Quadrupole formula

In general relativity, the quadrupole formula describes the gravitational waves that are emitted from a system of masses in terms of the (mass) quadrupole moment. The formula reads:

h

¯

i

j

(

t

,

r

)

=

2

G

c

4

r

I

¨

i

j

(

t

−

r

/

c

)

,

{\displaystyle {\bar {h}}_{ij}(t,r)={\frac {2G}{c^{4}r}}{\ddot {I}}_{ij}(t-r/c),}

where

h

¯

i

j

{\displaystyle {\bar {h}}_{ij}}

is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave.

G

{\displaystyle G}

is the gravitational constant,

c

{\displaystyle c}

the speed of light in vacuum, and

I

i

j

{\displaystyle I_{ij}}

is the mass quadrupole moment.

It is useful to express the gravitational wave strain in the transverse traceless gauge, by replacing the mass quadrupole moment

I

i

j

{\displaystyle I_{ij}}

with the transverse traceless projection

I

i

j

T

T

{\displaystyle I_{ij}^{TT}}

, which is defined as:

I

i

j

T

T

=

∫

ρ

(

r

)

[

r

i

r

j

−

r

n

(

r

i

n

j

+

r

j

n

i

)

+

1

2

r

n

2

(

n

i

n

j

+

δ

i

j

)

+

1

2

r

2

(

n

i

n

j

−

δ

i

j

)

]

d

3

r

{\displaystyle {I}_{ij}^{TT}=\int \rho (\mathbf {r} )\left[r_{i}r_{j}-r_{n}\left(r_{i}n_{j}+r_{j}n_{i}\right)+{\tfrac {1}{2}}r_{n}^{2}\left(n_{i}n_{j}+\delta _{ij}\right)+{\tfrac {1}{2}}r^{2}\left(n_{i}n_{j}-\delta _{ij}\right)\right]d^{3}r}

where

n

{\displaystyle \mathbf {n} }

is a unit vector in the direction of the observer,

r

n

≡

r

⋅

n

{\displaystyle r_{n}\equiv \mathbf {r} \cdot \mathbf {n} }

, and

r

2

≡

r

⋅

r

{\displaystyle r^{2}\equiv \mathbf {r} \cdot \mathbf {r} }

.

The total energy carried away by gravitational waves can be expressed as:

d

E

d

t

=

∑

i

j

G

5

c

5

(

d

3

I

i

j

T

d

t

3

)

2

{\displaystyle {\frac {dE}{dt}}=\sum _{ij}{\frac {G}{5c^{5}}}\left({\frac {d^{3}I_{ij}^{T}}{dt^{3}}}\right)^{2}}

where

I

i

j

T

{\displaystyle I_{ij}^{T}}

is the traceless mass quadrupole moment, which is given by:

I

i

j

T

=

∫

ρ

(

x

)

[

r

i

r

j

−

1

3

r

2

δ

i

j

]

d

3

r

.

{\displaystyle {I}_{ij}^{T}=\int \rho (\mathbf {x} )\left[r_{i}r_{j}-{\tfrac {1}{3}}r^{2}\delta _{ij}\right]d^{3}r.}

The formula was first obtained by Albert Einstein in 1918. After a long history of debate on its physical correctness, observations of energy loss due to gravitational radiation in the Hulse–Taylor binary discovered in 1974 confirmed the result, with agreement up to 0.2 percent (by 2005).