Some Ideas on Modied Theories of Classical
Gravity
Helios Sanchis Alepuz
Picture: Ute Kraus - http://www.spacetimetravel.org
Some references:
R. M. Wald, General Relativity, University of Chicago Press, 1984
C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, Freeman, 1973
S. Weinberg, Gravitation and Cosmology, Wiley, 1972
Open Questions in Cosmology, edited by Gonzalo J. Olmo, InTech Publishing, 2012
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Part I: Foundations.
Principle of Equivalence
Manifolds and vectors.
Derivatives and curvature.
Part II: Einstein's General Relativity.
Einstein equation.
Einstein-Hilbert lagrangian.
Friedmann-Robertson-Walker universe.
Schwarzschild solution.
Part III: Modied Gravities.
Metric vs Palatini formalism
f(R) anf f(R,Q).
Some selected results.
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Foundations
The geometric viewpoint:
From \R. Wald, General Relativity ":
Because motion is independent of the nature of the bodies, the paths of freely falling
bodies dene a preferred set of curves [...] just as in special relativity the paths in
spacetime of inertial bodies dene a preferred set of curves.
The paths of inertial bodies in special relativity are geodesics of the spacetime metric. Perhaps, then, the paths of freely falling bodies are always geodesics, but the spacetime
metric is not always that given by special relativity. What we think of as a gravitational
eld would then not be a new eld at all, but rather would correspond to a deviation
of the spacetime geometry from the at geometry of special relativity.
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Foundations
The other viewpoint:
From \S. Weinberg, Gravitation and Cosmology ":
I found that in most textbooks geometric ideas were given a starring role, so that a
student who asked why the gravitational eld is represented by a metric tensor, or why
freely falling particles move on geodesics [...] would come away with an impression that
this had something to do with the fact that space-time is a Riemannian manifold.
Of course, this was Einstein's point of view [...]. However, I believe that the geometrical
approach has driven a wedge between general relativity and the theory of elementary
particles [...] and too great an emphasis on geometry can only obscure the deep connections
between gravitation and the rest of physics.
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Foundations
Part I
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May 2015
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Foundations
Principle of Equivalence
Unlike Maxwell's equations, which are automatically compatible with Special Relativity,
Newton's gravity theory is not:
jF~12 j = G jr~1 (t )m1 mr~22(t )j2
Positions are at the same instant of time. Therefore, the equation is valid in one frame
only, contrary to the principle of relativity.
Principle of Equivalence:
Experimental fact: Inertial and gravitational masses are identical. That is,
acceleration can mimic and/or compensate for gravity.
Einstein's \gluckischste Gedanke seines Lebens": The gravitational eld has only a
relative existence, because observers falling freely in that gravitational eld would
feel no gravitational force.
Principle of Equivalence: There is no experiment (not only mechanical ones) that
can distinguish a uniform acceleration from a uniform gravitational eld. More
specically, experiments in a suciently small freely falling laboratory, over a
suciently small period of time, give results which are indistinguishable from
those of the same experiments in an inertial frame in empty space.
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Foundations
Consequences of the Principle of Equivalence
Gravity is geometry:
The principle of equivalence implies that in a gravitational eld one can choose an
observer a for which particles are at rest or uniform motion
d 2a
=0
d2
with proper time d 2 = d d with the Minkowski (at) metric.
In a non-freely falling frame x a the metric is not the Minkowski one!
@
@ dx g dx dx 2
d = dx
@x @x @ g = @
@ x @ x is, in general, a non-at metric.
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Foundations
Consequences of the Principle of Equivalence
Particles follow geodesics of a non-at geometry:
Again, for the freely falling observer a
d 2a
=0
d2
with proper time d 2 = d d In a non-freely falling frame x a the equation of motion becomes
@ d 2 x + @ 2 dx dx = 0
d @ dx =
d @x d @x d 2 @x @x d d A gravitational force has appeared
d 2 x @ x @ 2 dx dx + =0
d2
@ @ x @ x d d This is the geodesic equation on a Riemannian geometry with the Levi-Civita
@ x @ 2 (see later).
connection = @
@x @x The principle of equivalence relates the metric g and the connection , so that the
metric is the only dynamical eld.
