TOMMASO DE LORENZO - Agenda (Indico)
Transcript of TOMMASO DE LORENZO - Agenda (Indico)
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T O M M A S O D E L O R E N Z O
C O N S T R A I N T S O N GW W AV E F O R M SIN COLLABORAT ION W I TH ABHAY ASHTEKAR AND NEEV KHERA
CATAN IA , 3RD FLAG MEET ING , JUNE 14TH 2019
INST I TUTE FOR GRAV I TAT ION AND THE COSMOS , PENN STATE UN IVERS I TY
1906.00913 [GR-QC & ASTR O-PH .HE]
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R E S U LT
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R E S U LT
W H Y ?
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R E S U LT
W H Y ?
H O W ?
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FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !
W H Y ?
H O W ?
R E S U LT
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FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !
W H Y ?
H O W ?
LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL
R E S U LT
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FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !
W H Y ?
H O W ?
LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL
THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES
R E S U LT
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FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !
W H Y ?
H O W ?
LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL
THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES
IMPR OVED SENS I T I V I T Y AND AB UNDANCE OF EVENTS ASKS FOR BETTER CONTR OL !
R E S U LT
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FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !
W H Y ?
H O W ?
LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL
THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES
EXACT GR: ASYMPTOT I CALLY FLAT SPACET IMES
IMPR OVED SENS I T I V I T Y AND AB UNDANCE OF EVENTS ASKS FOR BETTER CONTR OL !
R E S U LT
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FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !
W H Y ?
H O W ?
LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL
THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES
EXACT GR: ASYMPTOT I CALLY FLAT SPACET IMES
BMS FLUXES + BOUNDARY COND I T IONS DEF IN ING CBC
IMPR OVED SENS I T I V I T Y AND AB UNDANCE OF EVENTS ASKS FOR BETTER CONTR OL !
R E S U LT
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T E M P L AT E B A N K S
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
INSP IRAL MER GER R INGDOWN
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
INSP IRAL MER GER R INGDOWN
BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE
2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT
INSP IRAL MER GER R INGDOWN
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT
3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT
INSP IRAL MER GER R INGDOWN
BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT
POST-NEWTON IAN NR
EOB(NR)/BOB
PHENOM
INSP IRAL MER GER R INGDOWN
PERTURBAT ION THEORY
BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE
3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT
POST-NEWTON IAN NR
EOB(NR)/BOB
PHENOM
INSP IRAL MER GER R INGDOWN
PERTURBAT ION THEORY
BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE
3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT
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T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR
2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT
POST-NEWTON IAN NR
EOB(NR)/BOB
PHENOM
INSP IRAL MER GER R INGDOWN
PERTURBAT ION THEORY
BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE
3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT
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A S Y M P T OT I C F L AT N E S S BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS
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A S Y M P T OT I C F L AT N E S S
i+
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS
FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ
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A S Y M P T OT I C F L AT N E S S
i+
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS
K INEMAT I CAL STRUCTURE ( ∘qab , ∘na)
FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ
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A S Y M P T OT I C F L AT N E S S
i+
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS
3-PARAMETERS FAM I LY OF BOND I -FRAMES RELATE BY ASYMPTOT I C BOOSTS
(∘qab = ω2 ∘qab ,
∘na = ω−1na)
ω = γ (1 − v /c ⋅ x) γ = (1 − (v/c)2)−1
KINEMAT I CAL STRUCTURE ( ∘qab , ∘na)
FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ
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A S Y M P T OT I C F L AT N E S S
i+
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS
3-PARAMETERS FAM I LY OF BOND I -FRAMES RELATE BY ASYMPTOT I C BOOSTS
(∘qab = ω2 ∘qab ,
∘na = ω−1na)
ω = γ (1 − v /c ⋅ x)
σ∘ = rhN = − ℒnσ∘ = − ·σ∘ NEWS = FREE DATA
γ = (1 − (v/c)2)−1
KINEMAT I CAL STRUCTURE ( ∘qab , ∘na)
DYNAM ICAL STRUCTURE
FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ
SHEAR OF ℓa
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A S Y M P T OT I C F L AT N E S S
i+
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS
3-PARAMETERS FAM I LY OF BOND I -FRAMES RELATE BY ASYMPTOT I C BOOSTS
(∘qab = ω2 ∘qab ,
∘na = ω−1na)
ω = γ (1 − v /c ⋅ x)
σ∘ = rhN = − ℒnσ∘ = − ·σ∘ NEWS = FREE DATA
γ = (1 − (v/c)2)−1
LEAD ING ORDER OF WEYL : Ψ∘2 = lim
r→∞r3Cabcd manbℓcmd
Ψ∘1 = lim
r→∞r4Cabcd ℓambℓcnd
KINEMAT I CAL STRUCTURE ( ∘qab , ∘na)
DYNAM ICAL STRUCTURE
FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ
SHEAR OF ℓa
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BMS S Y M M E T R I E S
𝒫 = 𝒯 ⋊ L
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
i+
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BMS S Y M M E T R I E S
𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
i+
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BMS S Y M M E T R I E S
𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L
SUPERTRANSLAT IONS 𝒮 ∈ ξa( f ) = f(θ, ϕ) ∘na
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
i+
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BMS S Y M M E T R I E S
𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L
SUPERTRANSLAT IONS 𝒮 ∈
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
B A L A N C E L A W
P( f )(u1) − P( f )(u2) = ℱ( f )(θ, ϕ)
i+
ξa( f ) = f(θ, ϕ) ∘na
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BMS S Y M M E T R I E S
𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L
SUPERTRANSLAT IONS 𝒮 ∈
B A L A N C E L A W
P( f )(u1) − P( f )(u2) = ℱ( f )(θ, ϕ)
P( f )(u) := −1
4πG ∮Cu
d2 ∘V f Re[Ψ∘
2 + ·σ∘σ∘]
ℱ( f )(u, θ, ϕ) :=1
4πG ∫u2
u1
du∮Cu
d2 ∘V f [ | ·σ∘ |2 − Re(ð2 ·σ∘)]
ℑ−
i−
uℑ+ ℑ+
u1
u2
naℓa
mai∘ i∘
ℑ−
qab
qabmana ℓa
i+
ξa( f ) = f(θ, ϕ) ∘na
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∫u2
u1
du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]
u2
u1
T H E R E S U LT
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
∫u2
u1
du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]
u2
u1
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
1- ENER GY CONSERVAT ION :
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
∫u2
u1
du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]
u2
u1
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
1- ENER GY CONSERVAT ION :
IN THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .
