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Announcements Please turn in Assignment 2 and pick up Assignment 3 You can also email assignments to the TAs: Ka Wa Tsang (kwtsang@nikhef.nl) Pawan Gupta (p.gupta@nikhef.nl) Pick up Visualization Project description

Last week’s big announcement: first image of black hole

Gravitational Wave Derivation and Astrophysical Sources Lecture 3: Gravitational Waves MSc Course

• Solving the Einstein Equations • Linearized Theory • Vacuum Solution • Solution with Source Term • Generation of Gravitational Waves • Effect of Gravitational Waves on Matter • LIGO & Virgo Astrophysical Sources • Coalescing Binaries • Continuous Waves • Transient Bursts • Stochastic Background • LISA & PTA Sources

The Einstein Equations 2 g µ ν R = 8 π G G µ ν = R µ ν − 1 c 4 T µ ν Given the source distribution , one can solve T µ ν this set of 10 coupled nonlinear partial differential g µ ν ( x ) equations for the metric

Methods Solving Einstein’s equations is difficult. They’re non-linear. In fact, the equations of motion are impossible to solve unless there is some symmetry present. In the absence of symmetry, there are two methods: 1. Numerical relativity (next time) 2. Approximation techniques For the approximation technique, we consider a metric very close to flat space with a small perturbation. And we consider only first order perturbations.

• Solving the Einstein Equations • Linearized Theory • Vacuum Solution • Solution with Source Term • Generation of Gravitational Waves • Effect of Gravitational Waves on Matter • LIGO & Virgo Astrophysical Sources • Coalescing Binaries • Continuous Waves • Transient Bursts • Stochastic Background • LISA & PTA Sources

Linearized Theory of Metric Field Consider the Minkowski metric - a combination of three dimensional Euclidean space and time into four dimensions. h µ ν Consider a small perturbation on flat space: | h µ ν | ⌧ 1 h µ ν so that higher orders of can be neglected when substituting in Einstein Field Equations (EFE)

Linearized Theory of Metric Field Can we make coordinate transformations under such systems? Yes, from one slightly curved one to another, aka “Background Lorentz transformation” So EFE are invariant under general coordinate transformations but invariance is broken as a result of background. h µ ν is an as yet unknown perturbation on flat space. We can make small changes in coordinates that leave η µ ν h µ ν unchanged but make small changes in We can only consider a sufficiently large specific reference frame where holds. In other words, we’re restricted in how much we can change the coordinates.

Linearized Theory of Metric Field We are restricted to a limited set of coordinate transformations called “gauge transformations” x µ → x 0 µ + ξ ( x µ ) If we transform the metric under this change of coordinates we find that the metric has the same form but with new perturbations given by h µ ν ( x ) → h 0 µ ν ( x 0 ) = h µ ν ( x ) − ( ∂ µ ξ ν + ∂ ν ξ µ )

Linearized Theory of Metric Field We can stream line some calculations by an appropriate choice of gauge conditions. We require a coordinate system in which Lorentz gauge (or harmonic gauge) holds ∂ µ ¯ h µ ν = 0 where we’ve defined the trace-reversed perturbation: h µ ν = h µ ν − h ¯ 2 η µ ν such that the trace has opposite sign: ¯ µ ≡ ¯ h µ h µ ν = − h

Linearized Theory of Metric Field The Riemann curvature tensor R µ ναβ = 1 2 ( ∂ µ ∂ α g νβ − ∂ ν ∂ α g µ β + ∂ ν ∂ β g µ α − ∂ µ ∂ β g να ) for a flat metric with a perturbation will become R µ νρσ = 1 2 ( ∂ ν ∂ ρ h µ σ + ∂ µ ∂ σ h νρ − ∂ µ ∂ ρ h νσ − ∂ ν ∂ σ h µ ρ ) Then substituting the trace-reversed perturbation, EFE takes form: h νρ = − 16 π G ∂ µ ∂ µ ¯ h µ ν + η µ ν ∂ ρ ∂ σ ¯ h ρσ − ∂ ρ ∂ ν ¯ h µ ρ − ∂ ρ ∂ µ ¯ T µ ν c 4 If we define the d’Alembertian operator: ⇤ ≡ ∂ µ ∂ µ h νρ = − 16 π G ⇤ ¯ h µ ν + η µ ν ∂ ρ ∂ σ ¯ h ρσ − ∂ ρ ∂ ν ¯ h µ ρ − ∂ ρ ∂ µ ¯ T µ ν c 4

Linearized Theory of Metric Field And impose the harmonic gauge, then the last three terms in previous equation vanish and we end up with the Linearized Einstein Equations h µ ν = − 16 π G ⇤ ¯ T µ ν c 4

• Solving the Einstein Equations • Linearized Theory • Vacuum Solution • Solution with Source Term • Generation of Gravitational Waves • Effect of Gravitational Waves on Matter • LIGO & Virgo Astrophysical Sources • Coalescing Binaries • Continuous Waves • Transient Bursts • Stochastic Background • LISA & PTA Sources

Solution in a Vacuum What happens outside the source, where ? T µ ν = 0 Then, the EFE reduces to ⇤ ¯ h µ ν = 0 ✓ ◆ � 1 c 2 ∂ t 2 + r 2 ¯ h µ ν = 0 Wave equation for waves propagating at speed of light c ! Solutions to wave equation can be written as superpositions ~ of plane waves traveling with wave vectors and frequency k � � � ~ ! = c k � � �

Solution in a Vacuum Plane wave solution: ⇣ ⌘ ! t − ~ h ( t ) = A µ ν cos k · ~ x Implications: Spacetime has dynamics of its own, independent of matter. Even though matter generated the solution, it can still exist far away from the source where T µ ν = 0

• Solving the Einstein Equations • Linearized Theory • Vacuum Solution • Solution with Source Term • Generation of Gravitational Waves • Effect of Gravitational Waves on Matter • LIGO & Virgo Astrophysical Sources • Coalescing Binaries • Continuous Waves • Transient Bursts • Stochastic Background • LISA & PTA Sources

Solution with Source Now allow for source. What would cause the waves to be generated? h µ ν = − 16 π G ⇤ ¯ T µ ν c 4 Solve using retarded Green’s function assuming no incoming radiation from infinity. The solution is ✓ ◆ x ) = 4 G 1 t − | ~ x 0 | Z x − ~ ¯ d 3 x 0 x 0 h µ ν ( t, ~ , ~ x 0 | T µ ν c 4 | ~ x − ~ c

Solution with Source We can utilize an additional gauge freedom by imposing the radiation gauge: , h 0 i = 0 h = 0 Combining the harmonic gauge and this radiation gauge, we can write the solution in the transverse traceless (TT) gauge ✓ ◆ x ) = 4 G 1 t − | ~ x 0 | Z x − ~ h TT d 3 x 0 x 0 ij ( t, ~ c 4 Λ ij,kl (ˆ n ) , ~ x 0 | T kl | ~ x − ~ c ~ n - direction of propagation of GW Λ ij,kl (ˆ n ) h µ ν is a tool to bring outside the source in the TT gauge.

Solution with Source Λ ij,kl (ˆ n ) h µ ν is a tool to bring outside the source in the TT gauge. n ) = P ik P jl − 1 Λ ij,kl (ˆ 2 P ij P kl P ij ≡ δ ij − n i n j h T T ij ( t, ~ x ) Then the perturbation can be evaluated outside the x 0 ~ ~ source at while is a point inside the source. x x 0 | /c, ~ x 0 ) 6 = 0 T kl ( t � | ~ x � ~ We’re looking at a distance r that is much larger than the size of the source d . Then we can expand x 0 · ˆ d 2 /r � � x = r − ~ n + O ∆ ~

Solution with Source Then we can write the TT solution as x 0 · ˆ ✓ ◆ x ) = 4 G Z 1 t − r c + ~ n h TT d 3 x 0 x 0 ij ( t, ~ c 4 Λ ij,kl (ˆ n ) , ~ n | T kl x 0 · ˆ | r − ~ c If the source is non-relativistic, v/c << 1 , then we can expand + x 0 i n i ✓ ◆ x 0 · ˆ c + ~ t − r n t − r @ 0 T kl + 1 ⇣ x 0 ⌘ 2 c 2 x 0 i x 0 j n i n j @ 2 x 0 , ~ c, ~ T kl = T kl 0 T kl + ... c c We can substitute this for T kl in the TT solution to get the multipole expansion x ) = 1 4 G  S kl + 1 S kl,m + 1 � S kl,mp + . . . c n m ˙ 2 c 2 n m n p ¨ h TT ij ( t, ~ c 4 Λ ij,kl (ˆ n ) r ret where ret is the retarded time t − r/c

• Solving the Einstein Equations • Linearized Theory • Vacuum Solution • Solution with Source Term • Generation of Gravitational Waves • Effect of Gravitational Waves on Matter • LIGO & Virgo Astrophysical Sources • Coalescing Binaries • Continuous Waves • Transient Bursts • Stochastic Background • LISA & PTA Sources

Generation of Gravitational Waves T ij Multipole moments of stress tensor Z S ij = d 3 xT ij ( t, ~ x ) Z S ij,k = d 3 xT ij ( t, ~ x ) x k Z S ij,kl = d 3 xT ij ( t, ~ x ) x k x l ... Multipole moments of the stress energy tensor are not physically intuitive.

Generation of Gravitational Waves We can express the multipole moments in terms of the mass moments and the momentum multipoles. Mass moments: momenta of energy density T 00 /c 2 M = 1 Z d 3 xT 00 ( t, ~ x ) c 2 M i = 1 Z d 3 xT 00 ( t, ~ x ) x i c 2 M ij = 1 Z d 3 xT 00 ( t, ~ x ) x i x j c 2 ...

Generation of Gravitational Waves We can express the multipole moments in terms of the mass moments and the momentum multipoles. T 0 i /c Momenta of momentum density P i = 1 Z d 3 xT 0 i ( t, ~ x ) c P i,j = 1 Z d 3 xT 0 i ( t, ~ x ) x j c P i,jk = 1 Z d 3 xT 0 i ( t, ~ x ) x j x k c ...

Generation of Gravitational Waves To leading order in v/c , we can eliminate the multipole moments in favor of the mass moments to get a solution of the form: quad = 1 2 G M kl ( t − r/c ) n ) ¨ h TT ⇥ ⇤ ij ( t, ~ x ) c 4 Λ ij,kl (ˆ r S ij = 1 ¨ M ij where we have used: 2 Mass quadrupole radiation!