Linearized Einstein Equations

The starting point in discussing gravitational waves cannot but come from the Einstein field equations, expressing the close equivalence between matter-energy and curvature

$$G_{\mu \nu} \equiv R_{\mu \nu } - \frac{1}{2} g_{\mu \nu } R = 8 \pi T_{\mu \nu} .$$ (59)

In the 10 linearly independent equations (59), $R_{\mu \nu }$ and $R$ are the Ricci tensor and scalar, respectively, $g_{\mu \nu}$ and $G_{\mu \nu }$ are the metric and Einstein tensors, respectively, while $T_{\mu \nu }$ is the stress-energy tensor of the matter in the spacetime considered.

Looking at the Einstein equations (59) as a set of second-order partial differential equations it is not easy to predict that there exist solutions behaving as waves. Indeed, and as it will become more apparent in this Section, the concept of gravitational waves as solutions of Einstein equations is valid only under some rather idealized assumptions such as: a vacuum and asymptotically flat spacetime and a linearized regime for the gravitational fields. If these assumptions are removed, the definition of gravitational waves becomes much more difficult. In these cases, in fact, the full nonlinearity of the Einstein equations complicates the treatment considerably and solutions can be found only numerically. It should be noted, however, that in this respect gravitational waves are not peculiar. Any wave-like phenomenon, in fact, can be described in terms of exact ``wave equations'' only under very simplified assumptions such as those requiring an uniform ``background'' for the fields propagating as waves.

These considerations suggest that the search for wave-like solutions to Einstein equations should be made in a spacetime with very modest curvature and with a metric line element which is that of flat spacetime but for small deviations of nonzero curvature, i.e.

$$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} + {\cal O}([h_{\mu \nu }]^2) ,$$ (60)

where

$$\eta_{\mu \nu } = {\rm diag}(-1,1,1,1) ,$$ (61)

and the linearized regime is guaranteed by the fact that

$$\vert h_{\mu \nu}\vert \ll 1 .$$ (62)

Is this condition verified in Cascina???

Fortunately, the conditions expressed by equations (60) and (62) are, at least in our Solar system, rather easy to reproduce and, in fact, the deviation away from flat spacetime that could be measured, for instance, on the surface of the Sun are

$$\vert h_{\mu \nu }\vert \sim \vert h_{00}\vert \simeq \frac{M_{\odot}}{R_{\odot}} \sim 10^{-6} .$$ (63)

Before writing the linearized version of the Einstein equations (59) it is necessary to derive the linearized expression for the Christoffel symbols. In a coordinate basis (as the one will will assume hereafter), the general expression for the affine connection is

$$\Gamma^{\mu }_{ \alpha \beta } = \frac{1}{2} g^{\mu \nu }(g_{\nu \alpha , \beta } + g_{\beta \nu , \alpha } - g_{\alpha \beta , \nu }) ,$$ (64)

where the partial derivatives are readily calculated as

$$g_{\nu \alpha , \beta } = \eta_{\nu \alpha , \beta } + h_{\nu \alpha , \beta } = h_{\nu \alpha , \beta } ,$$ (65)

so that the linearized Christoffel symbols become

$$\Gamma^{\mu }_{ \alpha \beta } = \frac{1}{2} \eta^{\mu \nu }(h_{\nu \alpha , \beta } + h_{\beta \nu , \alpha } - h_{\alpha \beta , \nu }) =$$

$$= \frac{1}{2} (h^{ \mu}_{\alpha , \beta } + h^{ \mu}_{\beta , \alpha } - h^{ ,\mu}_{\alpha \beta}) .$$ (66)

Note that the operation of lowering and raising the indices in expression (66) is not made through the metric tensors $g_{\mu \nu}$ and $g^{\mu \nu }$ but, rather, through the spacetime metric tensors $\eta_{\mu \nu }$ and $\eta^{\mu \nu }$. This is just the consequence of linearized approximation and, despite this, the spacetime is really curved!

