Definitions Flashcards
What are the foundations of general relativity?
Spacetime tells matter how to move, and matter tells spacetime how to curve.
What is general relativity?
GR is a generalisation of special relativity in which the laws of physics are valid in all inertial reference frames.
What are the postulates of relativity?
The speed of light in the vacuum will be the same for every inertial observer.
How are observers motion related?
Through lorentz transformations
Describe the spacetime interval between events.
It is independent of the observers reference frame.
What are the two forms of the equivalence principle?
The weak equivalence principle and the strong equivalence principle.
Describe inertial mass in Newtonian physics.
The inertial mass of a body is a measure of its resistance to acceleration.
What is the weak equivalence principle (WEP)?
The inertial mass and the gravitational mass are identically equal. Such that freely falling objects inhabits an inertial frame in which all gravitational forces have disappeared.
What is the local inertial frame (LIF)?
It is the reference frame inhabited by our freely falling object.
What is the strong equivalence principle (SEP)?
Locally all the laws of physics have their usual special relativistic form apart from gravity which disappears. Where there is no experiment that can distinguish between a LIF which is freely falling in a uniform gravitational field and an inertial frame which is in a region of the universe far from any gravitating mass.
What are the consequences of the equivalence principles?
- The empirically observed equality of gravitational and inertial mass is explained.
- The acceleration of a test mass in a gravitational field is entirely independent of its nature, mass and composition.
- The path of a light ray will be bent by the gravitational field of a massive body.
- A light ray emitted from the surface of a massive body will be redshifted - gravitational redshift - when its wavelength is measured by a distant observer
How can we describe space time rigorously?
Using physical quantities in terms of scalars and vectors to tensors.
What are geodesics?
The trajectories of freely falling particles in GR generalised to curved paths. They parallel transport their own tangent vectors.
What can geodesics not distinguish between?
Zero gravitational fields and a uniform gravitational field.
When do geodesic deviations accelerate?
They only accelerate for a non uniform or tidal gravitational field.
What are manifolds?
A manifold is a continuous space which is locally flat. They can be continuously parametrised.
What is a Riemannian manifold?
A differential manifold on which a distance, or metric has been defined.
What is a tensor?
A tensor of type (l,m) defined on an n dimensional manifold, is a linear operator which maps l one-forms and m vectors into a real number.
What is the covariant derivative?
A derivative which transforms covariently under a general coordinate transformation.
What do Christoffel symbols describe?
They describe how the basis vectors at different points in the manifold change as one moves across the manifold.
What does the Riemann Christoffel tensor or Riemann tensor describe?
The curvature of space time.
How can one derive the Riemann Christoffel tensor?
- by parallel transporting of a vector around a closed loop in our manifold
- by computing the deviation of two neighbouring geodesics in our manifold
What is the energy momentum tensor?
It is the source of space time curvature. Describing the presence and motion of gravitating matter.
What is a perfect fluid?
A mathematical idealisation but one which is a good approximate description of the gravitating matter in many astrophysical situations.
What is the simplest type of relativistic field?
Dust
What is the momentarily comoving rest frame?
The Lorentz frame where a collection of particles are all at rest.
What gives rise to pressure in the fluid?
The particles within a fluid element will face random motions giving rise to pressure in the fluid.
How can a fluid element exchange energy with its neighbours?
Via heat conduction
Describe viscous forces
Viscous forces are present between neighbouring fluid elements they are directed parallel to the interface between neighbouring fluid elements resulting in a shearing of the fluid
What is a perfect fluid?
A relativistic fluid element in its MCRF where the fluid element has no heat conduction or viscous forces,
What is an immediate consequence of the strong equivalence principle?
Any physical law which can be expressed as a tensor equation in SR has exactly the same form in a local inertial frame of a curved space time.
How can we describe the detection of gravitational waves?
With the weak field approximation. Gravitational wave results from deviations from the flat space time of SR which are small.
What do the free space solutions of the metric perturbations take the form of?
Free space solutions for the metric perturbations of a nearly flat space time take the form of a wave equation, propagating at the speed of light.
Give an example of a static metric
The schwarzschild solution for the space time exterior to a point mass is an example.
What are the conditions for a static metric?
- All metric components are independent of t
- The metric is unchanged if we apply the transformation t |-> -t
A weak gravitational field has what kind of space time?
‘Nearly’ flat space time
What explains the origin of the ‘transverse’ part of the transverse traceless gauge
There is no component of the metric perturbations in the direction of propagation of the wave
What are the two independent gravitational wave polarisation states
‘+’ and ‘x’
What is the polarisation of the distortion produced by a gravitational wave
It is quadrupolar due to the fact gravitational waves are produced by changes in the curvature of space time.
When is a gravitational wave invariant?
Under a rotation of 180 degrees about its propagation.