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Foundations
Manifolds
An n-dimensional manifold is a set made up of pieces that look like
Rn .
Manifold:
An n-dimensional C 1 real manifold M is a set together with a collection of subsets fO g
satisfying:
1 Each p 2 M lies at least in one O .
2 For each there is a one-to-one map
: O ! U , with U an open subset of
Rn . That is, we can dene coordinates for the sets fO g and a coordinate
system f g for the manifold.
3 If any two sets O , O overlap ( O \ O 6= ;), then the map
1
1
: [O \ O ] 2 U ! [O \ O ] 2 U is C . That is, we can
smoothly change coordinates.
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Foundations
Vectors
Vectors are dened in manifolds as directional derivatives over functions on M :
F.
Vector:
A tangent vector at a point p 2 M is a map v : F ! R satisfying:
1 Linearity: v (af + bg ) = av (f ) = bv (g ) ;
f ; g 2 F ; a; b 2 R.
2 Leibnitz rule: v (fg ) = f (p )v (g ) + g (p )v (f ).
They form a vector space at the point p : Vp . We can cover the whole manifold with vector
spaces, but there's no meaningful way (yet) to connect them.
Covector = Dual vector = Covariant vector:
Given a vector space V , we can dene a linear map v : V ! R. Dening usual addition
and scalar multiplication of this maps, they form a vector space V , called dual vector
space.
Tensors
A tensor T of type (k ; `) is a multilinear map T : V
: : xV} ! R.
| x :{z: : xV } x |Vx :{z
k
`
Metric
A metric g is a symmetric and non-degenerate type-(0; 2) tensor. It introduces the
concept of distance in a manifold.
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Foundations
Derivatives
To move vectors from one point to another, we need an extra structure. The notion of
how to (parallel-)transport a vector is equivalent to the knowledge of how to take
derivatives of vector elds.
Derivative operator:
A derivative operator r : (k ; `) ! (k ; ` + 1) is dened to have the usual properties:
Linearity
Leibnitz rule
plus:
Consistency with the notion of vectors as directional derivatives: t (f ) = t a ra f , with
f 2 F and t 2 Vp .
Torsion free : ra rb f = rb ra f
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Foundations
Derivatives
Example: Curvilinear coordinates in at space
In curvilinear coordinates, the basis vectors
change from point to point, like in curved
spaces.
vjj (x ) = v (x + dx ) + (x )v (x )t with d ~x = ~t (x ) are coecients that determine how
the vector v in x + dx is transported to x .
Derivative operator:
From the above example, we dene the covariant derivative in the direction of t , as
rt v (x ) =
v (x + dx )jtransp. to x
v (x )
This gives the following denition of covariant derivative or connection
r v = @@vx + v lim
To dene a connection we need to dene the coecients .
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Foundations
Given a derivative operator
vector along a curve.
Derivatives and parallel transport
r we can dene the notion of the parallel transport of a
Parallel transport:
Given a curve C with tangent vector t , a vector v is parallelly transported along the curve
if:
t a ra v b = 0
:::bk = 0
In general: t a ra Tcb11:::
cl
Levi-Civita connection:
A reasonable assumption is that the angle, or inner product, of two vectors is unchanged
when they are parallelly transported:
t a ra gbc v b w c = 0 ) t a v b w c ra gbc = 0 )
ra gbc = 0
This uniquely denes the derivative operator (connection). It is the Levi-Civita connec1 cd
c
ra v b = @a v b + bac v c with
@d gab ]
ab = g [@a gbd + @b gad
2
tion:
Geodesics:
Geodesics are curves that curve as little as possible, the straightest possible lines; curves
whose tangent vector t is parallelly transported along itself:
d 2x dx dx t a ra t b = 0 !
+ =0
with x (t ) coordinates of the geodesic.
2
dt
dt dt
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Foundations
Curvature
If spacetime is curved, it does matter the way we go from one point to another:
Two vectors, parallelly transported from
point A to point B along dierent curves
do not coincide.
picture: Misner-Thorne-Wheeler
Riemann curvature tensor:
A signal af curved geometry is the failure of covarariant derivatives to commute:
d
ra rb !c ra ra !c Rabc
!d
=@ R
Properties of the Riemann tensor:
1
2
3
4
d
d
Rabc
= Rbac
d
R[abc ] = 0
If rg = 0 then Rabcd = Rabdc
e
Bianchi identity: r[a Rbc
]d = 0
Helios Sanchis Alepuz (University of Giessen)
d
Rabc
: Riemann curvature tensor.
