Identification of Metric Theories of Gravitation using the SCHENBERG detector Nadja S. Magalhaes*, Claudemir Stellati†, Cesar H. Lenzi††, Carlos Frajuca and Fabio S. Bortoli *Federal University of Sao Paulo – Diadema Campus (UNIFESP), Brazil † Taubate University (UNITAU) , Brazil † † Technological Institute of Aeronautics (ITA) , Brazil Federal Institute for Education, Science and Technology of Sao Paulo (IF-SP, former CEFET-SP) , Brazil
In this poster results of a research on ways to make such a distinction using the SCHENBERG detector are presented. Through simulation it is shown how a detected signal is processed so that the correct metric theory becomes evident. The SCHENBERG gravitational wave detector • Sensitive to frequencies between 3.0 and 3.4 kHz • Antenna: CuAl(6%), solid, spherical, ∅=65cm, 1149.53 kg • Six two-mode transducers tuned to 3170.4 Hz (frequency of the free sphere’s quadrupole modes) and one two-mode transducer tuned to 7268.67 Hz (frequency of the free sphere’s monopole mode), all sensitive to radial motions of the antenna’s surface.
Metric Theories
N-P Parameters Φ22 = Ψ2 = Ψ3 = 0; Ψ4 ≠ 0
Vector-tensor (spins 1 and 2)
Φ22 = Ψ2 = 0; Ψ3 ≠ 0; Ψ4 ≠ 0
Escalar-tensor (spins 0 and 2)
Ψ3 = 0; Ψ4 ≠ 0; Φ22 ≠ 0 and/ or Ψ2 ≠ 0 Ψ3 ≠ 0; Ψ4 ≠ 0; Φ22 ≠ 0 and/ or Ψ2 ≠ 0
General relativity (GR)
0i 0 j
)
Im Ψ Re Ψ
4
4
−Φ
22 3
F1GW ( t ) = M χ quadrup ( R ) c 2 R ( 36 Re Ψ
4
)
F2GW ( t ) = M χ quadrup ( R ) c 2 R ( − 36 Im Ψ
4
)
F3GW ( t ) = M χ quadrup ( R ) c 2 R ( − 72 Im Ψ
F5GW ( t ) = M χ quadrup ( R ) c 2 R 12 3 ( Φ
Force on a solid body due to a gravitational wave • R0µ0ν are the “electric” independent components of the Riemann tensor d2 GW r ρ 2 xµ = f µ ( x , t ) = − ρ c 2 R0 µ 0 β ( t ) x β dt
T4 T3 T2 T1
Kaluza-Klein
Brans-Dicke
}
T6 T5 T4 T3 T2 T1
Transducers’ outputs depending on the metric theory of gravitation considered as the basis for the input gravitational wave.
F3GW ( ω ) = M χ quadrup ( R ) ω 2 R − 18h23 ( ω ) Im Ψ4
Tensor-vector Theory
General Relativity
Monop
F4GW ( ω ) = M χ quadrup ( R ) ω 2 R − 18h13 ( ω )
Mode 5 Mode 4
h11 ( ω ) + h22 ( ω ) 2 − h33 ( ω ) ( ω ) = M χ quadrup ( R ) ω R − 6 3 2
F5GW Ψ2
Mode 3 Mode 2
For the monopole mode: Re Ψ3
GW Fmonop (t) =
Im Ψ3 © C. F. Will
Im Ψ Re Ψ 2
4
(ω )
r qi ( xi , ω ) =
2
Kaluza-Klein
Monop Mode 5
M χ monop ( R ) c 2 R h11 ( ω ) + h22 ( ω ) + h33 ( ω )
Mode 4
Mode 1
5
∑
j= 1
Mode channels’ outputs depending on the metric theory of gravitation considered as the basis for the input gravitational wave.
{QijGW ( α , M R1 , M R 2 , M , ω 0 ) FjGW ( ω ) }
Continuity of this work Mode channels and MTG
(ω ) 2 Im Ψ 3 ( ω ) − 6Ψ 2 ( ω )
2
• Solution of the inverse problem for any metric theory of gravitation
Quadrupole modes, no noise.
3
Relating the Newmann-Penrose parameters to the dimensionless amplitude
ω2 Φ 22 ( ω ) = − 2 h11 ( ω ) + h22 ( ω ) 4c ω2 Im Ψ 4 ( ω ) = h ω) 2 12 ( 2c ω2 Re Ψ 3 ( ω ) = − 2 2h13 ( ω ) 8c ω2 Im Ψ 3 ( ω ) = 2 2h23 ( ω ) 8c ω2 Ψ 2 (ω ) = − h ω) 2 33 ( 12c
Brans-Dicke
Transducers’ outputs and MTG
2
ω h ω ) − h22 ( ω ) 2 11 ( 4c
2)
Quadrupole modes, no noise.
