Kepler’s Third Law

coordinates. For Schwarzschild spacetime we see that theKretschmann scalar R^µνρσR_µνρσ = 48G²M²

r⁶ (59)

blows up indicating that r= 0 represents an honest singularity.

The interpretation of M in the Newtonian limit is just the conventional Newto-nian mass leading to

ds² =−(1 + 2Φ)dt² + (1−2Φ)⁻¹dr² +r²dΩ², (60) where Φ is the gravitational potential defined in eq. (51). But for strong fields M also includes gravitational binding energies. Note also that if M vanishes, one recovers Minkowski spacetime. This makes sense, since Minkowski spacetime is pure vacuum. Besides that, also the limit r → ∞ leads to Minkowski spacetime. This property is called asymptotic flatness.

To finish this section about static spherical solutions of the EFE, let us briefly discuss Birkoff’s theorem [71]. It states that even if one starts with a time-dependent spherically symmetric line element of the form

ds² =−A(t, r)dt²+B(t, r)(dr²+r²dΩ²) + 2C(t, r)dtdr, (61) where A, B and C are some functions on the coordinates, the Schwarzschild metric is the unique vacuum solution. Although one starts just with spherical symmetry, the time-dependence drops out in the final result. One can view this as the relativis-tic generalization of Newton’s theorem which says that the external gravitational potential of a spherical mass (viz. eq. (51)) does not depend on the size of the mass.

Hence, even if the mass shrinks or expands, the metric stays time-independent.

3.6 Kepler’s Third Law

In this section we derive Kepler’s third law for a binary system on a circular path in the center-of-mass frame, which is discussed in more detail in Appendix D. In the center-of-mass frame, the two body system reduces effectively to a one body description of an object with reduced mass µ = m₁m₂/(m₁ +m₂) in the central potential of an object with a total mass m= m1+m2. To have a bound orbit the gravitational force exerted on the test particle has to be balanced by the centripetal force and hence we can write

µv²

R =E_pot⁰ (R), (62)

wherev is the velocity of the test particle,E_pot⁰ (R) = ∂_RE_pot(R) is the derivative of the gravitational potential energy E_potwith respect to distance between the objects R and

E_pot =−Gµm

R (63)

is the gravitational potential energy. Using v = 2πR/P, where P is the period for one orbit, and the time derivative of eq. (62) one can write

P˙ P =

R˙ 2R −

E˙_pot⁰

2E_pot⁰ , (64)

where the dot is the derivative with respect to time. Now we insert eq. (51) into eq. (64) and find

P˙

P =−3 2

|Eorbit|^·

|E_orbit|, (65)

where E_GR is the orbital energy

E_orbit =E_kin+E_pot =−Gµm

2R. (66)

4 Gravitational Waves

So far, we introduced the field equations of GR as nonlinear second-order partial differential equations for the metric tensor field. Without any approximation these equations are very complicated. We have seen in Sec. 3.5 that we can find exact solutions if we consider symmetries, like rotation invariance and stationarity which lead to the Schwarzschild solution. Besides that, most of the classical tests of GR are on the geodesics in Schwarzschild spacetime [16], but if we give up rotation invariance and stationarity, it is possible to study more complicated systems, which emit gravitational radiation. GWs are by definition weak gravitational fields. Hence, in Sec. 4.1 we derive the weak field expansion of the EFE, which is less restrictive than the Newtonian limit. In Sec. 4.1 we only keep the weak field approximation, but allow for relativistic motion of test particles. We will use this expansion to define the linearized theory and to calculate GWs in vacuum in Sec. 4.2 as well as GWs produced by quadrupole sources, like black hole or stellar binary systems in Sec. 4.3. For this reason, we need to fix the coordinate freedom to get rid of unphysical dofs²¹.

4.1 Expansion of the Einstein Field Equations

In this section we apply a weak field expansion to the EFE as the full analytic solution is too complicated and not known. We assume that it is possible to find a coordinate system in which the metric can be separated into a background part g_µν^B(x) and a small perturbation h_µν(x) with |h_µν| 1. We write

gµν =g_µν^B +hµν. (67) Further, we assume that the coordinates are chosen such that g^B_µν = O(1) and we introduce the notation h≡O(|h_µν|)²².