@ + Prominent related tensors:
1
2
3
b
Ricci curvature: Rac Rabc
(= Rca )
a
Scalar curvature: R Ra
Weyl (conformal) tensor:
Cabcd
Rabcd
2
n 2 ga[c Rd ]b gb[c Rd ]a
2
Rg g
+
(n 1)(n 2) a[c d ]b
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Foundations
Curvature
Some consequences of a non-zero curvature:
Geodesics deviation:
If the spacetime is curved, a family of geodesics that are
initially parallel, will fail to remain parallel. They accelerate
toward or away from each other at the rate:
a
aa = Rcbd
X bT c T d
d
a = 0 i Rabc
=0
Energy conservation:
Given Tab and an observer v , Ja
picture: Wald
Tab v b is the energy-momentum current.
Curved spacetime: ra Tab = 0, but is not, in general, energy conservation!.
Only if one can nd observers v such that ra vb = 0 R(or, equivalently r(a vb) = 0:
Killing equation), then: ra (Tab v b ) = ra Ja = 0 ! S J a dS a = 0
Weyl tensor:
The Weyl tensor contains information about the tidal forces in a gravitational eld.
In vacuum, only the Weyl tensor is non-zero. It dictates the propagation of
gravitational waves through empty space.
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Foundations
Curvature
Part II
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May 2015
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Einstein's General Relativity
Einstein equation
Motivation:
1
2
a
~ )r
~ $ Rcbd
Tidal acc. in Newton theory: (~x r
xbv c v d
Poisson equation: r2 = 4: Tab v a v b $ From here:
a
) Rcbd
v c v d $ @b @ a ! Rcd = 4Tcd
But this is wrong! It would lead to ra Tab 6= 0
a
Rcad
v c v d = 4 Tcd v c v d
It can be cured using, instead of the Ricci curvature, the Einstein tensor:
Gab
Then:
Gab = Rab [g ]
Rab
1
Rg
;
2 ab
1
R [g ]gab = 8 Tab [g ]
2
ra Gab = 0
Einstein's equation
(1)
(note that now it's 8 instead of 4 )
Einstein's General Relativity: Spacetime is a manifold M on which there is dened a
Lorentz metric gab . The curvature of gab is related to the matter distribution in spacetime
by Einstein's equation (1).
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Einstein's General Relativity
Einstein equation
Cosmological constant:
Gab + gab = 8 Tab
! Gab = 8Tab gab = T~ab
If T corresponds to a perfect uid: p~ = p
=8
~ = + =8
Lagrangian formulation:
The Einstein equation can be derived from the Einstein-Hilbert lagrangian:
LE
with:
g
SE
H [g ]
H
= det (gab )
=
Z
d 4 x LE
H
=
p
(+Lmatter )
gR
! gSab = p
gGab =
p
g Rab
1
Rg
2 ab
A couple of remarks:
p g contributes also to the variational calculation.
The gravity lagrangian (or the action) can be used to dene the
energy-momentum tensor:
S [g ] = SE
Helios Sanchis Alepuz (University of Giessen)
H [g ] + Smatter [g ]
!
Tab
1 1 Smatter
p g g ab
8
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Cosmological and Astrophysical solutions
Robertson-Walker geometry:
Robertson-Walker geometry:
At very large scales, the distribution of matter (clusters of galaxies) in the Universe
is homogeneous and isotropic (for a given observer). This is conrmed by the very
homogeneous and isotropic Cosmic Microwave Background (CMB). Thus, the metric has
to be also homogeneous and isotropic.
Symmetry suces to determine the metric, up to two factors:
ds 2 = d 2 + a2 ( )
dr 2
+ r 2 d 2 + sin d 2
2
1 kr
a ( )
:
k
a 2 ( )
:
Robertson-Walker metric
Scale factor
Open universe
(Spatially) Flat universe
Closed universe
8
< 1
0
k =
: +1
K (3) ( ) =
Spatial curvature
The values and evolution of a( ) are given by the Einstein equation.
