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