− 2 2 Re Ψ
4 ( ω ) − Φ 22 ( ω ) 2 Im Ψ 3 ( ω )
22
+ 12Ψ
Mode 2
−1 ∂2 ( t ) = 2 2 hµ ν ( t ) 2c ∂ t
)
)=
1 M χ monop ( R ) c 2 R ( 4Φ 2
Mode 1
Mode 3
ω r 2 β ( x , ω ) = − ρ c R0 µ 0 β ( ω ) x = − ρ hµ β ( ω ) x β 2
4(ω ) = −
T5
)
F2GW ( ω ) = M χ quadrup ( R ) ω 2 R − 18h12 ( ω )
g m ( ω ) 5× 1 =
The gravitational wave density force:
Re Ψ
2
In terms of the dimensionless amplitude
Relation to the Newmann-Penrose parameters:
Positions of the transducers tuned to the quadrupole mode
− 6Ψ
)
T6
Monop
The Riemann tensor:
f
22
{
Φ 22
Tensor-vector Theory
General Relativity
3)
F1GW ( ω ) = M χ quadrup ( R ) ω 2 R − 9 h11 ( ω ) − h22 ( ω )
In terms of the dimensionless amplitude, h µν
GW µ
3
F4GW ( t ) = M χ quadrup ( R ) c 2 R ( 72 Re Ψ
GW Fmonop (ω
− Re Ψ 4 ( ω ) − Φ 22 ( ω 2c 2 hij ( ω ) = 2 Im Ψ 4 ( ω ) ω − 2 2 Re Ψ 3 ( ω )
Different metric theories are expected to show clearly different signatures in SCHENBERG’s signal Monop
− 2 2 Re Ψ 3 2 2 Im Ψ 3 − 6Ψ 2
Re Ψ4
Results
In terms of the NewmannPenrose parameters
GW polarizations
R0 µ 0 β
• Four different kinds of waves were used, depending on the metric theory considered.
Force on the antenna due to a gw
Brans-Dicke: Ψ3 = 0; Ψ4 ≠ 0; Φ22 ≠ 0; Ψ2 = 0 Kaluza-Klein: Ψ3 ≠ 0; Ψ4 ≠ 0; Φ22 ≠ 0 Ψ2 = 0
2 2 Im Ψ
The displacements that each polarization mode allowed by metric theories of gravitation induces in a ring of particles are shown in this figure. The gravitational wave propagates in the +z direction.
r β v ( x ) x ρ d 3x µ ml
V
• To the lowest order in the perturbative expansion and for a gw arriving in the +z direction one finds:
(R
− c R0 µ 0 β ( t ) ∫ 2
For each of the five quadrupole modes:
• Located at Sao Paulo city, Brazil.
The SCHENBERG Detector
F
• Differences in spin show up in the six independent R0µ0ν
− Re Ψ 4 − Φ 22 = Im Ψ 4 − 2 2 Re Ψ 3
( t) =
GW ml
Exemples
Tensor (spin 2)
General (spins 0, 1 and 2)
• Using the mathematical model of the detector sinuisodal waves were simulated incident on the detector’s +z direction.
Sphere coupled to transducers, no noise.
Transducer Amplitude
With the use of appropriate methods, detected gw can be used to differentiate among metric theories of gravitation. The determination of the best metric theory that describes gravitation is an essential task.
Newmann-Penrose parameters ( Ψ2, Ψ3, Ψ4 and Φ22) allow for the identification of the spin content of a metric theory of gravitation
Transducer Amplitude
Gravitational waves (gw) are a phenomenon predicted by the general theory of relativity as well as other metric theories of gravitation. The most intense gw signals are expected to originate from astrophysical sources. The Mario SCHENBERG detector is one of the several detectors projected/constructed with the goal of detecting such waves.
Simulation of the SCHENBERG detector
Gravitational force and antenna’s modes
Metric theories of gravitation (MTG)
Transducer Amplitude
Introduction
GRAVITON GROUP
Transducer Amplitude
DIADEMA CAMPUS
Examples
4
∑
i= 2
GiGW ( α , M R1 , M R 2 , M , ω 0 ) F5GW ×1 ( ω
• Inclusion of noises in the system.
)
Mode Mode Mode Mode Mode Channel Channel Channel Channel Channel 1 2 3 4 5
Bibliography •
D. M. Eardley, D. L. Lee and A. P. Lightman, Phys. Rev. D8, 3308 (1973).
General relativity
Nonnull
Nonnull
0
0
0
Vector-tensor
Nonnull
Nonnull
Nonnull
Nonnull
0
Brans-Dicke
Nonnull
Nonnull
0
0
Nonnull
•
Kaluza-Klein
Nonnull
Nonnull
Nonnull
Nonnull
Nonnull
M.Bianchi et al. “Testing theories of gravity with a spherical gravitational wave detector”, CQG 13 (1996) 2865.
•
C. Stellati. “Classification of theories of gravitation using the gravitational wave detector Mario Schenberg”. PhD thesis. São Jose dos Campos: ITA, (2006).
•
N. S. Magalhaes et al. “The detection of gravitational waves as a test for theories of gravitation”. XVII ENFPC, Caxambu (1996).
Exemples General relativity: h11=-h22; h12 ≠ 0 h33 = h13=h23=0 Vector-tensor: h11= - h22; h33 = 0; h13,h23,h12 ≠ 0Brans-Dicke: h13 = h23 = 0; h11 ≠ -h22; h11 ≠ h22 h33 = 0; h12 ≠ 0 Kaluza-Klein: h33 = 0
Transducer’s output and MTG
See also
Monopole mode, no noise. qmonop ∝ [ 4Φ
22
+ 12Ψ
2
] ∝ [ h11 +
h22 + h33 ]
• Evidently null in the case of a vector theory (like GR) or a tensor-vector theory. • Non-zero in the case of scalar-tensor theories (like Brans-Dicke) or general theories (like Kaluza-Klein)
•
C. M. Will, “Theory and experiment in gravitational physics”. Cambridge: Cambridge Univ. Press, 1993.
•
C. M. Will, “The Confrontation between General Relativity and Experiment”, http://www.livingreviews.org/lrr-2006-3. •
F. Fucito, in “Gravitational waves”. Eds. I. Ciufolini et al. London: IOP, 2001. Sections 11.1 and 11.2.
NSM acknowledges FAPESP for the support through grant #06/07316-0, as well as IUPAP for the partial support.