The value of k is dictated by the matter content of the Universe.
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Cosmological and Astrophysical solutions
Robertson-Walker geometry:
Dynamics:
Tab = ua ub + P (gab + ua ub )
From Einstein's eq. !
Friedmann-Robertson-Walker cosmologies
3_a =a = 8 3k =a2
The Universe cannot be static!
3a=a = 4 ( + 3P )
2
2
v
Big Bang:
a < 0; a_ > 0
)
dR
d
the Universe was expanding faster
Open or closed Universe?
P 0 ! H 2 = 8=3
M 83
H2
In general:
Helios Sanchis Alepuz (University of Giessen)
k =a2
)
a_
= R
a
)
M > 1
M < 1
HR
At T < H
Hubble's law
1
ago: a = 0
Closed, k=1
Open, k=0,-1
R + M + k + = 1
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Cosmological and Astrophysical solutions
Robertson-Walker geometry:
Radiation lled (P = =3)
Dust lled (P = 0)
picture: Wald
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Cosmological and Astrophysical solutions
Schwarzschild geometry:
Schwarzschild geometry:
The Schwarzschild solution describes the exterior geometry of a static, uncharged and
spherically symmetric body (star) of mass M :
ds 2 =
1
2M
r
dt 2 + 1
2M
r
1
dr 2 + r 2 (d 2 + sin2 d 2 )
At r = 2M the metric is singular : It's a problem of coordinates!
At r = 0 the metric has a genuine singularity!
Rabcd R abcd =
48M 2
r6
this is coordinate-independent
For r < 2M , \r $ t " in terms of the metric signature: r=2M is the event horizon
(if the object is compressed to a size smaller than 2M rH , it becomes a Black
Hole)
For the Earth: rH 9 mm
For the Sun: rH 3 km
For the supermassive BH in the center of the Milky Way: rH
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12 10
6
km
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Cosmological and Astrophysical solutions
Schwarzschild geometry:
picture: Navarro-Salas, Fabbri;
Helios Sanchis Alepuz (University of Giessen)
Modelling Black Hole evaporation
May 2015
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Cosmological and Astrophysical solutions
Schwarzschild geometry:
(Schwarzschild geometry in Kruskal coordinates:)
picture: Navarro-Salas, Fabbri;
Helios Sanchis Alepuz (University of Giessen)
Modelling Black Hole evaporation
May 2015
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Cosmological and Astrophysical solutions
Hamiltonian description:
ADM decomposition:
A Hamiltonian formulation requires to specify a time variable, to dene the canonical
momenta pX @ L=@ X_ .
This seems to break dieomorphism invariance. The trick is to break it only formally
(ADM decomposition):
1. Assume that the manifold M has the topology M = R M (3) .
2. Dene a time function t such that M (3) M jt =const : , and a time ow t a such that
t a ra t = 1.
3. Decompose t a into normal and tangential parts:
t a = Nna + N a . This introduces the new elds N
(lapse) and N a (shift).
4. All spacetime quantities can now be splitted into
spatial and temporal parts. For instance:
gpab = hab pna nb = hab N
g =N h
r!D
Gab =
2
t N a )(t b N b )
( a
:::
picture: Thiemann,
Modern Canonical QG
(hab ; hab ),
5. The new canonical variables are the pairs
(N ; N ) and (N a ; Na ).
a
6. It can be seen that N and N are not dynamical variables. They appear as
Lagrange multipliers.
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Cosmological and Astrophysical solutions
Hamiltonian description:
Hamiltonian of GR:
Using the usual procedure to construct the Hamiltonian with the dynamical variables:
HGR = hab h_ ab LGR = N HN + Na Ha
Since N ; N a are not dynamical variables, N = 0 and Na = 0, and from Hamilton equations:
1 2
_ N = HGR = HN = 0 ! (3) R = h 1 ab ab h = 0 Di. constraint(2)
N
2
HGR
a
1=2 ab
a
_ N =
) = 0 Hamiltonian constraint
(3)
= H N = 0 ! Da ( h
Na
The GR Hamiltonian is constrained to vanish!
