sept. 2016 - p. 15/29
Example: Born-Infeld gravity
■
Let S =
◆
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
1 R
κ2 ε
● Why a wormhole?
● Geodesics in Born-Infeld
⇒
∇Lα
√ µν = 0 ↔ ∇Lα gµν = 0 ↔ Lλµν =
gg
gλρ
2
■
In GR
■
In BI gravity: ∇Γα
■
One finds gµν = |Σ|− 2 Σµ α qαν
■
Non-metricity caused by the energy-density of the matter fields.
● BH structure in f(R)
● Geodesics in f(R)
Conclusions
The End
√ µν qq
=0
∇Γα
● Curvature
Graphene Wormhole
√ −|gµν + εRµν (Γ)| − λ −g + Sm with κ2 = 8πG
Field equations (without torsion and with qµν ≡ gµν + εRµν (Γ) ):
√
|q|
◆ gµν ⇒ √ qµν − λgµν = −κ2 εT µν
◆ Γα
µν
● Black Holes as Geons
p
This is a Born-Infeld type extension of General Relativity.
GR is recovered at low energies: (here Λe f f = λ−1
ε )
i
h
1 R 4 √
ε
εR2
µν
limε→0 S = 2κ2 d x −g R − 2Λe f f + 4 − 2 Rµν R + . . . + Sm
|g|
● Wormhole structure
● Scattering of waves off WHs
d4x
√
qqµν = 0 ⇒ Γλµν =
1
qλρ
2
∂µ gρν + ∂ν gρµ − ∂ρ gµν
∂µ qρν + ∂ν qρµ − ∂ρ qµν
with Σµ ν = λδµ ν − κ2 εTµ ν
Matter fields play a role analogous to that of point defects in crystals.
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 15/29
Black Holes with charge in BI gravity
■
● Schwarzschild solution
The equations using qµν take on a very familiar form:
κ2
ν
ν
ν
Rµ (q) = |Σ|1/2 LBI δµ + Tµ , with Σµ ν = λδµ ν − κ2 εTµ ν and LBI =
|Σ|1/2 −λ
.
κ2 ε
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 16/29
Black Holes with charge in BI gravity
■
● Schwarzschild solution
● Troubles and implications
■
The equations using qµν take on a very familiar form:
κ2
ν
ν
ν
Rµ (q) = |Σ|1/2 LBI δµ + Tµ , with Σµ ν = λδµ ν − κ2 εTµ ν and LBI =
|Σ|1/2 −λ
.
κ2 ε
Coupling BI gravity to a static, spherically symmetric electric field one finds:
● What is a singularity?
● So . . . what is a singularity?
1
2
2
2 with:
ds2 = −A(x)dt 2 + A(x)σ
2 dx + r (x)dΩ
● Living with holes
+
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
◆
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
◆
A(x) =
1
σ+
1−
2M(r) 1
r σ1/2
−
,
.
where M(r) = M0 + M0 δ1 G(z) ,
● Geodesics in Born-Infeld
dr
dx
dG
dz
2
=
=
4
z4
z +1
√
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
◆
rS = 2M0 ,
rq2
= κ2 q2 /4π
,
ε = −2lε2
,
σ−
σ2+
rc2
z4 −1
4
, and σ± = 1 ± rrc4
, and z =
= rq lε , δ1 =
r
rc
.
rq2
2rS rc
.
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 16/29
Black Holes with charge in BI gravity
■
● Schwarzschild solution
● Troubles and implications
■
The equations using qµν take on a very familiar form:
κ2
ν
ν
ν
Rµ (q) = |Σ|1/2 LBI δµ + Tµ , with Σµ ν = λδµ ν − κ2 εTµ ν and LBI =
|Σ|1/2 −λ
.
κ2 ε
Coupling BI gravity to a static, spherically symmetric electric field one finds:
● What is a singularity?
● So . . . what is a singularity?
1
2
2
2 with:
ds2 = −A(x)dt 2 + A(x)σ
2 dx + r (x)dΩ
● Living with holes
+
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
◆
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
◆
● Black Holes as Geons
A(x) =
1
σ+
1−
2M(r) 1
r σ1/2
−
,
.
where M(r) = M0 + M0 δ1 G(z) ,
● Geodesics in Born-Infeld
dr
dx
dG
dz
2
=
=
4
z4
z +1
√
● Curvature
● Scattering of waves off WHs
◆
● BH structure in f(R)
● Geodesics in f(R)
rS = 2M0 ,
rq2
= κ2 q2 /4π
,
ε = −2lε2
,
σ−
σ2+
rc2
z4 −1
4
, and σ± = 1 ± rrc4
, and z =
= rq lε , δ1 =
r
rc
.
rq2
2rS rc
.
