General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29

ecture Spin (a quick look) In flat space-time, the axis of spin of a gyroscope is a constant. If the spin is represented by a 4-vector of the form s a = (0,s) then this implies an equation of motion of In non-flat space-time, we can generalize this expression to In the local inertial frame u.s=0 (in general, this is something that holds in all frames).

ecture Geodetic Precession A gyroscope in classical physics retains its orientation wrt to distant stars as it orbits a massive body. What about GR? Again, we consider an equatorial orbit (  =  /2). The only spatial part of the 4-velocity is For circular orbits,  = m/r 3, and the 4-velocity is (check u.u=1)

ecture Geodetic Precession and inserting the expression for u  then (algebra….)

ecture Geodetic Precession The geodetic precession per orbit is For the Earth, this is ~8.4”/yr. While this is small, this should be measurable by Gravity Probe B. With a rotating metric, there are additional precession effects.

ecture Gravitational Waves So far, the solutions to the field equations we have examined are static, making problems easier to solve. In reality, space-time should be dynamics, evolving as the distribution of matter/energy changes. Space-time can even be made to “wave” due to asymmetries in collapse or explosions (think of the electromagnetic waves produced as charges are accelerated). Large disturbances are difficult to calculate, requiring numerical computation of the evolving field equations.

ecture Wave hitting a Black Hole Harald Pfeiffer

ecture Gravitational Waves Small disturbances can be tackled analytically, revealing waves traveling at the speed of light. To do this, we need to employ the linearized field equations where  is a small, dimensionless parameter and h ab ! 0 as r ! 0. Defining (ignoring higher order terms) then

ecture Gravitational Waves Hence we can calculate the Christoffel symbols and the Riemann tensor is given by This satisfies the Bianchi identities R ab[cd,e] = 0

ecture Gravitational Waves Furthermore, the Ricci tensor and curvature scalar are where  cd h cd =h c c =h and the d’Alembertian is  =  ab  a  b The resulting Einstein tensor is given by

ecture Gauge Transformations As in EM, we can make a gauge transformation without changing the predictions of the theory. As GR does not prefer any particular coordinate system, we are free to make a coordinate transformation We can use this to transform g ab (and hence h ab ) to the new coordinates and find

ecture Gauge Transformations Calculating the curvature tensor and its contractions in the new coordinate system reveals that they are identical to those in the old coordinate system i.e. we have a gauge invariance. Given this, we can chose an appropriate coordinate system to simplify the linearized field equations. Introducing a new variable then

ecture Gauge Transformations We can impose the gauge condition that results in a wave equation of the form Hence the small perturbations on an otherwise flat metric can propagate as waves (See Hartle’s comparison of GR and EM on page 462 of his book).

ecture Plane Gravitational Waves If we assume substantial algebra (see textbook) reveals we can choose coordinates such that and the perturbation depends only on two independent functions.

ecture Polarization States If h 23 =0, the line element becomes the wave travels in the x-direction. For h 22 >0 then distances in the y-direction are decreased and those in the z-direction are increased. If h 22 =0, the line element becomes this is a rotation through 45 o of the case where h 23 =0.

ecture Polarization States Hence there are two polarization states, h + and h x at 45 o to one another. This is different from the EM case where polarization states are at 90 o. This is because EM is represented by a vector potential (A a, spin 1) whereas gravity is represented by a second rank tensor (h ab, spin 2).

ecture Detecting Gravitational Waves Gravitational waves should be formed in asymmetric core collapse supernova, or neutron star and black hole collisions. The expected waves would have h~ (or much less). Detecting these waves has a long history and in the 1960s Weber looked at the deformation of a large cylinder of aluminium although his results were inconclusive. More sensitive laser interferometers have been established, but to date, no waves have been seen!

ecture Pulsar Orbital Decay Indirect evidence for gravitational radiation comes from binary pulsar PSR whose orbit is decreasing as energy is carried away by gravitational waves. The quadrupole luminosity is where M is the mass of the stars in circular orbit, l o is their separation and  is their angular velocity (monopole and dipole radiation are forbidden due to spin 2 nature). The change in the period of the orbit is

ecture Nobels & Pulsars Taylor & Hulse won the 1993 Nobel prize for their measurement of the orbital decay of PSR1913. Energy is radiated away in the form of gravitational waves.

ecture Do Waves Carry Energy? Remember, we have solved the equation Where is the energy in the equation? Energy is used to form waves, and energy can be deposited by waves, but there is no localized energy identifiable in the waves. (The entire question of energy conservation in GR is a big one which does not appear close to being solved).