The condition |hµν| 1 does not unambiguously fix which part of gµν belongs to the background and which to the perturbation. In addition to this condition, we have to assume that the metric perturbation should only represent the GWs and not x-dependent parts of the Newtonian potential, for instance. Then it is

21The necessity to fix gauge dofs seems to be very clear after the development of gauge theories in particle physics. But in the beginnings of GR it was not clear at all that GWs even do exist. The reason for this was the confusion about the coordinate freedom and which coordinates to choose to calculate GWs. For a short history on this discussion, see [72].

22The inverse metric is given byg^µν =g_B^µν−h^µν can be derived from the conditiongµρg^ρν =δ^ν_µ. Further, note that indices can be pulled up or down with the background metric if we work to a certain order inh_µν.

4.1 Expansion of the Einstein Field Equations 21

clear that we are able to distinguish the metric perturbation from the background metric by their scale of change in time and space. We define the background to be a smooth, slowly varying function, whereas the metric perturbation oscillates very rapidly compared to the background metric. Thus, we introduce a typical frequency ω_B for the background metric and a characteristic frequency ω for the metric perturbation. Then, the condition

ω ω_B (68)

fixesh_µν for the chosen coordinate system. Analogously for the spatial scales we get

¯

λL_B, (69)

where L_B is the spatial scale of the background metric and ¯λ is the reduced wave-length of the metric perturbation. To distinguish GWs from the background it is enough that only one condition holds true²³. Hence, the following analysis can be either done with eq. (68) or with eq. (69) and for definiteness we will work with the frequency condition in eq. (68), which defines a small parameterω_B/ω (analogously eq. (69) defines ¯λ/LB).

The first step to expand the EFE is to insert eq. (67) in eq. (43) which leads to G^B_µν +G⁽¹⁾_µν +G⁽²⁾_µν +h³ =−8πG T_µν, (70) whereG^B_µν is the Einstein tensor constructed from g_µν^B,G⁽¹⁾µν depends linearly on h_µν and G⁽²⁾µν is of second order in hµν. We dropped the cosmological constant term, because effects of the cosmological expansion are assumed to be negligible for the analysis of GWs. Besides that, we are not interested in terms∼h³, because these are self-interactions of the gravitational field, which are source terms for nonlinearities of h_µν.

Now, we simplify eq. (70) by decomposing it into an equation for low frequencies and for high frequencies

G^B_µν =−8πG[T_µν]^low−

G⁽²⁾_µν^low

, (71a)

G⁽¹⁾_µν =−8πG[T_µν]^high−

G⁽²⁾_µνhigh

, (71b)

where ”low” and ”high” indicate the low and high frequency parts (analogously we can do this separation for long and short wavelengths). We shifted G⁽²⁾µν to the right-hand side, because its low-frequency part acts effectively as a source term for the background metric. This will become clear in Sec. 5.3. The high-frequency part is a source term for the metric perturbation, which will be neglected when we study the linearized theory in Sec. 4.2. Note that by definition G^B_µν only carries low frequencies, whereas G⁽¹⁾µν contains only high-frequency terms. Since G⁽²⁾µν is quadratic in hµν, it contains low and high frequency parts. This is because two rapidly oscillating modes with nearly identical frequencies could interfere such that their combination oscillates slowly.

In eq. (71a) and eq. (71b) we equate terms of different order in hµν. This

23Note: the relation between ¯λandω will be determined by the wave equation forh_µν (see eq.

(82)), but the relation betweenL_B andω_B is undetermined.

must be compensated by the other small parameter ωB/ω. Using ∂g_µν^B ∝ ωB and

∂h_µν ∝ωhthe low-frequency equation (71a) yields the relation

ω_B² ∼ω_M² +ω²h², (72) where ω_M is the characteristic frequency of the slowly oscillating matter contribu-tion. In vacuum this results in

h ∼ω_B/ω. (73)

On the other hand, if the background curvature is dominated by matter, we get

h ω_B/ω. (74)

From eq. (73) we learn that the condition h 1 is mandatory for the definition of GWs, because if we had h= O(1), we cannot distinguish the perturbation from the background since eqs. (73) and (74) lead to O(ωB) = O(ω). Further, a flat background metric does not oscillate at all, hence ω_B = 0 implies h 0. But this is in contradiction with the assumption that h_µν has a finite value and thus, the expansion in h_µν does not work in the linearized theory in flat spacetime.