Graphene Wormhole
■
This problem admits an exact analytical solution.
Conclusions
■
For not too small black holes (large mass and/or charge) and/or r ≫ rc :
The End
◆
Gonzalo J. Olmo
GR solution recovered: A(x) ≈ 1 −
rS
r
+
rq2
2r2
, σ± ≈ 1 , and r(x)2 ≈ x2
Lisbon, 12-15 Sept. 2016 - p. 16/29
Black Holes with charge in BI gravity
■
● Schwarzschild solution
● Troubles and implications
■
The equations using qµν take on a very familiar form:
κ2
ν
ν
ν
Rµ (q) = |Σ|1/2 LBI δµ + Tµ , with Σµ ν = λδµ ν − κ2 εTµ ν and LBI =
|Σ|1/2 −λ
.
κ2 ε
Coupling BI gravity to a static, spherically symmetric electric field one finds:
● What is a singularity?
● So . . . what is a singularity?
1
2
2
2 with:
ds2 = −A(x)dt 2 + A(x)σ
2 dx + r (x)dΩ
● Living with holes
+
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
◆
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
◆
● Black Holes as Geons
A(x) =
1
σ+
1−
2M(r) 1
r σ1/2
−
,
.
where M(r) = M0 + M0 δ1 G(z) ,
● Geodesics in Born-Infeld
dr
dx
dG
dz
2
=
=
4
z4
z +1
√
● Curvature
● Scattering of waves off WHs
◆
● BH structure in f(R)
● Geodesics in f(R)
rS = 2M0 ,
rq2
= κ2 q2 /4π
,
ε = −2lε2
,
σ−
σ2+
rc2
z4 −1
4
, and σ± = 1 ± rrc4
, and z =
= rq lε , δ1 =
r
rc
.
rq2
2rS rc
.
Graphene Wormhole
■
This problem admits an exact analytical solution.
Conclusions
■
For not too small black holes (large mass and/or charge) and/or r ≫ rc :
The End
◆
■
Gonzalo J. Olmo
GR solution recovered: A(x) ≈ 1 −
rS
r
+
rq2
2r2
, σ± ≈ 1 , and r(x)2 ≈ x2
Significant deviations arise as r → rc , near x → 0.
Lisbon, 12-15 Sept. 2016 - p. 16/29
Wormhole structure
■
From
dr
dx
2
=
σ−
σ2+
we find
r2 (x) =
x2 +
√
x4 +4rc4
2
, with a minimum at x = 0.
This is reminiscent of a wormhole geometry.
● Schwarzschild solution
rHxL
3.0
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
2.5
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
2.0
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
1.5
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
1.0
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
0.5
Conclusions
The End
-3
■
-2
-1
1
2
3
x
1
2
2
2
ds2 = −A(x)dt 2 + A(x)σ
2 dx + r(x) dΩ . D = 4 (solid)
+
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 17/29
Wormhole structure
■
From
dr
dx
2
=
σ−
σ2+
we find rD−2 (x) =
D−2
|x|
q
2(D−2)
+ |x|2(D−2) +4rc
2
, with a
minimum at x = 0. This is reminiscent of a wormhole geometry.
● Schwarzschild solution
● Troubles and implications
rHxL
3.0
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
2.5
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
2.0
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
1.5
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
1.0
● Geodesics in f(R)
Graphene Wormhole
0.5
Conclusions
The End
-3
■
-2
-1
1
2
3
x
1
2
2
2
ds2 = −A(x)dt 2 + A(x)σ
2 dx + r(x) dΩ . D = 4 (solid), D = 7 (dashed)
+
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 17/29
Why a wormhole?
■
The analogy with condensed matter systems and structure defects suggests that:
◆ Stress-energy density in space-time ↔ Density of point defects in crystals.
◆ Space-time regions with high energy density are analogous to regions with a
high concentration of vacancies.
■
The concentration of many vacancies makes a big hole.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 18/29
Black Holes as Geons
■
If there is a hole at the center, where are the sources???
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 19/29
Black Holes as Geons
■
■
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
If there is a hole at the center, where are the sources???
The lines of force of the electric field enter through one of the wormhole
mouths and exit through the other creating the illusion of a negatively charged
object on one side and a positively charged object on the other.
