Brief summaries on general relativity (II) Gungwon Kang (KISTI) July 26, 2015 at Int’l Summer School on NR & GW, KAIST in Korea

Outline I.Solutions in GR - Black holes - Relativistic stars - Cosmological models I.Gravitational waves - Weak gravitational fields - Detection principle

I. Solutions in GR

블랙홀 메트릭 : 1916 : 2 차 비선형 편미분 방정식 슈월츠촤일드 (Schwarzschild) 출처 : 과학동아

Causal structure: r=0 사건지평면 그림 출처 :

* Birkhoff’s theorem (1923): “The metric for a general spherically symmetric vacuum spacetime is locally isomorphic to the Schwarzschild metric.” - No propagation degrees of freedom with spherical symmetry in vacuum! - No monopole radiation as in the Maxwell’s case. - Does not hold for an axially symmetric star!

Rotating charged black holes: Newman et. al. (1965) where M: ADM mass J: Angular momentum (a=J/M) e: Charge i) Charged BH, i.e., a=0: Reissner-Nordstrom (1916, 1918) ii) Rotating BH, i.e., e=0: Kerr (1963)

Dragging effect in the rotating BH: J Ω Note: a=0.9 a=0.5 a=0

Draw the Penrose diagrams.

Interior solutions * TOV equations: “Spherically hydrostatic system” : EOS (Equation Of State) * Relativistic stars: “Compactness” ~ Mass/Size : 1/2~1

- A class of solutions parameterized by the central density: - The maximum mass exists for a given R, for instance, M < 4R/9 for constant density stars! N 설명 3/2 가벼운 별 내부 또는 무거운 별의 중심부의 대류가 주 로 일어나는 부분 0.5~ 1 중성자별 내부에 주로 쓰이는 값 3 온도가 굉장히 낮아 전자가 페르미 에너지 상태에 있 을 때 축퇴압 ( 상대론적 ) 3/2 온도가 굉장히 낮아 전자가 페르미 에너지 상태에 있 을 때 축퇴압 ( 비상대론적 ) ∞isothermal( 온도가 일정할 때 ) 상태방정식 Ex) Polytropic EOS: N: Polytropic index

For various EOS: Lattimer & Prakash

stronomy/sun/sunsize.shtml ~70000, 109, 9 times ∙ 20km Neutron star * Difference from Newtonian gravity:

Cosmological solutions: - Homogeneous & Isotropic - For the perfect fluid as matter, (Friedman eq.) (EOS) k=+1, 0 : “OPEN” universe, k=-1 : “CLOSED” universe

Gravitational waves - Weak gravitational fields - Detection principle -Generation, Detection instruments, Sources, Data analysis, etc.  School lectures

As a system of oscillating charges can generate electromagnetic waves, a system of time- dependent matter could produce a dynamical curvature around it. Spacetime wiggles propagate away, carrying a sort of “energy”.  Gravitational waves (GW)!

Weak gravitational fields A nice way to see the basic features of GWs is to consider weak gravitational fields perturbed from the flat Minkowski spacetime: * Linearized gravity: within a rectangular coordinate syst. Expand it up to the linear order.

Defining,,,

with * Gauge condition:

Even if provided thatis chosen such that So, in the Lorentz gauge condition, i.e., Finally, we havew/

In vacuum, i.e.,,, - Massless spin-2 field propagating in the flat spacetime: Fierz & Pauli (1939) - Propagation speed: c the speed of light! - Residual gauge in the case of vacuum: h=0, Traceless gauge - Transverse gauge: - TT gauge

To summarize: w/ z In vacuum, h = 0 and

–In a more general context, e.g., GWs propagating through curved ST, Isaacson (‘68) –GWs carry energy, angular momentum and momentum: : Higher order contributions Mass octupole moment Current quadrupole moment

Gravitational recoil or ‘kicking:

GW sources and strengths: So, extremely weak for most cases! Laboratory generation of GWs: Ex). A rotating dumbbell consisting of two masses (1ton, 2m & 1kHz) produces R ~ λ = 300km 