Once the linearized Christoffel symbols have been computed, it is possible to derive the linearized expression for the Ricci tensor which takes the form

$$R_{\mu \nu } = \Gamma^{\alpha }_{ \mu \nu ,\alpha } - \Gamma^{\alpha }_{ \mu \alpha ,\nu }$$

$$= \frac{1}{2} (h^{ \;\alpha }_{\mu \;,\nu \alpha } + h^{ \;\alp... ...u \alpha } - h^{ \;\alpha }_{\mu \nu , \alpha } - h_{, \mu \nu }) ,$$ (67)

where

$$h \equiv h^{\alpha }_{ \; \alpha } = \eta^{\mu \alpha } h_{\mu \alpha } ,$$ (68)

is the trace of the metric perturbations. The resulting Ricci scalar is then given by

$$R \equiv g^{\mu \nu } R_{\mu \nu } \simeq \eta^{\mu \nu } R_{\mu \nu } .$$ (69)

Making now use of (67) and (69) it is possible to rewrite the Einstein equations (59) in a linearized form as

$$h^{ \alpha }_{\mu \alpha , \nu } + h^{ \alpha }_{\... ...beta }_{\alpha \beta ,} - h^{ \alpha }_{,\alpha }) = 16 \pi T_{\mu \nu } .$$ (70)

Although linearized, the Einstein equations (70) do not seem yet to suggest a wave-like behaviour. A good step in the direction of unveiling this behaviour can be made if a more compact notation is introduced and which makes use of ``trace-free'' tensors defined as

$${\bar h}_{\mu \nu } \equiv h_{\mu \nu } - \frac{1}{2} \eta_{\mu \nu } h ,$$ (71)

where the ``bar-operator'' can be applied to any symmetric tensor so that, for instance, ${\bar R}_{\mu \nu } = G_{\mu \nu }$ and ${\bar {\bar h}}_{\mu \nu } = h_{\mu \nu }$³. Using this notation, the linearized Einstein equations (70) take the more compact form

$$-{\bar h}^{ \alpha }_{\mu \nu ,\alpha } -\eta_{\mu \nu }... ...+ {\bar h}^{ \; \alpha }_{\nu \alpha , \;\mu } = 16 \pi T_{\mu \nu } .$$ (72)

It is now straightforward to recognize that the first term on the right-hand-side of equation (72) is simply the Dalambertian (or wave) operator

$${\bar h}^{ \alpha }_{\mu \nu ,\alpha } = \Box {\bar h}_{... ...tial^2_t + \partial^2_x + \partial^2_y + \partial^2_z ) {\bar h}_{\mu \nu } ,$$ (73)

where the last equality is valid for a Cartesian $(t,x,y,z)$ coordinate system only. At this stage the gauge freedom inherent to General Relativity can (and should) be exploited to recast equations (73) in a more convenient form. A good way of exploiting this gauge freedom is by choosing the metric perturbations $h_{\mu \nu }$ so as to eliminate the terms in (72) that spoil the wave-like structure. Most notably, the metric perturbations can be selected so that

$${\bar h}^{\mu \alpha }_{ \;,\alpha } = 0 .$$ (74)

Making use of the gauge (74), which is also known as ``Lorentz'' (or Hilbert) gauge, the linearized field equations take the form

$$\Box{\bar h}_{\mu \nu } = - 16 \pi T_{\mu \nu } .$$ (75)

Despite they are treated in a linearized regime and with a proper choice of variables and gauges, Einstein equations (75) do not yet represent wave-like equations if matter is present (i.e. if $T_{\mu \nu } \ne 0$). A further and final step needs therefore to be taken and this amounts to consider a spacetime devoid of matter, in which the Einstein equations can finally be written as

$$\Box{\bar h}_{\mu \nu } = 0 ,$$ (76)

indicating that, in the Lorentz gauge, the ``gravitational field'' propagates in spacetime as a wave perturbing flat spacetime.

Having recast the Einstein field equations in a wave-like form has brought us just half-way towards analysing the properties of these objects. More will be needed in order to discuss the nature and features of gravitational waves and this is what is presented in the following Section.