Meaning of the constraints:
The situation is related to the dieomorphism (gauge) invariance of the theory, and to the
fact that we don't x the gauge. What do they tell us?
The Dieomorphism constraint (2) generates dieomorphisms on the spatial
hypersurface M (3) . Changing hab by the equivalence class h~ab of dieomorphism
related spatial metrics, then the constarint can be solved (i.e., h~ fullls the
constraint identically and this constraint is eliminated.).
The Hamiltonian constraint generates time dieomorphisms. It cannot be solved, so
it seems to be a genuine constraint of General Relativity ) Problem of time (no
true time evolution in GR).
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Cosmological and Astrophysical solutions
Hamiltonian description:
Canonical quantization:
The Hamiltonian constraint can be used to canonically quantize General relativity, a la
Gupta-Bleuer or BRST (Dirac 40's, Arnowitt-Deser-Misner 50's, Wheeler-DeWitt 60's):
^ [h; N ; N a ] j [h; N ; N a ]i = 0
HN = 0 ! H
Wheeler-DeWitt equation
It is a highly non-linear equation (essentially unsolvable).
It breaks explicit dieomorphism invariance, due to the ADM decomposition.
The rst problem is solved by using a dierent set of canonical variables ) Ashtekar
variables
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Cosmological and Astrophysical solutions
Hamiltonian description:
Part III
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Modied gravity:
Modied gravities
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May 2015
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Modied gravity:
f(R), f(R,Q), etc.:
p
The Einstein-Hilbert lagrangian
gR is the simplest, dieomorphism invariant term
that can be constructed out of the spacetime metric.
Nothing at the classical level (i.e. no renormalizability concerns) prevents to add arbitrary
terms, as long as they are dieomorphism invariant: R n ; Rab R ab Q ; R Q ; etc.
The simplest modication is then
p
gR
!p
gf (R ): f (R )-gravities.
f (R ) were proposed to explain the Universe acccelerated expansion without the need
for dark energy, for example:
RfR
R
2f
1
g
2
= 2 T
=
+
2 T
RfR f
g
fR
2fR
3
1
2
@ fR @ fR
g (@ fR ) +
2
2fR2
1
[r r fR g fR ]
fR
In vaccuum T = 0: R 12 g = g (R0 fR0 f0 )=2fR0 = e : g
The equation looks like GR with a cosmological constant (= Dark Energy).
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May 2015
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Modied gravity:
Metric and Palatini theories:
What are modied gravities good for?
First of all, there are two families of modied gravities. Curvature, in general, is dened
by the connection, not the metric (remember rst slides), and then:
Metric formalism: the connection is always the Levi-Civita connection rg = 0, it
depends only on the metric and then the curvature depends also on the metric. The
equations of motion have high order (not just second order) derivatives of the metric.
Palatini formalism: the connection is independent of the metric. Assume that
the new action is S (g ; ), take variations (independently) with respect to the metric
and the connection to obtain their equations of motion. The equations of motion
have only second order derivatives of the metric, like GR.
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Modied gravity:
Why have you never heard of the Palatini formalism?
In GR f (R ) = R and both formalisms give the same connection,
Helios Sanchis Alepuz (University of Giessen)
rg = 0!
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Modied gravity:
Why have you never heard of the Palatini formalism?
In GR f (R ) = R and both formalisms give the same connection,
rg = 0!
WARNING
Problem?
As we saw, the Principle of Equivalence implies that the connection is the Levi-Civita one,
rg = 0.
The Principle of equivalence describes the eect of gravity on matter
Use an independent connection in the gravity sector and a Levi-Civita one in the
matter sector
Satisfactory? Matter of taste...
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May 2015
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Modied gravity:
Late time expansion:
Do f (R )-theories solve the problem of dark energy?
These models are designed to modify GR at low curvatures (large scale Universe) but
recover GR at relatively high curvatures (Solar system).
Naively, this ensures a correct behaviour at all relevant scales. However, matter generates
curvature in way dierent from GR. In particular, it may depend not only on the density,
but on its derivatives.
