Keisuke Izumi(KMI & Dept. of Math., Nagoya U.)

Motivation from UVGeneral relativity (GR)cf. Gauge theories

BH Singularities
Initial singularity
quantization
Not renormalizableNot unitary

Classical singularity
at the source
quantization

Renormalizable
Unitary

GR should be modified in UV
to quantize gravity theory.
String theory, loop quantum gravity......

Motivation from IR
cf. In Solar system at end of 19c.

In present cosmology
Cosmic microwave background
Baryon acoustic oscillation
Galaxy rotational curve

Perihelion precessionof Mercury…

Dark Planet Vulcan???
(Invisible gravitational source)

Dark energy
Dark matter(Invisible gravitational source)

But, reality is
Newtonian gravity

There is a possibility:
Einstein GR
New theory of gravity??

Einstein GR
Scalar tensor theory, f(R),
Massive gravity, Bigravity,
Torsion gravity......

(end of 60s – beginning of 80s)

GR + standard model in particle physics① Causality (Cauchy problems)Gravitational waves (GWs) propagate to null direction(fastest speed of propagation is the same as that of photons).
This gives well-defined Cauchy developments.
② Positivity of total energy

In GR (with dominant energy condition)
ADM energy in asymptotically flat spacetime is positive.
Therefore, Hamiltonian is bounded from below andthe (Minkowski) vacuum is semiclassically stable.
③ Properties of Black Holes
Uniqueness of BHs
Penrose inequality

GR + standard model in particle physics+ modification term of gravity or/and exotic matters① Causality (Cauchy problems)Gravitational waves (GWs) propagate to null direction(fastest speed of propagation is the same as that of photons).
This gives well-defined Cauchy developments.
② Positivity of total energy

In GR (with dominant energy condition)
ADM energy in asymptotically flat spacetime is positive.
Therefore, Hamiltonian is bounded below andthe (Minkowski) vacuum is semiclassically stable.
③ Properties of Black Holes
Uniqueness of BHs
Penrose inequality

GR + standard model in particle physics+ modification term of gravity or/and exotic matters① Causality (Cauchy problems)Gravitational waves (GWs) propagate to null direction(fastest speed of propagation is the same as that of photons).
This gives well-defined Cauchy developments.
② Positivity of total energy

In GR (with dominant energy condition)
ADM energy in asymptotically flat spacetime is positive.
Therefore, Hamiltonian is bounded below andthe (Minkowski) vacuum is semiclassically stable.
③ Properties of Black Holes
Uniqueness of BHs
Penrose inequality

１、 Fastest propagations are null
(GR + Standard Model Particles)

Time evolution
time

Cauchydevelopment
Initial slice

Black holeSuperluminality

２、 Some propagations are faster than null.
(GR + Standard Model Particles + Gauss Bonnet term (string correction))
C. Aragone (1988)
Choquet-Bruhat (1988)

Time evolution

Black holetime

Cauchydevelopment
Initial slice

Information canescape from BH(Izumi 2014)

３、 Some propagations have infinite speed.
(Massive gravity or Torsion gravity)

Acausal(Izumi, Ong, Nester, Chen, Gu, 2013, 2014, 2015)

Time evolution
time

Initial slice
No Cauchy development
Acausal structures

Black hole

Existence of lowest energy state is importantfor the system to be stable.
The positivity of energy in GR was not so trivial.
Positive energy theorem
Deformation of minimal surface(Schoen, Yau 1979)

Spinor (Witten 1981)
Asymptotically AdS q
Bogomolny bound M õ Q2e + Q2m(Gibbons, Hull 1982)
Constraint on the scalar field potential(Townsend 1984)

In cosmology
scalar field with non-trivial kinetic term is discussed.
: k-essence L = f(röþröþ; þ)ö

But only canonical kinetic term röþr þmatches with Witten’s Positive energy proof.
(Nozawa,Shiromizu 2014)

BH is a good object to check the validity of GR
in strong gravitational field.
Stationary BH is characterized only by total mass
total angular momentum
total charge.
Suppose BHs are relaxed to be stationary,
we can know the spacetime structure of BH in reality.
We can test general relativity
from the observation of BH.
But the geometry on horizon never can be seen by definition.

Is there other object to investigate the strong gravity region?
Photon sphere

Photon Sphere (PS)
If the gravity field is enough strong,even photon can have circular orbits.

BH

Photon surface can be seen in principle.
Uniqueness of Photon sphere

Trajectory of photon(Yazadjiev, Stoytcho 2015)

Perturbative uniqueness of Photon sphere(Yoshino 2016)

Uniqueness of photon spherewith hair of conformally coupled scalar field.
(Tomikawa, Shiromizu, Izumi arXiv:1612.01228)

The size of BH is bounded from above.
Penrose inequality (Penrose 1973)

AAH ô 4ù(2mG)2(on maximal hypersurface)

Equality happens iff the spacetime is SchwarzschildThis is the condition for horizon, that we can never see.
Penrose inequality for “photon sphere”(Shiromizu, Tomikawa, Izumi, Yoshino, arXiv:1701.00564)

ALT ô 4ù(3mG)2(on maximal hypersurface)

Area of Loosely trapped hypersurface(LTH)
2Def: rarak (' d 2 det gij) õ 0drEquality happens iff the spacetime is Schwarzschildand then LTH is the photon sphere.

Many modified theories of gravity are proposed.

We should revisit the properties and theorems in GR.
It would give a hint for the real theory of gravity.

Thank you!