Announcements Please turn in Assignment 2 and Assignment 3 is - PowerPoint PPT Presentation

Announcements Please turn in Assignment 2 and Assignment 3 is uploaded to Piazza and the course website. DataNose has been updated to reflect the actual times of the Werkcolleges. Your app should now reflect this.

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 Using Bianchi identities, there are really only 6 independent equations.

Methods Solving Einstein’s equations is difficult. They’re non-linear. In fact, the equations of motion are impossible to solve analytically except for certain choices of metric (ex: flat Minkowski, Schwarzschild) In the absence of symmetry, there are two methods: 1. Numerical relativity (see slides from 2018) 2. Approximation techniques For solutions for weak gravitational fields , we consider an approximation with a metric very close to flat space but 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 among 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 the choice of background η µ ν 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 Thus, different choices for coordinates may give different forms for h µ ν In order to deal with this, we introduce gauge symmetry…

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 , in the direction k ~ of the vector with 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.