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The locally measured electric charge is defined by the flux Φ ≡
through any 2−surface S enclosing a wormhole mouth.
■
There is no need for sources in this scenario of self-gravitating fields.
■
Wheeler (1955) coined the term geon for regular, self-gravitating fields.
The End
Gonzalo J. Olmo
R
■
S ∗F
= 4πq
Lisbon, 12-15 Sept. 2016 - p. 19/29
Geodesic completeness in Born-Infeld
■
● Schwarzschild solution
● Troubles and implications
The equation that governs the evolution of geodesics in this space-time is:
2
dx
L2
1
2
= E −Ve f f , with Ve f f ≡ κ + r2 A(r) .
dλ
σ2
+
◆
● What is a singularity?
● So . . . what is a singularity?
◆
Where κ = 0 for null geodesics and κ = 1 for time-like geodesics.
L2 and E 2 are the angular momentum and energy per unit mass.
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 20/29
Geodesic completeness in Born-Infeld
■
The equation that governs the evolution of geodesics in this space-time is:
2
dx
L2
1
2
= E −Ve f f , with Ve f f ≡ κ + r2 A(r) .
dλ
σ2
+
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
◆
● So . . . what is a singularity?
Where κ = 0 for null geodesics and κ = 1 for time-like geodesics.
L2 and E 2 are the angular momentum and energy per unit mass.
● Living with holes
■
On the geometry of space-time
For null radial geodesics Ve f f = 0
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 20/29
Geodesic completeness in Born-Infeld
■
The equation that governs the evolution of geodesics in this space-time is:
2
dx
L2
1
2
= E −Ve f f , with Ve f f ≡ κ + r2 A(r) .
dλ
σ2
+
● Schwarzschild solution
◆
● Troubles and implications
● What is a singularity?
◆
● So . . . what is a singularity?
Where κ = 0 for null geodesics and κ = 1 for time-like geodesics.
L2 and E 2 are the angular momentum and energy per unit mass.
● Living with holes
■
On the geometry of space-time
For null radial geodesics Ve f f = 0
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
■
Null and time-like geodesics with L 6= 0: Ve f f
The End
Gonzalo J. Olmo
1
λ(x) ≈ ± 13 x ax 2
L2 Nq (δc −δ1 )rc
≈ − κ + r2 2N δ δ |x| .
- Vs - GR case:
c
λ(r) ≈ ± 32 r
c c 1
1
r
rS
2
◆
WH case:
.
◆
Main difference: x ∈] − ∞, +∞[ while r ∈]0, ∞[. Complete - Vs- Incomplete.
Lisbon, 12-15 Sept. 2016 - p. 20/29
Curvature
■
In GR, the Reissner-Nordström solution is characterized by:
RGR = 0 ,
QGR ≡ Rµν
Rµν
=
rq4
,
r8
KGR
≡ Rα
βµν Rα
βµν
=
12rS2
r6
−
24rS rq2
r7
+
14rq4
r8
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 21/29
Curvature
■
In GR, the Reissner-Nordström solution is characterized by:
RGR = 0 ,
QGR ≡ Rµν
Rµν
=
rq4
,
r8
KGR
≡ Rα
βµν Rα
βµν
=
12rS2
r6
−
24rS rq2
r7
+
14rq4
r8
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
■
In our case, defining rc ≡
48r8
R(g) ≈ − r10c
+O
rc9
r11
p
rq lε and rq2 = 2Gq2 , when r ≫ rc :
, Q(g) ≈
rq4
r8
144r r2 l 2
16lε2
1 − r2 + . . . , K(g) ≈ KGR + rS9 q ε + . . .
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 21/29
Curvature
■
In GR, the Reissner-Nordström solution is characterized by:
RGR = 0 ,
QGR ≡ Rµν
Rµν
=
rq4
,
r8
KGR
≡ Rα
βµν Rα
βµν
=
12rS2
r6
−
24rS rq2
r7
+
14rq4
r8
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
■
● So . . . what is a singularity?
● Living with holes
In our case, defining rc ≡
48r8
R(g) ≈ − r10c
On the geometry of space-time
+O
rc9
r11
p
rq lε and rq2 = 2Gq2 , when r ≫ rc :
, Q(g) ≈
rq4
r8
144r r2 l 2
16lε2
1 − r2 + . . . , K(g) ≈ KGR + rS9 q ε + . . .