Tests of f(R):
Cosmological speed-up (by design)
Solar system tests (they have to reproduce Solar system behaviour)
Microscopic tests (Matter is highly discontinous at the microscopical level: Large
derivatives of )
(Surprinsingly) Microscopic tests put the most stringent constraints on f (R ):
f (R ) = R
2(n+1) =R n ! jnj < 10
38
f (R ) R to pass all these tests!
f (R ), both in Palatini or metric formalism, are not good candidates for cosmic speed-up.
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May 2015
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Modied gravity:
Big bounce:
Again, what are modied gravities good for?
Modied gravities can give a classical eective description of the early-time
cosmology, so that they don't need to pass all those tests.
Any quantum correction of the Einstein-Hilbert lagrangian must be of the form f (R )
(or more complicated) to be dieomorphism invariant. Related to FRG approach to
gravity?
Helios Sanchis Alepuz (University of Giessen)
May 2015
34 / 38
Modied gravity:
Big bounce:
Again, what are modied gravities good for?
Modied gravities can give a classical eective description of the early-time
cosmology, so that they don't need to pass all those tests.
Any quantum correction of the Einstein-Hilbert lagrangian must be of the form f (R )
(or more complicated) to be dieomorphism invariant. Related to FRG approach to
gravity?
Why Palatini formalism?
Helios Sanchis Alepuz (University of Giessen)
May 2015
34 / 38
Modied gravity:
Big bounce:
Again, what are modied gravities good for?
Modied gravities can give a classical eective description of the early-time
cosmology, so that they don't need to pass all those tests.
Any quantum correction of the Einstein-Hilbert lagrangian must be of the form f (R )
(or more complicated) to be dieomorphism invariant. Related to FRG approach to
gravity?
Why Palatini formalism?
In the metric formalism, the equations of motion are no longer second order
dierential equations, but higher. This introduces many problems, spurious
solutions, etc. for which several \xes" have been designed
In the Palatini formalism the equations remain second order, so all the mathematical
apparatus and many of the theorems can be \recycled".
Helios Sanchis Alepuz (University of Giessen)
May 2015
34 / 38
Modied gravity:
Big bounce:
Again, what are modied gravities good for?
Modied gravities can give a classical eective description of the early-time
cosmology, so that they don't need to pass all those tests.
Any quantum correction of the Einstein-Hilbert lagrangian must be of the form f (R )
(or more complicated) to be dieomorphism invariant. Related to FRG approach to
gravity?
Why Palatini formalism?
In the metric formalism, the equations of motion are no longer second order
dierential equations, but higher. This introduces many problems, spurious
solutions, etc. for which several \xes" have been designed
In the Palatini formalism the equations remain second order, so all the mathematical
apparatus and many of the theorems can be \recycled".
Loop Quantum Cosmology predicts a Big Bounce instead of a Big Bang. There is
an f (R ) in Palatini formalism which describes exactly the LQC behaviour.
Bouncing solutions seem to be a general feature of f (R ) theories in Palatini. In
contrast, metric-f (R ) theories with a correct GR-limit, have a Big Bang singularity.
Helios Sanchis Alepuz (University of Giessen)
May 2015
34 / 38
Modied gravity:
Big bounce:
Again, what are modied gravities good for?
Modied gravities can give a classical eective description of the early-time
cosmology, so that they don't need to pass all those tests.
Any quantum correction of the Einstein-Hilbert lagrangian must be of the form f (R )
(or more complicated) to be dieomorphism invariant. Related to FRG approach to
gravity?
Why Palatini formalism?
In the metric formalism, the equations of motion are no longer second order
dierential equations, but higher. This introduces many problems, spurious
solutions, etc. for which several \xes" have been designed
In the Palatini formalism the equations remain second order, so all the mathematical
apparatus and many of the theorems can be \recycled".
Loop Quantum Cosmology predicts a Big Bounce instead of a Big Bang. There is
an f (R ) in Palatini formalism which describes exactly the LQC behaviour.
Bouncing solutions seem to be a general feature of f (R ) theories in Palatini. In
contrast, metric-f (R ) theories with a correct GR-limit, have a Big Bang singularity.
The may provide a singularity-free classical theory of gravity to be quantized.