Ex). From particle accelerators, e.g., LHC, where for In the LHC, v~ and 10^11 protons per bunch  h ~ 10^-43 !! Astrophysical sources of GWs, e.g., binary,: ~ 4km (~ ), r ~ 200km & R ~ 200Mpc  h ~  Ex). Neutron star binary Ex). Black hole binary ( ≫ ~10^-40)

Detection of GWs: –GW is predicted even before the full formulation of GR. –It was J. Weber (‘63) who tried for the first time the detection experiment by using a resonant-mass cylindrical bar at 1660Hz. –It was very sensitive h~10^-16, but still far from ~10^-22. Einstein (1916) Predicted by

–To overcome this gap (~10^6) interferometric detectors have been developed since ‘70. –LIGO (Laser Interferometer Gravitational-wave Observatory) performed its first operation, S1, in 2002, and achieved its designed sensitivity at S6 in 2010 (~10^-22). –Other GW detectors: Virgo, GEO, KAGRA (in construction), InLIGO (in plan). –There has been no GW signal detected so far. – LIGO has been upgraded to have a better sensitivity of ~10 times, and this advanced one (aLIGO) will be in operation in 2015.

Interferometric detector: Principles: -Measure the change of lengths by using a Michelson-type interferometer:  -O.K., but …… -Isn’t the light itself stretched as well due to passages of GWs?  Co-expansion of the arms and of the light wave!  The interferometer really works?

Maxwell’s eq. in a curved spacetime: Lorentz gauge: Perturbations: 1. Linearized approximations: Light interacting with gravitational waves Earth, seismic noise, acoustic wave, local matter, etc.

i) TT gauge: : slowly varying amplitude, : rapidly varying phase - Mirrors and the beam splitter remain at rest always! -How does light interact with GWs? w/  - Geometric optics: - Proper lengths: 2. Geometric optics approximation: - Kip Thorne’s lecture note at Caltech

-The solutions for the outwards and backwards from the beam splitter are, respectively, Therefore, we see an intensity modulation directly proportional to the GW perturbation!!

-The beam splitter is freely falling in the presence of a GW. -Let’s consider a local Lorentz frame co-moving with it; ii) Proper reference frame of the beam splitter: -Coordinate transformations with the TT gauge: -Note that the mirrors are moving, but the BS is at rest!

-Indeed, the locations of two mirrors are -In the TT frame, the phase shift arose from the interaction of the light with GWs, whereas, in the LL frame, it is due to the displacement of the mirrors and the effect of the interaction between light and GW is in the next order. -The LL frame analysis breaks down for, for instance, in LISA.

중력파 검출기 동영상 : 막스 - 플랑크 연구소 제작 (28:00~30:30)

Windows to the universe: A. Penzias & R. Wilson (1989) (2001)(2009) Figure courtesy: Wikipedia KVN Arecibo Chandra X-ray (1999) Super-K (1983) RENO ( 물첨, 2010 ) 1610 Galileo 1928 Palomar 1789 Herschel 1998 VLT 1997 Hubble in SM2 ~633 첨성대 1996 보현산

? Gravitational wave (LIGO) “GW astronomy”

Credit courtesy: Patrick Brady We are here Beginning of GW Astronomy

EM Waves Theory: Maxwell (1864) Detection: H. Hertz (1886) Gravitational Waves Theory: Einstein (1916) Detection: Not yet (??) Hulse & Taylor (1975) Weber (1960) ?

Prospect 중력파 (GW) = 일반상대론의 꽃 우주를 보는 새로운 창 : “ 중력파 천문학 ” (New window: “GW Astronomy”) 중력파 과학 (GW Sciences) 중력파 문명 (GW Civilization)

감사합니다 !

Back-up slides

stronomy/sun/sunsize.shtml 70000, 109, 9 배 ∙ 20km 중성자별 뉴튼 중력과의 차이 :

* Plane wave solution in vacuum: Let with the transverse gauge - Residual gauge freedom:

- Response to the + polarization: at z = 0 z-direction

M M 2R