1
2rS
r
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
■
But when z ≡ r/rc → 1 : δ1 =
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
h
rc2 R(g) ≈
rq3
lε
, δ2 =
h
16δc
δc
1
−4 + 3δ + O (z − 1) + . . . − 2δ 1 − δ
2
2
2
86δ
1
rc4 Q(g) ≈ 10 + 9δ21 − 52δ
+ . . . + 1 − δδc
3δ
2
2
1
1
√
rq lε
rS ,
1
(z−1)3/2
6δ2 −5δ1
3δ22 (z−1)3/2
and
−O
δc ≈ 0.572
√1
z−1
i
i
2 1
−...
+ . . . + 1 − δδc
8δ2 (z−1)3
1
2
2 2
88δ
2(2δ1 −3δ2 )
δc
δc
1
1
rc4 K(g) ≈ 16 + 9δ21 − 64δ
+
.
.
.
+
+
.
.
.
+
+...
1
−
1
−
2
2
3/2
3δ
δ
δ
4δ (z−1)3
2
2
1
3δ2 (z−1)
1
2
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 21/29
Curvature
■
In GR, the Reissner-Nordström solution is characterized by:
RGR = 0 ,
QGR ≡ Rµν
Rµν
=
rq4
,
r8
KGR
≡ Rα
βµν Rα
βµν
=
12rS2
r6
−
24rS rq2
r7
+
14rq4
r8
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
■
● So . . . what is a singularity?
● Living with holes
In our case, defining rc ≡
48r8
R(g) ≈ − r10c
On the geometry of space-time
+O
rc9
r11
p
rq lε and rq2 = 2Gq2 , when r ≫ rc :
, Q(g) ≈
rq4
r8
144r r2 l 2
16lε2
1 − r2 + . . . , K(g) ≈ KGR + rS9 q ε + . . .
1
2rS
r
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
■
h
But when z ≡ r/rc → 1 : δ1 =
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
rc2 R(g) ≈
rq3
lε
, δ2 =
h
16δc
δc
1
−4 + 3δ + O (z − 1) + . . . − 2δ 1 − δ
2
2
2
86δ
1
rc4 Q(g) ≈ 10 + 9δ21 − 52δ
+ . . . + 1 − δδc
3δ
2
2
1
1
√
rq lε
rS ,
1
(z−1)3/2
6δ2 −5δ1
3δ22 (z−1)3/2
and
−O
δc ≈ 0.572
√1
z−1
i
i
2 1
−...
+ . . . + 1 − δδc
8δ2 (z−1)3
1
2
2 2
88δ
2(2δ1 −3δ2 )
δc
δc
1
1
rc4 K(g) ≈ 16 + 9δ21 − 64δ
+
.
.
.
+
+
.
.
.
+
+...
1
−
1
−
2
2
3/2
3δ
δ
δ
4δ (z−1)3
2
2
1
3δ2 (z−1)
1
2
The End
Note that geodesics are complete despite curvature divergences!!!
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 21/29
Scattering of waves off (naked) WHs
■
Taking φ = 0 and φω,lm = e−iωt Ylm (θ, ϕ) fω,l (x)/r(x)
◆
with k ∝ (Nc [1 + l(l + 1)] − Nq ) , α = ω|k|−2/3 , Nc ≈ 16.5
◆
near the throat we have fy′ y′ + (α2 ± √1 ′ ) f = 0
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
|y |
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
■
If k < 0 the wave is almost totally transmitted (upper dotted curve).
■
if k > 0 , which happens as l grows, reflection may occur (lower curve).
■
For a given w, the transition occurs for l = lmax when α ∼ 1.5.
■
Curvature divergences neither affect wave propagation nor geodesic
completeness.
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 22/29
BH structure in f (R) = R − λR2
■
Consider a fluid like Tµ ν = diag[−ρ, −ρ, αρ, αρ] , where ρ(x) =
C
r(x)2+2α
.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 23/29
BH structure in f (R) = R − λR2
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
■
Consider a fluid like Tµ ν = diag[−ρ, −ρ, αρ, αρ] , where ρ(x) =
■
The line element in this space-time is of the form:
2M(x)
1
1
2
2
2
ds = fR −A(x)dt + A(x) dx + r2 (x)dΩ2 , with A(x) = 1 − x
and
● Living with holes
On the geometry of space-time
Palatini Gravity
◆ M(x) = M0 (1 + δ1 G(z)), δ1 =
rc3
8λM0 ,
z=
r
rc
and Gz =
α
1+ 2+2α
z
3/2
z2α f R
h
C
r(x)2+2α
1
1−α
1
.
