Gravity and action at a distance

Transcription

1 Gravitational waves

2 Gravity and action at a distance Newtonian gravity: instantaneous action at a distance Maxwell's theory of electromagnetism: E and B fields at distance D from charge/current distribution: - Change in E, B at time t depends on change in source distribution at time (t - D/c) - Reason is wave equation for E, B with propagation speed c Special relativity: speed of light c is the 'speed limit' for any kind of information transfer Expectation: weak gravitational fields should also propagate with speed c In particular, expect a wave equation with propagation speed c General relativity dynamical theory Does it imply such an equation?

3 Linearized general relativity Weak gravitational fields: Einstein equations: Metric covariant under general coordinate transformations: Invariance broken because of choice of background η μν, but residual freedom: "gauge transformations" where at most of the same order as

4 Linearized general relativity Riemann tensor to leading order in small h μν :... invariant under gauge transformations! Introduce Einstein equations to leading order: where Gauge transformations: d'alembertian Impose harmonic gauge: Linearized Einstein Equations:

5 Linearized general relativity Energy-momentum distribution T μν confined to spatially finite region V, then solution (from method of Green's functions): No instantaneous action at a distance!

6 Gravitational waves How is the information transmitted? Outside the source: Write out in full:... and since x 0 = ct :... which is a wave equation with propagation velocity c! Plane wave solution: Most general solution is superposition of plane waves :

7 Gravitational waves Electromagnetic waves: - 2 degrees of freedom (E, B) - 2 polarizations (rotation by 90 o ) What about gravitational waves? - A priori, has 10 independent components - Harmonic gauge eliminates 4, leaves 6 - Residual gauge freedom because harmonic gauge not spoiled by gauge transformations with ; hence we have 4 functions to eliminate 4 more components - Use to set - Use other 3, to set Harmonic gauge, μ = 0: - Constant part of is just Newtonian potential associated to total mass - Not dynamical, not measurable with GW detectors; set

8 Transverse-traceless gauge We have now used up all gauge freedom and are left with: Plane wave solutions: From harmonic gauge with μ = i:... hence perpendicular to the direction of motion (transverse) and also traceless "Transverse-traceless gauge" Plane wave in the z direction:

9 Effect of gravitational waves on matter Consider two point particles in free fall, small separation Geodesics: Take difference of the two, expand to leading order in : Local Lorentz frame: Non-relativistic motion: Relate to Riemann tensor: Riemann is invariant, can compute in any frame, e.g., TT frame: Tidal effect:

10 Effect of gravitational waves on matter Gravitational wave in the z direction: Relative displacements of particles in the (x,y) plane: δx, δy? If h x = 0: If h + = 0:

11 Effect of gravitational waves on matter h + polarization h x polarization

12 Energy and momentum carried by gravitational waves Solution to the linearized Einstein equations is typically not a solution to the full Einstein equations, nor even of the second order equations Einstein tensor Linearized vacuum Einstein equations Substitute this solution into second order equations: with a solution Introduce a further correction to the metric, Corrected second order vacuum Einstein equations: of order... which can be written as To second order, causes the same correction to the metric as would be caused by matter with energy-momentum tensor

13 Energy and momentum carried by gravitational waves is symmetric and since satisfies linearized Einstein equations, Interpret as the energy-momentum tensor of gravitational field itself? Problem: not invariant under gauge transformations: General relativity has no local notion of energy density of the gravitational field! But, can take average over small spatial volume surrounding a point and redefine : Average implies spatial integral in which one can integrate by parts, neglecting the surface terms. Then use gauge conditions and linearized Einstein equations to find: Gauge invariant!

14 Energy and momentum carried by gravitational waves Gauge invariant energy density:... or in terms of the + and x polarizations: Similarly momentum density:

15 Energy and momentum carried by gravitational waves Energy in spatial volume bounded by a large sphere of radius r: Energy emitted in gravitational waves: Energy passing through sphere : Similarly for momentum:

16 The generation of gravitational waves Linearized Einstein equations: Solve by the method of Green's functions: Solution for given an energy-momentum tensor : "Retarded" Green's function (no incoming radiation from infinity)... hence:

17 The generation of gravitational waves If linearized Einstein equations already in the form... then is in the harmonic gauge, Want metric perturbation in the transverse-traceless gauge This can be done by applying a projection operator where unit vector perpendicular to wave front:... hence

18 The generation of gravitational waves Want to know what happens far from the source Suppose source has characteristic size d, then... so that to good approximation... and we can expand Note that in Fourier expansion, and can be viewed as a characteristic speed Expansion in

19 The generation of gravitational waves This leads to multipole expansion: where "ret" refers to retarded time and we have multipole moments

20 The generation of gravitational waves Multipole moments of the stress tensor not physically intuitive But, can be expressed in terms of multipole moments of energy and momentum density... Mass multipoles: Momentum multipoles: To leading order: Mass quadrupole radiation

21 The generation of gravitational waves No monopole radiation: No dipole radiation: - Mass dipole zero (hence constant) in center of mass frame - No momentum monopole contribution:

22 The generation of gravitational waves Without loss of generality: study radiation in the z direction - Metric perturbation: - Right hand side: Exercise... hence... evaluated at retarded time