Describe the graviton
A spin, S = 2 particle
What is the dominant form of radiation from a moving charge in electromagnetic theory?
The electric dipole followed by magnetic dipole and electric quadrupole radiation.
What is the gravitational analogue of the electric dipole moment?
The mass dipole moment, d
What does a gravitational ‘mass dipole’ luminosity of zero mean?
There can be no mass dipole radiation from any sort of gravitational source.
The quadrupole from a spherically symmetric mass is zero. What does this suggest?
Metric perturbations which are spherically symmetric do not produce gravitational radiation. The collapse of a spherically symmetric star will generate no gravitational waves.
A non zero curvature gives rise to what?
Acceleration of geodesic deviation which gives rise to a non uniform gravitational field.
What are the characteristics of the traceless transverse gauge?
- Most components of the amplitude tensor vanish
- The trace reverse perturbation reduces to the perturbation
Why do we need to look at celestial bodies in order to appreciate gravitational forces?
Because gravitational forces are very weak.
What are gravitational waves emitted by?
Asymmetric, accelerating distributions of mass where M1 cannot equal M2.
How can we achieve appreciable amplitudes?
Using very large masses
Is it possible to generate gravitational waves?
No
What are examples of compact sources?
- Binary black hole mergers
- Binary Neutron star mergers
What is an example of a burst source?
- Core collapse supernovae
What is a pulsar?
A highly magnetised rotating compact star that emits beams of electromagnetic radiation out of its magnetic poles.
Describe the resonant bar detector.
A resonant bar detector was the first kind of gravitational wave detector. It used the natural frequencies of large aluminium bars which could be excited by passing gravitational waves. They are narrow band sensitive.
What is narrow band sensitivity?
Sensitivity is limited to the natural frequencies of the metallic bar. Detector is not sensitive at other frequencies.
How can interferometry be used to detect gravitational waves?
A laser is split equally down two perpendicular vacuum tubes. End test mass mirrors reflect laser light. It is thought that passing gravitational waves would warp the position of mirrors giving rise to constructive interference a measurable signal.
What are potential hazards with LIGO?
1.Alligators
2.Bad drivers
3.Tumbleweeds
4.Ravens
What does lower noise give?
Higher sensitivity
What is the optimum arm length?
Gravitational wave detection is a length measurement. The longer arms, the bigger the effect. The optimum arm length is when the wave stays in the detector for a half a wavelength for the entire round trip. Each arm length should be a quarter of the wavelength.
What are gravitational waves?
Perturbations in the metric of space time travelling at the speed of light.
Describe current gravitational wave detectors.
Current detectors are based on interferometers which can achieve a sensitivity of ~ x 10^{-19}
What was the size of the first detected gravitational wave signal?
ΔL ~ 10^{-18}
which is approximately ~ 1/1000th a proton diameter
What noise dominates at ultra-low frequencies?
Seismic noise
What noise dominates at frequencies where the detectors are most sensitive?
Mirror coating thermal (Brownian) noise.
What are the two parts of quantum noise?
Radiation pressure noise and photon shot noise
What are gravitational wave detectors mirrors made of?
High quality substrates of polished glass with a coating stack ontop to make them reflective where they are suspended with high quality glass.
How do you make highly reflecting coatings?
By stacking layers of different materials with different refractive indices.
What happens if the difference in the refractive index of the two materials becomes larger?
The more light gets reflected at each interface. Hence we can make the coating thinner.
What was indirect verification of the existence of atoms and molecules?
‘Jittering’ motion of pollen grains and dust suspended in water.
What is the origin of brownian motion?
The first time that random fluctuations, here random displacement of pollen grains, had been linked to a dissipative process, here the loss of energy due to the fluid viscosity.
What is the origin of Nyquist and Johnson noise?
Fluctuation-dissipation-theorem (FDT) which gives a link between spectral density of thermal fluctuations and mechanical impedance of the system.
Describe homogeneous loss?
It sums up thermal motion from all modes of the system to give the total thermal noise
What is fluctuating thermal energy?
Brownian thermal noise.
What is fluctuating temperature?
Thermo-elastic, thermo-refractive noise
What is mechanical loss?
The phase-lag between stress and strain in a real solid. The amount of energy lost when something vibrates.
How do we measure mechanical loss?
It is measured on disk resonators before and after coating by isolating them from external influences, and then vibrating them. Ringdown experiments.
Can we improve these coatings by coolings?
No, it causes serious limitations in achievable sensitivity in a cryogenically cooled GWD.
What do higher refractive indices offer
thinner coatings, thermal noise reduction and promising low loss and thermal noise reduction.
What is the proposed coating for the einstein telescope?
aSi based coating is employed to meet the ET coating thermal noise requirements.
What does thermal noise depend on?
Temperature and thickness