− 2z2+2α
i
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 23/29
BH structure in f (R) = R − λR2
■
Consider a fluid like Tµ ν = diag[−ρ, −ρ, αρ, αρ] , where ρ(x) =
■
The line element in this space-time is of the form:
2M(x)
1
1
2
2
2
ds = fR −A(x)dt + A(x) dx + r2 (x)dΩ2 , with A(x) = 1 − x
and
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
◆ M(x) = M0 (1 + δ1 G(z)), δ1 =
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
■
rc3
8λM0 ,
z=
r
rc
and Gz =
α
1+ 2+2α
z
3/2
z2α f R
h
C
r(x)2+2α
1
1−α
1
.
− 2z2+2α
i
These solutions have wormhole structure (here α = 1/10, 1/2, 4/5):
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 23/29
Geodesics in f (R) = R − λR2
■
The relevant equation for geodesics in f (R) theories is
1
f R2
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
dx 2
dλ
with gtt ≈ − 8δ
= E 2 + gtt
L2
k + r2 (x)
,
rc2
δ1
2
2
2 (1−α ) (r−rc )
as r → rc .
● Living with holes
On the geometry of space-time
Palatini Gravity
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 24/29
Geodesics in f (R) = R − λR2
■
The relevant equation for geodesics in f (R) theories is
1
f R2
● Schwarzschild solution
● Troubles and implications
dx 2
dλ
with gtt ≈ − 8δ
● What is a singularity?
● So . . . what is a singularity?
= E 2 + gtt
L2
k + r2 (x)
,
rc2
δ1
2
2
2 (1−α ) (r−rc )
as r → rc .
● Living with holes
■
On the geometry of space-time
Palatini Gravity
For time-like geodesics,
◆
dx
dλ
= 0 before reaching the wormhole at x = 0.
Massive observers never reach the curvature divergence.
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
● Wormhole structure
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 24/29
Geodesics in f (R) = R − λR2
■
The relevant equation for geodesics in f (R) theories is
1
f R2
● Schwarzschild solution
● Troubles and implications
dx 2
dλ
with gtt ≈ − 8δ
● What is a singularity?
● So . . . what is a singularity?
= E 2 + gtt
L2
k + r2 (x)
,
rc2
δ1
2
2
2 (1−α ) (r−rc )
as r → rc .
● Living with holes
■
On the geometry of space-time
For time-like geodesics,
◆
Palatini Gravity
dx
dλ
= 0 before reaching the wormhole at x = 0.
Massive observers never reach the curvature divergence.
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
■
For null, radial geodesics,
● Wormhole structure
1
f R2
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
±Eλ(z) = − √
z
1−z−2(α+1)
+ 2z 2 F1
dx 2
dλ
= E 2 , we get
1
1
1
−2(α+1)
2 , − 2(α+1) ; 1 − 2(α+1) ; z
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
Graphene Wormhole
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 24/29
Geodesics in f (R) = R − λR2
■
The relevant equation for geodesics in f (R) theories is
1
f R2
● Schwarzschild solution
● Troubles and implications
dx 2
dλ
with gtt ≈ − 8δ
● What is a singularity?
● So . . . what is a singularity?
= E 2 + gtt
L2
k + r2 (x)
,
rc2
δ1
2
2
2 (1−α ) (r−rc )
as r → rc .
● Living with holes
■
On the geometry of space-time
For time-like geodesics,
◆
Palatini Gravity
dx
dλ
= 0 before reaching the wormhole at x = 0.
Massive observers never reach the curvature divergence.
● Metric-affine gravity
● Example: BI gravity
● BHs in BI gravity
■
For null, radial geodesics,
● Wormhole structure
1
f R2
● Why a wormhole?
● Black Holes as Geons
● Geodesics in Born-Infeld
● Curvature
±Eλ(z) = − √
z
1−z−2(α+1)
+ 2z 2 F1
dx 2
dλ
= E 2 , we get
1
1
1
−2(α+1)
2 , − 2(α+1) ; 1 − 2(α+1) ; z
● Scattering of waves off WHs
● BH structure in f(R)
● Geodesics in f(R)
■
Graphene Wormhole
Conclusions
■
As z → 1:
±Eλ(z) ≈ − √2α+21 √z−1 ≈ − |1x̃| .
■
Geodesically complete space.
The End
Gonzalo J. Olmo
As z → ∞: ±Eλ(z) ≈ z ≈ x .