Helios Sanchis Alepuz (University of Giessen)
May 2015
34 / 38
Modied gravity:
Big bounce:
picture: Olmo G.J. , Int.J.Mod.Phys. D20 (2011)
Helios Sanchis Alepuz (University of Giessen)
May 2015
35 / 38
Modied gravity:
Big bounce:
Some remarks:
The bouncing solutions are studied in homogenous, isotropic universes.
Anisotropies in the matter distribution can lead to problems. In many cases
non-singular solutions still exist, but in others, singularities reappear.
f (R ; Q ) theories like R + `P (aR 2 + R R ), have a stable Big bounce.
No general proof of the absence of Big Bang singularities for arbitrary f (R ),
f (R ; Q ), etc.
Helios Sanchis Alepuz (University of Giessen)
May 2015
36 / 38
Modied gravity:
Compact objects
What about Black Hole singularities?
It has been found that in some cases there exist static solutions (like the
Schwarzschild solution) with no central singularities.
In particular, in a theory of the type R + `P (aR 2 + R R ) and with charged matter it
has been found (Olmo,Rubiera-Garcia Eur.Phys.J. C72 (2012) 2098)
For certain charge congurations, instead of central singularity, an ultracompact
central core develops. The area of this core is quantized, if the electric charge is
quantized.
External horizon almost like in GR
picture: Olmo G.J. , Int.J.Mod.Phys. D20 (2011)
Helios Sanchis Alepuz (University of Giessen)
May 2015
37 / 38
Modied gravity:
Compact objects
What about Black Hole singularities?
It has been found that in some cases there exist static solutions (like the
Schwarzschild solution) with no central singularities.
In particular, in a theory of the type R + `P (aR 2 + R R ) and with charged matter it
has been found (Olmo,Rubiera-Garcia Eur.Phys.J. C72 (2012) 2098)
For certain charge congurations, instead of central singularity, an ultracompact
central core develops. The area of this core is quantized, if the electric charge is
quantized.
External horizon almost like in GR
In some cases, the external horizon might disappear, revealing its central core.
Ultracompact ( 1000 moles) and charged astrophysical objects?
Helios Sanchis Alepuz (University of Giessen)
May 2015
37 / 38
Modied gravity:
Compact objects
What about Black Hole singularities?
It has been found that in some cases there exist static solutions (like the
Schwarzschild solution) with no central singularities.
In particular, in a theory of the type R + `P (aR 2 + R R ) and with charged matter it
has been found (Olmo,Rubiera-Garcia Eur.Phys.J. C72 (2012) 2098)
For certain charge congurations, instead of central singularity, an ultracompact
central core develops. The area of this core is quantized, if the electric charge is
quantized.
External horizon almost like in GR
In some cases, the external horizon might disappear, revealing its central core.
Ultracompact ( 1000 moles) and charged astrophysical objects?
Again, no general theorems. We don't have singularity theorems like in GR, but
it looks like the absence of singularities is a general feature, while in GR we
necessarily have singularities.
Helios Sanchis Alepuz (University of Giessen)
May 2015
37 / 38
Modied gravity:
Canonical Quantization of modied gravities?
Since Palatini theories lead to second-order equations of motion, they are probably
easier to quantize than metric theories.
For quantization `a la LQG, we need the Hamiltonian of the theories. Very
complicated because one needs the time-space decomposition of the connection as
well
Cheat: Olmo, HSA Phys.Rev. D83 (2011) 104036
f (R ) theories equivalent, at
classical level, to GR plus a scalar eld coupled to the gravity sector. Easier to obtain
the Hamiltonian. Same constraints as in GR. Good enough to use for quantization?
Helios Sanchis Alepuz (University of Giessen)
May 2015
38 / 38
Modied gravity:
Final Remarks
Many other interesting and exotic solutions have been found (Wormhole solutions,
self-gravitating objects, etc.)
Once we go beyond GR, there is no restriction to which functional form the gravity
lagrangian should have.
That makes the derivation of general results an extremely dicult, if not impossible
task.
Are the famous gravitational singularities, sued as a major argument for the need of a
quantum theory of gravity, just an artifact of using the Einstein's-Hilbert lagrangian?
A lot more work is necessary, but the eld (using the Palatini formalism) is rather
new.
Helios Sanchis Alepuz (University of Giessen)
May 2015
38 / 38