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Mackenzie Turvey
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for 𝜕! 𝑡!" = 0. But one notices that this is not gauge invariant. We can correct this by going back and noticing that if we average over 3-­‐volume for a few wavelengths over a small enough region, that this becomes gauge invariant (to a low enough order). By small enough, we mean smaller than the curvature of the background, but large enough for a few wavelengths. Only keeping first derivatives, we see, 1
1
!
!
𝑡!" =
𝜕! ℎ!"
𝜕! ℎ ! !" − 𝜕! ℎ ! 𝜕! ℎ ! − 𝜕! ℎ ! !" 𝜕! ℎ!"
32𝜋𝐺!
2
!
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− 𝜕! ℎ
𝜕! ℎ!"
1
! !"
! !!
𝜕! ℎ !!
𝜕! ℎ!"
=
32𝜋𝐺!
This is the energy-­‐momentum tensor (to 2nd order) of the gravitational wave itself. Now that we have this, we can calculate the energy and power radiated away. 𝑃=
!!
𝑡!" 𝑛! 𝑟 ! 𝑑Ω where 𝑛! is the normal to the sphere Σ. Relating back to the perturbed metric: 2𝐺 𝑑 ! 𝐼!" 𝑡 − 𝑟
𝜕! 𝜕! ℎ!" =
𝑟
𝑑𝑡 !
the radial differential can be evaluated at 𝑟 = ∞. For the radial-­‐temporal part of the energy-­‐momentum tensor, 𝐺 𝑑 ! 𝐽!" 𝑑 ! 𝐽!"
1
; 𝐽!" = 𝐼!" − 𝜂!" 𝐼!" 𝜂!" 𝑡!" = −
!
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8𝜋𝑟 𝑑𝑡 𝑑𝑡
2
𝐺
𝑃 = − 𝐽!" 𝐽!" 5
If we consider the gravitational potential of energy radiating out of a sphere, 𝐸 𝐺𝑃 𝐺𝑀
3𝑐 !
!
~
=
<
𝑐
→
𝑃
<
𝐺
𝑐 ! 3𝑐 !
𝑅
we see that the power radiated is limited, otherwise the system would collapse into a black hole.