2- ASYMPTOT I C STAT IONAR I TY :
∂uΨ∘1 → 0 AS
AS
u → − ∞u → + ∞
∫u2
u1
du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]
u2
u1
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
1- ENER GY CONSERVAT ION :
2- ASYMPTOT I C STAT IONAR I TY :
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘
2 + ·σ∘σ∘]u=−∞
u=+∞
I N THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .∂uΨ∘
1 → 0 AS
AS
u → − ∞u → + ∞
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘
2]u=−∞
u=+∞
1- ENER GY CONSERVAT ION :
2- ASYMPTOT I C STAT IONAR I TY :
IN THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .∂uΨ∘
1 → 0 AS
AS
u → − ∞u → + ∞
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘
2]u=−∞
u=+∞
1- ENER GY CONSERVAT ION :
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
IN THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .
2- ASYMPTOT I C STAT IONAR I TY :
∂uΨ∘1 → 0 AS
AS
u → − ∞u → + ∞
Ψ∘2
SPHER I CALLY SYMMETR I C
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
1- ENER GY CONSERVAT ION :
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
IN THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .
2- ASYMPTOT I C STAT IONAR I TY :
∂uΨ∘1 → 0 AS
AS
u → − ∞u → + ∞
PAST REST-FRAME limu→−∞
Ψ∘2 = − GMi∘
FUTURE REST-FRAME limu→+∞
Ψ∘2 = − GMi+
Ψ∘2
SPHER I CALLY SYMMETR I C
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘
2]u=−∞
u=+∞
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T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
1- ENER GY CONSERVAT ION :
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
IN THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .
2- ASYMPTOT I C STAT IONAR I TY :
∂uΨ∘1 → 0 AS
AS
u → − ∞u → + ∞
PAST REST-FRAME limu→−∞
Ψ∘2 = − GMi∘
FUTURE REST-FRAME limu→+∞
Ψ∘2 = − GMi+
limu→+∞
Ψ∘2 =
−GMi∘
γ3 (1 − vc ⋅ x)
3
Ψ∘2
SPHER I CALLY SYMMETR I C
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘
2]u=−∞
u=+∞
![Page 39: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/39.jpg)
T H E R E S U LT
LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS
1- ENER GY CONSERVAT ION :
u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS
IN THE PAST REST-FRAME , AND
IN THE FUTURE REST-FRAME .
2- ASYMPTOT I C STAT IONAR I TY :
∂uΨ∘1 → 0 AS
AS
u → − ∞u → + ∞
PAST REST-FRAME limu→−∞
Ψ∘2 = − GMi∘
FUTURE REST-FRAME limu→+∞
Ψ∘2 = − GMi+
limu→+∞
Ψ∘2 =
−GMi∘
γ3 (1 − vc ⋅ x)
3
Ψ∘2
SPHER I CALLY SYMMETR I C
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
![Page 40: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/40.jpg)
C O N C L U S I O N S
DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .
F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
![Page 41: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/41.jpg)
C O N C L U S I O N S
F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!
F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .
![Page 42: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/42.jpg)
C O N C L U S I O N S
ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .
F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!
F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .
![Page 43: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/43.jpg)
C O N C L U S I O N S
A NOVEL TEST OF GR? WORK IN PR OGRESS .
ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .
F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!
F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .
![Page 44: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/44.jpg)
C O N C L U S I O N S
A NOVEL TEST OF GR? WORK IN PR OGRESS .
ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .
F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!
F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !
E I THER SUPERTRANSLAT ION CONSERVAT ION OR SUPERR OTAT ION?
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .
![Page 45: TOMMASO DE LORENZO - Agenda (Indico)](https://reader030.fdocument.pub/reader030/viewer/2022012605/619a7e453074096081419731/html5/thumbnails/45.jpg)
C O N C L U S I O N S
A NOVEL TEST OF GR? WORK IN PR OGRESS .
ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .
F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!
F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !
E I THER SUPERTRANSLAT ION CONSERVAT ION OR SUPERR OTAT ION?
∫+∞
−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −
GMi+
γ3 (1 − vc cos θ)
3
DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .
T H A N K Y O U !