Lisbon, 12-15 Sept. 2016 - p. 24/29
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Graphene Wormhole
● Wormholes with graphene layers
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 25/29
Wormholes with graphene layers
■
Graphene is a versatile 2D material that allows the construction of wormholes:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
● Wormholes with graphene layers
Conclusions
The End
◆
◆
Gonzalo J. Olmo
Take two layers of graphene and join them with a graphene nanotube.
To join them, make a junction with heptagonal carbon rings (defects).
Lisbon, 12-15 Sept. 2016 - p. 26/29
Wormholes with graphene layers
■
Graphene is a versatile 2D material that allows the construction of wormholes:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
◆
Graphene Wormhole
● Wormholes with graphene layers
Conclusions
The End
Gonzalo J. Olmo
◆
■
Take two layers of graphene and join them with a graphene nanotube.
To join them, make a junction with heptagonal carbon rings (defects).
A continuum description of this geometry is possible:
◆ Take the radius of the throat much larger than the nanotube length.
◆ From a distance, the wormhole appears as the surface of two planes
connected at the boundary of a common hole.
◆ In this limit, the wormhole curvature diverges at the throat.
Lisbon, 12-15 Sept. 2016 - p. 26/29
Wormholes with graphene layers
■
Graphene is a versatile 2D material that allows the construction of wormholes:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
◆
Graphene Wormhole
● Wormholes with graphene layers
Conclusions
◆
■
A continuum description of this geometry is possible:
◆ Take the radius of the throat much larger than the nanotube length.
◆ From a distance, the wormhole appears as the surface of two planes
connected at the boundary of a common hole.
◆ In this limit, the wormhole curvature diverges at the throat.
■
Besides curving the geometry, the heptagons (defects) generate an effective
SU(2) gauge flux concentrated at the WH throat.
◆ In our gravitational examples, U(1) gauge fields generate wormholes.
◆ The maximum energy density is attained at the throat.
The End
Gonzalo J. Olmo
Take two layers of graphene and join them with a graphene nanotube.
To join them, make a junction with heptagonal carbon rings (defects).
Lisbon, 12-15 Sept. 2016 - p. 26/29
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Summary and Conclusions
Conclusions
● Summary and Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 27/29
Summary and Conclusions
■
● Schwarzschild solution
● Troubles and implications
GR predicts the existence of singular space-times:
◆ Geodesic incompleteness is the criterion to detect singularities.
◆ Curvature pathologies are neither necessary nor sufficient conditions.
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
● Summary and Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 28/29
Summary and Conclusions
■
GR predicts the existence of singular space-times:
◆ Geodesic incompleteness is the criterion to detect singularities.
◆ Curvature pathologies are neither necessary nor sufficient conditions.
■
Addressing the issue of defects in the space-time microstructure, the
introduction of new geometric elements (non-metricity) has allowed to prevent
both incomplete geodesics and infinities in the matter sector:
◆ Central singularity of charged black holes replaced by a wormhole.
◆ These wormholes have been discovered, not designed.
◆ The matter energy density is regularized (geon - Vs - point-like).
◆ The WH guarantees the extendibility of geodesics in different ways.
◆ Scattering of waves is well-behaved despite curvature divergences.
◆ The meaning and implications of curvature divergences is still uncertain.
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
● Summary and Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 28/29
Summary and Conclusions
■
GR predicts the existence of singular space-times:
◆ Geodesic incompleteness is the criterion to detect singularities.
◆ Curvature pathologies are neither necessary nor sufficient conditions.
■
Addressing the issue of defects in the space-time microstructure, the
introduction of new geometric elements (non-metricity) has allowed to prevent
both incomplete geodesics and infinities in the matter sector:
◆ Central singularity of charged black holes replaced by a wormhole.
◆ These wormholes have been discovered, not designed.
◆ The matter energy density is regularized (geon - Vs - point-like).
◆ The WH guarantees the extendibility of geodesics in different ways.
◆ Scattering of waves is well-behaved despite curvature divergences.
◆ The meaning and implications of curvature divergences is still uncertain.
■
Conclusion:
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Conclusions
● Summary and Conclusions
The End
The avoidance of singularities can be achieved with simple models in
classical geometric scenarios with independent metric and affine structures.
What are the implications for quantum gravity?
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 28/29
● Schwarzschild solution
● Troubles and implications
● What is a singularity?
● So . . . what is a singularity?
● Living with holes
On the geometry of space-time
Palatini Gravity
Graphene Wormhole
Thanks!
Conclusions
The End
Gonzalo J. Olmo
Lisbon, 12-15 Sept. 2016 - p. 29/29