Gravitational Waves (& Gravitons ?)
-Vishal Kasliwal
Classical Electromagnetism
.E 0
.B 0
E B
t
B E
t
VacuumMaxwell Field Equations E 0
B 0
where
2
t2
2
x2
2
y2
2
z2
Light!!
Electromagnetic waves
Quantum ElectromagnetismHamiltonian of Quantized
Field
H
k 3 k
k N k 1
2 30where
(Harmonic oscillators !!)
Eigenstates and Energy
k n k
nk 1
2 30 3 k n k
Photons!!Particles
nk is the occupation number of the mode a ie. we have n photons with momentum k.
Photons are spin one Bosons because they arise from the quantization of a vector field.
General Relativity
• Set of 10 coupled, non-linear, hyperbolic-elliptical PDEs•Linearize•Weak field, low velocity limit
g hs
1
2h
1
3 h
curvedpart
flatpart
smallperturbation
function of smallperturbation
v v 0
Geodesics
When no ‘forces’ are acting on a body, it travels along a geodesic!Fancy Newton’s 1st Law
These replaceGalilean Transforms
Lorentz Transforms
Lorentz matrices
x' 'x
'
Einstein Field Equation
‘Or how we create gravity!’ – Replaces Newton’s Law of Universal Gravitation
‘Force’ = Geometry
G g 8T
Classical Gravity
G-Light?
Gravitational waves
s = 0
VacuumEinstein Field Equations
G 0
Weak field – low velocityLinearized
•1993 Noble Prize – Russell Hulse & John Taylor• PSR B1913+16: Binary star with pulsar as one component.•Strong emitter of gravitational radiation.•Orbit decays as predicted by loss of energy due to G-waves inaccordance with GR
Do They Exist?
Gravitational Waves
•Amplitude (h): Fraction of stretching or squeezing.•Frequency (n): 1/Time between two max squeezes.•Wavelength (l): Distance along wave between two max squeezes.•Speed (c): Speed of light.
+ polarization
× polarization
Produced by•Binary systems with neutron stars or black holes•Supernova Detected by•Weber Bar type detectors: 60s, 70s but still investigated – MiniGrail (The Netherlands)•Interferometric type: LIGO (US), CLIO (Japan), GEO 600 (Germany), VIRGO (Europe – Italy), TAMA 300 (Japan)
Gravitons•Quantization of gravitational field (hasn’t been done yet) produces gravitons.•Massless tensor-gauge bosons.•Carry very little energy – might be impossible to detect. •G-wave is wave packet of gravitons just as EM-wave is wave packet of photons.•Polarization of gravitational waves invariant under rotations of 180 degrees.•Spin = 360/(Rotation angle under which polarization modes are invariant) = 360/180 = 2• Applies to EM waves (S = 360/360 = 1)
•Not much known – more research required!
Fin
Planck Units
•Length & Time•Energy & Mass•Simplifies forms of equations.
c = G = = k = 1
4= 1
e =
Quantized Hamiltonian of EM FieldOscillator Hamiltonian
k 1
2a k , ta
†k , t a†k , tak, t
ak, t 1
2x k , t pk, tRaising Operator
Metric
t x0
x x1
y x2
z x3
g
1 0 0 00 1 0 00 0 1 00 0 0 1
s2 x xg x0 2 x1 2 x2 2 x3 2
Curvature I
v t•s•v TS R
Parallel Transport !
Covariant Derivative
x x
x
Christoffel Symbol
1
2g g g g
R
Reimann Curvature Tensor
G R 1
2Rg
R R
R R
Ricci Tensor
Scalar Curvature
Einstein Tensor
Curvature IITensor Symbol Indep. Comps
Reimann 20
Ricci 10
Scalar Curvature 1
Einstein 10
R
R
G
R
Energy-Momentum TensorT
Typically derived from action principle or equation of motion
G
Einstein Tensor
Measures volume distortions
Geodesics
CosmologicalConstant
Vacuum energy of free space.
Weber Detector
MiniGrail
Interferometric Detector